course notes for osu me 8260 advanced …...rl harne me 8260, adv. eng. acoust. 2019 the ohio state...

RL Harne ME 8260, Adv. Eng. Acoust. 2019 The Ohio State University

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Course Notes for OSU ME 8260 Advanced Engineering Acoustics Prof. Ryan L. Harne*

Department of Mechanical and Aerospace Engineering, The Ohio State University, Columbus, OH 43210, USA

*Email: [email protected]

Last modified: 2019-01-04 13:31


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Table of contents

1 Course introduction 6

1.1 Scope of acoustics 6

1.2 Mathematical notation 6

2 Mathematics review 7

2.1 The harmonic oscillator 7

2.2 Initial conditions 8

2.3 Energy of vibration 9

2.4 Complex exponential method of solution to ODEs 10

2.5 Damped oscillations 12

2.6 Forced oscillations 14

2.7 Mechanical power 19

2.8 Linear combinations of simple harmonic oscillations 20

2.9 Fourier's theorem, series, and transform 24

3 Wave equation analysis 28

3.1 Transverse waves on a string 28

3.1.1 General solution to the one-dimensional wave equation 29

3.1.2 Wave propagation at string boundaries 32

3.1.3 Forced vibration of a semi-infinite string 34

3.1.4 Forced vibration of a finite length string 36

3.1.5 Normal modes of the fixed-fixed string 40

3.2 Longitudinal waves along a one-dimensional rod 43

3.3 Transverse waves along a beam 47

3.4 Dispersion, phase velocity, group velocity 50

3.5 Forced excitation of beams 53

3.5.1 Point force excitation of infinite beam 53

3.5.2 Arbitrary force excitation of finite beam 55

3.6 Waves in membranes 58

3.6.1 Free vibration of a rectangular membrane with fixed edges 59

3.6.2 Free vibration of a circular membrane 62


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3.7 Vibration of thin plates 65

3.7.1 Free vibration of thin plates 66

3.7.2 Forced vibration of thin plates 67

3.8 Impedance and mobility functions of structures 67

4 Acoustic wave equation 70

4.1 Assembling the components to derive the acoustic wave equation 70

4.1.1 Deriving the acoustic wave equation 74

4.2 Harmonic, plane acoustic waves 75

4.2.1 Impedance terminology clarification 77

4.3 Plane progressive shock waves 77

4.4 Acoustic intensity 79

4.5 Harmonic, spherical acoustic waves 80

4.5.1 Spherical wave acoustic intensity and acoustic power 81

4.6 Comparison between plane and spherical waves 82

4.7 Decibels and sound levels 84

4.8 Point source and source strength 85

4.9 Acoustic reciprocity 87

5 Reflection and transmission 88

5.1 Normal incidence wave propagation at a fluid interface 88

5.2 Normal incidence wave propagation through a fluid layer 91

5.3 Oblique incidence wave propagation at a fluid interface 94

5.4 Oblique incidence wave propagation at a thin partition interface: the mass law 98

5.5 Method of images 99

5.6 Acoustic metamaterials 102

5.6.1 One-dimensional monatomic lattice 103

5.6.2 One dimensional diatomic lattice 107

5.6.3 Metamaterial continua 109

5.6.4 Computational methods of analysis 110

6 Sound radiation from structures 122

6.1 Monopole and point source 122


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6.2 Two point sources 123

6.2.1 Dipole 125

6.3 Line source in the free field 125

6.3.1 Integral methods of analysis for radiating acoustic sources 130

6.4 Directivity 130

6.5 Rigid, circular, baffled piston source 132

6.5.1 On-axis acoustic pressure from baffled piston source 133

6.5.2 Far field acoustic pressure from baffled piston source 134

6.5.3 Generalization of the Rayleigh's integral for planar, baffled acoustic sources in the far field 137

6.6 Radiation impedance 139

6.7 Acoustic arrays 140

6.7.1 Line array 140

6.7.2 Arrays of directional sources 142

6.8 Spatial Fourier transform 143

6.8.1 Line source far field acoustic pressure response by spatial Fourier transform method 143

6.8.2 Forced, baffled rectangular plate far field acoustic pressure response by spatial Fourier transform method 145

6.8.3 Arrays and far field acoustic pressure design by spatial Fourier transform method 148

7 Enclosures and waveguides 149

7.1 Normal modes in acoustic enclosures 149

7.1.1 Rectangular enclosure 149

7.1.2 Cylindrical enclosure 151

7.2 Acoustics waveguides of constant cross-section 153

7.2.1 Rectangular waveguide 153

7.3 Pipes 155

7.3.1 Rigid termination of pipes 159

7.3.2 Short, closed volume 161

7.3.3 Open-ended pipes 161

7.3.4 Helmholtz resonator 163

7.4 Filtering properties of lumped acoustic impedances 164


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7.4.1 Low pass filter 167

7.4.2 High pass filter 168

7.4.3 Band stop filter 169


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1 Course introduction

Acoustics is defined as "a science that deals with the production, control, transmission, reception, and effects of sound", while sounds are defined as "mechanical radiant energy that are transmitted by longitudinal pressure waves in a material medium (as air) and are the objective cause of hearing" [Merriam-Webster dictionary]. Alternative definitions recognize that the context of acoustics is broader than merely "sound". Contemporary use of the term acoustics now often follows the definition denoting "a science [dealing with] the generation, transmission, and reception of energy as vibrational waves in matter" [1] where matter may be solid, gas, or fluid. This course will adopt the contemporary definition to undertake rigorous treatments of theories pertaining to wave propagation in matter. Yet, a greater balance of attention will be given to problems of fluid-borne waves, although they may be generated by waves or vibrations of structures.

1.1 Scope of acoustics

Acoustics is an encompassing science. Lindsay's wheel of acoustics is shown in Figure 1, and it is an exemplary picture of how acoustical subjects impact myriad scientific disciplines and even the liberal arts. The shared core principles, at the center of the wheel, are what unite the diverse disciplines. This course will dive deep into these fundamental principles of physical acoustics and will consider many of the extensions of such foundations as applied throughout engineering practices.

1.2 Mathematical notation

In the course notes, the following mathematical notations will be used.

We use j to denote the imaginary number, 1j = − .

Bold mathematics denote complex numbers, for example 8.0 2.3j= +k , j tDe ω=d . If written out by hand,

we use an underline to denote a complex number, for example 8.0 2.3k j= + , j td De ω= .

Mathematics with overbar denote vectors, for example 2.1 1.2 2.3v i j k= + + , x x y yu u e u e= + .

Boldandbarred mathematics denote complex vectors, for example ( ) j tx yu i u j e ω= +u .

Figure 1. Lindsay's wheel of acoustics, adapted from R. B. Lindsay, J. Acoust. Soc. Am. 36, 2242 (1964).


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2 Mathematics review

2.1 The harmonic oscillator

We first review the foundations of mechanical vibrations analysis.

Consider Figure 2 which shows the schematic of a mass-spring oscillator having one dimension of displacement, x (SI units [m]), with respect to the fixed ground. There are no gravitational influences to account for by considering the statically stable equilibrium as the reference point of 0x = .

The spring exerts a force on the mass m (SI units [kg]) according to the deformation of the spring with respect to the ground. Assuming that the spring has no undeformed length, the spring force is f sx= −

where the spring stiffness s has units [N/m]. Applying Newton's second law of motion, we determine the governing equation of motion for the mass

2

2

d xm f sxdt

= = − (2.1.1)

Rearranging terms and using the notation that that overdot indicates the /d dt operator, we have that

0mx sx+ = (2.1.2)

Based on the fact that stability conditions require m and s to be >0, we can define a new term 20 /s mω =

such that the governing equation (2.1.2) becomes

20 0x xω+ = (2.1.3)

Figure 2. Mass-spring oscillator.

Solution to the second-order ordinary differential equation (ODE) (2.1.3) determines the displacement x of the mass for all times. Solving ODEs is commonly accomplished by assuming trial solutions and verifying their correctness. We formulate assumed solutions to the ODEs by intuition and experience. For instance, consider a mass at the end of a Slinky and how that mass may harmonically move given an initial extension of the Slinky. Given the analogy, we hypothesize that a suitable trial solution to (2.1.3) is

1 cosx A tγ= (2.1.4)

By substitution, we find that (2.1.4) is a solution to (2.1.3) when 0γ ω= . Likewise, we also discover that

an alternative trial solution to (2.1.3) is


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2 0sinx A tω= (2.1.5)

When an ODE has multiple solutions, the total solution is the superposition of the individual solutions. Thus, the general solution to (2.1.3) is

( ) 1 0 2 0cos sinx t A t A tω ω= + (2.1.6)

The term 0ω is the natural angular frequency and has units [rad/s]. Thus, the mass will exhibit oscillatory

motion at 0ω . There are 2π radians in a cycle, which gives that the frequency in cycles per second is

0 / 2ω π . We refer to this as the natural frequency 0 0 / 2f ω π= which has units [Hz = cycles/s]. In practice,

we observe the mass oscillation over a duration of time. The period T of one oscillation is therefore

01 /T f= [s].

2.2 Initial conditions

To determine the unknown constants 1A and 2A , two initial conditions are required because this is a second-

order ODE. Thus, if the mass displacement at an initial time 0t = is ( ) 00x t x= = and the initial mass

velocity is ( ) 0 00x t x u= = = we can determine the constants by substitution.

0 1 0 2 0 1cos 0 sin 0x A A Aω ω= + = (2.2.1)

0 0 1 0 0 2 0 0 2sin 0 cos 0u A A Aω ω ω ω ω= − + = (2.2.2)

Using this knowledge, the general solution to (2.1.3), and the mass displacement described for all time, is

( ) 00 0 0

0

cos sinux t x t tω ωω

= + (2.2.3)

Alternatively, we express these two sinusoidal functions using an amplitude and phase

( ) [ ]0cosx t A tω φ= + (2.2.4)

where ( )1/222

0 0 0/A x u ω = + and 0 0 0tan /u xφ ω= − .

The time derivative of displacement is velocity

( ) ( ) [ ] [ ]0 0 0sin sinx t u t A t U tω ω φ ω φ= = − + = − + (2.2.5)

where we define the speed amplitude 0U Aω= . Likewise, the acceleration is

( ) [ ]0 0cosa t U tω ω φ= − + (2.2.6)

From these results, we see that the velocity leads the displacement by 90°, and that the acceleration is 180° out-of-phase with the displacement, Figure 3. This response is fully dependent upon the initial conditions of displacement and velocity. Such response is referred to as the free response or free vibration.


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Figure 3. Free vibration response of mass-spring oscillator.

2.3 Energy of vibration

From physics, the potential energy associated with deforming a spring is the integration of the spring force across the path of deformation

2

0

1d2

x

pE sx x sx= =∫ (2.3.1)

Here, we assume our integration constant is zero, which means that our minimum of potential energy pE

is set to zero. If we substitute (2.2.4) into (2.3.1), we find that

[ ]2 20

1 cos2pE sA tω φ= + (2.3.2)

By definition from physics, the kinetic energy of the mass is

2 21 12 2kE mx mu= = (2.3.3)

Likewise, by substitution of (2.2.5) into (2.3.3) we have

[ ]2 20

1 sin2kE mU tω φ= + (2.3.4)

The total energy of this dynamic system is the sum of potential and kinetic energies

2 2 2 20

1 1 12 2 2p kE E E m A mU sAω= + = = = (2.3.5)

where we have used the definition 20 /s mω = and the identity 2 2sin cos 1α α+ = .

The total energy (2.3.5) is independent of time, which is a statement of the conservation of energy. The total energy is equal to both the peak elastic potential energy (when kinetic energy is zero) and the peak

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-50

-40

-30

-20

-10

0

10

20

30

40

50

time [s]

resp

onse

10x displacement [m]velocity [m/s]

acceleration [m/s2]


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kinetic energy (when potential energy is zero). Figure 4 illustrates the instantaneous exchange between potential and kinetic energies over the course of the mass-spring system oscillation.

Figure 4. Exchange of energy between potential and kinetic forms as mass-spring oscillates.

2.4 Complex exponential method of solution to ODEs

A useful technique in harmonic analysis of engineering problems is the complex exponential form of

solution. In this course, we will use the engineering notation j te ω where the imaginary number is 1j = −

. This notation is in contrast to the notation j te ω− used in physics and mathematics. When confronted with a derivation performed according to the alternative notation, the conversion to the engineering notation requires taking the complex conjugate. In other words, one replaces the j by j− everywhere to recover

the engineering notation. Table 1 consolidates the primary acoustics textbook references which use the complex exponential notations.

Table 1. Primary acoustics textbook usage of the complex exponential notation.

j te ω notation [1] [2] [3] [4] [5]

j te ω− notation [6] [7] [8] [9] [10]

Recalling the equation (2.1.3), the more general solution method is to assume

( ) tt e= γx A (2.4.1)

where we use the boldface type to represent complex numbers, with the exception of the explicit description

of the imaginary number 1j = − . By substituting (2.4.1) into (2.1.3), one finds that 2 20ω= −γ and thus

0jω= ±γ . Since two solutions are obtained, the general solution to (2.1.3) by the complex exponential

notation is the superposition of both terms associated with 0jω= ±γ

( ) 0 01 2

j t j tt e eω ω−= +x A A (2.4.2)

Recall the initial conditions, ( ) 00 x=x and ( ) 0 00 x u= =x . By substitution we find

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

5

10

15

20

25

30

time [s]

ener

gy [J

]

total energypotential energykinetic energy


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0 1 2x = +A A (2.4.3)

0 0 1 0 2u j jω ω= −A A (2.4.5)

from which we find

01 0

0

12

ux jω

= −

A , and 0

2 00

12

ux jω

= +

A (2.4.6)

We see that the unknown constants 1A and 2A are complex conjugates. Substituting (2.4.6) into (2.4.2)

and using Euler's identity cos sine θ θ θ± = ± , we find

0 00 0 00 0 0 0 0

0 0 0

1 1 cos sin2 2

j t j tu u ux j e x j e x t tω ω ω ωω ω ω

− = − + + = +

x (2.4.7)

The result in (2.4.7) is the same as (2.2.3). Thus, although we used a complex number representation of the assumed solution (2.4.2), satisfying the initial conditions, which are both real, led to an elimination of the imaginary components of the assumed solution. In general, there is no need to perform this elimination process, because the real part of the complex solution is itself the complete general solution of the real differential equation.

For example, we could have alternatively assumed

( ) 0j tt e ω=x A (2.4.8)

Given that, in general, a jb= +A , we have that

[ ] 0 0Re cos sina t b tω ω= −x (2.4.9)

Then we satisfy the initial conditions

0 0x a= (2.4.10)

0 0u bω= − (2.4.11)

Thus, 0a x= and 0 0/b u ω= − . Then by substitution of a and b into (2.4.9), we again arrive at (2.4.7). This

confirms the conclusion that the real part of the complex solution is the complete general solution.

We use complex exponential assumed solution forms similar to (24) throughout this course. According to the relation between displacement, velocity, and acceleration, we summarize

00 0

j tj e jωω ω= =u A x (2.4.12)

02 20 0

j te ωω ω= − = −a A x (2.4.13)

The term 0j te ω is a unit phasor that rotates in the complex plane, while A is a complex function that modifies the amplitude of the rotating amplitude and shifts it in phase according to the complex component of A .


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Thus, 2 2a b= +A and tan /b aφ = such that we have [ ] ( )00cos Re j tt e ω φω φ + + = A A . Figure 5

illustrates how the complex exponential form representation may be considered in the complex plane. The magnitude-scaled phasor rotates in the plane with angular rate 0ω as time changes, while the Real

contribution of the phasor oscillates between positive to negative values. By considering (28), the velocity phasor leads the phasor rotation of displacement by 90° while the acceleration phasor is out-of-phase with the displacement phasor by 180°. The phasor rotate in the counterclockwise direction, for 0ω >0 and positive

increase in time t .

Figure 5. Physical representation of a phasor.

2.5 Damped oscillations

All physical systems are subjected to phenomena that dissipate kinetic energy as time advances. In general, such damping phenomena decrease the instantaneous amplitude of free oscillations as time increases. Unlike forces characterized with potential energy and inertial forces associated with kinetic energy, damping forces associated with energy dissipation are often identified empirically since the exact mechanisms of energy dissipation occur on length and time scales several orders of magnitude less than the measured physical system. This means that sequences of tests are conducted to study the rate at which energy decays in the system according to changes in the damping element's parameters, and a damping model with unknown coefficients is fit to the data to identify the respective damping constants.

A common form of damping observed in mechanical systems is viscous damping. Viscous damping exerts a force proportional to the velocity of the mass away from its equilibrium, Figure 6.

r mdxf Rdt

= − (2.5.1)

The damping constant mR has a positive value and SI unit [N.s/m]. We refer to this damping constant as

the mechanical resistance. Typical dampers with viscous damping realize such velocity-proportional force behavior by using turbulent fluid flow through orifices and channels. Such technique is the convention for energy dissipation in dashdot dampers that serve as automotive shock absorbers.


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Figure 6. Damped harmonic oscillator schematic.

Considering the damped mass-spring oscillator schematic of Figure 6, we apply Newton's laws to find

0mmx R x sx+ + = (2.5.2)

By incorporating our definition of the natural angular frequency, we express

20 0mRx x x

mω+ + = (2.5.3)

To solve this governing equation for the mass motion, we resort to the complex exponential solution method assuming that

( ) tt e= γx A (2.5.4)

By substituting (2.5.4) into (2.5.3), we find

2 20 0j tmR e

mω + + =

γγ γ A (2.5.5)

The only non-trivial solution to (2.5.5) requires that the terms in brackets be equal to zero. Thus

( )1/22 20β β ω= − ± −γ (2.5.6)

where we have introduced / 2mR mβ = which has units of [1/s]. Often, the damping is small such that

0ω β>> . We can then consider rearranging the radicand by defining the damped natural angular

frequency

2 20dω ω β= − (2.5.7)

such that the term γ is given by

djβ ω= − ±γ (2.5.8)

Considering the other common conventions of denoting damping, we recognize that

02mRm

ζω= (2.5.9)

where ζ is termed the damping ratio. From this, we recognize that 0β ζω= , and 20 1dω ω ζ= − .


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Thus, after substitution, the general solution for the damped mass-spring oscillator free vibration equation of motion is

1 2d dj t j tte e eω ωβ− = + x A A (2.5.10)

As described and demonstrated previously, only the real part of (2.5.10) is the complete, general solution to (2.5.5). Thus, we express (2.5.10) as

( ) [ ]costdx t Ae tβ ω φ−= + (2.5.11)

The constants A and φ are determined by substituting (2.5.11) into the initial conditions. The second order

ODE (2.5.2) necessitates that two initial conditions must be known, oftentimes for the initial displacement and initial velocity.

Unlike the undamped harmonic oscillator considered in Sec. 2.1, the vibrations of the damped oscillator

decay in amplitude with increase of time due to the term te β− . To assess the rate at which this decay occurs, we recognize that the term β has units of [1/s]. Thus, we can therefore define a relaxation time 1 /τ β=

which characterizes the decay of the oscillation amplitude as time increases. The term tAe β− is essentially the envelope of the vibration response that is exemplified in Figure 7.

Figure 7. Damped free oscillation. 0/ 0.1β ω = , and the same additional parameters as those to generate Figure 3.

2.6 Forced oscillations

In contrast with free vibration, forced oscillations are those induced by externally applied forces ( )f t ,

Figure 8. This modifies the governing equation (2.5.2) to yield

( )mmx R x sx f t+ + = (2.6.1)

For linear response by the harmonic oscillator, the total response is the summation of the individual responses. Thus, by recalling Fourier's theorem that any periodic function may be described using an infinite series of sinusoids (even when the period extends for an infinitely great duration), we consider that the force

( )f t is composed according to ( ) ( )ii

f t f t=∑ where each ( )if t accounts for one of the harmonic,

0 1 2 3 4 5 6 7 8-1

-0.5

0

0.5

1

1.5

time [s]

disp

lace

men

t [m

]


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sinusoidal components of force. Based on this, by linear superposition we only need to solve for the response of the oscillator when subjected to harmonic forces occurring at single frequencies. Once we solve

for the individual displacement responses ( )ix t corresponding to their respective harmonic forces ( )if t ,

we then obtain the total response by the superposition of the individual displacement responses.

Figure 8. Schematic of forced damped harmonic oscillator.

Therefore, in this course we will often consider harmonic driving inputs at single frequencies, such as

( ) cosf t F tω= . When excited by such a force from a state accounting for unique initial conditions of

displacement and velocity, the oscillator will undergo two responses: transient response associated with the initial conditions and a steady-state response associated with the periodic forcing function.

The forced response is the solution to the ODE (2.6.1) considering the initial conditions to be zero-valued during the solution process. The forced response is mathematically termed the particular solution of the ODE. When the forced response of a linear system is due to a harmonic driving input, the forced response is termed the steady-state response.

The free response associated with transient dynamic behavior is the solution to the ODE (2.5.2) accounting for the initial conditions and a zero-valued forcing function. For systems that possess even an infinitesimal amount of damping, the free response decays in amplitude with increase in time. At times sufficiently greater than the relaxation time 1 /τ β= , the transient response is insignificantly small when compared to

the steady-state response induced by the forcing function. The mathematical term for the free response is the homogeneous solution to the ODE.

In many engineering contexts, and often in acoustical engineering, we will be concerned with the steady-state response. In these cases, the complex exponential solution approach to the ODE (2.6.1) will be

favorable. Consider that the real driving force ( ) cosf t F tω= is replaced with the complex driving force

( ) j tt Fe ω=f . Then, the equation (2.6.1) becomes

j tmm R s Fe ω+ + =x x x (2.6.2)


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Since the real part of the driving force represents the actual driving force, similarly, the real part of the

complex displacement [ ]Re x that solves the equation (2.6.2) represents the actual displacement in resulting

from the force.

To proceed with the solution, we assume j te ω=x A and by substitution into (2.6.2) we have

2 j t j tmm j R s e Feω ωω ω − + + = A (2.6.3)

In general, the application of the above approach refers to the assumption of steady-state, time-harmonic response.

We then solve for the complex displacement coefficient A and substitute that back into the assumed solution form to determine the complex displacement

( ) ( )2

1/

j t j t

mm

Fe Fej R j m ss m j R

ω ω

ω ω ωω ω= =

+ −− +x (2.6.4)

Similar to the case of free vibration for (2.4.12), the complex steady-state velocity of the mass is jω=u x

( )/

j t

m

FeR j m s

ω

ω ω=

+ −u (2.6.5)

We introduce the complex mechanical input impedance mZ

m m mR jX= +Z (2.6.6)

where we define the mechanical reactance /mX m sω ω= − . The magnitude of the mechanical impedance j

m mZ e Θ=Z is

( )1/222 /m mZ R m sω ω = + − (2.6.7)

while the corresponding phase angle of the mechanical impedance is determined from

/tan m

m m

X m sR R

ω ω−Θ = = (2.6.8)

Considering (2.6.6), the dimensions of the mechanical impedance are the same as the mechanical resistance, [N.s/m].

Considering (2.6.5), we see that the mechanical impedance is equal to the ratio of the driving force to the harmonic response velocity

mfZ =u

(2.6.9)


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From (2.6.9), we see that the complex mechanical impedance mZ is the ratio of the complex driving force

f to the complex velocity u of the system. The interpretation of (2.6.9) is important.

• First, note that (2.6.9) is a transfer function between f and u in the frequency domain. • Second, for large mechanical input impedance magnitudes, (2.6.9) indicates that large force is

required to achieve a given system velocity, all other factors remaining the same. In contrast, for small mechanical impedance magnitudes it is relatively easy to apply the harmonic driving force to obtain considerable system velocity in oscillation.

• It is important to recognize that the mechanical input impedance can be measured by collocated force transducer and velocity transducer on the mechanical system. Because the transfer function of mechanical input impedance is in the frequency domain, it can be computed by taking the ratio of the Fast Fourier transforms of the force and velocity measurements.

Thus knowing the mechanical input impedance, one may compute the complex velocity for a different

harmonic forcing function according to / m=u f Z . By virtue of the assumed solution form j te ω=x A , the

complex displacement becomes / mjω=x f Z . In other words, determining mZ is analogous to solving the

differential equation of motion for the linear system.

As a result, the actual displacement is the real part

[ ] [ ] ( ) [ ]Re Re / / sinm mj x F Z tω ω ω= = = −Θx f Z (2.6.10)

whereas the actual speed is

[ ] ( ) [ ]Re / cosmu F Z tω= = −Θu (2.6.11)

From (2.6.11), for a constant amplitude of the harmonic force, it is seen that the speed of the mass is maximized when the impedance magnitude is minimized. This occurs when

0/ 0 /m s s mω ω ω ω− = → = = (2.6.12)

In other words, when the harmonic excitation frequency corresponds to the natural frequency, the speed of the mass oscillation will be maximized. This phenomenon is called resonance. Thus, at resonance, the impedance is minimized and purely real, such that the speed is

j t

m

F eR

ω≈u and [ ]Re cosm

Fu tR

ω= =u (2.6.13)

As seen in (2.6.13), for a given harmonic force amplitude, when driven at resonance the system response is damping- or resistance-controlled. In other words, the damping is the principal determinant for the amplitude of the system velocity at frequencies close to resonance. The (2.6.13) also indicates that when driven at resonance, the oscillator velocity is perfectly in phase with the harmonic driving force, which is otherwise not generally true.


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When the excitation frequency is significantly less than the natural angular frequency 0ω ω<< , we write

[ ]

2 20

20

/ /

/ /

/ /

m m

m m

m m m

m R m j

m R m j

R j s js

ω ω ω ω

ω ω ω

ω ω

= + − ≈ + −

≈ + − → ≈ −

Z

Z

Z Z

(2.6.14)

Thus, when the excitation frequency is much less than the natural frequency, the mass velocity is

j tj F es

ωω≈u (2.6.15)

from which we see that the complex displacement is

j tF es

ω≈x (2.6.16)

yielding that the actual mass displacement is

[ ]Re cosFx ts

ω= =x (2.6.17)

This shows that at excitation frequencies considerably less than the natural frequency, the system response is stiffness-controlled.

Similarly, for high frequencies, 0ω ω>> , this routine will yield that the complex acceleration is

j tF em

ω≈a (2.6.18)

given a real acceleration response of

[ ]Re cosFa tm

ω= =a (2.6.19)

which shows that at high excitation frequency the system response is mass-controlled.

Summarizing the findings from the above derivations:

• Around resonance 0ω ω≈ the system is damping- or resistance-controlled and the velocity is

independent of frequency, although the band of frequencies around which this occurs is typically narrow for lightly damped structures in many engineering contexts

• For harmonic excitation frequencies significantly below the natural angular frequency 0ω ω<< ,

the system is stiffness-controlled and the displacement is independent of the excitation frequency • For excitation frequencies much greater than the natural angular frequency 0ω ω>> , the system is

mass-controlled and the acceleration is independent of the excitation frequency

These results are summarized by the example plot shown in Figure 9.


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Figure 9. Harmonic force excitation of damped mass-spring oscillator. mR =0.377 [N.s/m], m =1 [kg], s =39.5 [N/m], F =1

[N].

2.7 Mechanical power

In acoustical engineering applications, the power and energy associated with sound sources and wave radiation are critical factors to assess in the determination of acoustic performance and quality. To first introduce these concepts, we consider the power relations for the mechanical oscillator.

The instantaneous mechanical power iΠ delivered to the harmonic oscillator is determined by the product

of the instantaneous driving force and the corresponding instantaneous speed

[ ] [ ] [ ]2

cos cos Re Reim

F t tZ

ω ωΠ = −Θ = f u (2.7.1)

In general, the average power Π delivered to the system is the more relevant engineering quantity to consider. The average power is the instantaneous power averaged over one cycle of oscillation

[ ]{ }

{ }

2 2 /

0 0

2 2 / 2

0

2

1 d cos cos d2

cos cos cos sin sin d2

cos2

T

im

m

m

Ft t t tT Z

F t t t tZ

FZ

π ω

π ω

ω ω ωπ

ω ω ω ωπ

Π = Π = −Θ

= Θ + Θ

= Θ

∫ ∫

∫ (2.7.2)

Recalling (2.6.8),

/ sintancos

m

m m

X m sR R

ω ω− ΘΘ = = =

Θ (2.7.3)

which shows that cos /m mR ZΘ = and leads to the result that the average power delivered to the mechanical

oscillator is

2*

2

1 Re2 2m

m

R FZ

Π = = f u (2.7.4)

10-2

10-1

100

101

102

10-4

10-2

100

102

frequency [Hz]

ampl

itude

velocity [m/s]displacement [m]acceleration [m/s]


20

The units of power, whether instantaneous or average, are Watts, [W=J/s].

When the oscillator is driven at resonance 0ω ω= , the average power becomes 2 / 2 mF RΠ = . This shows

that the peak average power delivered to the oscillator is damping-controlled. Based on the in-phase relationship between the input force and oscillator speed at resonance, we find that peak average power delivery to the oscillator occurs when speed and force are perfectly in phase. This conclusion is also intuitively arrived at by considering the procedures of (2.7.4).

In addition, the power delivered to the oscillator is proportional to the resistance. The presence of reactance diminishes the mechanical power delivered from the force to the moving mass. As will become clearer through additional impedance relations in the study of wave propagation, the real part of the impedance, the resistance, is associated with energy transfer or power delivery, while the imaginary part of the impedance, the reactance, is associated with energy exchange (alternatively termed reciprocating energy) which does not deliver power.

2.8 Linear combinations of simple harmonic oscillations

In numerous vibration and acoustics contexts, it is needed to determine the total response associated with a combination of individual responses. We consider only linear systems here, so the principle of linear superposition applies.

Consider two harmonic oscillations at the same angular frequency ω . These two harmonic oscillations are termed coherent or correlated due to the sharing of frequency.

( )11 1

j tA e ω φ+=x and ( )22 2

j tA e ω φ+=x (2.8.1)

The linear combination 1 2= +x x x is therefore

( ) ( )1 21 2

j t j j j tAe A e A e eω φ φ φ ω+ = + (2.8.2)

Determining the magnitude A and the total phase φ is accomplished trigonometrically from considering

representative phasors in the complex number coordinate plane, Figure 10

( ) ( )1/22 2

1 1 2 2 1 1 2 2cos cos sin sinA A A A Aφ φ φ φ = + + + (2.8.3)

1 1 2 2

1 1 2 2

sin sintancos cos

A AA A

φ φφφ φ+

=+

(2.8.4)


21

Figure 10. Phasor combination of 1 2= +x x x .

The real response displacement is

[ ] [ ] [ ]1 2Re Re cosx A tω φ= + = +x x (2.8.5)

As will be seen throughout later portions of this course, many response metrics of interest in acoustics involve mean-square and root-mean-square quantities. The mean-square is computed by

[ ]( )

[ ]

2 /22

1 1 10

2 /2 2

1 10

cos d2

cos d2

x A t t

A t t

π ω

π ω

ω ω φπ

ω ω φπ

= +

= +

∫

∫ (2.8.6)

Keep in mind that the name "mean-square" indicates the operators occur to the expression from the right-to-left: thus, square the expression first and secondly take the mean. Using trigonometric identities on (2.8.6) yields

2 /2 2 21 1 1

0

1 1cos 2 d2 2 2

x A t t Aπ ωω ω

π = + = ∫ (2.8.7)

The root-mean-square (RMS) is the square root of the mean-square quantity, and is typically denoted by

rmsx . Likewise, operate on the expression from the right-to-left: square the expression, take the mean, and

then take the square root of the mean result. Thus, continuing the derivation from (2.8.7), it is seen that the RMS oscillation displacement is


22

11, 2rms

Ax = (2.8.8)

so as to relate the response amplitude 1A to the RMS value of the harmonic component 1,rmsx via

1, 12 rmsx A= .

From the perspective of complex exponential representations (2.8.2), the RMS of the oscillation is likewise the amplitude of the response divided by the square-root of 2.

For the summation of two harmonic oscillations having the same frequency (2.8.6), the mean-square is

found by steps. First, the square [ ]2 2Re x=x is

[ ] [ ] [ ] [ ]

[ ]( ) [ ]( ) [ ] [ ]( )

2 2 2 2 21 1 2 2 1 2 1 2

2 21 1 2 2 1 2 1 2 1 2

cos cos 2 cos cos1 11 cos 2 2 1 cos 2 2 cos 2 cos2 2

x A t A t A A t t

A t A t A A t

ω φ ω φ ω φ ω φ

ω φ ω φ ω φ φ φ φ

= + + + + + +

= + + + + + + + + + − (2.8.9)

where trigonometric identities are used to transition from the first to second line of (2.8.9). Then, we

compute the mean-square 2x and find

[ ]

[ ]

2 2 21 2 1 2 1 2

2 21 2 1 2 1 2

1 1 cos2 2

2 cos

x A A A A

x x x x

φ φ

φ φ

= + + −

= + + − (2.8.10)

Thus, for harmonic oscillations occurring at the same frequency ω , a significant variation in the total

response may occur as relates to the mean-square output. For instance, if 0 1 2A A A= = and if 1 2 0φ φ= = ,

then the mean-square displacement is 2 2 2 2 20 0 0 0

1 1 22 2

x A A A A= + + = so as to quadruple the mean-square

result with respect to an individual harmonic oscillation: 20

12

A . On the contrary, if the 0 1 2A A A= = and

2 1 0φ φ π= + = , then we find [ ]2 2 2 2 2 20 0 0 0 0

1 1 cos 02 2

x A A A A Aπ= + + − = − = which shows that out-of-phase

oscillations destructively interfere so as to eliminate the mean-square measure. Of course, this occurs for direct oscillation summation, as well, but this also confirms that the mean-square quantity likewise is eliminated.

Again adopting the perspective of complex exponential representations (2.8.2), the RMS is computed from comparable steps. The results of (2.8.3) and (2.8.4) have indeed already yielded the bulk of the derivation:

( ) ( ) ( ) ( )1 2 1 21 2 1 2

j t j t j tj j j tA e A e A e A e e Aeω φ ω φ ω φφ φ ω+ + += + = + =x (2.8.11)

where A and φ are defined in (2.8.3) and (2.8.4), respectively. Considering what was shown in (2.8.9) to

(2.8.11), the RMS of x is


23

2rmsAx = (2.8.12)

One may perceive a contradiction between (2.8.12) and the example cases above that demonstrated that the RMS of a summation of harmonic oscillations at the same frequency can potentially constructively or destructively interfere. Yet, recalling (2.8.3), the amplitude in (2.8.12) varies according to such phase differences between the oscillations. Therefore, for instance in the event of perfect destructive interference by waves of the same amplitude but out-of-phase, we find 0A→ , which was shown above.

Note that in general, j te ω=x A where the contribution of the phase is included within the complex

amplitude A . Thus, 2

2

2rmsx =A .

Two oscillations that occur at different frequencies are termed incoherent or uncorrelated. When the oscillations do not occur at the same frequency, there is no further simplification to adopt for

[ ] [ ]1 2 1 1 1 2 2 2cos cosx x x A t A tω φ ω φ= + = + + + (2.8.13)

For the mean-square quantity, the computation shows

2 2 21 2x x x= + (2.8.14)

Also, the RMS quantities are the summation of the individual terms

2 2 21, 2,rms rms rmsx x x= + (2.8.15)

The same results for the mean-square and RMS quantities would be obtained by applying the complex exponential form of the response.

Finally, by linear superposition, the procedures outlined above for two harmonic oscillations extend to any number of oscillations. In particular, for coherent or correlated sinusoid summation, it is found that

( ) ( )1/22 2

cos sinn n n nA A Aφ φ = + ∑ ∑ (2.8.16)

sintan

cosn n

n n

AA

φφ

φ= ∑∑

(2.8.17)

Thereafter, following the computation of (2.8.16)), the mean-square and RMS values follow naturally from computations of (2.8.7) and (2.8.12), respectively.

For incoherent or uncorrelated sinusoid summation, there are no general simplifications available to employ for the resulting oscillation itself, although the mean-square and the RMS quantities are respectively computed from

2 2nx x=∑ (2.8.18)


24

2 2,rms n rmsx x=∑ (2.8.19)

2.9 Fourier's theorem, series, and transform

Linear combinations of oscillations is a foundational means of predicting wave propagation phenomena throughout many studies in acoustics. Conversely, in the acquisition of acoustic data sets, it is regularly needed to decompose an intricate time series measurement into frequency components that have greater significance in the interpretation of the data. To make this leap from a combination response to components, we draw from the Fourier's theorem.

Fourier's theorem is a statement that any single-valued, periodic function may be expressed as a summation of simple harmonic components whose frequencies are integral multiples of the periodicity of the whole function.

For a function ( )f t that has a period T , Fourier's series expands this function by

( ) 0

1

2 2cos sin2 n n

n

a n nf t a t b tT Tπ π∞

=

= + + ∑ (2.9.1)

where the angular frequency of repetition is 2 / Tω π= . The coefficients of the series are determined from

( )2 2cos ; 0,1,2,...t T

n t

na f t t dt nT T

π+ = = ∫ (2.9.2)

( )2 2sin ; 1,2,3,...t T

n t

nb f t t dt nT T

π+ = = ∫ (2.9.3)

In this way, the function ( )f t with fundamental periodicity rate ω is represented by sine and cosine

components of rates nω , 1,2,3,...n = , and a constant term 0a that is independent of frequency.

The specific constitution of the time series function ( )f t determines the number N , 1,2,...,n N= ,

required in the superposition in order to faithfully reproduce the original function, arbitrary though it may

be. Of course, odd functions, ( ) ( )f t f t= − − , are expressed using only sines. In a similar way, even

functions, ( ) ( )f t f t= − , require only cosine functions for reproduction. The constant term 0a vanishes

only when the function is symmetric about f =0.

Looking further, functions that are mostly sinusoidal in effect at a handful of lower-order harmonics, such as that observed in the generation of "notes" by certain musical instruments, require only a few terms, e.g.

6N ≤ , for accurate reproduction. On the other hand, the presence of sharp changes in a time series response demand a large number of harmonic terms in order to reconstruct the sudden change in the function. This is representative of significant high frequency contributions, where high is relative to the fundamental frequency of the periodic function.


25

Example: Determine the number of terms in a Fourier series required such that a pulse train ( )f t may be

reproduced with 5% or less RMS error. The pulse train period is 2T π= .

( )1;01; 2

tf t

tπ

π π< <

= − < <

Answer:

The function is odd, such that cosine terms are anticipated to be absent from the series. Similarly, the pulse train function is symmetric about f =0, such that the constant term is anticipated to be absent. Indeed,

[ ]{ } [ ]{ } [ ] [ ]2

2

00

1 1 1 1 1 11cos 1cos sin sin 0na nt dt nt dt nt ntn n

π ππ π

πππ π π π

= + − = + − = ∫ ∫

for all 0,1,2,...n = , thus confirming the anticipation. For the sine terms of the series,

[ ]{ } [ ]{ }

[ ] [ ]

2

0

2

0

1 11sin 1sin

1 1 1 1cos cos

4 4 4,0, ,0, ,...3 5

nb nt dt nt dt

nt ntn n

π π

π

π π

π

π π

π π

π π π

= + −

= − +

=

∫ ∫

In other words, for ( ) 0

1

2 2cos sin2 n n

n

a n nf t a t b tT Tπ π∞

=

= + + ∑ , 0na = for all n , while 4 /nb nπ= for

n odd but nb =0 for n even.

The plot at left in Figure 11 shows a reconstruction of the pulse train using 172 terms in the Fourier series. This number of terms is determined from the plot at right in Figure 11 as approximately the point at which the RMS error is 5%. The code used to generate these results is given in Table 2. The overshoot observed at the discontinuity in the pulse train function at t π= is referred to as Gibbs phenomenon.

0 1 2 3 4 5 6-1.5

-1

-0.5

0

0.5

1

1.5

time [s]

func

tion

f(t)

black solid curve, exactred dashed curve, Fourier series 172 terms

Figure 11. At left, time series reconstruction of pulse train using 172 terms in the Fourier series. At right, RMS error convergence by increasing number of terms in Fourier series.

100

101

102

103

10-2

10-1

100

number of Fourier series coefficients

erro

r of F

ourie

r ser

ies

expa

nsio

n to

act

ual


26

Table 2. Code to generate Figure 11

% pulse train. f(t)=1; 0<t<pi; f(t)=-1; pi<t<2*pi. T=2*pi. time=linspace(0,2*pi,1e4); f=zeros(length(time),1); f(1:length(time)/2)=1; f(length(time)/2:end)=-1; numcoef_all=172; clear error for ooo=1:length(numcoef_all) numcoef=numcoef_all(ooo); % number of fourier series coefficients fest=zeros(length(time),1); for iii=1:2:numcoef, fest=fest+4/pi/iii*sin(iii*time)'; end; error(ooo)=rms(fest-f); disp(num2str(error,'%1.5f ')); end figure(1); clf; plot(time,f,'k') hold on plot(time,fest,'--r'); xlabel('time [s]'); ylabel('function f(t)'); xlim([min(time) max(time)]); legend('black solid curve, exact','red dashed curve, Fourier series 172 terms') numcoef_all=logspace(0,3,401); clear error for ooo=1:length(numcoef_all) numcoef=numcoef_all(ooo); % number of fourier series coefficients fest=zeros(length(time),1); for iii=1:2:numcoef, fest=fest+4/pi/iii*sin(iii*time)'; end; error(ooo)=rms(fest-f); end figure(2); clf; plot(numcoef_all,error); xlabel('number of Fourier series coefficients'); ylabel('error of Fourier series expansion to actual'); set(gca,'yscale','log'); set(gca,'xscale','log'); xlim([min(numcoef_all) max(numcoef_all)]);

Like all integrals, Fourier's (integral) transform is the limiting case of the Fourier's series. The Fourier transform pair of the transform and inverse transform are given in (2.9.4) and (2.9.5), respectively.

( ) ( ) j tF f t e dtωω+∞ −

−∞= ∫ (2.9.4)

( ) ( )12

j tf t F e dωω ωπ

+∞

−∞= ∫ (2.9.5)

The integral pair relate the time series of the function ( )f t to its continuous representation in the frequency

domain ( )F ω , in contrast to the discretized frequency domain representation given in the Fourier series.


27

As before, 2 / Tω π= . Note that the signs in the complex exponentials of (2.9.4) and (2.9.5) can be interchanged without loss of generality.


28

3 Wave equation analysis

3.1 Transverse waves on a string

The string represents one of the most elementary vibrating bodies that permits the propagation of waves according to the distributed nature of the system. This contrasts with the lumped parameter mechanical vibrations considered in Sec. 2. Thus, the string serves as a suitable introduction into the formation of a wave equation that governs the motion of the distributed parameter system and will consequently enable exploration of wave propagation principles.

Consider the infinitesimal element of taut, stretched string shown in Figure 12. The string is under tension T , units [N], and possesses a uniform linear density Lρ , units [kg/m]. The stiffness of the string is

presumed to be negligible and gravitational forces are omitted by virtue of considering a tension T significantly great enough such that no slack in the string occurs. The element of string is of sufficient small extent ds that the tension is nearly constant along the element. Also, the vertical displacement of the string from one end of the element to another, from an equilibrium displacement y , is also assumed to be small.

Comparatively, the horizontal displacements of the string are negligibly small.

Figure 12. Schematic of taut string element.

By virtue of such assumptions, the vertical force present between the two ends of the infinitesimal string element due to the tension is

( ) ( )sin siny x dx xdf T Tθ θ+

= − (3.1.1)

where θ is the angle between the tangent to the string and the x axis. Also, ( )sin x dxT θ+

is the value of

sinT θ at the location x dx+ , while ( )sin xT θ is the corresponding value at location x . A Taylor series

expansion is used to approximate ( )sin x dxT θ+

( ) ( ) ( )sinsin sin x

x dx x

TT T dx

xθ

θ θ+

∂≈ +

∂ (3.1.2)

Consequently, the vertical force is approximately


29

( )siny

Tdf dx

xθ∂

≈∂

(3.1.3)

For small vertical displacements of the string, the angle tangent to the x axis is approximately sin tan /dy dxθ θ≈ ≈ . Thus, the vertical force exerted on the string by the tension along the element length

is

2

2yy ydf T dx T dx

x x x∂ ∂ ∂ = = ∂ ∂ ∂

(3.1.4)

The string element mass is Ldxρ . Newton's 2nd law of motion then relates the acceleration 2

2

yt

∂∂

of this

mass to the net vertical force according to

2 2

2 2Ly yT dx dx

x tρ∂ ∂

=∂ ∂

(3.1.5)

which is reduced to

2 2

2 2 2

1y yx c t∂ ∂

=∂ ∂

(3.1.6)

where the constant is defined 2 / Lc T ρ= , units [m/s]. The (3.1.6) is the one-dimensional wave equation.

3.1.1 General solution to the one-dimensional wave equation

Jean-Baptiste le Rond d'Alembert (1717-1783) derived the general solution to the one-dimensional wave equation (3.1.6). He found that a complete solution to (3.1.6) required

( ) ( ) ( )1 2,y x t f ct x f ct x= − + + (3.1.1.1)

To show the satisfaction of (3.1.6) by (3.1.1.1), one applies the differential operators of (3.1.6) on (3.1.1.1).

( ) ( )2

1 212

f ct xc f ct x

t∂ − ′′= −

∂ (3.1.1.2)

( ) ( )2

112

f ct xf ct x

x∂ − ′′= −

∂ (3.1.1.3)

where ( )ff

αα

∂′ =

∂. Performing the operations also on ( )2f ct x+ , by substitution we find

2 21 2 1 22

1f f c f c fc

′′ ′′ ′′ ′′+ = +

(3.1.1.4)

which confirms the satisfaction of (3.1.6) by (3.1.1.1).


30

It is important to recognize that the arguments of functions 1f and 2f are arbitrary. We investigate this

general solution (3.1.1.1) by considering the result of a single function 1f at different times, 1t and 2t ,

Figure 13. A triangular pulse waveform is considered. Because the wave shape remains constant in the absence of dissipative or dispersive terms in the one-dimensional wave equation, we find that

( ) ( )1 1 1 1 2 2, ,f x t f x t= . As a result, the arguments of the functions must yield the same outcome. Consider

that the wave at 1x at time 1t is the same as the wave response at 2x at time 2t . Thus, the wave function

arguments between these two times must yield the same result

2 11 1 2 2

2 1

x xct x ct x ct t−

− = − → =−

(3.1.1.5)

As a result, it is clear that the parameter c is the speed at which a wave propagates. It is termed the phase speed because this is the rate of travel for a point of constant phase of the wave, units [m/s].

If we repeat the analysis for ( )2f ct x+ , we find that in increasing time the wave travels "to the left". Thus,

comparing the arguments ct x− and ct x+ , we see that waves travel towards increasing x values for increasing time according to the argument ( ct x− ), while waves travel towards decreasing x values for increasing time according to the argument ( ct x+ ).

Figure 13. Wave nature of the general solutions of the one-dimensional wave equation.

For a string of significantly great length, we may consider the one-dimensional wave equation as an initial-value problem (IVP) rather than as an initial-boundary-value problem (IBVP). In the former case, we have

the initial conditions, ( ) ( ),0y x U x= and ( ) ( ),0y x V x= . We then evaluate how our two functions for the

general solution (3.1.1.1) accommodate the initial conditions by direct substitution. To this end, it may be shown that we can rearrange the order of the functions without loss of generality. Thus, in the following,

we use ( )1f x ct− and ( )2f x ct+ having the same meanings as before [3]. Thus, applying the initial

conditions

( ) ( ) ( ) ( )1 2,0y x f x f x U x= + = (3.1.1.6)


31

( ) ( ) ( ) ( )1 2,0y x c f x f x V x ′ ′= − + =

(3.1.1.7)

We then integrate the equation for the initial condition on velocity.

( ) ( ) ( )1 21 x

bf x f x V S dS

c− = − ∫ (3.1.1.8)

Using (3.1.1.6) and (3.1.1.7), we solve to find ( )1f x and ( )2f x .

( ) ( ) ( )11 12 2

x

bf x U x V S dS

c= − ∫ (3.1.1.9)

( ) ( ) ( )21 12 2

x

bf x U x V S dS

c= + ∫ (3.1.1.10)

Then, rather than only the argument x , the full arguments x ct± are inserted into the functions and they are summed together to yield the general solution.

( ) ( ) ( ) ( ) ( )1 1,2 2

x ct x ct

b by x t U x ct U x ct V S dS V S dS

c+ − = − + + + − ∫ ∫ (3.1.1.11)

( ) ( ) ( ) ( )1 1,2

x ct

x cty x t U x ct U x ct V S dS

c+

−

= − + + + ∫ (3.1.1.12)

The implications of (3.1.1.11) are that an initial displacement of the string results in one-half of the displacement shape propagating towards increasing x values along the string length, while the other half of the initial displacement shape propagates towards decreasing x values along the string length. The rate of propagation is c . Figure 14 shows a representative case of initial-value wave propagation when the initial velocity of the string is zero. The initial displacement profile is shown in the top frame of the figure. As time elapses, the wave breaks into forward and backward traveling wave components. By the time

2 /t a c= , it is clear that the initial displacement shape is equally propagating forward and backward at one-half of the original shape amplitude.


32

Figure 14. Snap-shots of one-dimensional wave propagation considering the initial displacement condition shown in the top-most panel.

3.1.2 Wave propagation at string boundaries

Two basic string boundaries are worth considering since they, and their analogs in fluid-borne wave propagation, are representative of the two key limiting cases of boundary interactions involving waves.

Consider a string rigidly supported at 0x = . For all time, the rightward and leftward x propagating waves must always sum to zero value at this boundary constraint. In other words,

( ) ( ) ( )1 20, 0 0y t y ct y ct= − + + (3.1.2.1)

One way in which this may be satisfied is if the rightward propagating wave is always negated by its leftward propagating wave which is mirrored over 0y = . Thus,

( ) ( ) ( )1 1,y x t y ct x y ct x= − − + (3.1.2.2)

This is illustrated in Figure 15(a) where, for a constant phase speed c , the summation of propagating waves

at 0x = always yields a zero value for ( )0,y t , leading to the summation result in the bottom figure panel.

The second common boundary condition is a free end. In this case, the string may displace at ( )0,y t but

has no transverse force at its constraint, sin 0T θ = . As shown in the derivation of the one-dimensional


33

wave equation, this is comparable to 0yTx∂

=∂

. Since the tension must remain finite, the string slope must

be zero at the free end boundary condition.

One way for the slope to the zero at the boundary condition is for rightward and leftward propagating waves to possess opposite slopes at 0x = , such that their summation is zero. Thus, the leftward propagating wave

( )2y ct x+ can be a mirror of ( )1y ct x− about 0x = .

( ) ( ) ( )1 1,y x t y ct x y ct x= − + + (3.1.2.3)

This cancellation of slopes of the string vibration at the free boundary is illustrated in Figure 15(b).

Figure 15. Illustrations of wave reflection phenomena at (a) rigid and (b) free supports. The top row shows the two traveling wave components while the bottom row shows the actual result which is the superposition of traveling waves in the domain

0x ≥ .

Table 3. Code used to generate Figure 15

x_l=linspace(-1,1,401); % create string x coordinate x_r=linspace(-1,1,401); % create string x coordinate y_r=(x_r-1.23).^1/4.*cos(4*pi*(x_r-.86)).*exp(-3.*(x_r.^2-3.1)); % create rightward propagating wave component y_l=-fliplr(y_r); % create leftward propagating wave component figure(1); clf; plot(x_r,y_r,x_l,y_l,x_r,y_r+y_l); xlabel('x'); ylabel('y(x,t)') legend('+x propagating wave','-x propagating wave','superposition of waves'); title('string reflection at rigid support') y_l=fliplr(y_r); % create leftward propagating wave component figure(2); clf plot(x_r,y_r,x_l,y_l,x_r,y_r+y_l); xlabel('x'); ylabel('y(x,t)') legend('+x propagating wave','-x propagating wave','superposition of waves'); title('string reflection at free support')


34

3.1.3 Forced vibration of a semi-infinite string

Consider a string that extends without end in the x+ axis starting from 0x = . The string is excited

transversally at 0x = by a harmonic force, ( ) cosf t F tω= . As a result, waves only propagate in the string

in the x+ direction away from the excitation boundary condition at 0x = . This hypothetical situation could be realized simply by consideration of a string of significantly greater length than the wavelength (to be defined below) under study, such that inherent dissipation mechanisms in the string prevent x− traveling waves from developing.

Using the complex exponential mathematical convention, the harmonic force is denoted

( ) j tt Fe ω=f (3.1.3.1)

The string response is therefore also most generally complex,

( ) ( )1,x t ct x= −y y (3.1.3.2)

Application of the boundary condition at 0x = yields the general expression

( )0, j tt e ω=y A (3.1.3.3)

where the unknown A must be determined from the boundary condition. Prior to this, we apply (3.1.3.3) to (3.1.3.2) to relate the general result to the rightward propagating wave component,

( ) ( )1

jk ctct e=y A (3.1.3.4)

Here, we introduce the wavenumber

/k cω= (3.1.3.5)

For all ct x− , we have for (3.1.3.2) that

( ) ( ) ( ), jk ct x j t kxx t e e ω− −= =y A A (3.1.3.6)

The top of Figure 16 illustrates the real part of the waveform (3.1.3.6), which is the physical and observable part, at several points in time along the string length. It is clear that for a given time t , the elements of the

string undergo harmonic oscillation along the string length, with an amplitude of A . The distance from

one peak of this oscillation to the next is the wavelength

2 / kλ π= (3.1.3.7)

Considering (114), for every 2k nλ π= with 1,2,...n = the waveform repeats in x .

As shown in the inset of Figure 16, for a fixed location of the string, harmonic oscillation of the

displacement ( )0 ,y x t occurs with an amplitude of A . The time elapsed for a point of constant phase on

the wave to repeat is T , which is the period. The period is related to the frequency via / 1 /T fπ ω= 2 =

where f is in [Hz]=[cycles/s].


35

Figure 16. Schematic of wave propagation in string due to harmonic force excitation at 0x = .

Because the waveform moves one wavelength λ in a time equal to one period T , we find that

c fλ= (3.1.3.8)

Summarizing, the wavelength λ is the spatial duration of the harmonic waveform. The wavenumber k is the amount of phase change of the waveform that occurs per unit distance. Analogously, the period T is the time duration of the harmonic waveform. The angular frequency ω is the amount of phase change of the waveform that occurs per unit time.

One may also refer to the angular frequency ω as the time rate of change of the waveform for a given spatial location. Likewise, the wavenumber k is the spatial rate of change of the waveform for a given time.

The complex amplitude A remains to be determined. The transverse force at the harmonically excited end results in the boundary condition

0x

Tx =

∂ = − ∂

yf (3.1.3.9)

By substitution of (3.1.3.1) and (3.1.3.6), we find

( )j t j tFe T jk eω ω= − − A (3.1.3.10)

indicating that the amplitude is

F FjjkT kT

= = −A (3.1.3.11)

As a result, the waveform of the string is completely described by


36

( ) ( ), j t kxFx t j ekT

ω −= −y (3.1.3.12)

The transverse speed of a string element particle is / t= ∂ ∂u y , so that

( ) ( ), j t kx

L

Fx t ec

ω

ρ−=u (3.1.3.13)

Like for mechanical oscillators described in Sec. 2.6, impedance measures are valuable in the analysis and development of systems supporting wave propagation. Thus, we introduce the input mechanical impedance

0mZ and define it as the ratio of the harmonic driving force to the string element particle speed at the

location of the driving force,

( )0 0,m t=

fZu

(3.1.3.14)

For the semi-infinite string, we find that

0m Lcρ=Z (3.1.3.15)

We find that the input mechanical impedance of the semi-infinite string is purely real. As described in Sec. 2.6, the real part of the impedance is associated with power delivery. Likewise, the units of the real part of impedance are plainly associated with damping mechanisms (of course, the units of Lcρ are [kg/s], the

same as mR ). Thus, for the harmonically driven semi-infinite string, the wave propagation away from the

driving end at 0x = acts to remove the energy injected into the driven end in a way analogous to a damping mechanism.

The instantaneous power input to the string is

[ ] [ ]0Re Rei x=Π = f u (3.1.3.16)

By substitution, we have

( )cos cosiL

FF t tc

ω ωρ

Π = (3.1.3.17)

while the time average of the power gives the average power delivered

220

1 12 2 L

L

F cUc

ρρ

Π = = (3.1.3.18)

where the input transverse speed amplitude is ( )0 0, / LU t F cρ= =u .

3.1.4 Forced vibration of a finite length string

When the string has finite length, rightward and leftward propagating waves must be taken into consideration and boundary conditions play significant roles on the resulting dynamic behaviors. Assuming


37

that transient responses associated with temporary disturbances or the start-up of the driving force are sufficiently decayed due to small inherent damping mechanisms and the passage of time, we conclude that two harmonic, steady-state waves must be employed to satisfy the wave equation and boundary conditions

( ) ( ) ( ), j t kx j t kxx t e eω ω− += +y A B (3.1.4.1)

3.1.4.1 Forced, fixed string

Consider the string to be driven by a harmonic force at 0x = and fixed at the opposite end at x L= . At the driven end,

0

j t

x

Fe Tx

ω

=

∂ = − ∂

y (3.1.4.1.1)

By substitution of (3.1.4.1) into (3.1.4.1.1), we obtain the equation

( )T jk jk F− =A B (3.1.4.1.2)

At the fixed end, we have the equation

0jkL jkLe e− + =A B (3.1.4.1.3)

Solving (3.1.4.1.2) and (3.1.4.1.3) yields

2 cos

jkLFejkT kL

= −A (3.1.4.1.4)

2 cos

jkLFejkT kL

−

= +B (3.1.4.1.5)

Then, by substitution, we express the general solution for the string motion in two ways, (3.1.4.1.6) and (3.1.4.1.7).

( ) ( ) ( ){ },2 cos

j t k L x j t k L xFx t j e ekT kL

ω ω+ − − − = − −y (3.1.4.1.6)

( )( )sin

,cos

j tk L xFx t ekT kL

ω − =y (3.1.4.1.7)

The (3.1.4.1.6) and (3.1.4.1.7) are mathematically equivalent, and are transformed from one to another by application of Euler's identity. Yet, they indicate a striking phenomenon associated with wave motion in systems of finite spatial extent. The (3.1.4.1.6) describes the resulting waveform as a combination of rightward and leftward traveling waves, where the waves have the same wavelength and amplitude. The (3.1.4.1.7) describes the resulting waveform in terms of a standing wave that oscillates harmonically by

j te ω . Thus, in systems of finite spatial extent, the resulting wave motion may be conceived as either waves traveling in opposite directions superposing to a unique response, or conceived as a single, standing wave shape that oscillates harmonically.


38

Considering the standing wave perspective in further detail, depending on the frequency ω , the numerator

term may possess zeros. In other words, ( )sin k L x − may have zeros called nodes associated with zero

displacement at all times. These nodes are found to exist according to the relation ( )k L x nπ− = where

0,1,2,.... /n kL π= ≤ . The resulting node locations are

/ 2nx L nλ= − ; 0,1,2,..., 2 /n L λ= ≤ (3.1.4.1.8)

A representative standing wave is shown in the top panel of Figure 17, with node locations shown by the locations along the string that retain zero displacement at all times, such as around 0.37x ≈ [m]. Conversely, antinodes are observed around locations 0.22x ≈ [m], 0.525x ≈ [m] , and 0.80x ≈ [m]. Antinodes are locations of local peak displacement in a standing waveform. The standing wave wavelength spans two nodes or two antinodes.


L=1; % [m] string length c=1e3; % [m/s] phase speed x=linspace(0,L,301); % [m] spatial duration of string F=1.123; % [N] harmonic driving force amplitude T=0.81; % [N] string tension omega=7.853981633974483e+03; % [rad/s] angular frequency of harmonic force k=omega/c; % [1/m] wavenumber times=2*pi/omega.*[1/8 1/7 1/6 1/5]; % [s] times for iii=1:length(times) y(:,iii)=F/k/T/cos(k*L)*sin(k*(L-x))*exp(j*omega*times(iii)); end figure(1); clf; plot(x,real(y)) xlabel('length along string [m]'); ylabel('string transverse displacement'); legend('1/8 period','1/7 period','1/6 period','1/5 period','location','best'); title(['c=' num2str(c) ' [m/s]. omega=' num2str(omega) ' [rad/s]. L=' num2str(L) ' [m].']); ylim(1.1*[min(min(real(y))) max(max(real(y)))])


39

Another feature is revealed from (3.1.4.1.7). The denominator becomes zero according to cos 0kL = when

2 12

nkL π−= ; 1,2,3,...n = (3.1.4.1.9)

Thus, the frequencies of the harmonic force associated with these conditions are

2 14n

n cfL

−= (3.1.4.1.10)

which are referred to as resonance frequencies. When the string is driven by the force at frequencies identified by (3.1.4.1.10), the string is driven at resonance such that the string transverse displacement will exhibit infinite-valued peaks at the antinodes. Thus, for even vanishingly small input force amplitudes F , the string response at the antinodes becomes unbounded. An example of this resonant excitation of the string is shown in the bottom panel of Figure 17. Note the large difference in vertical axis range comparing the off-resonant and on-resonant excitation of the string from the top to bottom panels of the figure. When the forced-fixed string is driven at resonance, there is always an antinode at the driven end, such that the string transverse speed is maximized.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

-0.1

-0.05

0

0.05

0.1

length along string [m]

strin

g tra

nsve

rse

disp

lace

men

t

c=1000 [m/s]. omega=10000 [rad/s]. L=1 [m].

1/8 period1/7 period1/6 period1/5 period

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

-4

-2

0

2

4

x 1014

length along string [m]

strin

g tra

nsve

rse

disp

lace

men

t

c=1000 [m/s]. omega=7853.9816 [rad/s]. L=1 [m].


Figure 17. Top panel, example of off-resonance excitation of forced-fixed string. Bottom panel, example of on-resonance excitation. Note the relative amplitudes of the string transverse displacement in comparison from top to bottom panels to exemplify the resonance condition in the bottom panel example.


40

Conversely, when cos 1kL = ± , the denominator of (3.1.4.1.7) is at its greatest value, such that string motion is minimized along its length. The frequencies at which these conditions occur are

2mm cf

L= ; 1,2,3,...m = (3.1.4.1.11)

which are termed antiresonance frequencies. Under such conditions, the string possesses a node at the driven end, such that the string transverse speed is zero.

The input mechanical impedance of the string is determined as before using (3.1.3.14). By substitution of (3.1.4.1.7) into (3.1.3.14), it is found that

0 cotm Lj c kLρ= −Z (3.1.4.1.12)

In this case, (3.1.4.1.12) shows that the input mechanical impedance of the forced-fixed string is purely imaginary, thus only reactance contributes to the impedance. For a purely lossless (undamped) string, there is no way for energy to leave the system and thus with the passage of time the string vibration should always become unbounded regardless of on- or off-resonance excitation. In reality, inherent damping mechanisms in real strings result in a balance of energy dissipated and energy injected by the harmonic force. Resonant excitation examples result in significantly larger amplitudes of such string motion than off-resonant excitation, but the energy balance between dissipation and energy input occurs as well.

Considering (3.1.4.1.12), substitution of the values kL corresponding to resonance, (3.1.4.1.9), reveals that the input mechanical impedance vanishes. Because (3.1.4.1.12) is a pure reactance, the more general statement to conclude is that for any mechanical system, resonant excitation is associated with a vanishing of the input mechanical reactance.

3.1.5 Normal modes of the fixed-fixed string

Consider a string fixed at both ends, 0x = and x L= . Thus, ( ) ( )0, , 0y t y L t= = . While we established

the efficacy of the assumed solution (3.1.4.1) for the case of harmonic excitation of the string, in the absence of the harmonic force we now take a step back and address the one-dimensional wave equation (3.1.6) by a more general approach often used to analyze PDEs.

According to the method of separation of variables, we assume that a separable solution to (3.1.6) may be found whereby

( ) ( ) ( ), X Ty x t x t= (3.1.5.1)

where the response y is purely real and is determined as the product of two functions ( )X x and ( )T t that

are dependent only on their respective space and time variables. Substitution of (3.1.5.1) into (3.1.6) yields

( ) ( ) ( ) ( )2

1X T X Tx t x tc

′′ = (3.1.5.2)

The corresponding boundary and initial conditions are (3.1.5.3) and (3.1.5.4), respectively.


41

( ) ( )X 0 T 0t = ; ( ) ( )X T 0L t = (3.1.5.3)

( ) ( ) ( )X T 0x U x= ; ( ) ( ) ( )X T 0x V x= (3.1.5.4)

The prime operator indicates derivatives with respect to x and the overdot operator indicates derivatives with respect to t . The (3.1.5.2) is rearranged to be

( )( )

( )( )2

X T1X T

x tx c t

′′=

(3.1.5.5)

It is seen that the left-hand side of (3.1.5.5) is independent of t while the right-hand side of (3.1.5.5) is independent of x . As a result, for (3.1.5.5) to be true for all x and t , both sides of the equation must be equal to the same constant, σ .

( )( )

XX

xx

σ′′

= (3.1.5.6)

( )( )2

T1T

tc t

σ=

(3.1.5.7)

The (3.1.5.6) is then the BVP of this IVBP, while (3.1.5.7) is the IVP of this IVBP.

Considering (3.1.5.6),

( ) ( )X X 0x xσ′′ − = (3.1.5.8)

with the boundary conditions

( ) ( )X 0 X 0L= = (3.1.5.9)

it is discovered that the nature of (3.1.5.8) is governed by the unknown constant. Investigations of such equations shows that solutions exist only when 0σ < [11]. For later convenience, we select the constant to

be of the form 2σ φ= − . Thus,

( ) ( )2X + X 0x xφ′′ = (3.1.5.10)

Past experience shows that solutions to (3.1.5.10) are the summation of two sinusoidal functions,

( ) 1 2X sin cosx A x A xφ φ= + (3.1.5.11)

Satisfying the fixed-fixed boundary conditions, (3.1.5.9), indicates that

2 0A = ; 1 sin 0A Lφ = (3.1.5.12)

where non-trivial solutions to (3.1.5.12) occur only when sin 0Lφ = , thus


42

nnLπφ = ; 1,2,3,...n = (3.1.5.13)

Consequently, the total solution to (3.1.5.10) is the superposition of an infinite number of terms

X sinn nnA xLπ =

(3.1.5.14)

Since the constants nφ are now known, they may be substituted into the IVP (3.1.5.7).

( ) ( )2

2T T 0nt c tLπ + =

(3.1.5.15)

Like before, the general solution to (3.1.5.15) is a summation of sinusoidal functions. Now, we consider

the thn of these solutions which is

( ) ( ) ( )1 2T sin cosn n nn c n ct A t A t

L Lπ π = +

(3.1.5.16)

where the superscripts in (3.1.5.16) are notational and do not refer to power operations.

Taking the product of the thn solutions for the IVP and BVP, we find the general thn solution to (3.1.6) for the fixed-fixed string is

( ), sin sin cos sinn n nn c n n c ny x t a t x b t x

L L L Lπ π π π = +

(3.1.5.17)

where the constants na and nb are the products of the constants ( )1n nA A and ( )2

n nA A , respectively.

The general response of the string is therefore

( )1

, sin sin cos sinn nn

n c n n c ny x t a t x b t xL L L Lπ π π π∞

=

= +

∑ (3.1.5.18)

The initial conditions (3.1.5.4) are applied to determine the constants na and nb .

( )1

sinnn

nU x b xLπ∞

=

=

∑ (3.1.5.19)

The functions ( )sin /n x Lπ are orthogonal, such that we take advantage of the orthogonality property

0

1 ;1 sin sin 20;

L

mnm nn mx x dx

L L L m n

π π δ = = =

≠∫ (3.1.5.20)

Thus, the constants nb are found by multiplying (3.1.5.19) by ( )sin /m x Lπ and integrating over the length,


43

( )0

2 sinL

nnb U x x dx

L Lπ =

∫ (3.1.5.21)

A similar procedure is used in the application of the second initial condition, on the string velocity along

its length ( ) ( ),0y x V x= , (3.1.5.4).

( ) ( )0 0

2 2sin sinL L

nn

n na V x x dx V x x dxn c L L L

π ππ ω

= = ∫ ∫ (3.1.5.22)

Because there is no harmonic force excitation, these results indicate that the string response only occurs at

frequencies /n n c Lω π= and in spatial distributions ( )sin /n x Lπ . The extents to which the thn of these

frequencies and spatial distributions are activated are determined by the initial conditions through the constants (3.1.5.21) and (3.1.5.22).

The shapes ( )sin /n x Lπ are referred to as the normal modes of vibrations. The frequencies /n n c Lω π=

are referred to as the natural frequencies. Thus, when given initial disturbances and in the absence of excitation mechanisms, the string will undergo free vibration at the natural frequencies each of which is associated with a normal mode. Reflecting upon the steps used to determine the normal modes and the natural frequencies, the application of the boundary conditions yields both such information while application of the initial conditions via Fourier's theorem identifies the exact contribution of normal modes to a given free vibration response.

The orthogonality property (3.1.5.20) indicates that the normal modes are linearly independent. Thus, if the string is given an initial condition of displacement identical to one of the normal modes, the string will respond only in that particular mode for all time.

One arrives at the same results as (3.1.5.18), (3.1.5.21), and (3.1.5.22) if a complex exponential form of assumed solution is adopted, like (3.1.4.1). For instance, see [1] Sec. 2.10. The utilization of the method of separation of variables here demonstrates an alternative and general means by which these conclusions may be reached regarding the free vibration response of systems that permit wave propagation.

3.2 Longitudinal waves along a one-dimensional rod

Unlike the string displacements which are transverse to the axis of wave motion, longitudinal waves involve harmonic compression and rarefaction of particles in the axis of wave propagation. Longitudinal waves occur in myriad structural and acoustic contexts and we first encounter them in the structure of the one-dimensional rod.

Consider a rod whose lateral dimensions are small compared to the length, such that each cross-sectional plane of the rod moves identically according to the stresses acting on the elemental face. Figure 18 provides a schematic of the rod. The cross-sectional area is S units [m2], while the rod density and Young's modulus are, respectively, ρ [kg/m3] and Y [N/m2].


44

Figure 18. Schematic used to formulate equation of motion for the longitudinal waves propagating in a one-dimensional rod.

The free-body diagram of the differential element of the rod is shown in Figure 18(b). The forces on the faces of the cross-sectional areas are determined via Hooke's law. Considering the left face of the differential element, we find

( ) ( ), ,F x t S x tσ= (3.2.1)

where ( ),x tσ is the stress on the face, and we adopt the convention that the stress is positive in compression

and negative in tension. Stress is related to the material strain via the constitutive relation

( ) ( ), ,x t Y x tσ ε= − (3.2.2)

In (3.2.2), the positive compressive stress is a result of a negative strain; because the Young's modulus Y is positive, the right-hand side of (3.2.2) is likewise a positive value. Although this convention is the reverse of that used by materials scientists, it is convention in the context of studying acoustics because positive pressure changes correspond to decreases in the volume of a fluid.

Finally, we use the strain-displacement relation for the compressive stress

( ) ( ),,

x tx t

xξ

ε∂

=∂

(3.2.3)

All together, the force on the differential element face is

( ) ( ),,

x tF x t SY

xξ∂

= −∂

(3.2.4)

The (3.2.4) is Hooke's law for the rod. The force on the right face of the differential element is

( ) ( ),,

x dx tF x dx t SY

xξ∂ +

+ = −∂

(3.2.5)

A Taylor series approximation for the displacement is taken to yield


45

( ) ( ) ( ),, ,

x tx dx t x t dx

xξ

ξ ξ∂

+ ≈ +∂

(3.2.6)

This approximation is illustrated in Figure 18(c). Of course, for small deformations from the original element, the approximation is reasonably accurate. For all purposes in this course, we will consider such small amplitude perturbations from an equilibrium which form the underpinnings of a significant proportion of topics in acoustics.

By using (3.2.6), we find that (3.2.5) is

( ) ( ) ( ) ( ) ( )2

2

, , ,, ,

x t x t x tF x dx t SY x t dx SY SY dx

x x x xξ ξ ξ

ξ ∂ ∂ ∂∂

+ ≈ − + = − − ∂ ∂ ∂ ∂ (3.2.7)

The average acceleration of the differential element is illustrated in Figure 18(c) such that the inertial force is

( )2

2

,x tS dx

tξ

ρ∂

∂ (3.2.8)

Using Newton's second law, we obtain the governing equation of motion for the rod

( ) ( ) ( )2

2

,, ,

x tF x t F x dx t S dx

tξ

ρ∂

− + =∂

(3.2.9)

( ) ( ) ( ) ( )2 2

2 2

, , , ,x t x t x t x tSY SY SY dx S dx

x x x tξ ξ ξ ξ

ρ ∂ ∂ ∂ ∂

− − − − = ∂ ∂ ∂ ∂ (3.2.10)

( ) ( )2 2

2 2

, ,x t x tSY S

x tξ ξ

ρ∂ ∂

=∂ ∂

(3.2.11)

The (3.2.11) is then rearranged

2 22

2 2cx tξ ξ∂ ∂=

∂ ∂ (3.2.12)

The (3.2.12) is also a one-dimensional wave equation. There is clearly a fundamental shared physics between the propagation of transverse waves in a stretched string and the propagation of longitudinal waves in rods. Indeed, further shared physics will be seen in the continued study of other elastic and acoustic systems throughout this course.

Similar to the case of strings, we likewise refer to term c as the phase speed. Now the relation of the material properties of the rod to the phase speed is

/c Y ρ= (3.2.13)

In SI units, the phase speed is [m/s].


46

Because the longitudinal deformations of the rod are governed by the one-dimensional wave equation, we use the general form of solution to satisfy (3.2.12). Thus,

( ) ( ) ( )1 2,x t ct x ct xξ ξ ξ= − + + (3.2.14)

The complex exponential form of assumed solution is here

( ) ( ) ( ), j t kx j t kxx t e eω ω− += +ξ A B (3.2.15)

Because (3.2.12) is mathematically equivalent with (3.1.6), we promptly (and rightly) conclude that a fixed-fixed rod exhibits the same normal modes and natural frequencies a a fixed-fixed string. As a result, we study a different example of boundary conditions applied to the rod.

Consider an example case of a rod with free boundaries at both 0x = and x L= . While a finite length string with both ends free can be realized using two slider ends, such as that illustrated in Figure 15(b), it is much easier to realize a free-free rod in practice (e.g. rest a rod on a soft foam bed) and such an elementary structure may be used in the study of vibration or wave control.

A free boundary on the rod must have no force acting on the cross-sectional area element. Thus,

( ) ( )0,0, 0

tF t SY

xξ∂

= = −∂

; ( ) ( ),, 0

L tF L t SY

xξ∂

= = −∂

(3.2.16)

It is observed then that

( ) ( )0, ,0

t L tx x

ξ ξ∂ ∂= =

∂ ∂ (3.2.17)

Inserting the assumed solution (3.2.15) into the two components of (3.2.17) yields

0jk jk− + =A B ; B = A ; ( ), 2 cosj tx t e kxω=ξ A (3.2.18)

2 sin 0k kL− =A (3.2.19)

where (3.2.19) indicates that responses are permitted only when

sin 0 ; 1,2,3,...nn ckL n

Lπω= → = = (3.2.20)

The natural frequencies of the free-free rod are the same as those of the fixed-fixed rod. On the other hand, the normal modes are cosines and not sines. The normal modes of the free-free rod are

cos cosnnk x xLπ =

(3.2.21)

Thus, the thn mode response of the free-free rod vibrates according to

( ), 2 cosnj tn n nx t e k xω=ξ A (3.2.22)


47

Also recognize that in the process of steps (3.2.18) we simplified the general expression of assumed solution (3.2.15) using Euler's identity and implicitly transitioned our perspective from referring to oppositely traveling waves to standing waves. Yet, the mathematical procedures all ultimately yield the same results once computed. Thus, the free-free rod undergoing vibration in the normal modes may be perceived as oscillating in standing waves or as permitting a pair of oppositely traveling waves along its length.

3.3 Transverse waves along a beam

While longitudinal waves and vibrations are permitted in many engineering structures, a more common form of response due to arbitrary loads is bending, which is a motion transverse to the axis of the structure. The same rod that allows longitudinal waves to travel along its length is likely to oscillate in bending if an excitation acts off the center of the axis. In an opposite extreme of excitation to the rod, excitations that are normal to the axis will induce only bending motions.

To differentiate bending vibrations and waves from longitudinal counterparts, we refer to the elementary structure here as the beam although, as described above, a realistic structure that permits longitudinal compressions along the primary axis will also permit bending motions transverse to its axis. Consider the schematic shown in Figure 19(a). The beam has a uniform cross-section S and is symmetric through its thickness. The x coordinate denotes positions along the beam length while y denotes the transverse

displacement, similar to our convention in deriving the wave equation for the string. The ξ denotes

longitudinal stretching of a filament of the beam along its length dimension in x .

Figure 19. Schematic of beam bending.

When the beam is bent, the lower part of the beam becomes compressed while the upper part is under tension, as exemplified in the schematic of Figure 19(a). The neutral axis length remains unchanged from the undeformed state. The bending is characterized by the radius of curvature R of the neutral axis. The

strip of the beam r from the neutral axis is stretched (or compressed) by an amount ( )/x x dxδ ξ= ∂ ∂ . As

a result, a longitudinal force develops for this strip of the beam given by

xdf YdS YdSdx xδ ξ∂

= − = −∂

(3.3.1)


48

The sign convention according to Figure 19 is that the positive stretch xδ denotes a tensile force, which is shown by the negative sign of the force in (3.3.1). Geometrically, we have

dx x dx x rR r R dx R

δ δ+= → =

+ (3.3.2)

Thus, we have that the force is

Ydf rdSR

= − (3.3.3)

The bending moment M in the beam develops by virtue of the distortion of the infinitely many beam strips, so that

2YM rdf r dSR

= = −∫ ∫ (3.3.4)

A constant is introduced as

2 21 r dSS

κ = ∫ (3.3.5)

where κ is referred to as the radius of gyration. For a beam, the radius of gyration is often given according

to its second moment of area I and cross-sectional area S according to 2 /I Sκ = . For beams of cross-sectional area of width b and height h bent along their height dimension, the relevant moment of area is

the common expression 3 /12I bh= .

Thus, the bending moment is simplified in expression to be

2YSMRκ

= − (3.3.6)

When the displacements of the beam from its undeformed state are small / 1y x∂ ∂ << such that the radius

of curvature is large, we may use the approximation

3/22

2 2

2 2

11

yx

Ry y

x x

∂ + ∂ ≈ ≈∂ ∂∂ ∂

(3.3.7)

which enables the approximation of the bending moment as

2 22

2 2

y yM YS YIx x

κ ∂ ∂= − = −

∂ ∂ (3.3.8)

The (3.3.8) relates the bending moment to the out-of-axis (transverse) displacement y .


49

Shear forces are also developed in a beam in bending. Consider Figure 19(b) that denotes the establishment of upward shear force yF at x while a downward shear force is created at x dx+ along the beam length.

For a beam in equilibrium, no net moment of the beam segment must be induced. Thus, taking moments about x , we have the constraint that

( ) ( ) ( )yM x M x dx F x dx dx− + = + (3.3.9)

Expanding the moments using a linearized Taylor series, assuming a small beam segment and small moments, yields the expression

3 32

3 3yM y yF YS YIx x x

κ∂ ∂ ∂= − = =

∂ ∂ ∂ (3.3.10)

The net force on the beam segment in the y (vertical) axis is therefore

( ) ( )4 4

24 4

yy y y

F y ydF F x F x dx dx YS dx YI dxx x x

κ∂ ∂ ∂

= − + = − = − = −∂ ∂ ∂

(3.3.11)

Newton's 2nd law then equates this force (3.3.11) to the net inertial force. Considering a beam volumetric density given by ρ , we have that

( )4 4 2

24 4 2

y YI y ycx S x t

κρ

∂ ∂ ∂− = − =

∂ ∂ ∂ (3.3.12)

where the constant is defined 2 /c Y ρ= .

Eq. (3.3.12) is not a wave equation. In particular, the general function ( )1f ct x− is not a solution. This is

evidence that transverse waves do not travel along beams with unchanging shape.

To solve (3.3.12), we apply the method of separation of variables, now assuming time-harmonic form for the time-dependent component. Thus,

( ) ( ), j tx t x e ω=y Y (3.3.13)

Then (3.3.13) is substituted into (3.3.12), yielding

44

4

ddx v

ω =

Y Y (3.3.14)

( )2v cω κ= (3.3.15)

where (3.3.15) states that the frequency ω is a function of cκ , and v has units of speed. Assuming a form

of ( ) xx e= γY , substitution into (3.3.14) reveals that this assumed solution is valid so long as ( )/ vω= ±γ

and ( )/j vω= ±γ . The four possible forms of γ are anticipated due to the fourth-order differential equation

(3.3.14). Thus, we define a new term


50

24/ Sg v

YIρ ωω= = (3.3.16)

so that the complete solution to (3.3.14) is

( ) gx gx jgx jgxx e e e e− −= + + +Y A B C D (3.3.17)

( ) ( ) ( ) ( ), j t gx j t gxgx gx j tx t e e e e eω ωω + −−= + + +y A B C D (3.3.18)

It is apparent that the unknown g has units of inverse length, which reveals that g is the wavenumber for

the transverse bending waves along the beam. The general solution (3.3.18) is associated with two traveling waves, associated with complex amplitudes C and D , and two standing waves with spatial attenuation coefficients, associated with complex amplitudes A and B . The real part of (3.3.18) is the actual beam response

( ) [ ] ( ), cosh sinh cos sin cosy x t A gx B gx C gx D gx tω φ= + + + + (3.3.19)

while the unknown constants are related to the complex unknown constants in (3.3.18) through application of boundary and initial conditions. Application of the boundary conditions alone, at a minimum, provides definition of the wavenumber values g for free vibration.

Interestingly, (3.3.15), (3.3.16), and (3.3.18) show that the wave propagation in the beam is frequency dependent. This leads to a discussion of dispersion in wave propagation.

3.4 Dispersion, phase velocity, group velocity

Waves traveling in media may exhibit two types of speeds. The more common terminology to use here are velocities, and we temporarily employ this nomenclature.

Consider a monofrequency waveform ( ) ( ), cosy x t Y t kxω= − . The wavenumber is 2 /k π λ= and the

angular frequency is 2 / Tω π= . The phase velocity is therefore / /pc k Tω λ= = . This represents the

velocity at which points of constant phase on the wave travel.

Now consider a wave composed of two frequencies, 1ω and 2ω , but the same amplitude 0Y

( ) ( ) ( )0 1 1 0 2 2, cos cosy x t Y t k x Y t k xω ω= − + − (3.4.1)

1 1; ;k k k ω ω ω ω ω= + ∆ = + ∆ ∆ << (3.4.2)

2 2; ;k k k k kω ω ω= − ∆ = − ∆ ∆ << (3.4.3)


51

Figure 20. Wave packet travel. Also see https://en.wikipedia.org/wiki/Group_velocity

Trigonometric identities enable us to write (3.4.1) as

( ) ( ) ( )0, 2 cos cosy x t Y t kx t kxω ω= − ∆ − ∆ (3.4.4)

The term in brackets in (3.4.4) is the wave modulation. This phenomenon is shown in Figure 20. The distance between individual waveform peaks corresponds to the period 2 /T π ω= , while the distance

between the slow modulation nulls is 0 /T π ω= ∆ .

Considering the phase velocity definition, the modulation velocity is / kω∆ ∆ . This is referred to as the group velocity

g gc ck kω ω∆ ∂

= → =∆ ∂

(3.4.5)

The group velocity is the rate at which a wave packet travels. Because a series of wave phase information is encapsulated in the packet, the group velocity may be considered as the rate at which information in the wave travels.

Dispersion is the phenomenon in which phase velocity depends on frequency. A consequence of this is that the group and phase velocities are no longer the same. For non-dispersive media, p gc c= . For dispersive

media, the ω is a nonlinear function of k .

Consider the longitudinal wave propagation along a one-dimensional rod. We found that

2 22

2 2cx tξ ξ∂ ∂=

∂ ∂ (3.4.6)

With a time-harmonic assumed solution currently considering wave propagation only in the x+ direction,

( ) ( ), j t kxx t e ω −=ξ A (3.4.7)

we have that

0 0.5 1 1.5 2 2.5 3

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

x

y(x,

t)


52

2 2 2c k kcω ω − = − → = (3.4.8)

The angular frequency is a linear function of the wavenumber. Thus, the phase velocity and group velocity are (3.4.9) and (3.4.10), respectively.

/pc k cω= = (3.4.9)

gc ckω∂

= =∂

(3.4.10)

Thus, the longitudinal wave propagation in a rod is non-dispersive, p gc c= . Formally, the dispersion

relation is the expression relating the frequency to the wavenumber, although here the relation is ( )k kcω =

which is linear such that longitudinal wave propagation in a rod is non-dispersive.

Now consider the transverse bending wave propagation in a beam. We found

( )4 2

24 2

y ycx t

κ ∂ ∂− =

∂ ∂ (3.4.11)

With a time-harmonic assumed solution currently considering wave propagation only in the x+ direction,

( ) ( ), j t kxx t e ω −=y A (3.4.12)

we have that

( )2 4 2 2c k k cκ ω ω κ− = − → = (3.4.13)

The angular frequency ω is a nonlinear function of k . The dispersion relation for transverse bending wave

propagation is ( ) 2k k cω κ= .

Thus, the phase velocity and group velocity are (3.4.14) and (3.4.15), respectively.

/pc k k cω κ= = (3.4.14)

2 2g pc k c ckω κ∂

= = =∂

(3.4.15)

Thus, transverse bending wave propagation in a beam is dispersive, p gc c≠ .

Contextually, it is often the group velocity that is more important since the group velocity governs the transmission rate of the information or wave packet while the phase velocity is the transmission rate of points of constant phase on the wave. Such points of constant phase are not inherently associated with information since information requires a series of phase data to establish the wave packet.


53

Considering (3.4.13) in another light, the dispersion of the waves is manifest by the fact that /k cω κ=

, which reveals that higher frequency waves move at faster rates than lower frequency waves. A comparison of non-dispersive and dispersive wavenumber-frequency relationships is given in Figure 21.

Figure 21. Dispersive and non-dispersive wavenumber-frequency relationships.

3.5 Forced excitation of beams

Examination of several cases of forced excitation of beams is useful to appreciate how the injected energy contributes to waves propagation in dispersive structures.

3.5.1 Point force excitation of infinite beam

Considering a time-harmonic form of driving force ( ) j tt Fe ω=f at a central point 0x = on a beam of

infinite spatial extent in x± directions, we use the notation of the transverse beam bending equation of motion that employs the alternative relations to the radius of gyration, thus

( )4

24YI S F x

xρ ω δ∂

− =∂

y y (3.5.1.1)

where ( )xδ is the Dirac delta function. For the unforced beam, we identify 2

4f

SkYI

ρ ω= ± following

substitution of the comparable terms for radius of gyration into (3.4.13). For completeness, also 2

4f

Sk jYI

ρ ω= ± .

For the forced system, we introduce the spatial Fourier transform to assess (3.5.1.1). The spatial Fourier transform uses a pair of integral relations to make the connection between spatial frequency (wavenumber) and displacement. This integral pair is analogous to the traditional Fourier transform that relates functions of frequency and functions of time. Thus, the spatial Fourier transform pair is

1000 2000 3000 4000 5000 60000

0.2

0.4

0.6

0.8

1

1.2

1.4

frequency [rad/s]

wav

enum

ber [

1/m

]

k=ω/c

k=[ω/c]1/2


54

( ) ( ) jkxk x e dx+∞

−∞= ∫Y y (3.5.1.2)

( ) ( )12

jkxx k e dxπ

+∞ −

−∞= ∫y Y (3.5.1.3)

Multiplication of (3.5.1.1) by jkxe and integration from negative to positive infinity yields

( ) ( )4 4f

FkYI k k

=−

Y (3.5.1.4)

Analogous to Laplace transform operations, the inverse transform (3.5.1.3) is used to convert the spatial frequency domain (wavenumber domain) response (3.5.1.4) back into the response in the spatial domain. With considerable simplifications [12], we obtain for the response 0x >

( ) ( )3,4

f fjk x k x j t

f

jFx t e je eYIk

ω− −= − −y (3.5.1.5)

A similar result is obtained for waves in the 0x < domain. The result (3.5.1.5) shows that two waves propagate away from the driving point force into the x+ direction. One wave travels, having a complex

exponential term fjk xe− , while a second wave decays exponentially away from the force, having the

standard exponential term fk xe− . The prior is of course a traveling wave while the latter is referred to as an evanescent wave. This is evidence of the dispersion in the media: that the wave shape does not remain constant in the course of wave travel. Yet, it should be emphasized that evanescent waves are not only found in dispersive media. Indeed, evanescent waves appear prominently in the study of fluid-borne wave propagation through multiple fluid media, and fluids are non-dispersive. Finally, we note that the real part of (3.5.1.5) is the actual or measurable bending wave response in the beam.

An example of the combined nature of the traveling and evanescent waves for the x+ domain of the infinite beam is given in Figure 22. The evanescent components of the wave decay relatively quickly away from the excitation point at 0x = as evidenced by the relatively quick convergence of the waveforms (at different time snap shots) to common waveform shapes for increasing values of x .


55

Figure 22. Evidence of evanescent and traveling waves at a snap shot of time along an infinite beam forced at its center

point 0x = . 1F = [N], 70 9Y e= [N/m2], 2 3eρ = [kg/m3], 0.109S = [m2], 1 3I e= − [m4], 9 3eω = [rad/s].

3.5.2 Arbitrary force excitation of finite beam

Consider again a harmonic force excitation of the beam, but now the force is arbitrary in spatial distribution

( ) ( ), j tx t F x e ω=f and the beam is of finite extent with simply-supported boundary conditions 0x = and

x L= . The governing equation of motion is

( )44

4j t

f

F xk e

x YIω∂

− =∂

y y (3.5.2.1)

while the associated boundary conditions are

( ) ( )0, , 0t L t= =y y (3.5.2.2)

( ) ( )2 2

2 2

0, ,0

t L tx x

∂ ∂= =

∂ ∂y y

(3.5.2.3)

Substitution of the assumed solution ( ) ( ) ( ), j t kx j t kxx t e eω ω− += +y A B into the homogeneous form of (3.5.2.1)

and into the associated boundary conditions reveals the normal modes (3.5.2.4), and natural frequencies and wavenumbers (3.5.2.5) of the system.

( ) ( )sinn n fny x Y k x= (3.5.2.4)

1/2 2

nYI n

S Lπω

ρ =

; fn

nkLπ

= (3.5.2.5)

In this way, the total bending response of the simply-supported beam is

0 0.5 1 1.5 2 2.5 3-8

-6

-4

-2

0

2

4

6x 10

-11

x

y(x,

t)



56

( ) ( )1

, sin j tn fn

nx t Y k x e ω

∞

=

=∑y (3.5.2.6)

For the inhomogeneous form of (3.5.2.1), in other words with the harmonic force ( ) j tF x e ω , we hypothesize

that the response of the beam will also be a linear superposition of the normal modes where the contribution from each mode to the total response is computed in a way similar as for the initial condition contribution in the free response, such as for the string Sec. 3.1.5.

Substituting (3.5.2.6) into (3.5.2.1), we find

( )

( )

42

1

4 4

1

sin

sin

nn

fn f nn

F xn S nY xL YI L YI

F xnk k Y xL YI

π ρ πω

π

∞

=

∞

=

− =

− =

∑

∑ (3.5.2.7)

Taking advantage of the orthogonality property of the normal modes, we multiply (3.5.2.7) by sin m xLπ

and integrate over the beam length, to find the modal amplitudes.

( ) ( )4 4 0

2 sinL

nfn f

nY F x x dxLYIL k kπ = − ∫ (3.5.2.8)

To investigate (3.5.2.8), a form of the force distribution ( )F x must be examined. Consider then a point

force acting on the beam at location 0x , where 00 x L< < . The force is therefore ( ) ( )0 0, j tx t F x x e ωδ= −f

. Evaluation of the integral (3.5.2.8) yields

04 4

2 1 sinnfn f

F nY xYIL k k L

π = − (3.5.2.9)

Then, the total response is determined by (3.5.2.6),

( ) ( ) ( )04 41

2 1, sin sin j tfn fn

n fn f

Fx t k x k x eYIL k k

ω∞

=

=−∑y (3.5.2.10)

recalling the distinction that ( )44 /fnk n Lπ= and 4 2 /fk S YIρ ω= .

Recognizing an embodiment of Euler's identity using ( )sin / 2fn fnjk x jk xfnk x j e e−= − , (3.5.2.10) is again

indicative of the duality of perceiving the oscillations of a system of finite spatial extent as being composed of standing waves or a pair of oppositely traveling waves. When nω ω≠ , the response is more associated

with a pair of traveling waves along the beam. Alternatively, when nω ω≈ the denominator of (3.5.2.10)

nearly vanishes and there is strong reinforcement of the normal mode response which would commonly appear as more of a significant standing wave in practice.


57

Examples of such off- and near-resonant point force excitation of the simply-supported beam are shown in Figure 23 using 20 modes in the linear superposition, and a point force located at one-third the distance along the beam length. For the off-resonant excitation case shown in the top panel, the off-center point force excitation is evident by virtue of the significant asymmetry in response that is weighted towards the excitation location / 3L . Yet, this feature is not readily observed when the beam is excited near to resonance, as shown in the bottom panel since the normal mode motion dominates.


num_modes=20; % number of modes to use in summation L=1; % [m] beam length x0=L/4; % [m] point force excitation point along beam length F=1; % [N] point force amplitude Y=72e9; % [N/m^2] young's modulus of beam I=1e-5; % [m^4] second moment of area rho=2e3; % [kg/m^3] density S=0.1; % [m^2] cross-sectional area omega=1.00001*sqrt(Y*I/rho/S)*(3*pi/L)^2; % [rad/s] excitation frequency time=2*pi/omega*[1/8 1/7 1/6 1/5]; % [s] times x=linspace(0,L,401); % [m] define beam spatial extent y=zeros(length(x),length(time)); % pre-allocate beam bending vector for ooo=1:length(time) for iii=1:num_modes

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-6

-4

-2

0

2

4x 10

-10

length along beam [m]

y(x,

t), b

eam

ben

ding

dis

plac

emen

t [m

]

frequency 6555.3912 [rad/s], 1.23 times the third natural frequency


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-1

-0.5

0

0.5

1x 10

-5

length along beam [m]

y(x,

t), b

eam

ben

ding

dis

plac

emen

t [m

]

frequency 5329.6397 [rad/s], 1 times the third natural frequency


Figure 23. Top panel, example of slightly off-resonant point force excitation of simply-supported beam. Bottom panel, example of near-resonant excitation.


58

y(:,ooo)=y(:,ooo)+1/((iii*pi/L)^4-(rho*S*omega^2/Y/I)).*sin(iii*pi/L*x0).*sin(iii*pi/L*x)'.*exp(j*omega*time(ooo)); end end figure(1); clf; plot(x,2*F/Y/I/L*real(y)) xlim([min(x) max(x)]); xlabel('length along beam [m]'); ylabel('y(x,t), beam bending displacement [m]'); legend('1/8 period','1/7 period','1/6 period','1/5 period','location','best') title(['frequency ' num2str(omega) ' [rad/s], ' num2str(omega/(sqrt(Y*I/rho/S)*(3*pi/L)^2)) ' times the third natural frequency']);

3.6 Waves in membranes

Having evaluated numerous examples of wave propagation in one-dimensional systems, characteristics of wave propagation in two-dimensional systems is a logical next step for study. We first consider the membrane, which is the two-dimensional analog of the string.

We assume that the membrane is thin, is stretched uniformly in directions x and z , and undergoes small

displacements ( ), ,y x z t away from equilibrium. A schematic of the membrane element under consideration

is shown in Figure 24. The membrane is stretched with tension τ per unit length [N/m] and possesses a

linear density of Sρ [kg/m2]. Forces exerted by the tensions in x and z axes displace the membrane out-

of-the-page, respecting the view of Figure 24.

Figure 24. Schematic of membrane element.

Such net upward force induced by the opposing tensions dzτ is

2

2x dx x

y y ydz dxdzx x x

τ τ+

∂ ∂ ∂ − = ∂ ∂ ∂ (3.6.1)

where a linearized Taylor series expansion is used for x dx

yx +

∂ ∂

. Similarly, the net upward force induced

by the opposing tensions dxτ is


59

2

2z dz z

y y ydx dxdzz z z

τ τ+

∂ ∂ ∂ − = ∂ ∂ ∂ (3.6.2)

The summation of these forces must be equal to the net inertial force, by Newton's 2nd law of motion. As a result, the governing equation for the membrane is

2 2 2

2 2 2 2

1y y yx z c t∂ ∂ ∂

+ =∂ ∂ ∂

(3.6.3)

where the constant is defined 2 / Sc τ ρ= . The more general expression of (3.6.3) is

22

2 2

1 yyc t

∂∇ =

∂ (3.6.4)

where 2∇ is the Laplacian operator, which is dependent upon the coordinate system considered. The (3.6.4) is the two-dimensional wave equation.

The Laplacian operator is ordinarily selected for convenience based on the system geometry under study. For a system that is mostly rectangular in geometry, a Cartesian coordinate system is convenient. Thus the Laplacian for Cartesian coordinates is

2 22

2 2x z∂ ∂

∇ = +∂ ∂

(3.6.5)

For systems that possess a circular geometry, the polar coordinate system ( ),r θ is preferable such that the

Laplacian becomes

2 22

2 2 2

1 1r r r r θ∂ ∂ ∂

∇ = + +∂ ∂ ∂

(3.6.6)

Assumption of a general time-harmonic response encourages a separable assumed solution to (3.6.4) of

j te ω=y Y (3.6.7)

where the complex component Y is a function only of space. Substitution of (3.6.7) into (3.6.4) yields the Helmholtz equation

2 2 0k∇ + =Y Y (3.6.8)

Solution to (3.6.8) is the means to compute the normal modes of the system subject to known boundary conditions.

3.6.1 Free vibration of a rectangular membrane with fixed edges

Consider a rectangularly-shaped membrane with edges fixed at 0x = , xx L= , 0z = , and zz L= .

( ) ( ) ( ) ( )0, , , , ,0, , , 0x zy z t y L z t y x t y x L t= = = = (3.6.1.1)


60

Following the logic of (3.6.7), we assume that the membrane responds according to

( ) ( ), , , j tx z t x z e ω=y Y (3.6.1.2)

By substitution of (3.6.1.2) into (3.6.4), we find

2 22

2 2 0kx z

∂ ∂+ + =

∂ ∂Y Y Y (3.6.1.3)

The method of separation of variables is then employed. We assume independence of the x and z

responses in the complex function ( ),x zY . Thus

( ) ( ) ( ),x z x z=Y X Z (3.6.1.4)

Therefore, substituting (3.6.1.4) into (3.6.1.3) leads to

2 22

2 2

1 1 0d d kdx dz

+ + =X Z

X Z (3.6.1.5)

In order for (3.6.1.5) to be true, the first and second terms must be equal to constants.

2 2 2 0x zk k k− − + = (3.6.1.6)

As a result, separating (3.6.1.5) leads to the set of equations

22

2 0xd kdx

+ =X X ;

22

2 0zd kdz

+ =Z Z (3.6.1.7)

Satisfying (3.6.1.7) requires a pair of general sinusoidal functions, leading to

( ) ( ) ( ), , sin sin j tx x z zx z t k x k z e ωφ φ= + +y A (3.6.1.8)

where the wavenumber components and phases are determined by the boundary conditions.

For the membrane fixed according to (3.6.1.1), the phases xφ and zφ must be zero to ensure the sine

functions are zero. Also, it is seen that

; 1,2,3,...xx

mk mLπ

= = (3.6.1.9)

; 1,2,3,...zz

nk nLπ

= = (3.6.1.10)

Therefore the standing waves on the membrane are expressed by

( ), , sin sinj tx zx z t e k x k zω=y A (3.6.1.11)

Considering (3.6.1.9), (3.6.1.10), and (3.6.1.6), the free vibration of the membrane may occur only at particular frequencies


61

1/22 2

2 2mn

mnx z

c m nfL L

ωπ

= = +

(3.6.1.12)

where each combination ( ),m n is associated with a normal mode, sin sinx zk x k z . This analogous to the

one-dimensional case for the string vibration response. Figure 25 shows several examples of the lower-order, normal modes of the fixed-fixed rectangular membrane. Clear nodal lines occur where the membrane possesses no displacement. Thus, if excited only in a given mode, additional fixed supports could be positioned along nodal lines without affecting the response of the membrane.


m=2; % mode number in x direction n=2; % mode number in z direction Lx=pi/2; % [m] dimension of x extent of membrane Lz=1; % [m] dimension of z extent of membrane x=linspace(0,Lx,61); % [m] x extent z=linspace(0,Lz,51); % [m] z extent kx=m*pi/Lx; % [1/m] x wavenumber kz=n*pi/Lz; % [1/m] z wavenumber clear y y=zeros(length(x),length(z));

00.5

11.5

00.5

1-0.5

0

0.5

x [m]

mode (1,1) of fixed-fixed rectangular membrane

z [m]

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

00.5

11.5

00.5

1-0.5

0

0.5

x [m]


z [m]

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

00.5

11.5

00.5

1-0.5

0

0.5

x [m]


z [m]

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

00.5

11.5

00.5

1-0.5

0

0.5

x [m]


z [m]

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

Figure 25. Examples of normal modes of fixed-fixed rectangular membrane.


62

for iii=1:length(z) y(:,iii)=0.5*sin(kx*x)*sin(kz*z(iii)); end figure(1); clf; surf(x,z,y','linestyle','none'); xlabel('x [m]'); ylabel('z [m]'); colorbar axis equal zlim([-.5 0.5]) view(-45,15) title(['mode (' num2str(m) ',' num2str(n) ') of fixed-fixed rectangular membrane']);

3.6.2 Free vibration of a circular membrane

Circular drum heads are one of the most common surface geometries employed for percussive instruments. This is effectively a circular membrane with fixed edge at the radius, by our assumptions employed to arrive at (3.6.4). Consider a membrane of radius a . Using the appropriate Laplacian operator, the associated Helmholtz equation is

2 22

2 2 2

1 1 0kr r r r θ

∂ ∂ ∂+ + + =

∂ ∂ ∂Y Y Y Y (3.6.2.1)

where /k cω= according to the time-harmonic assumption. A separable solution is again assumed

( ) ( ) ( ),r rθ θ= ΘY R (3.6.2.2)

satisfying the boundary condition

( ) 0a =R (3.6.2.3)

A constraint must be imposed on the azimuthal function ( )θΘ , namely that it is smooth and continuously

a function of θ . Substituting (3.6.2.2) into (3.6.2.1) yields

2 22

2 2 2 0kr r r r θ

∂ Θ ∂ ∂ ΘΘ + + + Θ =∂ ∂ ∂

R R R R (3.6.2.4)

Rearrangement and pre-multiplication by 2rΘR

yields

2 2 22 2

2 2

1 1r d d dk rdr r dr dθ

Θ+ + = − Θ

R RR

(3.6.2.5)

Like for prior implementations of the method of separation of variables, the independent sides of the

(3.6.2.5) must be equal to the same constant, 2m . Considering the right-hand side,

22

2 0d mdθΘ+ Θ = (3.6.2.6)

which is solved using


63

( ) ( )cos mmθ θ γΘ = + (3.6.2.7)

The terms mγ are associated with azimuthal dependence in the initial conditions. The second equation to

result from an application of the method of separation of variables to (3.6.2.5) is Bessel's equation, (3.6.2.8).

2 22

2 2

1 0d d mkdr r dr r

+ + − =

R R R (3.6.2.8)

Solutions (3.6.2.8) are Bessel functions of order m of the first and second kind, respectively, ( )mJ kr and

( )mY kr . Thus,

( ) ( ) ( )m mr J kr Y kr= +R A B (3.6.2.9)

Consider the polar coordinate system to have as its origin the center of the membrane. For a membrane with bounded displacements at its center, the terms 0=B for all m . Therefore,

( ) ( )mr J kr=R A (3.6.2.10)

The boundary condition (3.6.2.3) requires ( ) 0mJ ka = . We denote the arguments of the function as mnj

that cause the Bessel function of the first kind to be zero. Therefore, the wavenumbers associated with these arguments are /mn mnk j a= . Appendix A5 of Ref. [1] provides a list of these zeros of the Bessel function,

which enable the computation of the associated wavenumber and reconstruction of the normal modes of the circular membrane that is fixed at its edge. An abbreviated version of the tabulated results is given in Table 7.

Collectively, the response of the ( ),m n mode is

( ) ( ) ( ), , cos j tmn mn m mn mnr t J k r m e ωθ θ γ= +y A (3.6.2.11)

where mn mnk a j= . The natural frequency associated with the ( ),m n mode is

2mn

mnj cf

aπ= (3.6.2.12)

Table 7. Zeros of the Bessel function of the first kind, ( ) 0m mnJ j =

mnj

m n 0 1 2 3

0 0 2.40 5.52 8.65

1 0 3.83 7.02 10.17

2 0 5.14 8.42 11.62


64

3 0 6.38 9.76 13.02


a=.1; % [m] radius of membrane c=3e3; % [m/s] speed constant m=1; % mode number in theta direction n=2; % mode number in r direction bzeros=[0 2.4 5.52 8.65;0 3.83 7.02 10.17;0 5.14 8.42 11.62;0 6.38 9.76 13.02]; % bessel zeros r=linspace(0,a,51); % [m] create radial coordinate theta=pi*linspace(-1,1,41); % [rad] create azimuthal coordinate y=zeros(length(r),length(theta)); % pre-allocate for displacement for iii=1:length(theta) y(:,iii)=0.1*besselj(m,(r*bzeros(m+1,n+1)/a))*cos(m*theta(iii)); end for ooo=1:length(r) for iii=1:length(theta) x(ooo,iii)=r(ooo)*cos(theta(iii)); w(ooo,iii)=r(ooo)*sin(theta(iii)); z(ooo,iii)=y(ooo,iii);

-0.1-0.05

00.05

0.1

-0.1

0

0.1-0.1

-0.05

0

0.05

0.1

x

mode (0,1) of fixed-rim circular membrane

y -0.1-0.05

00.05

0.1

-0.1

0

0.1-0.1

-0.05

0

0.05

0.1

x


y

-0.1-0.05

00.05

0.1

-0.1

0

0.1-0.1

-0.05

0

0.05

0.1

x


y -0.1-0.05

00.05

0.1

-0.1

0

0.1-0.1

-0.05

0

0.05

0.1

x


y

Figure 26. Examples of normal modes of fixed-edge circular membrane.


65

end end figure(1); clf; surf(x,w,z,'linestyle','none'); axis equal xlabel('x'); ylabel('y'); zlim(0.1*[-1 1]) view(-25,25) title(['mode (' num2str(m) ',' num2str(n) ') of fixed-rim circular membrane']);

3.7 Vibration of thin plates

While the restoring forces in membranes arise from the tension applied to maintain their shape, plates develop restoring forces by virtue of their inherent bending stiffness. To assess the vibrations of plates as a next stage in development of standing wave principles, we move straight to the governing equation of motion for plates leaving the detailed derivation to reference [13]. Consider a plate of small thickness with respect to the span-wise dimensions extending in the x y− plane. The displacement of the plate out of this

plane from an undeformed configuration is ( ), ,w x y t .

( ) ( ) ( )2

42

, ,, , , ,

w x y tYI w x y t h p x y t

tρ

∂∇ + = −

∂ (3.7.1)

where I is the moment of inertia per unit width, ( ), ,p x y t is an applied external pressure or load, ρ is the

volumetric density, and h is the plate thickness. The moment of inertia is thus ( )3 2/12 1I h ν= − where

the Poisson's ratio is ν .

The biharmonic operator is 4∇ and its selection depends on the coordinate system. Numerous engineering systems are composed of rectangular-shaped plates and so we consider the Cartesian form of the biharmonic operator here.

4 4 44

4 2 2 42x x y y∂ ∂ ∂

∇ = + +∂ ∂ ∂ ∂

(3.7.2)

Considering the homogeneous form of (3.7.1) for a plate with infinite span in the x y− plane, we assume

that a wave propagates with an angle α measured counterclockwise from the x axis. This wave adopts the form

( ) ( ), , x yj t k x k yx y t e ω − −=w A (3.7.3)

Substitution of (3.7.3) into the homogeneous form of (3.7.1) reveals

( )22 2 2 0x yYI k k hρ ω+ − = (3.7.4)

Consequently, we define

2 2 2x y fk k k+ = (3.7.5)


66

24f

hkYI

ρ ω= (3.7.6)

Similarly, the relations between the wavenumber components to (3.7.6) are

cosx fk k α= (3.7.7)

siny fk k α= (3.7.8)

The perspective of (3.7.7) and (3.7.8) is illuminating since the assumed solution alternatively adopts the form

( ) ( )cos sin, , f fj t k x k yx y t e ω α α− −=w A (3.7.9)

which plainly indicates that the wave propagates as a common front in a direction α away from the x axis

in the x y− plane. Thus, fk is the amplitude of a wavenumber vector f x yk k x k y= + where the overbars

indicate vectors, and x and y are respective unit vectors.

3.7.1 Free vibration of thin plates

Consider the homogeneous form of (3.7.1) for a plate that is simply-supported at all edges. For a simply-supported plate, the boundary conditions on the edges require

( ), , 0w x y t = along all edges (3.7.1.1)

( )2

2

, ,0

w x y tx

∂=

∂ along edges 0x = and xx L= (3.7.1.2)

( )2

2

, ,0

w x y ty

∂=

∂ along edges 0y = and yy L= (3.7.1.3)

An assumed solution to (3.7.1) for the simply-supported plate may adopt our established experience that sine functions satisfy separable equations pertaining to the normal modes. Thus, we assume

( ) ( ) ( ) ( ), , , sin sinj t j tm nx y t x y e k x k y eω ω= =w W A (3.7.1.4)

Substitution of (3.7.1.4) into (3.7.1), and satisfaction of the boundary conditions (3.7.1.1), (3.7.1.2), and (3.7.1.3) gives non-trivial results so long as (3.7.1.5) and (3.7.1.6) are satisfied.

; 1,2,3,...mx

mk mLπ

= = (3.7.1.5)

; 1,2,3,...ny

nk nLπ

= = (3.7.1.6)

The associated natural frequencies of the plate are therefore


67

2 22 21/2 1/21

2 2mnx y x y

YI m n YI m nfh L L h L L

π π ππ ρ ρ

= + = + (3.7.1.7)

with 2mn mnfω π= .

3.7.2 Forced vibration of thin plates

For the inhomogeneous form of (3.7.1), the pressure load on the plate is presumed to be harmonic by

( ) ( ), , , j tx y t P x y e ω=p . In the way performed for beams, we assume that the response of the thin plate is a

superposition of its normal modes weighted by modal coefficients.

( ) ( ) ( ) ( )1 1 1 1

, , , sin sinj t j tmn mn m n

m n m nx y t e x y e k x k yω ω

∞ ∞ ∞ ∞

= = = =

= =∑∑ ∑∑w W A (3.7.2.1)

The (3.7.2.1) is substituted into (3.7.1), then the normal modes are multiplied by the result and integrated over the plate domain. In this way, the modal amplitudes mnA are determined.

( ) ( ) ( ) ( )2 2 0 0

4 , sin sinx yL L

mn m nx y mn

P x y k x k y dxdyhL Lρ ω ω

=− ∫ ∫A (3.7.2.2)

When the plate is excited by a point force at ( )0 0,x y ,

( ) ( ) ( ) ( )0 0 0 0 0, , , j t j tx y t P x y e P x x y y eω ωδ δ= = − −p (3.7.2.3)

the evaluation by (3.7.2.1) gives the modal amplitudes

( ) ( )( )

0 002 2

sin sin4 m nmn

x y mn

k x k yPhL Lρ ω ω

=−

A (3.7.2.4)

Summarizing, the forced vibration of the thin plate to a point force at ( )0 0,x y is

( ) ( ) ( )( ) ( ) ( )0 00

2 21 1

sin sin4, , sin sinm nj tm n

m nx y mn

k x k yPx y t e k x k yhL L

ω

ρ ω ω

∞ ∞

= =

=−

∑∑w (3.7.2.5)

As emphasized in Sec. 3.5.2, by use of Euler's identity on the normal modes, we see the duality of (3.7.2.5) as perceived either as standing waves associated with modes of vibration or as oppositely traveling waves in the x and y axes.

3.8 Impedance and mobility functions of structures

As described in Sec. 2.6, determination of the impedance for the system is equivalent to solving the differential equation of motion. This referred to the case of an ordinary differential equation. The question may be asked whether this broad statement of differential equation solution applies to partial differential equations that govern the motions of numerous and diverse engineering structures.


68

In fact, the first questions to ask are whether the impedance of a distributed parameter system is defined and whether it is useful in the analysis of that system's dynamic response.

To the first part of the question, for distributed parameter systems, we must distinguish between drive point (or input) impedance and transfer impedance.

Recall the infinite beam driven by a harmonic point force at the beam center. The beam response due to this force was determined to be (3.5.1.5), which is repeated for convenience and expanded for both x± domains.

( ) ( )3,4

f fjk x k x j t

f

jFx t e je eYIk

ω− −= − −y ; for 0x > (3.8.1)

( ) ( )3,4

f fjk x k x j t

f

jFx t e je eYIk

ω+ += − −y ; for 0x < (3.8.2)

The drive point impedance of the beam is

( )( )0 0,

tt

=f

Zy

(3.8.3)

Given the time-harmonic assumptions, we find that the drive point impedance is

( )

3

0

41

fYIkjω

=−

Z (3.8.4)

Eq. (3.8.4) characterizes the resistance and reactance, i.e. respectively energy transfer and energy exchange, of the harmonic energy attempted to be injected into the infinite beam by the point force. It is clear that (3.8.4) is complex and frequency dependent.

The transfer impedance from the force point 0x , here 0x =0, to the spatial location x along the beam is

( )( )0 ,x

tx t

=f

Zy

(3.8.5)

Evaluating, we find

( )3

0

4f f

fx jk x k x

YIk

e jeω=

−Z

(3.8.6)

The transfer impedance characterizes the ability for energy to be injected at location A and be delivered (or returned from) location B. Here, (3.8.6) shows that the impedance is complex, exhibits decaying reactance components as one considers beam locations further and further removed from the point location, and is also frequency dependent like the drive point impedance.


69

Analysis of drive point and transfer impedances in structures is a critical part of engineering system development and evaluation throughout design processes. This analysis is useful in the determination of sensitive paths of vibration or wave energy transfer through built-up systems. As a result, impedance-based analysis of structural, mechanical, and (as we will find) acoustic systems is common in many engineering practices for its value in first approximations of key sub-systems that govern the dynamics [14].

Drive point and transfer mobilities are the inverse relationships of drive point and transfer impedances. Thus, the transfer mobility from the force point 0x to the spatial location x along the beam is

( ) ( )0 , /x x t t=Y y f (3.8.7)

Expressions for impedances and mobility of various infinite and finite dimensioned elementary structures are available in texts [5] [14].


70

4 Acoustic wave equation

Towards deriving the acoustic wave equation, we first define acoustic variables and constants. In the following, we assume that the fluid is lossless and inviscid meaning that viscous forces are negligible, and that the fluid undergoes small (linear), relative displacements between adjacent particles prior to a pressure change. These fluid particles imply an infinitesimal volume of fluid large enough to contain millions of fluid molecules so that within a given small fluid element, the element may be considered as a uniform, continuous medium having constant acoustic variables (defined below) throughout the element.

Of course, at smaller and smaller length scales, the individual fluid molecules that make up the fluid element are in constant random motion at velocities considerably in excess of the particle velocities associated with acoustic wave propagation. Thus within any given fluid element, these molecules may in fact leave the element in an infinitesimal duration of time. Yet, they are replaced by other molecules entering the element. The consequence is that evaluating the relatively slow motions associated with wave propagation does not need to account for the very small time-scale dynamics of the individual fluid molecules.

Because we consider only linear relative displacements between fluid particles, our theoretical development is limited to accurate treatment of linear acoustic phenomena. The same applies to the consideration of pressure and density fluctuations which are very small with respect to ambient pressure and ambient density. While focused on linear contexts, these linear acoustic phenomena are the core of a significant proportion of all acoustic events in air-borne and water-borne acoustic applications. The linearity of acoustic phenomena also corresponds to nearly lossless wave propagation, which indicates that thermal losses associated with fluid particle motion are negligible. In other words, the acoustic processes are isentropic.

Finally, at this stage we consider only the acoustic free field which means we examine only unbounded fluid media. Also, we consider that this acoustic field contains no sources, and instead aim to characterize the type of wave functions that satisfy the acoustic wave equation itself, irrespective of the specific source that may be present.

4.1 Assembling the components to derive the acoustic wave equation

Acoustic pressure ( ),p x t is the pressure fluctuation around the equilibrium (atmospheric) pressure 0P :

( ) ( ) 0, ,p x t P x t P= − . The SI units of pressure are Pascals [Pa=N/m2]. The atmospheric pressure is 0P ≈

100 [kPa] near sea level. Here, x is the spatial dimension, limited currently to one dimension. The fluid

particle displacement is ( ),x tξ while the fluid particle velocity is ( ) ( ) ( ),, ,

x tu x t x t

tξ

ξ∂

= =∂

. The

equilibrium (atmospheric) fluid density is 0ρ , while the instantaneous fluid density is ( ),x tρ .

To derive the acoustic wave equation, three components are required:

• the equation of state • the equation of continuity


71

• the Euler's equation Equation of state. The fluid must be compressible to yield changes in pressure. Therefore, the thermodynamic behavior of the fluid must be considered. All fluids may be described by an appropriate equation of state that relates pressure, density, and temperature. For isentropic acoustic processes, the pressure is a function of density and for small pressure changes this can be expressed via a Taylor series

( ) ( )0 0

22

0 0 02

1 ...2

P PP Pρ ρ

ρ ρ ρ ρρ ρ∂ ∂

= + − + − +∂ ∂

(4.1.1)

Because the pressure and density fluctuations within the fluid element associated with a significant proportion of acoustic sound pressure levels are so small, the terms in (4.1.1) of order 2 and greater are insignificantly small, thus

( )0

0 0PP P

ρ

ρ ρρ∂

≈ + −∂

(4.1.2)

which is the linear approximation of the pressure change from ambient condition. From (4.1.2) we rearrange to yield

0

00 0

0

PP Pρ

ρ ρρ

ρ ρ

−∂ − ≈ ∂ (4.1.3)

The term on the right-hand side of (4.1.3) in brackets [ ] is called the condensation

0

0

s ρ ρρ−

= (4.1.4)

The condensation is the relative deviation of fluid element density from a reference value. The term on the right-hand side of (4.1.3) in the curly brackets { } is the adiabatic bulk modulus B .

0

0PB

ρ

ρρ∂

=∂

(4.1.5)

Finally, as already defined, the term on the left-hand side is the acoustic pressure.

p Bs= (4.1.6)

Equation (4.1.6) is the equation of state. It may be thought of as "Hooke's law for fluids", since it relates a fluid "stress" (pressure) to the fluid "strain" (condensation) through the fluid "stiffness" (bulk modulus). A particular mathematical difference from the solid mechanics analogy is that (4.6) is a scalar equation since pressure is not a vector.


72

Figure 27. Schematics for deriving (a) equation of continuity and (b) Euler's equation, respectively, giving attention to constant volume element and constant fluid element.

Equation of continuity. Within an unchanging, defined volume of space, pressure changes cause mass to flow in and out of the volume. By conservation of mass, the net rate with which mass flows into the volume through the enclosing surfaces of the volume must be equal to the rate with which the mass increases within the volume. Figure 27(a) illustrates the scenario. A constant volume element dV dxdydz= is fixed in space.

The mass flow rate in the volume in the x axis from the left/rear face of area dydz is

xdydz uρ (4.1.7)

where the particle velocity u component in the x axis is xu . The mass leaving the volume on the front/right

face is approximated by a linearized Taylor series expansion to be

( )xx

udydz u dx

xρ

ρ ∂

+ ∂ (4.1.8)

The net rate of mass influx in the x axis is therefore

( ) ( )x xx x

u udydz u dydz u dx dV

x xρ ρ

ρ ρ ∂ ∂

− + = − ∂ ∂ (4.1.9)

Similar expressions are obtained for the net influx of mass in the y and z directions. Consequently,

( ) ( ) ( ) ( )yx zuu udV u dV

x y zρρ ρ

ρ ∂∂ ∂ − + + = −∇ ⋅ ∂ ∂ ∂

(4.1.10)

The rate at which the mass in the control volume changes is equal to

dVtρ∂∂

(4.1.11)

Thus, by conservation of mass, we obtain the equation of continuity (4.1.12)

( )0 0 0s ut

ρ ρ∂+∇ ⋅ =

∂ (4.1.12)


73

where the dependence of the density ( )0 1 sρ ρ= + is assumed to be a weak function of time and the

condensation is a small value, to permit the substitution.

The continuity equation states that the time rate of change of mass in the control volume is equal to the net influx of mass in the same time duration.

For small (linear) changes of acoustic pressure and weak dependence of the equilibrium density 0ρ on

space, the linearized equation of continuity is

0s ut∂

+∇ ⋅ =∂

(4.1.13)

Euler's equation. Unlike for the continuity equation, we now focus on a fluid element that is deformed in consequence to the pressure differences between opposing faces, see Figure 27(a,b,c). The fluid element moves with the remaining, host fluid media in which it exists. The pressure on the left-most face is

( ), , ,P x y z t while, by a linearized Taylor series approximation, the pressure on the right-most face is

( ) ( ), , ,, , ,

P x y z tP x y z t dx

x∂

+∂

. Thus, the force difference between left and right faces is

( ) ( ) ( ) ( ), , , , , ,, , , , , ,

P x y z t P x y z tP x y z t P x y z t dx dydz dV

x x ∂ ∂ − + = − ∂ ∂

(4.1.14)

Analogous expressions to (4.1.14) are found for the net forces in the y and z axes. Because the force is

directional, the total force vector is therefore

df PdV= −∇ (4.1.15)

The total acceleration of the fluid particle is [1]

( )x y zu u u u ua u u u u ut x y z t

∂ ∂ ∂ ∂ ∂= + + + = + ⋅∇∂ ∂ ∂ ∂ ∂

(4.1.16)

The net force due to the pressure differentials (4.1.15) balances the net inertial force acting on the fluid particle, which is determined by the product of the total acceleration (4.1.16) and the fluid particle mass

dVρ . Thus, by Newton's 2nd law, we find

( )uPdV u u dVt

ρ ∂ −∇ = + ⋅∇ ∂ (4.1.17)

In the absence of acoustic excitation, consideration of small absolute condensation values, and weak spatial dependence on the particle velocity vector in comparison to the time dependence, we simplify (4.1.17) to be

0upt

ρ ∂−∇ =

∂ (4.1.18)


74

The (4.1.18) is the linear Euler's equation.

4.1.1 Deriving the acoustic wave equation

To summarize,

equation of state p Bs= (4.1.1.1)

equation of continuity 0s ut∂

+∇ ⋅ =∂

(4.1.1.2)

Euler's equation 0upt

ρ ∂−∇ =

∂ (4.1.1.3)

To derive the acoustic wave equation, we first take the gradient of (4.1.1.3) and assume that the equilibrium is a weak function of time and space

20

upt

ρ ∂ −∇ = ∇ ⋅ ∂ (4.1.1.4)

Second, we take the time derivative of (4.1.1.2), use the method of mixed partials to change the order of derivatives on the particle velocity, and multiply by the equilibrium density

2

0 02 0s ut t

ρ ρ∂ ∂ + ∇ ⋅ = ∂ ∂ (4.1.1.5)

Then, rerranging the terms of (4.1.1.5), we finally add (4.1.1.4) and (4.1.1.5) to yield

22

0 2

spt

ρ ∂∇ =

∂ (4.1.1.6)

Using (4.1.1.1), we obtain the acoustic wave equation (4.1.1.7)

22

2 2

1 ppc t

∂∇ =

∂ (4.1.1.7)

where the sound speed is 0/c B ρ= .

To summarize the developments and assumptions, the (4.1.1.7) is the linear (small pressure changes), lossless (inviscid) acoustic wave equation.

The adiabatic bulk modulus is

0

0 0PB P

ρ

ρ γρ∂

= =∂

(4.1.1.8)

where γ is the ratio of specific heats which is specific to a given gas. In general 0 0/P ρ is almost

independent of pressure so that the sound speed is primarily a function of temperature. Thus, an alternative derivation of the sound speed for air yields


75

0 1 / 273K Cc rT c Tγ= = + (4.1.1.9)

where CT is the temperature in degrees [°C]. 0c is the sound speed in air at 0 [°C] which is approximately

0c =331.5 [m/s] at 1 [atm] pressure (sea level) and at 0 [°C]. Appendix A10 of [1] provides a table of

material properties for common gases.

Mathematically, the curl of the gradient of a function must vanish, 0f∇×∇ = , so that by Euler's equation

the particle velocity is irrotational, 0u∇× = . This directs us to introduce a new function, which is a scalar, Φ :

u = ∇Φ (4.1.1.10)

The function Φ is termed the velocity potential. Substituting (4.1.1.10) into (4.1.1.3) and assuming the equilibrium pressure is a weak function of space shows that

0 0pt

ρ ∂Φ ∇ + = ∂ (4.1.1.11)

Consequently, we find

0pt

ρ ∂Φ= −

∂ (4.1.1.12)

Therefore, the velocity potential Φ satisfies the acoustic wave equation by substitution of (4.1.1.12) into (4.1.1.7).

22

2 2

1c t

∂ Φ∇ Φ =

∂ (4.1.1.13)

Thus, solving the (4.1.1.14) permits the subsequent determination of the fluid particle velocity and acoustic pressure by (4.1.1.10) and (4.1.1.12), respectively.

The purpose of introducing the potential is to relate the particle velocity to the acoustic pressure via a common function, here a non-physical scalar. Thus, the acoustic pressure is related to the time rate of change of this scalar quantity at a location in space (4.1.1.12), while the particle velocity is related to the gradient of this scalar quantity at the same location (4.1.1.10). This is a general set of relations and is not limited in its interpretation to the coordinate system or waveforms of interest (e.g. whether plane, cylindrical, spherical waves).

4.2 Harmonic, plane acoustic waves

When the acoustic variables are constant in the cross section spanning dy and dz , the Laplacian operator

is 2 2 2/ x∇ = ∂ ∂ . Thus, in one Cartesian dimension, the acoustic wave equation is

2 2

2 2 2

1p px c t

∂ ∂=

∂ ∂ (4.2.1)


76

A scenario that realizes this one-dimensional variation of acoustic variables would be a duct stretching along the x axis and considering low frequencies of harmonic oscillations, respecting the duct cross-sectional dimensions.

Of course, (4.2.1) is similar to the one-dimensional wave equation determined for transverse oscillations of the string (3.1.6) as well as the wave equation for the longitudinal harmonic deformations of the rod (3.2.12). As a result, we anticipate that the complex exponential form of harmonic solution that satisfied (3.1.6) (3.2.12), via the (3.1.4.1) (3.2.15), will also satisfy (4.2.1).

Therefore, we assume that the time-harmonic form

( ) ( ) ( ), j t kx j t kxx t e eω ω− += +p A B (4.2.2)

is a solution to (4.2.1). These are harmonic plane acoustic waves that occur at angular frequency ω and respectively travel in the x± directions according to the arguments kx in the exponentials.

The particle velocity is determined from Euler's equation (4.1.1.3)

0p ux t

ρ∂ ∂− =∂ ∂

(4.2.3)

0 0j jk cx

ωρ ρ∂− = =∂p u u (4.2.4)

( ) ( ) ( ) ( ) ( )

0 0

1, j t kx j t kx j t kx j t kxjkx t e e e ej c

ω ω ω ω

ωρ ρ− + − + = − − + = − u A B A B (4.2.5)

The (4.2.4) exemplifies the time-harmonic form of the Euler's equation for plane waves.

Using that /k cω= and expressing ( )j t kxe ω −+ =p A and ( )j t kxe ω +

− =p B , the particle velocity components in

the positive + and negative - directions are found to be

0/ cρ± ±= ±u p (4.2.6)

Breaking up (4.2.6), we see

0/ cρ+ +=u p ; 0/ cρ− −= −u p (4.2.7)

The specific acoustic impedance is the ratio of acoustic pressure to fluid particle velocity and, like for mechanical vibrations, is a measure of resistance and reactance of the fluid media to inhibit or assist the propagation of waves:

z = p / u (4.2.8)

Therefore, for plane waves,

00/

cc

ρρ±

= ±±

pz =p

(4.2.9)


77

the specific acoustic impedance is purely real and dependent upon the direction of wave travel. The interpretation of a purely real impedance for traveling waves is that energy is transferred without return from a previous source or origin. For air at 20 [°C], the specific acoustic impedance is 0cρ =415 [Pa.s/m].

Because the product of atmospheric density and sound speed constitutes the impedance, their combination is typically more important in acoustics applications than the individual values alone. Appendix A10 of [1] reports the impedances of numerous fluid media.

4.2.1 Impedance terminology clarification

There are several variations of impedance used throughout the study of acoustics. The terminology must be clarified since each variant possesses different units and has different contexts of appropriate use.

• The mechanical impedance is the ratio of force to velocity, units [N.s/m]. • The specific acoustic impedance is the ratio of acoustic pressure to particle velocity, units [N.s/m3]. • The acoustic impedance is the ratio of acoustic pressure to volume velocity of the fluid, units

[N.s/m5].

For plane waves, where the acoustic pressure is constant over a constant cross-section S , the acoustic impedance is the specific acoustic impedance normalized by the area S , while the mechanical impedance is the specific acoustic impedance multiplied by the area S .

4.3 Plane progressive shock waves

A detailed derivation of the nonlinear acoustic wave equation is considerably beyond the scope of this course. Interested individuals should consult [3] or [15] for such derivation. It suffices us to consider the equation directly:

( ) ( ) ( ) ( )22

22 2 2 2 20 02

1 112 2

c ut t t

γγ∂ ∇Φ∂ Φ ∂Φ − ∇ Φ − = + − ∇ Φ +∇Φ ⋅ ∇Φ ∇ Φ − + ∇ ∇Φ ∂ ∂ ∂

(4.3.1)

where 0c is the linear sound speed, γ is the ratio of specific heats and 0u is the fluid medium mean speed.

If the fluid is not flowing in this way, then 0 0u = . Of course, should all of the nonlinear terms on the right-

hand side of (4.3.1) vanish, then (4.3.1) returns to the linear acoustic wave equation as expressed according to the velocity potential, (4.1.1.13).

No assumptions as to the significance of the acoustic variable perturbations are made in the derivation of (4.3.1). Thus (4.3.1) holds for all acoustic variable fluctuations for lossless fluid media.

The (4.3.1) is derived by use of two equations, which under the assumptions of plane wave propagation reduce to (4.3.2) and (4.3.3).

1 02

c c uu ct x x

γ∂ ∂ − ∂+ + =

∂ ∂ ∂ (4.3.2)


78

2 01

u u cu ct x xγ

∂ ∂ ∂+ + =

∂ ∂ − ∂ (4.3.3)

The (4.3.2) and (4.3.3) govern the time- and space-dependent sound speed ( ),c x t and particle velocity

( ),u x t , respectively. The dependencies on the sound speed are of course unique with respect to linear

acoustic wave disturbances. Integral evaluations of (4.3.2) and (4.3.3), and further manipulations, reveal

01

2c c uγ −= + (4.3.4)

Then by substitution of (4.3.4) into (4.3.3), it is seen that

( )0 0u uc ut x

β∂ ∂+ + =

∂ ∂ (4.3.5)

where the constant is

( )1 12

β γ= + (4.3.6)

Solving (4.3.5) is accomplished with relative ease, compared to (4.3.1). First, the linearized result when uux

β ∂∂

is negligible is

0

xu f tc

= −

(4.3.7)

which is of course the general form of wave solution argument for waves propagating in the x+ direction.

We then consider that replacement of 0c with 0c uβ+ in (4.3.7) is a way to satisfy the full form of (4.3.5).

By substitution, indeed this is the case. Thus,

0

xu f tc uβ

= − +

(4.3.8)

The argument of (4.3.8) has units [s]. For harmonic, lossless, progressive plane waves, the argument is assumed to be in units [rad] to accommodate a sinusoidal profile. Thus,

( )0

, sin xu x t U tc u

ωβ

= − +

(4.3.9)

The (4.3.9) is a nonlinear function of the particle velocity and its interpretation is revealing to the formation of shock waves. For great enough speed amplitude U , the propagation of the wave to increasing values x

causes the crests of the wave (traveling at 0c Uβ+ ) to overtake the troughs of the wave (traveling at

0c Uβ− ). Figure 28 shows a representative case of this trend. For increasing x a point is reached at which

the solution to the nonlinear (340) results in a multi-valued speed. In reality this is the onset of a shock wave


79

and an "N"-shaped wave profile is measured in practice. Thus, a point in the acoustic field is subjected to a near-instantaneous increase in acoustic pressure from a resting, equilibrium value. In some contexts, shock waves are referred to as sonic booms, particularly in aerospace applications. A measured shock wave from a line charge (explosion) is shown in Figure 29. The "N" shape to the waveform is very apparent.

Figure 28. Example of plane progressive shock wave formation.

Figure 29. Measured shock wave. Courtesy of M.C. Remillieux of Ref. [16].

4.4 Acoustic intensity

As observed in (4.2.9) by the lack of imaginary terms, acoustic pressure and the particle velocity are in phase for plane waves. As a result, acoustic power is transmitted.


80

The instantaneous intensity ( )I t pu= of an acoustic wave is the rate per unit area at which work is done

by one element of the acoustic fluid on an adjacent fluid element. Note that in general, the particle velocity

is a vector u and thus the intensity is also vector quantity. The units of ( )I t are [W/m2]. The average

acoustic intensity I is the time average of ( )I t

( )0

1 dT

TTI I t pu pu t

T= = = ∫ (4.4.1)

where, for harmonic waves with angular frequency ω , the period is 2 /T π ω= , and the integration considers the real components of pressure and particle velocity.

In general throughout this course, unless otherwise specified, use of the term acoustic intensity refers to the time-averaged version (4.4.1) and not the instantaneous measure of intensity.

For plane waves, the intensity is one-dimensional so that we drop the overbar, thus recognizing that "travel" in the opposing direction is indicated by a change of sign:

22

00

pI c u

cρ

ρ= = (4.4.2)

Using the relations for mean-square, RMS, and amplitude, the intensity may be written as

2 2

0 02rms

rms rmsp PI p u

c cρ ρ= = = (4.4.3)

In general, when using the complex exponential representation of acoustic variables, thus considering time-harmonic oscillations, the time-harmonic [average] intensity is computed from

[ ] [ ] *

0

1 1Re Re d Re2

T

I tT

= = ∫ p u pu (4.4.4)

where the asterisk denotes the complex conjugate, the vector notation on particle velocity is retained, and the 1/2 multiple results from the time-harmonic integration. This time-harmonic intensity is only a vector if u spans more than one dimension.

Note that for a complex function x , 2* 21 1Re2 2rmsx x ⋅ = = x x .

4.5 Harmonic, spherical acoustic waves

For spherically symmetric sound fields, use of the appropriate Laplacian operator in (4.1.1.7) and use of the chain rule of differentiation allows us to express the wave equation as

( ) ( )2 2

2 2 2

1rp rpr c t

∂ ∂=

∂ ∂ (4.5.1)


81

It is apparent that this equation is similar to (4.1.1.7) with the variable mapping p rp↔ . Thus, we use the

same general solution to the wave equation

( ) ( )1 2rp f ct r f ct r= − + + (4.5.2)

Then, we express the acoustic pressure, without loss of generality as

( ) ( ) ( )1 21 1,p r t f ct r f ct rr r

= − + + (4.5.3)

which is valid for all 0r > . For the first term of (4.5.3), the waves are outgoing, from the origin 0r = , and spread in greater and greater areas from the origin. For the second term of (4.5.3), the waves are incoming and increase in intensity approaching an origin. In this second case, there are few examples in linear acoustics that pertain to such self-focusing of sound energy by a wavefield converging to an origin. So we hereafter neglect the second term of (4.5.3).

By and large, the most important spherically-spreading acoustic waves are harmonic, in which case we assume a solution to (4.5.1) of the form

( ) ( ), j t krr t er

ω −=Ap (4.5.4)

Using the Euler's equation (4.1.1.3) in time-harmonic form for spherically symmetric wave propagation

0 0j jk cr

ωρ ρ∂− = =∂p u u (4.5.5)

the particle velocity is then derived to be

( ) ( )0

,1, 1r t

r t jkr cρ

= −

pu (4.5.6)

Using (4.5.4) and (4.5.6), the specific acoustic impedance for spherical waves is

( )( ) ( )

2

0 0 2 21 1 1

krjkr krc c jjkr kr kr

ρ ρ

= = = + + + +

pzu

(4.5.7)

The specific acoustic impedance of spherical waves contains resistive and reactive components. A comparison of this characteristic to plane waves is made in Sec. 4.6.

4.5.1 Spherical wave acoustic intensity and acoustic power

The acoustic intensity for spherical waves is computed in the same manner as for plane waves, it is the time average of work per unit area that fluid particles exert on neighboring particles. Using the time-harmonic

equation form with ( ) ( ), j t krAr t er

ω −=p and (4.5.6), we have


82

( ) ( )*

0

1 1 1 1Re Re 12 2

j t kr j t krA AI e j er kr c r

ω ω

ρ− − − = = +

pu (4.5.1.1)

2

20

12

AI Ic rρ

= = (4.5.1.2)

where the overbar is dropped due to the uni-axial (radial) propagation of the waves. Expressing (4.5.1.2) using /P A r= , we have

2

02PI

cρ= (4.5.1.3)

The (4.5.1.3) is equivalent to (4.4.3) for plane waves, but the interpretation of P is distinct between (4.5.1.3) and (4.4.3). For harmonic plane waves, P is independent of the spatial coordinate x . Thus, in lossless acoustic media, energy transfer persists without diminishing in intensity during propagation. In contrast, for spherical waves the change of P is in inverse proportion to the change in radial coordinate r. Thus, spherical wave intensity changes in proportion to the square of the radial distance of travel 2r for the wave. This means that the acoustic energy radiated to a field point that is a distance r from the spherical wave origin (source) is reduced in proportion to the squared distance between source and point, and explains why sounds decay in amplitude as they travel from their origins.

In general, the acoustic power is the sound energy per time radiated by an acoustic source, and is defined according to the intensity via

dS

I n SΠ = ⋅∫ (4.5.1.4)

where S is a surface that encloses the sound source and n is the unit normal to the surface. Thus the dot product in (4.5.1.4) refers to the component of the intensity vector that is normal to the enclosing surface under consideration. For a given radial distance from a source of sound, if the intensity has the same magnitude, then the acoustic power is simply ISΠ = .

For plane and spherical waves, only one direction of wave propagation is considered, either a single axis of one-dimensional motion or a radial direction spreading, respectively. For spherical waves, the area through

which the sound spreads is 24 rπ . Thus, the acoustic power is

22 2

0

4 4 rmspr I rc

π πρ

Π = = (4.5.1.5)

4.6 Comparison between plane and spherical waves

A comparison of the plane wave acoustic pressure and particle velocity with those corresponding values

for spherical waves shows significant differences. It is instructive to consider the term ( )2 /kr rπ λ= . In

other words, when 1kr < the observation radial location of the acoustic variables is approximately within one acoustic wave wavelength from the origin of radial wave propagation.


83

Thus, when 1kr < , the imaginary component of the spherical wave particle velocity (4.5.6) is as important or more significant than the real component. This means there is a phase difference between acoustic pressure (4.5.4) and particle velocity (4.5.6) for spherical waves near the origin of the sound source from which the waves propagate. As a result, the specific acoustic impedance of spherical waves near the sound source contains real and imaginary components, Figure 30 and (4.5.7).

For radial distances far from the acoustic origin, 1kr >> , the imaginary component of the particle velocity (4.5.6) is insignificantly small. As a result, for 1kr >> the spherical wave particle velocity is in phase with the acoustic pressure. Figure 30 shows this trend that the imaginary contribution of the specific acoustic impedance of spherical waves converges to zero for 1kr >> while the real component approaches that of plane waves.

Figure 30. Specific acoustic impedance normalized by plane wave specific acoustic impedance.

Comparing plane and spherical acoustic waves, we refer to Table 9.

Table 9. Comparison between plane and spherical acoustic waves

plane waves spherical waves

radiate in direction x radiate in direction r

pressure amplitude is constant as distance x changes pressure amplitude decreases as 1 / r∝

acoustic intensity is proportional to 2rmsp acoustic intensity is proportional to 2

rmsp and decreases as

21 / r∝

pressure and particle velocity are in phase pressure and particle velocity are only in phase for 1kr >> ,

and in general are not in phase

the specific acoustic impedance is real the specific acoustic impedance is real for 1kr >> , and in

general is complex

Thus, for point acoustic sources (infinitesimally small radiating spheres), we refer to the acoustic far field as the location in space where spherical wave characteristics approach those of plane waves. The acoustic far field satisfies 1kr >> . In the acoustic far field, spherical waves have the following characteristics

0 1 2 3 4 5 6 7 8 9 100

0.2

0.4

0.6

0.8

1

kr, normalized distance from acoustic source origin

spec

ific

acou

stic

impe

danc

e no

rmal

ized

by ρ 0

c

plane wavereal(spherical wave)imag(spherical wave)


84

• the acoustic pressure and the particle velocity are in-phase • the acoustic intensity is in the radial direction r • the specific acoustic impedance is purely real and equal to 0`cρ

In general, the acoustic far field refers to a span of many acoustic wavelengths between the receiving point

of acoustic pressure waves and the acoustic wavelength. This is evident by ( )2 / 2 / 1kr r rπ λ π λ= = >> .

4.7 Decibels and sound levels

In acoustics, we often use logarithmic scales to characterize the amplitude of acoustic pressures to which we are subjected. The sound pressure level is defined

2

10 210log rms

ref

pSPLp

= (4.7.1)

where refp is the reference sound pressure and is often taken to be refp =20 μPa for sound in air. The units

of SPL are decibels [dB]. Equation (4.7.1) is also written

1020log rms

ref

pSPLp

= (4.7.2)

Recall that / 2rmsp P= where P is the amplitude of the acoustic pressure in units [Pa].

In addition to sound pressure level, we are also often interested in sound power level

1010logref

LΠ

Π=

Π (4.7.3)

where refΠ =10-12 [W]. The sound power level for spherical waves may be also expressed using

2 2

10 100

10

420log 10log

20log 11

refrms

ref ref

r ppLp c

SPL r

πρΠ = +

Π

= + +

(4.7.4)

To combine incoherent acoustic signals, the total RMS pressure of the incoherent frequencies iω is

computed from (4.7.5), after which the SPL is determined from (4.7.2). An alternative computation using the individual SPL values at the incoherent frequencies is given in (4.7.6).

( )2 2irms rms i

ip p ω=∑ (4.7.5)

( )/101010log 10 i iSPL

iSPL ω= ∑ (4.7.6)


85

4.8 Point source and source strength

The point source is the fundamental acoustic source constituent employed in the study of more complex and realistic sources of acoustic waves. This is because, by summation or integral forms according to the Huygens' principle, arbitrary-shaped sources of sound may be analyzed by virtue of the appropriate combination of point sources.

Consider that a sphere of radius a oscillates radially with a velocity of j tUe ω , Figure 31, and therefore has a displacement amplitude /U ω . This acoustic source is called the monopole. To satisfy the boundary condition at the sphere surface, the particle velocity of the acoustic fluid must be equal to the surface normal velocity of the sphere, which is everywhere U . Therefore, we have that

( ) ( )

( )

0

0

,1, 1

11

j t

j t ka j t

a ta t j Ue

ka c

j e Ueka a c

ω

ω ω

ρ

ρ−

= − =

= − =

pu

A (4.8.1)

0 1jkajkacUa e

jkaρ=

+A (4.8.2)

Consequently, the pressure field generated by a monopole is

( )0 1

j t k r aa jkacU er jka

ωρ − − =+

p (4.8.3)

By substitution, the particle velocity is

( ) ( ), j t k r aar t U er

ω − − =u (4.8.4)

Note that (4.8.4) meets the boundary condition that r a= , we have ( ), j ta t Ue ω=u . Note also that (4.8.4)

is only valid r a> .

At distances satisfying 1ka << , which is equivalent to ( )2 / 1aπ λ << , the pressure is

( ) ( )0

j t kraj cU ka er

ωρ −=p (4.8.5)

We call such spherical acoustic sources (4.8.5) that are small relative to the acoustic wavelength as point sources.


86

Figure 31. Monopole in spherical coordinate system showing location vector r denoting point (grey) in field.

The complex source strength Q is computed from the integral of the surface normal velocity over the surface.

j t

Se ndSω = ⋅∫Q u (4.8.6)

Considering the development defining the monopole, the complex source strength is equal to the product of the uniform velocity amplitude and the sphere surface area.

24Q a Uπ= =Q (4.8.7)

The units of source strength are illuminating: [m3/s]. Thus, the source strength characterizes the rate at which fluid volume is harmonically pumped back and forth at the source surface. The expression (4.8.6)

using j te ωQ is referred to as the complex source volume velocity, and is used regularly in the analysis and evaluation of sound power delivery from acoustic sources.

Substituting (4.8.7) into the expression for the acoustic pressure emitted by the point source gives

( ) ( )

( )

( )

0 2

0

0

4

4

2

j t kr

j t kr

j t kr

Q aj c ka ea r

Qj c ker

Qj c er

ω

ω

ω

ρπ

ρπ

ρλ

−

−

−

=

=

=

p

(4.8.8)

While (4.8.8) is valid only for field points many acoustic wavelengths from the source center, the expression for source strength (4.8.7) may be substituted into the full relation between the monopole surface normal velocity amplitude and the acoustic pressure emitted (4.8.1) since the determination of (4.8.6) is an integral evaluation over the source surface.


87

4.9 Acoustic reciprocity

A detailed establishment of the principle of acoustic reciprocity is beyond the scope of this book. Interested individuals should consult Ref. [7] for further details and the relevant integral formulations. It suffices us to understand what acoustic reciprocity means and its impacts.

Acoustic reciprocity is the principle that under time-invariance in non-flowing, unbounded fluid, and regardless of the presence of scattering objects, the positions of an acoustic source and receiver may be interchanged without altering the received acoustic pressure [1] [17]. There are numerous implications to this principle that will be encountered later in this course. Nevertheless, at this time, the unique symmetry of acoustics embodied in reciprocity is to be appreciated.


88

5 Reflection and transmission

Acoustic waves incident on interfaces between two media are partially reflected and partially transmitted. The theories employed in this portion of the course are applicable for media interfaces between multiple fluids under all circumstances of wave incidence while the theories are applicable for media interfaces between fluids and solids only under special cases. Such factors will be elaborated upon at the appropriate times in the following sections.

Figure 32. Schematic of normally incident, reflected, and transmitted pressure waves through two fluid media.

5.1 Normal incidence wave propagation at a fluid interface

Consider a normally-incident, plane acoustic pressure wave ip with complex amplitude iP arriving at the

boundary between a source fluid (fluid 1) of specific acoustic impedance 1 1 1r cρ= and a second fluid (fluid

2) of specific acoustic impedance 2 2 2r cρ= , as shown schematically in Figure 32. A reflected, plane wave

rp with amplitude rP is induced back within the source fluid while a transmitted, plane wave tp with

amplitude tP is induced in the second fluid.

The pressure reflection coefficient and the pressure transmission coefficient are defined, respectively, using (5.1.1) and (5.1.2).

r

i

=PRP

(5.1.1)

t

i

=PTP

(5.1.2)

The pressure reflection and transmission coefficients are most generally complex. The intensity of plane

waves is 2 / 2P r , where P is the amplitude of the complex pressure and as above r is the specific acoustic impedance. The intensity reflection coefficient and the intensity transmission coefficient are, respectively, computed by (5.1.3) and (5.1.4).

2rI

i

IRI

= = R (5.1.3)


89

21

2

tI

i

I rTI r

= = T (5.1.4)

The power reflection coefficient and the power transmission coefficient are computed by (5.1.5) and (5.1.6), respectively. Note that the incident and reflected areas are the same iA while the transmitted area may be

of a different cross-section tA for instance in the event of a duct of changing cross-section at the fluid

interface.

2IR RΠ = = R (5.1.5)

21

2

t tI

i i

A A rT TA A rΠ = = T (5.1.6)

Based on the normalization of the constants and by virtue of the conservation of energy, we have that

1R TΠ Π+ = (5.1.7)

Considering the interface plane between fluids 1 and 2 to be given at 0x = , as shown in Figure 32, we define the incident, reflected, and transmitted acoustic pressures.

( )1j t k xi ie

ω −=p P (5.1.8)

( )1j t k xr re

ω +=p P (5.1.9)

( )2j t k xt te

ω −=p P (5.1.10)

Because the waves must occur at the same frequency to maintain contact, the wavenumbers between fluids 1 and 2 are related by 1 1/k cω= and 2 2/k cω= . Thus, the rates of which points of constant phase in the

waves travel are different from fluid 1 to fluid 2. Correspondingly, this indicates that the acoustic wavelength in each fluid is unique. In fluid 1, the wavelength is 1 1 /c fλ = while in fluid 2 the wavelength

is 2 2 /c fλ = .

In order for the fluids to remain in contact at 0x = , two boundary conditions must be satisfied. Continuity of pressure keeps the interface at 0x = , otherwise a net force would produce a shift in the boundary in the x direction. Continuity of normal particle velocity keeps the fluid particles at the interface in contact. Applying these two boundary conditions leads to

i r t+ =p p p (5.1.11)

i r t+ =u u u (5.1.12)

A third, derivative boundary condition is observed by virtue of dividing (5.1.11) by (5.1.12). The continuity of normal specific acoustic impedance is therefore generally given by


90

i r t

i r t

+=

+p p pu u u

(5.1.13)

By the notations of this section, plane wave pressure and particle velocity are related by / r= ±p u .

Therefore, we make appropriate substitutions for the particle velocities in (5.1.13) to find

1 2i r

i r

r r+=

−p pp p

(5.1.14)

Rearrangement of (5.1.14) gives the expression for the pressure reflection coefficient between the two fluid media.

2

2 1 1

22 1

1

1

1

rr r r

rr rr

−−

= =+ +

R (5.1.15)

Division of (5.1.11) by ip shows that 1+ =R T . Therefore, the pressure transmission coefficient is found

to be

2

2 1

22 1

1

22

1

rr r

rr rr

= =+ +

T (5.1.16)

The intensity reflection and transmission coefficients for normal incidence acoustic pressure between the two fluids are

22

2

2 1 1

22 1

1

1

1I

rr r rR rr r

r

− − = = + +

(5.1.17)

( )

2

1 2 12 2

2 1 2

1

44

1I

rr r rT

r r rr

= =+

+

(5.1.18)

Studying the pressure reflection coefficient, we first observe that R is always real. When 1 2r r< , the

reflected pressure is in phase. Thus, the pressure near to the interface on the side of incidence is greater in amplitude than the amplitude of the incident wavefront itself. This is schematically shown in Figure 33(a) and would be similar to the case of a normally incident wave in air arriving at an air-water interface.


91

Figure 33. Examples of reflected and transmitted pressures based on the impedance difference between two fluids.

When 1 2r r> , similar to a water-air interface, Figure 33(c), the reflected pressure is out of phase with the

incident pressure. Thus, the pressure near to the interface on the side of incidence is less in amplitude than the amplitude of the incident wavefront itself. Of course, when 1 2r r= , there is no "interface" between fluids

because they are the same and no reflected pressure wave is induced, Figure 33(e).

The pressure transmission coefficient is always real and positive, so that transmitted waves are always in phase with the incident wavefront. Yet, the transmitted waves will either be reduced or magnified in amplitude based on the conditions 1 2r r> or 1 2r r< , respectively. The transmitted pressure trends are shown

in Figure 33(b,d,f), including the case when 1 2r r= and the pressure transmission coefficient takes on a unit

value.

In the limit of 1 2/ 0r r = , a rigid boundary is realized. The transmitted pressure is doubled although the

power transmission coefficient is zero.

In the limit of 1 2/r r = ∞ , a pressure release boundary is realized. As a result, there is neither pressure

transmitted nor power transmitted.

Importantly, there is no difference in the intensity reflection and transmission coefficients in the consideration of the direction of the incident wave.

5.2 Normal incidence wave propagation through a fluid layer

Consider a fluid layer of specific acoustic impedance 2 2 2r cρ= and thickness L that is bounded by a fluid

with incident wavefront ip and specific acoustic impedance 1 1 1r cρ= and by a final fluid layer permitting

a transmitted wavefront tp and specific acoustic impedance 3 3 3r cρ= . The situation is shown in Figure 34.


92

Figure 34. Schematic of wave transmission through a fluid layer.

Incident and reflected waves in fluid 1 are

( )1j t k xi ie

ω −=p P (5.2.1)

( )1j t k xr re

ω +=p P (5.2.2)

while the waves transmitted into fluid 2 (the fluid layer) are

( )2j t k xa e ω −=p A (5.2.3)

( )2j t k xb e ω +=p B (5.2.4)

while the transmitted wave into the final fluid 3, which does not have bound for continued increase in x , is

( )3j t k xt te

ω −=p P (5.2.5)

Continuity of normal specific acoustic impedance at 0x = and x L= yields (5.2.6) and (5.2.7), respectively.

2

1

i r

i r

rr

+ +=

− −P P A BP P A B

(5.2.6)

2 2

2 2

3

2

jk L jk L

jk L jk L

re ee e r

−

−

+=

−A BA B

(5.2.7)

Following considerable manipulation of (5.2.6) and (5.2.7), one obtains the pressure reflection coefficient

1 2 12 2

3 3 2

1 2 12 2

3 3 2

1 cos sin

1 cos sin

r r rk L j k Lr r rr r rk L j k Lr r r

− + −

= + + +

R (5.2.8)

In addition, the intensity and power transmission coefficients, which are the same presuming the cross-sectional areas of the fluid interfaces are all the same, are


93

22 23 1 31 2

2 221 3 1 3 2

4

2 cos sinIT T

r r rr rk L k Lr r r r r

Π= =

+ + + +

(5.2.9)

Evaluating special cases of the most general result of (5.2.9) provides illumination to the opportunities for wave transmission and reflection.

When 1 3r r= , we have that

222 1

21 2

1

11 sin4

Tr r k Lr r

Π =

+ −

(5.2.10)

Continuing, when the layer has specific acoustic impedance that is significantly greater than the fluid impedances bounding it, 2 1r r>> , (5.2.10) becomes (5.2.11).

222

21

1

11 sin4

Tr k Lr

Π =

+

(5.2.11)

The transmission of normally incident pressure waves through such a near-rigid media is comparable to sound transmission from one room to another through most walls of moderate thickness. In fact, considering

the realistic case that ( )2 1 2/ sin 2r r k L >> when the fluid is air and the cross-sectional dimensions of the

wall layer are much greater than the thickness, we obtain a further-reduced form of (5.2.11) that approximately holds for normally incident sound transmission through a wall

2

1

2 2

2sin

rTr k LΠ

=

(5.2.12)

Considering practical building designs, unless the frequency of consideration is extremely high, (5.2.12) reduces further by virtue of 2 2sin k L k L≈

2

1

2 2

2 rTk L rΠ

=

(5.2.13)

Therefore, the transmitted pressure amplitude is inversely proportional to the wall thickness while the transmitted power is inversely proportional to the square of the thickness. This is effectively a re-statement of the mass law in characterization of the sound insulation capabilities of panels. In the mass law, it is seen that the transmitted pressure amplitude is inversely proportional to the mass per unit area (product of density and wall thickness).


94

A case of particular interest in the general form of the power transmission coefficient (5.2.9) occurs when

212

k L n π = −

where 1,2,3..n = . As such, we find

1 32

1 32

2

4r rTr rrr

Π =

+

(5.2.14)

As a result, when the frequency of the incident wave is near 212 2

cf nL

= −

and the specific acoustic

impedance of the separating fluid layer is the geometric mean of the bounding media impedances, 2 1 3r r r=

, there is total transmission 1TΠ ≈ of the acoustic energy. This corresponds to a form of acoustic resonance.

For instance, by expressing these frequencies in terms of the acoustic wavelength 2λ , we find that the

lengths of the separating fluid layer for which this phenomenon occurs are

( ) 2 2 2 21 1 3 52 1 ; ; ; ; ...4 4 4 4

L n L L Lλ λ λ λ= − → = = = (5.2.15)

The use of quarter-wavelength devices to permit large wave transmission through interfacing layers is a means to permit high efficiency power transmission between media. This technique is effectively demanded in medical ultrasonics where piezoelectric-based acoustic radiators must interface with the human body, which has a specific acoustic impedance similar to water [18]. Thus, several such "matching layers" are employed whereby each layer is approximately the geometric mean of its adjacent neighbors. Note that this resonant phenomenon is only encountered for wave propagation at select frequencies.

A similar resonant phenomenon occurs for 2k L nπ= . In such events, the power transmission coefficient

becomes ( )21 3 1 34 /T r r r rΠ = + which takes on a unit value when the bounding fluid media are the same,

1 3r r r= = . Thus, we find that half-wavelength layers, 2 / 2L nλ= , permit perfect sound transmission at the

frequencies associated with this through-thickness resonance, 2 / 2f nc L= .

5.3 Oblique incidence wave propagation at a fluid interface

Consider the incident plane wave impinging on the interface between two fluid media at an angle of incidence iθ where the interface is at 0x = . Figure 35 gives a schematic of the situation. The incident,

reflected, and transmitted waves are expressed by

( )1 1cos sini ij t k x k yi ie

ω θ θ− −=p P (5.3.1)

( )1 1cos sinr rj t k x k yr re

ω θ θ+ −=p P (5.3.2)

( )2 2cos sint tj t k x k yt te

ω − −= θ θp P (5.3.3)


95

There is no fluid interface in the 0y = plane such that the assumed directions of wave propagation within

fluid 1 in the y axis are correct as shown in Figure 35 according to our finding of Sec. 5.1. The angle of

the reflected wave to the normal is rθ while a more general, complex angle form is used for the transmitted

wave tθ for reasons to be discovered through derivation.

Figure 35. Schematic of obliquely incident, reflected, and transmitted pressure waves through two fluid media.

Continuity of pressure at 0x = gives

1 21sin sinsini trjk y jk yjk yi r te e eθ θ− −−+ = θP P P (5.3.4)

The result of (5.3.4) must be true for all values y which requires the exponents of (5.3.4) to all be equal.

Thus, first we conclude that

sin sini rθ θ= (5.3.5)

which requires i rθ θ= . Also, the exponents from left- to right-hand sides of (5.3.4) must be equal so that

Snell's law is obtained.

1 2

sin sini t

c cθ

=θ (5.3.6)

In addition, division of (5.3.4) by the incident acoustic pressure also relates the pressure reflection to pressure transmission coefficients.

1+ =R T (5.3.7)

Continuity of the normal component of particle velocity must be satisfied lest the two fluids lose contact at 0x = . Therefore,

cos cos cosi i r r t tθ θ+ =u u u θ (5.3.8)

The (5.3.8) is written in terms of the pressure amplitudes and specific acoustic impedances of the fluids to yield

1

2

cos1cos

t

i

rr θ

− =θR T (5.3.9)


96

Together, (5.3.7) and (5.3.9) may be manipulated to give the pressure reflection coefficient

2 2 1

1

2 12

1

coscos cos coscos

cos coscos

t

i t i

t

t ii

r r rr

r rrr

θ θ

θθ

− −= =

++

θθR θθ

(5.3.10)

The (5.3.10) is referred to as Rayleigh's reflection coefficient. In addition, rearrangement of Snell's law (5.3.6) gives

222

1

cos 1 sint icc

θ

= −

θ (5.3.11)

Considering the (5.3.10) and (5.3.11), the angle of wave transmission is strongly dependent on the relative difference between the sound speeds of the fluids.

If 1 2c c> the angle of the transmitted wave tθ is real and less than the incident angle iθ . Figure 36(a) gives

an example of this outcome which can happen at an interface separating water and fluid domains, with the incident wave coming from the water domain. In this case, the transmitted waves are bent towards the normal to the interface.

If 1 2c c< and i cθ θ< , the transmitted angle tθ is real but greater than the angle of incidence. In this case,

the transmitted waves are bent away from the normal to the interface. The angle cθ is termed the critical

angle. By virtue of (414), the critical angle is computed from

1

2

sin ccc

θ = (5.3.12)

The pressure reflection coefficient is unity when incident waves impinge on the interface at the critical angle. Thus, the reflected waveform has the same amplitude and is in phase with the incident waveform. By (5.3.7), the transmitted acoustic pressure (and power) under such circumstances is zero.

Figure 36. Examples of oblique incidence acoustic pressure wave reflection and transmission phenomena between two fluids.


97

When 1 2c c< and i cθ θ> , the cosine of the transmitted angle cos tθ is imaginary, while by Snell's law the

sine of the transmitted angle sin tθ is real. Consequently, the transmitted pressure is alternatively written

( )1 sin ij t k yxt te e ω θγ −−=p P (5.3.13)

where the constant of exponential decay is

222

21

sin 1ickc

γ θ

= −

(5.3.14)

From (5.3.13), it is seen that the y axis component of transmitted wave propagation persists while the

component normal to the interface, in the x axis, decays in amplitude exponentially away from the interface. This evanescent wave consequently does not transmit acoustic energy into the fluid 2 (beyond a very short distance in x ). This example is shown in Figure 36(b) and could occur for such "steep" angles of incidence at an air-water interface where the incident wave arrives from the air fluid domain. In this case

i cθ θ> , the pressure reflection coefficient (5.3.10) becomes alternatively expressed using Euler's identity

and manipulations

je φ=R (5.3.15)

2

1

2

cos1tan 12 cos

c

i

θρφρ θ

= −

(5.3.16)

As clearly shown by (5.3.15), the reflected wave has the same amplitude as the incident wave. Yet, the phase of the reflected wave is shifted respecting the incident waveform. For angles of incidence slightly exceeding the critical value, the reflection coefficient is near R =+1 such that the reflected wave is of the same amplitude and in phase with the incident wave, which is similar to the special case indicated above and is comparable to a rigid boundary. As the incident angle approaches grazing incidence, 90θ → ° , the reflection coefficient approaches R =-1. Therefore, the reflected wave has the same amplitude but is perfectly out of phase to the incident wave, similar to a pressure release boundary.

In the general case, the power transmission coefficients are

2

12

2

1

cos4cos

coscos

t

i

t

i

rrT

rr

θ

θ

Π =

+

θ

θ; when tθ is real (5.3.17)

0TΠ = ; when tθ is imaginary (5.3.18)

which confirms the aforementioned claims that no acoustic power is transmitted when the incident angle exceeds criticality due to the decaying exponential factor that is a function of x in (5.3.13).


98

An interesting phenomenon occurs when 2 1/ cos / cost ir r θ= θ . Then, 1TΠ = . In other words, all incident

acoustic power is transmitted into the second fluid. Manipulating (5.3.17) under such conditions leads to the means to compute the angle of intromission Iθ :

2 2

2 1

1 22 2 2

2 2 1

1 1 2

1 1sin

1I

r rr r

r cr c

θρρ

− −

= =

− −

(5.3.19)

This phenomenon enables the complete transfer of obliquely incident acoustic energy from one fluid media to another, regardless of the frequency. Intromission thus creates perfect acoustic transparency between two distinct media. Intromission requires either [ 2 1r r> and 2 1c c< ] to occur, or [ 2 1r r< and 2 1c c> ] to occur.

While the combinations of these properties are not met in natural bulk materials, there are ways of developing or engineering artificial materials to achieve perfect acoustic transparency from an air-material interface. For example, see Ref. [19] and select references therein.

5.4 Oblique incidence wave propagation at a thin partition interface: the mass law

In building or architectural acoustics, a formula known as the mass law serves as a ballpark estimate of the significance of sound insulation from one room to another provided by a thin partition. The mass law may be derived directly by the methods undertaken in this chapter pertaining to oblique incidence sound transmission through multiple media.

Similar to Figure 34, consider an obliquely incident wave from an unbounded air domain impinging on a thin, flexible partition material at 0x = of thickness L , on the other side of which is another unbounded region of air. Thus, the difference between Figure 34 and this scenario is that the incident wavefront is obliquely incident to the intermediate layer.

When the partition is thin with respect to the acoustic wavelength, 2 1k L << . Therefore, an application of

Snell's law reveals that the angle of transmission tθ is the same as the angle of incidence iθ . As a result,

the normal component of particle velocity at the thin interface is

i r t+ =u u u (5.4.1)

The surface density of the partition is S Lρ ρ= where ρ is the volumetric density. Newton's 2nd law

normalized by area is thus an equivalence between the pressure balance from both sides of the layer to the area-normalized inertial force.

cosi r t S tjωρ θ+ − =p p p u (5.4.2)

Multiplying (5.4.1) by the specific acoustic impedance (recalling that there is only one fluid) yields


99

i tri r t

i r t

+ =−

p ppu u uu u u

(5.4.3)

and division by the incident acoustic pressure on (5.4.3) and (5.4.2), respectively yields (5.4.4) and (5.4.5).

1− =R T (5.4.4)

1 cosSjr

ωρθ+ = +R T T (5.4.5)

Solving for the power transmission coefficient gives

( ) 2

1

1 cos2

S

r

θωρ θ

ΠΤ = +

(5.4.6)

The transmission loss TL is computed from the power transmission coefficient by

11010logTL T −

Π= (5.4.7)

Therefore, the transmission loss of a flexible partition in the moderate frequency range, i.e. for most frequencies associated with speech and music, the mass law is

2

1010log 1 cos2

STLr

ωρθ

= + (5.4.8)

In many cases, the denominator of (5.4.6) is overwhelmed in magnitude by the second term, so long as the angle of incidence is not significant. Thus, considering (5.4.6) and (5.4.8), each doubling of surface density reduces the power transmission coefficient by approximately four times, while the TL correspondingly

increases by 6 [dB], ( )1010log ~ 4 6≈ . A common reality in building and architectural acoustics is that mass

between two rooms is the one assured means to inhibit sound transmission. The mass law can guide decision-making in such practices.

5.5 Method of images

If the wavefront incident on a plane interface is spherical, such as at the 0z = plane interface in Figure 37, we can use the method of images to evaluate the waves received at locations elsewhere in the acoustic media from where the waves originated. The exact plane interface type, e.g. air-to-water or water-to-air, does not matter so long as the boundary conditions pertaining to the interface are satisfied, thus satisfying the acoustic wave equation.

The method of images replaces the reflected wavefront by an imaged wavefront that hypothetically emanates from the fluid below the plane 0z = . The superposition of incident and imaged wavefronts are determined so as to satisfy the boundary conditions at the interface. Consequently, the acoustic field above


100

the plane surface, in the domain of the real point source, is correctly reconstructed. On the other hand, the acoustic field below the surface is not correctly reconstructed.

Figure 37. Schematic of physical point source 1 above a plane interface at 0z = , and an image source below the plane.

Consider the example shown in Figure 37. A point source 1 of spherical waves is at location ( ) ( ), 0,x z d=

above a perfectly rigid plane 0z = . The field point is at ( ), ,x y z , where positive y locations are into the

page. The direct pressure wave received at the field point as emitted by the point source 1 is

( )j t kri

A er

ω −−

−

=p (5.5.1)

( )2 2 2r z d y x− = − + + (5.5.2)

We then imagine that the fluid domain from where the waves arrive continues without interupt by the plane. Then, we introduce a hypothetical, coherent image point source at a location below the plane interface that is identical to the distance above the plane of the real point source 1. The image point source has the same amplitude as the point source 1.

( )j t krr

A er

ω +−

+

=p (5.5.3)

( )2 2 2r z d y x+ = + + + (5.5.4)

The rigid plane boundary condition requires that the normal component of particle velocity (in the z axis) at the 0z = plane must vanish. Because the sources have the same pressure amplitude and are identical distances away from the now-hypothetical plane interface, it is clear that the normal component of particle velocity vanishes at 0z = .

Thus, instead of a rigid plane interface, the direct and reflected pressures in the fluid domain 0z > are recreated by an unbounded fluid domain possessing original and imaged point sources. The acoustic field


101

in 0z > is correctly reconstructed by this approach while the field in 0z < is not correctly reconstructed by this approach.

Simplifying the problem and considering the schematic of Figure 37, we assume that the field point is a

radial distance r from ( )0, ,0y such that cosr d θ>> . Thus, we find that the superposition of original and

imaged point source spherical wavefronts at the field point gives

( ) ( ), ,1 1

jk r jk rj t krA e ep r t e r rr

r r

ωθ∆ − ∆

−

≈ + ∆ ∆ − +

(5.5.5)

where sinr d θ∆ = .

When the field point is many times r∆ away from the origin line ( )0, ,0y , r r>> ∆ , the inclusion of this

term into the denominators of (5.5.5) may be safely disregarded. On the other hand, the presence of r∆ in the complex exponentials may not be disregarded since, by Euler's identity, the function arguments repeat every 2π . Thus, by use of Euler's identity and with the simplifications described above, we have that

( ) ( ) [ ], , 2 cos sinj t krAp r t e kdr

ωθ θ−≈ (5.5.6)

When the source is very near to the rigid plane with respect to the wavelength 1kd << , the pressure received

at the field point is ( ) ( ), , 2 j t krAp r t er

ωθ −≈ . In other words, a point source near a rigid plane projects twice

the amplitude of acoustic pressure into the field as the same point source in the free field. Simple calculations may show that the intensity in the real (not imaged) acoustic field increases 4 times in consequence to the rigid plane while the acoustic power increases 2 times. In practice, a baffle is the term used to describe the rigid plane onto or into which an acoustic source is placed in order to enhance its projection of sound power into the field.

If the interfacing surface is a pressure release boundary, rather than rigid boundary, then it can be shown that the imaged source must be of the same amplitude but of opposite phase as the original point source.

The resulting pressure superposition at the plane satisfies the boundary condition of ( ), , 0 0p x y z = =

while, using polar coordinates, the pressure in the field is

( ) ( ) [ ], , 2 sin sinj t krAp r t j e kdr

ωθ θ−≈ (5.5.7)

To apply the method of images, the acoustic waves need not be single frequency since the superposition of

general spherical wave solutions for the direct and image sources, according to the form ( )1 f ct rr

= −p ,

satisfies the boundary condition where the normal component of particle velocity vanishes at the plane


102

defined by ( ), ,0x y . Thus, the method of images is applicable for all waveforms: single frequency, transient,

stochastic, etc.

Should the receiver and original point source be switched, so that the receiver is now located at ( )0, ,y d

and the point source is located at ( ), ,x y z , then the application of the method of images reveals that the

resulting pressure received at the field point receiver is the same. This is illustrated by the two cases shown in Figure 38. The result exemplifies that the principle of acoustic reciprocity holds regardless of the presence of boundaries. In other words, as subject to the assumptions of the linear acoustic wave equation, switching of source and receiver locations does not influence the acoustic variables at the receiving point.

Figure 38. Switch of point source and receiver locations to demonstrate that the principle of acoustic reciprocity holds regardless of the presence of boundaries. Demonstrated by application of the method of images.

5.6 Acoustic metamaterials

Metamaterials are engineered structures, often composed of strategically assembled or periodic constituents, that exhibit unprecedented bulk properties by virtue of the composition. Because such properties are unconventional with respect to materials, the term metamaterials is warranted, despite the fact that the engineered systems are often realized in structural forms. An illustration of the concept of metamaterial is given in Figure 39(a) where a rod-like object in the bulk (top) is actually composed of a mass-spring lattice, although such microarchitecture may not be observable from a macroscopic perspective.

Studies on metamaterials were mostly concentrated in photonic and electromagnetic applications in early years [20], while more recent decades have seen an acceleration of acoustic metamaterials research as it pertains to waves in fluids and solids [21] [22]. Because the term acoustic waves is often interchangeably used to refer to any waves satisfying a linear wave equation, as we have seen in Sec. 3 there are numerous solid or structural media that have equivalent wave equations as that derived for fluid-borne wave propagation, Sec. 4. Thus, the use of the term acoustic metamaterials is used in the scientific communities to refer to metamaterials that have unusual elastic and fluid-borne wave propagation properties [23].


103

Figure 39. Schematics of fundamental metamaterial architectures. (a,b) show respectively atomic and diatomic lattice representations while (c) shows a fluid media representation analogous to the diatomic lattice of (b).

While our studies to date have focused upon wave propagation in systems governed by PDEs, the analysis of wave propagation through periodic-types of metamaterials is best understood by first considering discrete mass-spring lattices, such as that shown in Figure 39(a).

5.6.1 One-dimensional monatomic lattice

The perfect periodicity of the composition of the lattice in Figure 39(a) warrants its name the monatomic

lattice. When the thn mass m of the lattice in Figure 39(a) is displaced an amount nw from equilibrium,

forces are exerted upon it by the neighboring elements of the monatomic lattice. Thus, a force acts on the thn mass to restore it to equilibrium according to

( ) ( )1 1n n n n nF s w w s w w+ −= − − − (5.6.1.1)

Therefore, by Newton's 2nd law, the equation of motion for the thn mass m of the lattice is

( )1 1 2n n n nmw s w w w+ −= + − (5.6.1.2)

Due to the periodicity of this mass-spring composition, we refer to these constituents as the unit cell of the monatomic lattice. We seek one-dimensional traveling wave solutions to (5.6.1.2) according to an assumed form

( )j t knan e ωθ +=w (5.6.1.3)

where na , termed the propagation constant, is a quantity similar to the continuous spatial coordinate x in the study of plane wave propagation in an elastic or fluid medium. Due to the lumped-parameter formulation


104

and the spatial periodicity of (5.6.1.2), there is no particular need to distinguish the jkna± exponential

arguments for "forward" or "backward" propagating waves.

This form of assumed solution (5.6.1.3) to wave propagation in a periodic media is an expression of Bloch's theorem. In other words, time-harmonic waves permitted in periodic media are modulated according to the period of the media. As a result, the study of the unit cell, respecting its location within the periodic media, enables the full characterization of the wave propagation characteristics of the whole periodic media. Thus, the use of (5.6.1.3) as an assumed solution, and similar forms for multi-dimensional wave propagation, to study time-harmonic wave propagation is termed Bloch wave analysis for periodic media.

Substituting (5.6.1.3) into (5.6.1.2), we find

( )2 2jka jkam s e eω −− = + − (5.6.1.4)

Using Euler's identity, we find that (5.6.1.4) yields non-trivial solutions for the wave frequency ω only so long as

( ) 4 1sin2

sk kam

ω = ± (5.6.1.5)

The (5.6.1.5) is the dispersion relation ( )kω for the monatomic lattice, relating the frequency of wave

propagation ω to the wavenumber k (or, phase change per unit length) characterizing the propagation. By (5.6.1.5), it is evident that the monatomic lattice is dispersive since the frequency is a nonlinear function of the wavenumber. The dispersion relation for the monatomic, one-dimensional mass-spring lattice is shown in Figure 40. Negative frequencies are not shown because they are non-physical, purely mathematical outcomes of (5.6.1.5). The dependence of ω on k is symmetric about 0k = , and the maximum frequency

that is permitted to propagate through the lattice is 4 /m s mω = which occurs when /mk aπ= ± . This

indicates that the maximum frequency that may propagate corresponds to adjacent masses oscillating

perfectly out of phase. In other words, most generally we have 1/ jkan nw w e−

+ = , such that when mk k= , we

find 1/ 1n nw w + = − . Considering small values of the wavenumber, (5.6.1.5) becomes

( )/ sk a kam

ω π<< ≈ . Considering the density to be mass per length to be /m aρ = and considering the

equivalent elastic stiffness B sa= , the aforementioned limit shows B k ckωρ

≈ = , defining

/ /c B a s mρ= = . This shows that the monatomic mass-spring lattice permits wave propagation in an

identical manner as a continuous media satisfying the linear wave equation when the waves are at low frequencies respecting the spatial periodicity of the lattice.


105

Figure 40. Dispersion relation for monatomic mass-spring lattice

Several factors about the elementary, monatomic mass-spring lattice are illuminating as they pertain to the broader class of acoustic metamaterials and periodic structures studied in recent research. These factors, described below, are largely the outcome of realizing periodic media that in some respects are analogous to continuous media from a macroscopic perspective and in some respects are different than continuous media.

5.6.1.1 Periodicity

Values of wavenumber greater than / aπ repeat the trends from / /a k aπ π− < < + . This indicates that for a given frequency of wave propagation, the wavelength, by 2 /k π λ= , is not uniquely defined. In fact, this corresponds to a spatial aliasing phenomenon in lattices that also governs the direction of the wave propagation that will occur.

The wavenumber-frequency space for this lattice [ ]/ , /k a aπ π∈ − is referred to as the first Brillouin zone,

(FBZ) while the truly unique space [ ]0, /k aπ∈ is referred to as the irreducible Brillouin zone (IBZ). These

areas of the dispersion relation are named in honor of an applied mathematician Léon Brillouin whose textbook on the subject of wave propagation in periodic structures has stood the test of time and continues to be cited regularly almost 80 years after its publication [24]. Conveniently, the text is an easy and good read. The unique frequencies of wave propagation permitted in the monatomic lattice are encapsulated in the IBZ. Similar domains are identified for multi-dimensional lattices in the respective spatial extents of the lattice composition, and hence in wave propagation directions.

5.6.1.2 Dispersion

The dispersion relation for the lattice is given in (5.6.1.5). The phase velocity of the monatomic lattice is

1sin1 4 1 2sin 12

2

p

kasc ka c

k k m ka

ω = = =

(5.6.1.2.1)

-5 -4 -3 -2 -1 0 1 2 3 4 50

0.5

1

1.5

2

ka/π, normalized wavenumber [dim]

ω/(s

/m)(1

/2) n

orm

aliz

ed fr

eque

ncy

[dim

]


106

The phase velocity of the lattice is distinctly a nonlinear function of the wavenumber, unlike the phase velocity of one-dimensional linear wave propagation in a rod or in fluids. Thus, waves at high frequencies (within the band of frequencies able to be propagated in the lattice) travel slower than lower frequencies. Yet, for long wavelengths, the limit 1ka << shows pc c≈ , showing a parallel to linear wave propagation

in a rod or fluids.

Similarly, the group velocity of the monatomic lattice is

1 1cos cos2 2g

sc a ka c kak mω∂ = = = ∂

(5.6.1.2.2)

which shows a vanishing of group velocity as the wavelength approaches the limits of the IBZ, / 1ka π ≈ . This indicates that information ceases to travel as the frequency of the wave increases. As shown in Sec. 5.6.1, the adjacent masses merely oscillate out-of-phase when excited at this frequency. This provides a physical explanation for why "information" cannot travel in such a periodic lattice at high frequency: the energy of vibration is merely exchanged between each adjacent oscillating mass, as a transition between kinetic and potential energy, so that no energy remains to flow through the system. The trends of variation in the phase and group velocity are plotted in Figure 41.

Figure 41. Phase and group velocity of monatomic lattice.

5.6.1.3 Propagating and attenuation bands

The dispersion relation (5.6.1.2.1) does not permit waves at frequencies above 4 /m s mω = to propagate.

This is evidence of a stopband or bandgap. These are frequency ranges in which waves are not permitted to travel by virtue of the fact that the dispersion relation does not have a fully real-valued relationship between wavenumber and frequency in such a parametric range. Specifically, stopbands refer to ranges of frequencies greater than some value in which waves do not propagate, while bandgaps refer to a finite

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1


red solid, normalized phase velocity cp/(a(s/m)(1/2)) [dim]blue dash, normalized group velocity cg/(a(s/m)(1/2)) [dim]


107

bandwidth of frequencies in which waves do not propagate. The monatomic lattice possesses a stopband while the diatomic lattice, as shown in Sec. 5.6.2, possesses a bandgap and stopband.

Considering the forced excitation of the lattice at frequencies mω ω< , waves will propagate through the

lattice with phase speeds (5.6.1.2.1) and according to the spatial variation of mass displacement of

1/ jkan nw w e−

+ = . Yet, when the forcing frequency exceeds mω , there are no corresponding real-valued

wavenumbers obtained from the dispersion relation. In this case, the waves from the driven point of the lattice will be evanescent and will not propagate. The development and exploitation of bandgaps in metamaterials is a commonly used means to manipulate elastic and acoustic waves [21] [22].

To characterize the significance of the wave attenuation in the stopband or bandgap, the dispersion relation

is written as the wavenumber as a function of the frequency 21cos 12

mkasω= − and solved for a range of

frequency ω . The resulting computation yields real rk and imaginary ik components of the wavenumber

r ik k jk= + , as shown in Figure 42. Because the wavenumber is complex, the assumed solution is

consequently characterized by propagating and attenuating components ( )ri j t k najk nan e e ωθ +−=w exemplifying

the inability for the monatomic lattice to propagate energy in such frequency ranges.

Figure 42. Real and imaginary wavenumbers computed from dispersion relation.

5.6.2 One dimensional diatomic lattice

Consider the diatomic lattice shown in Figure 39(b). The unit cell consists of two masses and two springs. In the diatomic lattice, masses m are located at positions with odd indices, while masses M are located at positions with even indices.

Application of Newton's 2nd law for the thn unit cell reveals

-0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

2

2.5

3


ω, wave propagation frequencyred solid, real component of wavenumber. blue dash, imaginary component of wavenumber


108

( )2 2 1 2 1 22n n n nMw s w w w+ −= + − (5.6.2.1)

( )2 1 2 2 2 2 12n n n nmw s w w w+ + += + − (5.6.2.2)

Similar to the methodology used to analyze the monatomic lattice, we assume that the lattice permits wave propagation according to

( )22

j t nkan e ωθ +=w (5.6.2.3)

[ ]( )2 12 1

j t n kan e ωφ + ++ =w (5.6.2.4)

Substituting (5.6.2.3) and (5.6.2.4) into (5.6.2.1) and (5.6.2.2) yields

( )( )

2

2

2

2

jka jka

jka jka

M s e e s

m s e e s

ω θ φ θ

ω φ θ φ

−

−

− = + −

− = + − (5.6.2.5)

The (5.6.2.5) has unique solutions for θ and φ so long as the determinant of coefficients is zero.

2

2

2 2 cos0

2 cos 2s M s ka

s ka s mω

ω− −

=− −

(5.6.2.6)

Solving for the wave propagation frequency yields

1/22 22 1 1 1 1 4sin kas s

M m M m mMω

= + ± + −

(5.6.2.7)

This pair of dispersion relations is shown in Figure 43 within the irreducible Brillouin zone. Two branches in the plot are shown.

Figure 43. Dispersion relations for diatomic lattice.


109

At low frequencies, it may be shown that θ φ≈ , such that the masses oscillate in phase with nearly the

same amplitude. As we know from Sec. 4, this is how waves propagate through acoustic fluids: the fluid molecules within a fluid particle are so numerous that from one molecule to the next the motions are nearly identical. Thus, in this diatomic lattice, at low frequencies waves propagate in a way similar as in acoustic fluids, warranting the term for this portion of the dispersion relation as the acoustic branch.

At high frequencies, it may be shown that / /m Mθ φ ≈ − . Thus, the masses oscillate perfectly out of phase

with an amplitude ratio proportional to the mass ratio. Phonons exhibit this wave propagation phenomenon, which has led to the use of the term optical branch to denote the dispersion relation component corresponding to high frequency wave propagation.

A bandgap is apparent from the dispersion relation. Frequencies of waves from 2 / 2 /s M s mω< < will not propagate through the lattice, which is evidence of a bandgap. High frequencies greater than

1 12sM m

ω > +

will also not propagate, which is evidence of a stopband.

5.6.3 Metamaterial continua

The periodic system of continuous elastic or fluid layers, such as the two-layer media shown in Figure 39(c), is analyzed using similar methods as that utilized in Sec. 5.6.2 but now the attention is drawn to the original wave equations of motion and associated boundary conditions. Namely, (5.6.3.1) must be satisfied for each domain

2 2

2 2 2

1 ; 1,2i i

i

p p ix c t

∂ ∂= =

∂ ∂ (5.6.3.1)

where /i i ic B ρ= . Assuming time-harmonic waves and by invoking Bloch's theorem [25], we presume

that the pressure response is periodic with spatial periodicity d , ( ) ( )i ip x p x d= + . Consequently, we take

assumed solutions to be

( ) ( ) ( )1 11 1 2

x n njk jkp a e a eξ ξ−= + ; 10 dξ< < (5.6.3.2)

( ) ( ) ( )2 22 1 2

x n njk jkp b e b eξ ξ−= + ; 1d dξ< < (5.6.3.3)

where x ndξ = + , 1 1/k cω= , and 2 2/k cω= . In (5.6.3.2) and (5.6.3.3), the shared time-harmonic j te ω

terms are dropped for brevity.

At 0ξ = and 1dξ = , the boundary conditions between fluid layer 1 to 2 are applied to yield a set of four

equations in terms of the amplitude coefficients ( )1

na , ( )2na , ( )

1nb , and ( )

2nb . The determinant of this coefficient

matrix must be zero in order for non-trivial coefficients and corresponding wave propagation amplitudes to exist. The result of this determinant is a dispersion relation given by


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1 21 1 2 2 1 1 2 2

2 1

1cos cos cos sin sin2

r rkd k d k d k d k dr r

= − +

(5.6.3.4)

where 2 1d d d= − . Recalling that 1 1/k cω= and 2 2/k cω= , (5.6.3.4) defines an implicit relationship of

the frequency ω on the wavenumber k .

Figure 44 shows representative results of the dispersion relation for a periodic fluid system composed of air and steam layers as fluids 1 and 2, respectively. The corresponding parameters are 1c =343 [m/s], 1ρ

=1.21 [kg/m3], 2c =404 [m/s], 2ρ =0.6 [kg/m3], 1d =100 [mm], and d =150 [mm]. The method of solving

(5.6.3.4) is either to prescribe a range of frequency ω and solve for the corresponding wavenumber, or prescribe a range of wavenumber k (often normalized as /kd π ) and solve for the corresponding wave frequencies. Taking the latter approach requires determination of the non-unique responses and may be more challenging from an algorithm-coding perspective.

While diatomic lattices possess only two "branches", due to the infinite-dimensional nature of the fluid continua and the partial differential governing equations of motion, an infinite number of dispersion branches occur in periodic continua. It is only a matter of how many eigenfrequencies associated with the branches, as determined by the dispersion relation (5.6.3.4), are ultimately examined in a plot. In Figure 44, four of the branches are shown, revealing bandgaps around 1.2 [kHz], 2.3 [kHz], and 3.7 [kHz].

Figure 44. Dispersion plot of waves through periodic two-fluid-media unit cell.

5.6.4 Computational methods of analysis

While lumped parameter periodic systems are useful, simplified representations of practical periodic metamaterials, more realistic models incorporate the continua representations of the domains, as exemplified in Sec. 5.6.3. Yet, the closed-form analysis of metamaterials becomes significantly

0 0.2 0.4 0.6 0.8 10

1

2

3

4

5

normalized wavenumber, kd/π

frequ

ency

[kH

z]


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cumbersome for more intricate architectures and when the metamaterials possess a greater number of physical dimensions. Consequently, computational methods are well-established means of investigating acoustic metamaterials. The method herein described is based on the finite element method, as used via COMSOL Multiphysics software. Alternative computational methods are available [23], but the ease of learning to appropriately use COMSOL and its wide implementation in the research communities warrants its consideration here.

A concise introduction to the effective use of COMSOL Multiphysics is provided here. This software is best used by undertaking the following steps

• Define parameters • Define geometry • Define materials and assign domain to prescribed materials • Associate physics with domains and assign boundary conditions • Define mesh with suitable refinement • Construct parameter study (as applicable) and other relevant study characteristics • Plot and assess results

The following overview uses a one-dimensional Pressure Acoustics model as an example.

5.6.4.1 Define parameters

Inserting parameters as "hard coded" numbers into a COMSOL model is tedious when one inevitably needs to change that parameter. Thus, defining parameters using the model-wide Global Definitions is wise. A representative set of parameters for a one-dimension acoustic model is shown in Figure 45.

Figure 45. Defining parameters in COMSOL.


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5.6.4.2 Define geometry

The geometry is defined entirely according to the parameters set aside in the Global Definitions. Importantly, one should retain the original default units of [m] and [deg], as shown in Figure 46. Figure 47 shows an example of geometry definition for a line interval.

Figure 46. COMSOL units definition.

Figure 47. Definition geometry in COMSOL.


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5.6.4.3 Define materials and assign domains

Once a geometry is defined, the various domains must be assigned material properties. Figure 48 shows an example of such definitions.

Figure 48. Assigning parameters to material properties and assigning domains to the material.

5.6.4.4 Associate physics with domains and assign boundary conditions

In models with multiple physics, the domains must be associated with the appropriate physics. For instance, in a fluid-structure interaction problem, only the fluid domain should be associated with the Pressure Acoustics physics, while the Solid Mechanics physics should only be associated with the structural domains. In our one-dimensional acoustics model, both intervals (domains in one dimension) are assigned to the Pressure Acoustics physics.

The boundary conditions for determining the dispersion relation are the Periodic Condition, by Floquet periodicity. The destination boundary is arbitrary in a one-dimensional model, but is usually the "right" or "top" boundary in the model. The Bloch's theorem is alternatively termed the Floquet-Bloch theorem, which explains the term used in COMSOL. Figure 49 shows the assignment of the edge points of the domains to this boundary condition and its definition according to the wavenumber k that was previously identified as a parameter above. Figure 50 shows an example of prescribing the destination boundary.


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Figure 49. Floquet periodicity boundary condition assignment and definition.

Figure 50. Destination definition for Floquet-periodicity.

5.6.4.5 Define mesh with suitable refinement

In any finite element or boundary element model of wave processes, it is the established convention that at least 6 elements are required per wavelength in order to yield reasonably accurate results [26]. This is because the spatial discretization is similar to time discretization. Thus, in data acquisition, at least 6 points in time are needed to viably reconstruct a sinusoid. Similarly, at least spatial sampling points are required


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to viably reconstruct a sinusoidal wave in space. Yet, here we are determining the frequency associated with a given wavenumber. As a result, we will first need to run the model with a moderate to excessive spatial discretization (high refinement), assess the resulting frequencies respecting the associated wavelengths and mesh discretization, and then decide whether further mesh refinement is desired for greater accuracy. Figure 51 shows one example of the mesh definition. It is recommended to always define a mesh with respect to key spatial parameters instead of using the COMSOL-based mesh refinements of "Coarse", "Fine", "Finer", and so on.

Figure 51. Mesh discretization definition.

5.6.4.6 Construct the study

Here, a parameter study will be used to iterate through potential wavenumbers and solving the associated finite element-based eigenvalue problem. This will yield the dispersion relation. Figure 52 shows the

definition of a parameter sweep for the normalized wavenumber [ ]0,1k∈ that interacts with the Periodic

Boundary condition. The Eigenfrequency study step here solves for the lowest 4 eigenfrequencies. This small number is used for conciseness of results presentation and speed of computation. In general, it is better to compute a greater number of eigenfrequencies so long as the mesh discretization is refined enough for accurate computations, according to the requirements described in Sec. 5.6.4.5.


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Figure 52. Parameter sweep definition for the eigenfrequency study.

5.6.4.7 Plot and assess results

Here, we use a Global plot to evaluate the eigenfrequency as a function of the sweep parameter, the normalized wavenumber k . The subplot shown in Figure 53 appears very similar to the analytical plot shown in Figure 44, which is to be expected because the analytical model (computed by MATLAB) employed the same parameters as the COMSOL model. The final check is the assessment of suitable mesh discretization. The highest eigenfrequency determined is near 5 [kHz]. The shortest wavelength at 5 [kHz] between the two fluid media included in the study is around 69 [mm]. The mesh discretization defined the largest element size to be /100 1.5d = [mm], see Figure 51. Thus, 6 elements would span approximately 9 [mm]. In other words, the existing mesh discretization is appropriate for the frequency range computed in this one-dimensional acoustics study.


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Figure 53. Example Global plot for the eigenfrequencies determined according to each parameter in the sweep.

5.6.4.8 Nuances for two-dimensional dispersion relation COMSOL study

The parameter and periodic boundary condition definitions are unique for two-dimensional models. As observed in Sec. 3.7, two-dimensional wave propagation may be defined according to a wavenumber vector that includes components in the x and y axes. This is the concept employed in the study of two-

dimensional acoustic metamaterials composed of continuous media, rather than lumped parameter constituents. Interested individuals should consult Refs. [23] [27] for further details. In COMSOL, the implementation of the wavenumber vector requires the explicit definition of the wavenumber components in x and y axes as parameters.

For a rectangular periodic domain, the irreducible Brillouin zone (IBZ) of a two-dimensional acoustic metamaterial is described according to the region outlined in Figure 54 by the label range X MΓ − − . This is the conventional notation used in the literature [23]. For points evaluated at Γ , the wavenumber vector components are 0x yk k= = which is the rigid body mode. For points evaluated at X , the wavenumber

vector components are /xk dπ= [1/m] and yk =0, which corresponds to wave propagation only in the x

axis. For points evaluated at M , the wavenumber vector components are /xk dπ= [1/m] and /yk dπ=

[1/m], which corresponds to plane wave propagation at an angle 45 ° from both x and y axes. The

variation of the wavenumber vector components from these limits of the IBZ yield the full range of corresponding wavenumbers and hence associated frequencies that are permitted within the periodic domain, while the symmetry of the IBZ warrants the omission of the wavenumber vector point that would correspond to xk =0 and /yk dπ= .


118

Figure 54. Irreducible Brillouin zone outlined by X M YΓ − − − of two-dimensional acoustic metamaterial with rectangular domain.

To implement this computational procedure in COMSOL, the wavenumber vector components are defined according to if/else criteria. In other words, by sweeping the wavenumber k from values arbitrarily

assigned from 0 to 1, the wavenumber vector components xk and yk are swept from the values

corresponding to Γ to those corresponding to X . Then, the process repeats by sweeping the wavenumber k from values arbitrarily assigned from 1 to 2, wherein the wavenumber vector components xk and yk are

swept from the values corresponding to X to those corresponding to M . Then a sweep from M to Y is accomplished. From X to M to Y , the relative contribution of wavenumbers in x and y axes is assessed,

thus characterizing wave propagation at angles through the lattice respecting the x y− plane. Then the final

sweep from Y to Γ is undertaken. From Y to Γ and from Γ to X , wave propagation strictly in y or x

axes is evaluated, respectively. The parameter definition in COMSOL is shown in Figure 55 and Figure 56 where it is observed that one set of material properties is similar to steel, while Figure 57 shows the representative geometry where in scaled circle (ellipse) is the steel domain. The specific parameter definitions for xk and yk , written in the COMSOL if/else style, are

xk , if(k<1,pi/d*k,if(k<2,pi/d,if(k<3,pi/d*(3-k),0)))

yk , if(k<1,0,if(k<2,(k-1)*pi/d,if(k<3,pi/d,(4-k)*pi/d)))

The corresponding boundary condition definition is given in Figure 58. The parameter sweep, in Figure 59,

now indicates the wavenumber [ ]0,4k∈ . The resulting plot of results is also given in Figure 59 within the

subplot. Note from the results that the dispersion relation is periodic from the plot limits of k =0 and 4k =. Typically, the axes of the plots are not labeled numerically, but are instead labeled Γ , X , M , and Y and are correspondingly "stretched" to accommodate the different spatial lengths associated with the right triangular space of the IBZ. Based on the symmetries of the geometry, the IBZ of a given two-dimensional periodic lattice structure may be reduced or expanded, such as due to hexagonal arrangements or circular symmetries [28]. Also, the output from COMSOL orders the eigenfrequencies from low to high values


119

although the spatial motion, the eigenmode, associated with each frequency is distinct. Thus, there are "crossover" points in the plot of Figure 59 which are only due to the ordering provided by COMSOL. Thus, an eigenmode plot of points below and above such points in the wavenumber space will reveal the connections between eigenmotion and eigenfrequency.

Figure 55. Two-dimensional COMSOL parameter definitions for dispersion analysis, kx.


120

Figure 56. Two-dimensional COMSOL parameter definitions for dispersion analysis, ky.

Figure 57. Two-dimensional geometry definition.


121

Figure 58. Two-dimensional periodic boundary condition definition.

Figure 59. Two-dimensional wavenumber parameter sweep definition and representative dispersion plot results.


122

6 Sound radiation from structures

A significant proportion of acoustic pressure fluctuations in myriad engineering and science applications occur in consequence to vibrating structures. If the fluid impedance is much less than the relevant structural impedance, such as a bending wave impedance, then it is permissible to assume that the structural component is equivalent to a moving boundary acting on the fluid without otherwise being acted upon by the fluid. Analyzing such a one-way transfer of energy from mechanical vibrations into acoustic pressure fluctuations is a first, and oftentimes quite accurate, start to understanding how structures radiate sound. We explore this area of acoustics by first introducing the elementary acoustic radiators and then establish integral principles that build from the concept.

6.1 Monopole and point source

The monopole was introduced in Sec. 4.8. It is a sphere of radius a that harmonically oscillates in its radial

direction with a velocity of j tUe ω . By satisfying the boundary condition between the acoustic fluid and sphere radius, we may determine the acoustic pressure (6.1.1) and particle velocity (6.1.2).

( )0 1

j t k r aa jkacU er jka

ωρ − − =+

p (6.1.1)

( ) ( ), j t k r aar t U er

ω − − =u (6.1.2)

These results are only valid for r a> . Considering our more general assumed solution for spherically-symmetric wave propagation

( ) ( ), j t krr t er

ω −=Ap (6.1.3)

we see from (6.1.1) that the amplitude of the acoustic pressure is indeed complex, such that we associate

0 1jkajkacUa e

jkaρ=

+A (6.1.4)

At distances much greater than the oscillating sphere radius, r a>> , the point source exhibits an acoustic pressure variation expressed by

( ) ( )0

j t kraj cU ka er

ωρ −=p (6.1.5)

where the complex amplitude of the pressure is likewise evident, ( )0j cUa kaρ=A . Recall that a baffled

point source projects twice this amplitude of acoustic pressure into the acoustic field above the rigid plane in which it rests, Sec. 5.5.

The point source is the jumping-off point for a considerable number of investigations in acoustics that pertain to the far field. The far field is characterized by three distinguishing traits [29]:


123

1. The pressure amplitude decreases monotonically in inverse proportion to the distance from the source center.

2. The directivity properties of the source, by angular variation in pressure amplitude, do not vary for further increase in the distance from the effective source center. Thus, the directivity is fully developed.

3. The specific acoustic impedance is equal to the plane wave impedance.

To be in the far field, one must meet the following three radial range criteria, where d is the characteristic dimension of the acoustic source. For a point source, d a= . These criteria are not explicitly related to the distinguishing traits given above.

(a) ( )2 / 1kr rπ λ= >> , "acoustic far field"

(b) / 1r d >> , "geometric far field"

(c) 2 1rd dλ

π>> , accounts for the variation in delivery of acoustic waves from one region of an effective

source to another when the source has considerable characteristic dimension respecting the acoustic wavelength

Note that the radial range criterion (b) is employed in the definition of the point source.

The near field exists within a few acoustic wavelengths of the effective source center. In the studies of our course, we will focus attention on the far field. Interested individuals are encouraged to consult Ref. [29] for additional details about the distinctions between near and far fields emitted by sources of sound.

6.2 Two point sources

Consider two point sources in the free field emitting acoustic pressure waves at angular frequency ω and

in phase to the far field point located at ( ),r θ where the net acoustic pressure is ( ), ,r θ ωp , Figure 60. This

can be considered as the most elementary array: two point sources emitting sound together to yield a total acoustic field at a common point.

The total acoustic pressure at the field point is written as

1 2= +p p p (6.2.1)

where ( )ij t krii

i

A er

ω −=p . In the far field, the distances ir are approximated by 1 sin2dr r θ≈ − and

1 sin2dr r θ≈ + because the distances 1,2r are nearly parallel to r . Then, by substitution, we have

sin sin2 2

sin sin2 2

d dj t k r j t k rA Ae ed dr r

ω θ ω θ

θ θ

− − − + = +− +

p (6.2.2)


124

In the far field, we have that sin2dr θ>> . Thus, the denominators of the above are approximately r . On

the other hand, the complex exponential requires the full information since this is phase-related and repeats every 2π , by Euler's identity. Simplifying and grouping terms, we have

( ) sin sin2 2d djk jkj t krA e e e

rθ θω −−

= +

p (6.2.3)

Figure 60. Two point sources emitting sound to the far field point.

Using Euler's identity on the component in brackets to obtain a trigonometric relation, we have

sin sin2 2 cos sin sin sin cos sin sin sin

2 2 2 2

2cos sin2

d djk jk d d d de e k j k k j k

dk

θ θθ θ θ θ

θ

− + = + + − =

(6.2.4)

Consequently, we find that the complex acoustic pressure at the field point is

( )2 cos sin2

j t krA de kr

ω θ− = p (6.2.5)

The amplitude of the complex acoustic pressure is

( ), 2 cos sin2

A dr kr

θ θ =

p (6.2.6)

In general, arbitrary coherent sources of sound are characterized as possessing far field acoustic pressure amplitudes expressible in the form

( ) ( ) ( ), , ,axr P r Dθ φ θ φ=p (6.2.7)

where θ is the elevation angle and φ is the azimuthal angle, used in the conventional spherical coordinate

system. The ( )axP r is denoted the far field axial pressure, while the term ( ),D θ φ is denoted as the beam

pattern. These components of the far field acoustic pressure amplitude often change according to the

frequency ω . Note that the beam pattern is ( )0 , 1D θ φ≤ ≤ .


125

Considering the results for the two point source array (6.2.6), the component of this amplitude that relates

to the far field axial pressure is 2 Ar

. The leading 2 is the outcome of (6.2.4) which goes into determining

the beam pattern ( ),D θ φ . Thus, attributing the leading 2 to the far field axial pressure component is due

to the need to modify the beam pattern to meet the limits ( )0 , 1D θ φ≤ ≤ . The component of the amplitude

(6.2.6) that relates to the beam pattern is cos sin2dk θ

. Finally, note that while the baffled point source

emits the same amplitude of pressure to the far field, it does not possess angular variation in the amplitude like the two point source array in the free field

6.2.1 Dipole

When the two point sources of the array operate 180° out of phase, small changes are required in the derivation to yield the new far field acoustic pressure amplitude:

( ), 2 sin sin2

A dr kr

θ θ =

p (6.2.1.1)

This is termed the doublet. The dipole is the term used to denote the doublet when only low frequencies are considered respecting the characteristic dimension d . Thus, for 1kd << , we have the far field acoustic pressure amplitude

( ), sinAr kdr

θ θ=p (6.2.1.2)

Comparing the pressure amplitude of a single point source

( )0a AcU kar r r

ρ= ≈ =A

p (6.2.1.3)

to that of the dipole, it is evident that the dipole is an inefficient radiator of sound at low frequencies, 1kd << , when compared to the single point source, regardless of the direction to which the sound is headed

according to the elevation angle θ in the far field.

6.3 Line source in the free field

The line source is serves as our introduction to integral formulations to determine the net sound pressure received by points in the field resulting from a large distribution of elemental sources.

Consider the line source in the free field shown in Figure 61. The radius of the cylindrical line source is a. The length of the source is L such that the ( ),x y Cartesian coordinate system is defined with the x axis

coincident with the line source axis and the y axis is 0y = at the center point of the line source. This

cylindrical source oscillates with radial surface speed j tUe ω=u .


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Figure 61. Schematic of line source emitting acoustic pressure waves to far field point.

As shown in Figure 61, we examine the line source according to its composition of a very large number of small cylindrical sources of incremental length dx . Following the procedure of Sec. 4.8, each of these differential sources is an unbaffled simple source of strength

2dQ aUdxπ= (6.3.1)

leading to an incremental acoustic pressure, following Sec. 4.8, of

( )

( )

0

0

2

4

j t kr

j t kr

dQd j c er

dQj er

ω

ω

ρλ

ρ ωπ

′−

′−

=′

=′

p (6.3.2)

where r′ is the radial distance from the differential source to the field point. By substitution of (6.3.1) into (6.3.2), the total acoustic pressure at the field point is found by integrating the contributions from these many differential simple sources over the length of the line source.

( ) ( )/2

0 /2

1 1, ,2

L j t kr

Lr t j cUka e dx

rωθ ρ ′−

−=

′∫p (6.3.3)

Putting attention on the far field, the radial distance to the field point from the coordinate reference

( ) ( ), 0,0x y = , r , is very nearly the same as the radial distance r′ . As a result, the amplitude deviations

induced by substituting r for r′ in the denominator of (6.3.3) are negligible. On the other hand, by Euler's identity the complex exponential is a term that repeats every 2π and as such a direct replacement of r for


127

r′ in the complex exponential would lead to large errors. Instead a more accurate, but still incomplete, approximation is made for the complex exponential: sinr r x θ′ ≈ − .

( ) ( ) /2 sin0 /2

1, ,2

Lj t kr jkx

L

kar t j cU e e dxr

ω θθ ρ −

−= ∫p (6.3.4)

The integral in (6.3.4) may be evaluated with relative ease to yield

( ) ( )0

1sin sin1 2, , 12 sin

2

j t krkL

ar t j cU kLer kL

ωθ

θ ρθ

−

=p (6.3.5)

Recalling the general form by which to express the far field acoustic pressure amplitude of arbitrary acoustic sources

( ) ( ) ( ), , ,axr P r Dθ φ θ φ=p (6.3.6)

We see that for the line source, the far field axial pressure is ( ) 0 / 2axP r j cUakL rρ= while the directive

factor is ( ) ( )sin /D θ α α= where ( )sin / 2kLα θ= .

Plots of the line source beam pattern, ( )D θ , in a decibel [dB] scale, are shown in Figure 62 for several

values of non-dimensional line source length /kL L λ∝ . (Note, one should neglect the numbers shown for the angular variation on the polar plots. MATLAB does not currently have flexible means of modifying

polar plots. Thus, the actual angular variation shown in the plot is from [ ]/ 2, / 2θ π π∈ − + [rad] and is

oriented similar to the line source schematic of Figure 61). When the acoustic wavelength is similar in dimension with the line source length, such as the examples shown in the top row of the figure, there is not a significant angular variation in the acoustic pressure at the field point due to the beam pattern.


128

Figure 62. Polar plots of line source beam patterns.


kL=[1 5 10 50]; % non-dimensional size of line source theta=pi/2*linspace(-1,1,402); % [rad] define theta range beam=nan(length(theta),length(kL)); % pre-allocate beam pattern matrix for iii=1:length(kL) beam(:,iii)=abs(sin(1/2*kL(iii)*sin(theta))./(1/2*kL(iii)*sin(theta))); end addon=40; % [dB] add on dB in order to have positive values in polar plot figure(1); clf; for iii=1:length(kL) kL_select=iii; subplot(2,2,iii) polar(theta'+pi/2,10*log10(beam(:,kL_select))+addon,'r') titlename=['kL= ' num2str(kL(kL_select)) ' non-dimensional line source length']; title(titlename); end

On the other hand, as the acoustic wavelength becomes smaller than the source length, such as in the bottom left example 10kL = , lobes and nodes of acoustic pressure appear in the beam pattern. Likewise, such

features appear in the acoustic pressure amplitude ( ), ,r tθp since the beam pattern directly modifies the

overall amplitude. Lobes of acoustic pressure are local maxima, while nodes of acoustic pressure are local minima. Depending on the exact non-dimensional length kL chosen and the exact refinement of the angular

range θ chosen for computation, the nodes may or may not be true zeros computed from ( )D θ . The lobe

centered at the axis normal to the line (or plane, in two dimensions) of the acoustic source is termed the major lobe and the direction at which it points is termed broadside. Here, the broadside line source position is 0θ = ° , and indeed the elevation angle coordinate is oftentimes defined as zero for consistency in defining

20

40

30

210

60

240

90

270

120

300

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broadside for a given acoustic source. Of course, as the acoustic wavelength becomes smaller still with respect to the line source length, such as for that plot shown in the bottom right of Figure 62 where 50kL =, the number of lobes and nodes increases greatly while the major lobe at broadside becomes much more like a beam of sound directed at those locations in the far field. The lobes of sound pressure that occur at elevation angles away from broadside are referred to as side lobes, while the nodes are occasionally referred to as notches.

Occasionally, beam patterns are presented as surface contours, although this is ordinarily more useful for the researcher him/herself than for the audience. Figure 63 shows such surface contours for the line source using the same non-dimensional lengths as those in Figure 62. The color shading corresponds to the [dB] of the beam pattern. By using the code available in Table 11, it is more valuable to "spin around" the plots to understand the correlations between the beam pattern "slices" in Figure 62 and the full surfaces that would exist in all three-dimensions since the line source, like all acoustic sources, exists in such 3R .

Figure 63. Corresponding surface contours of the beam pattern for the line source, whose polar plots are shown in Figure 62.


kL=[1 5 10 50]; % non-dimensional size of line source theta=pi/2*linspace(-1,1,92); % [rad] define theta range beam=nan(length(theta),length(kL)); % pre-allocate beam pattern matrix for iii=1:length(kL) beam(:,iii)=abs(sin(1/2*kL(iii)*sin(theta))./(1/2*kL(iii)*sin(theta)));

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end addon=40; % [dB] add on dB in order to have positive values in polar plot figure(2); clf; phi=pi*linspace(-1,1,71); % [rad] define rotation around cylinder axis x=nan(length(theta),length(phi)); % pre-allocate x y=nan(length(theta),length(phi)); % pre-allocate y z=nan(length(theta),length(phi)); % pre-allocate z z_amp=nan(length(theta),length(phi)); % pre-allocate z_amp for iii=1:length(kL) kL_select=iii; for ooo=1:length(theta) x(ooo,:)=(10*log10(beam(ooo,kL_select))+addon)*cos(theta(ooo)).*cos(phi); y(ooo,:)=(10*log10(beam(ooo,kL_select))+addon)*sin(theta(ooo)).*ones(length(phi),1); z(ooo,:)=(10*log10(beam(ooo,kL_select))+addon)*cos(theta(ooo)).*sin(phi); z_amp(ooo,:)=(10*log10(beam(ooo,kL_select))+addon).*ones(length(phi),1); end subplot(2,2,iii) surf(x,y,z,z_amp,'linestyle','none') axis equal xlabel('z'); ylabel('x'); zlabel('y'); caxis([10 40]) colorbar titlename=['kL=' num2str(kL(kL_select)) ' non-dimensional line source length. [dB] of beam pattern']; title(titlename); end

6.3.1 Integral methods of analysis for radiating acoustic sources

The integral approach used to compute the overall sound pressure received at a field point in consequence to the projection of an acoustic source of arbitrary shape is common in acoustics analysis. Taking the bottom portion of (6.3.2), the differential pressure for the free field acoustic pressure is integrated over the surface of the vibrating radiator that interfaces with the acoustic fluid. Each differential radiator is effectively a infinitesimally small simple source such that the surface integration accurately accounts for the collective influence of the many radiators. The 4π in the denominator of the bottom portion of (6.3.2) is representative of the fact that the source exists in the free field. As will be undertaken in a subsequent section, when the source is in a baffle, the denominator term becomes 2π because the integration is only over a hemisphere and not around a full sphere of the differential source.

6.4 Directivity

The directivity of an acoustic source is properly defined as [3]

max

ave

D= II

(6.4.1)

where maxI is the maximum intensity at point r projected by the acoustic source to the angular location

maxθ and maxφ , which correspond to the angular location of peak intensity, that is oftentimes the broadside

location. Finally, aveI is the average intensity if the total power radiated by the source were uniformly

spread over a spherical area of radius r .


131

Substituting in the appropriate values into (6.4.1), and following series of simplification steps, we find that

the directivity D is defined according to the beam pattern ( ),D θ φ as

( )( )

2

2

max max

4D,,S

r

DdS

D

π

θ φθ φ

=

∫ (6.4.2)

where S is the surface over which the acoustic source exists. This is effectively the area of vibration of the

source. Because the maximum of the beam pattern is oftentimes unity, ( )max max, 1D θ φ = , (6.4.2) may be

simplified to be

( )

2

2

4D,

S

rD dS

π

θ φ≈∫

(6.4.3)

The directivity is useful in the determination of sources that meet demands for projecting a certain SPL to a given field point at location r .

For example [3], consider that we have an amplifier that may output Π acoustic power and we must meet a required reqSPL at a field point r distant from an acoustic source center. The source has not yet been

designed, so we are able to tailor its directivity in such a way to achieve the reqSPL . By (4.5.1.5), the RMS

squared pressure of an omnidirectional (i.e., not directive) acoustic source is

2, 02

14rms omnip c

rρ

π= Π (6.4.4)

while the corresponding SPL of this source is

2,

10 210log rms omniomni

ref

pSPL

p= (6.4.5)

The required reqSPL is then rewritten according to

2,

10 2

22,,

10 102 2,

10log

10log 10log

rms reqreq

ref

rms reqrms omni

ref rms omni

pSPL

p

ppp p

=

= +

(6.4.6)

Considering (6.4.1) and (6.4.5), we find that the last expression of (6.4.6) is alternative expressed by

1010log Dreq omniSPL SPL= + (6.4.7)

As a result, if the available sound power is known Π , we can design the source directive features in order to achieve the reqSPL at a field point located at r .


132

6.5 Rigid, circular, baffled piston source

The oscillating, rigid piston in a baffle is another fundamental acoustic source, like the line source, that is amenable to analytical study via integral formulations. Consider the schematic of the baffled, rigid piston

in Figure 64. The speed of the piston is j tUe ω at angular frequency ω . The circular piston radius is a .

Figure 64. Schematics of rigid piston source in baffle. (a) Axial response coordinates, (b) far field response coordinates.

Consider the surface of the piston to be discretized into a large number of infinitesimal surfaces dS that are each baffled simple sources. The incremental source strength is now dQ UdS= . Consequently, the

differential pressure generated by each baffled source is

( )

( )

0

0

2

2

j t kr

j t kr

dQd j er

Uj e dSr

ω

ω

ρ ωπ

ρ ωπ

′−

′−

=′

=′

p (6.5.1)

The total acoustic pressure at the field point is consequently the integral over this area

( )0

12

j t kr

S

Uj e dSr

ωρ ωπ

′−=′∫p (6.5.2)

The (6.5.2) is an exemplary realization of Rayleigh's integral. Rayleigh's integral determines the sound pressure in the field due to the vibration of a baffled radiator of arbitrary shape, which may not vibrate

uniformly over its surface. Considering (6.5.2), we note that j U Aω = which is the surface acceleration.

Consequently, we may express Rayleigh's integral as

( )0

12

j t kr

S

A e dSr

ωρπ

′−=′∫p

(6.5.3)

which is further generalized presuming that the surface acceleration is not uniform over the radiator

( ) ( )0

,12

j t kr

S

A x ye dS

rωρ

π′−=

′∫p

(6.5.4)


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The (6.5.4) is illuminating as it pertains to acoustically radiating bodies, specifically for their surfaces that interface with an acoustic fluid. Having consolidated the terms most compactly, it is clear that the sound radiation is a consequence of an accelerating surface and not strictly a surface moving with prescribed velocity.

For arbitrarily-shaped radiators, Rayleigh's integral is impossible to solve in a closed form manner. Yet, in a few, and oftentimes relevant and important, cases, the integral may be directly evaluated. For further background on integral formulations of acoustics, interested individuals are encouraged to refer to [7].

6.5.1 On-axis acoustic pressure from baffled piston source

The on-axis response of the baffled piston is one example that yields an integral able to be solved directly. Considering the schematic of Figure 64(a) and using simple trigonometry, the Rayleigh's integral for the on-axis response of the baffled piston is

( )2 2

0 0 2 2, 0, 2

2

jk raj tU er t j e dr

σωθ ρ ω πσ σ

π σ

− +

= =+

∫p (6.5.1.1)

Evaluating the integral, we find

( ) ( ) ( )2 2

0,0, 1jk r a r j t krr t cU e e ωρ

− + − − = −

p (6.5.1.2)

The magnitude of (6.5.1.2) for the on-axis acoustic pressure provided by the baffled piston is

( ) ( )20

1,0, 2 sin 1 / 12

r t cU kr a rρ = + − p (6.5.1.3)

It is seen that the absolute value term of (6.5.1.3) may exhibit nulls and unit values when

( )21 1 / 1 ; 0,1,2,..2 2

kr a r m mπ + − = = (6.5.1.4)

due to the interference of waves arriving at the common field point on-axis as delivered from different regions of the vibration piston surface. In other words, the on-axis acoustic pressure delivery from the piston can result in perfect constructive or destructive interference due to the integration of all infinitesimal radiating surfaces over the area of the circular piston. Figure 65 illustrates the concept encountered. Note that the figure is not technically correct since the consideration of complete constructive or destructive interference requires the full integral evaluation and not only the interference effects from two distinct locations on the piston surface. Figure 66 presents examples of the normalized on-axis acoustic pressure, revealing the complete nulls and unit values in consequence to the significant interference effects.


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Figure 65. Illustrations of constructive and destructive interference at field point.

Figure 66. On-axis acoustic pressure delivered from baffled, rigid piston.

As a limiting case of (6.5.1.3), when r a>> and / / 2r a ka>> , the pressure amplitude on the axis of the baffled piston is

( ) 01,0,2

ar t cU kar

ρ≈p (6.5.1.5)

Clearly, at sufficiently long distances away from the piston surface, the response trends to the well-known

result of the far field that 1r

∝p . This proportionality is evident in Figure 66.

6.5.2 Far field acoustic pressure from baffled piston source

As indicated in Sec. 6.1, a significant number of contexts in engineering and scientific investigations of acoustics are interested in the far field. Consequently, the analyze the acoustic pressure delivered to the field point in the far field, we utilize a different schematic notation. The relevant schematic is shown in Figure 64(b). The piston is now considered to be composed of a large number of small, baffled line sources, whose lengths are from one side of the piston to the other. The length of each line source is 2 sina φ , where

φ is the azimuthal angle. The relevant source strength is consequently

2 sindQ Ua dxφ= (6.5.2.1)

0 1 2 3 4 5 6 7 8 9 100

0.2

0.4

0.6

0.8

1

r/a

norm

aliz

ed o

n-ax

is a

cous

tic p

ress

ure,

p/(2ρ c

cU)

ka=[10 20 30] plotted as red dotted, green solid, blue dashed


135

where dx is the incremental width of the line source in the x axis. The differential pressure emitted by each such line source to the field point is

( )0 sin j t krUd j c ka e dx

rωρ φ

π′−=

′p (6.5.2.2)

When the field point is far away from the piston surface with respect to the characteristic dimension, r a>>

, we have that sin cosr r a θ φ′ ≈ − is a useful approximation for the complex exponential, while r r′ ≈ in

the denominator of (6.5.2.2). Thus, the total acoustic pressure received by the integration of the many line sources is

( )

( )

sin cos0

sin cos 20 0

sin

sin

aj t kr jka

a

j t kr jka

Uj c kae e dxr

U aj c kae e dr

ω θ φ

πω θ φ

ρ φπ

ρ φ φπ

+−

−

−

=′

=

∫

∫

p (6.5.2.3)

Using Euler's identity on the complex exponential, it is found that the imaginary components vanish by

symmetry in the integration from [ ]0,φ π∈ . In contrast, the real components do not vanish and are instead

evaluated in terms of a Bessel function of the first kind and first order.

( ) ( ) ( )10

2 sin1, ,2 sin

j t kr J kaar t j cU kaer ka

ω θθ ρ

θ−

=

p (6.5.2.4)

Considering the amplitude of (6.5.2.4), we may easily identify the far field axial pressure

( ) 012ax

aP r cU kar

ρ= (6.5.2.5)

and the beam pattern

( ) ( )12 sinsin

J kaD

kaθ

θθ

= (6.5.2.6)

where the leading factor of 2 is retained since as sin 0kaα θ= → , ( )12 / 1J α α → .

Representative beam patterns for the baffled piston are shown in Figure 67 as polar plots and in Figure 68 as surface contours. It is observed that for similar non-dimensional lengths, the baffled piston exhibits greater angular directionality in its delivery of acoustic pressure amplitude to the far field when compared to the line source.


136

Figure 67. Polar plots of baffled piston source beam patterns.

Figure 68. Corresponding surface contours of the beam pattern for the baffled piston source, whose polar plots are shown in Figure 67.

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Table 12. Code used to generate Figure 67 and Figure 68

ka=[1 5 10 50]; % non-dimensional size of line source theta=pi/2*linspace(-1,1,102); % [rad] define theta range beam=nan(length(theta),length(ka)); % pre-allocate beam pattern matrix for iii=1:length(ka) alpha=ka(iii)*sin(theta); beam(:,iii)=abs(2*besselj(1,alpha)./alpha); end addon=40; % [dB] add on dB in order to have positive values in polar plot figure(1); clf; for iii=1:length(ka) ka_select=iii; subplot(2,2,iii) polar(theta'+pi/2,10*log10(beam(:,ka_select))+addon,'r') titlename=['ka=' num2str(ka(ka_select)) ' non-dimensional baffled piston size. [dB] of beam pattern']; title(titlename); end figure(2); clf; phi=pi*linspace(-1,1,71); % [rad] define rotation around cylinder axis x=nan(length(theta),length(phi)); % pre-allocate x y=nan(length(theta),length(phi)); % pre-allocate y z=nan(length(theta),length(phi)); % pre-allocate z z_amp=nan(length(theta),length(phi)); % pre-allocate z_amp for iii=1:length(ka) ka_select=iii; for ooo=1:length(theta) x(ooo,:)=(10*log10(beam(ooo,ka_select))+addon)*sin(theta(ooo)).*cos(phi); y(ooo,:)=(10*log10(beam(ooo,ka_select))+addon)*sin(theta(ooo)).*sin(phi); z(ooo,:)=(10*log10(beam(ooo,ka_select))+addon)*cos(theta(ooo)).*ones(length(phi),1); z_amp(ooo,:)=(10*log10(beam(ooo,ka_select))+addon).*ones(length(phi),1); end subplot(2,2,iii) surf(x,y,z,z_amp,'linestyle','none') axis equal xlabel('x'); ylabel('y'); zlabel('z'); caxis([10 40]) colorbar titlename=['ka=' num2str(ka(ka_select)) ' non-dimensional baffled piston size. [dB] of beam pattern']; title(titlename); end

6.5.3 Generalization of the Rayleigh's integral for planar, baffled acoustic sources in the far field

We can generalize the results of Sec. 6.5.2 to accommodate the study of far field acoustic pressure delivered from an arbitrarily shaped planar radiator, Figure 69. Consider the Rayleigh's integral using the notation that the normal surface velocity may be a function of the radiator area

( ) ( )0

,12

j t kr

S

U x yj e dS

rωρ ω

π′−=

′∫p (6.5.3.1)

where dS dx dy′ ′= is the differential area of an infinitesimally small, baffled radiating element of the

source, located at ( ), ,0x y′ ′ . Using the geometry of Figure 69, we have that the distance r′ is

( )2 2 2 sin cosr r r rr θ φ α′ = + − − (6.5.3.2)


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For attention given to far field sound radiation, r r>> , a binomial expansion may be used on (6.5.3.2) to result in

( )sin cosr r r θ φ α′ ≈ − − (6.5.3.3)

Figure 69. Arbitrarily shaped, baffled planar acoustic source

One further identifies that cosx r α′ = and siny r α′ = . Consequently, the radial distance from the

differential baffled sound source to the field point is

sin cos sin sinr r x yθ φ θ φ′ ′ ′≈ − − (6.5.3.4)

The far field acoustic pressure delivered from the whole of the arbitrarily shaped radiator is therefore determined from Rayleigh's integral

( )

( ) ( )sin cos sin sin0

1 ,2

j t krjk x y

S

ej U x y e dSr

ωθ φ θ φρ ω

π

−′ ′+′ ′= ∫ ∫p (6.5.3.5)

Defining wavenumber components

sin cosxk k θ φ′ = ; sin sinyk k θ φ′ = (6.5.3.6)

by substitution, it is found that

( )

( )01 ,

2yx

j t krk yjk x

S

ej U x y e e dx dyr

ω

ρ ωπ

′′

−

′ ′ ′ ′= ∫ ∫p (6.5.3.7)

Two things are apparent in the (6.5.3.7). Firstly, the integral contains components strictly associated with

the beam pattern ( ),D θ φ while the terms leading the integral are those that contribute to the far field axial

pressure, prior to its normalization. Secondly, it is also apparent that the integral of (6.5.3.7) is a two-dimensional Fourier transform between space and wavenumber of the normal surface velocity. Consequently, the established mathematics and relations among Fourier transform pairs directly assists the study and design of planar, baffled radiators with velocity distributions or spatial distributions that meet


139

prescribed beam pattern demands. This is a powerful, first look of the influence of straightforward Fourier transforms on the development of acoustic sources for prescribed far field radiation beam patterns.

6.6 Radiation impedance

Impedance characterizes the relative proportions of bulk energy transfer and reciprocal energy exchange from one location to another. For radiating structures, the radiation impedance is used to the same effect as it pertains to the transfer and/or exchange of mechanical vibration energy to acoustic energy in the field.

The normal force upon the element of a radiator of differential area dS that vibrates with normal velocity u is Sdf . This normal force is due to the acoustic pressure as well as due to the other differential areas of

the radiator.

The radiation impedance is defined as the integration of the ratio of differential force to the normal velocity:

Sr

d= ∫

fZu

(6.6.1)

Figure 70. Schematic used to compute radiation impedance from baffled piston.

An example helps to illustrate the outcomes and implications of (6.6.1). For the baffled, circular piston, we consider that the amplitude of the normal velocity is everywhere uniform as U . Thus, considering the schematic of Figure 70, the total force on the piston differential area dS ′ is

( )S A dS ′= ∫f p (6.6.2)

The term ( )Ap denotes the complex acoustic pressure at the point A . This pressure is evaluated from

(6.5.2). In other words, the ( )Ap that is found in the integration of (6.6.2) is itself an integration where the

field point, originally at location r′ from the differential surface dS ′ is now itself at the point A . Moreover, considering the schematic of Figure 70, acoustic reciprocity indicates that the acoustic pressure at dS ′ due to the differential radiator element dS is the same as the acoustic pressure at dS due to the differential


140

radiator element dS ′ . Thus, our integration limits need only to evaluate the force between each differential element pair once, while we compensate for the second, mutually reciprocal, evaluation by multiplying the result by two.

Ultimately, the integration is expressed

2 /2 2 cos

0 0 0 /2 02

aj t jkrS

Uj c e e drd d dπ π σ θω

πρ σ θ ψ σ

λ+ −

−= ∫ ∫ ∫ ∫f (6.6.3)

where dS d dσ σ ψ′ = considering the schematic of Figure 70. Thus, the radiation impedance is

( ) ( )1 120

2 2 2 21

2 2r

J ka H kac a j

ka kaρ π

= − +

Z (6.6.4)

The normalized resistance 1R and normalized reactance 1X are, respectively, the bracketed and

parenthetical terms in (6.6.4).

Assessment of the radiation impedance is left to the interested individual. A valuable tactic in such assessment is the use of the Maclaurin series to approximate the resistance and reactance.

( ) ( ) ( )2 4 6

1 2 2 2 2 2 2 ...1 2 1 2 3 1 2 3 4ka ka ka

R ≈ − + −⋅ ⋅ ⋅ ⋅ ⋅ ⋅

(6.6.5)

( ) ( )3 5

1 2 2 2

2 24 2 ...3 3 5 3 5 7

ka kakaXπ

≈ − + −

⋅ ⋅ ⋅ (6.6.6)

6.7 Acoustic arrays

Arrays of acoustic radiators are often used in practice to enhance the delivery of acoustic energy to points far in the field. In other words, when a required SPL is needed at a field point and an omnidirectional source is insufficient to meet this target level, then, by virtue of the theory on directivity established in Sec. 6.4, an array of simple sources may be assembled that results in the constructive interference effects to make up the difference.

6.7.1 Line array

Consider the line array of N point sources shown in Figure 71. The sources emit acoustic pressure at frequency ω in phase and at the same amplitude A [Pa/m]. Considering the field point to be in the far

field, the 0th source emits acoustic pressure to the field point of

( )0

j t krA er

ω −=p (6.7.1.1)

The 1st− source emits acoustic pressure


141

[ ]( )sin 21 0

j t k r d jA e er

ω θ φ− + −− = =p p (6.7.1.2)

where we define 1 sin2

kdφ θ= . The 2nd− source emits acoustic pressure

[ ]( )2 sin 42 0

j t k r d jA e er

ω θ φ− + −− = =p p (6.7.1.3)

Similarly, the acoustic pressures delivered to the field point from the other point sources in the line array are derived.

Figure 71. Schematic of line array of point sources.

Similar findings are made for the remaining sources in the array. All together, the acoustic pressure at the field point is

( ) 4 2 2 40, , ... 1 ...j j j jr t e e e eφ φ φ φθ − − = + + + + + + p p (6.7.1.4)

Following simplification and association with geometric series, we find that the pressure is

( ) 0sin, ,sin

Nr t φθφ

=p p (6.7.1.5)

On-axis, θ =0, the amplitude of the acoustic pressure is

( ) 0,0,r t N=p p (6.7.1.6)

We normalize the (6.7.1.5) to be

( ) 0sin, ,

sinNr t N

Nφθφ

=

p p (6.7.1.7)


142

( ) sinsinaND

Nφθφ

= (6.7.1.8)

The aD (6.7.1.8) is the array factor which is analogous to the beam pattern for individual sources.

Similar to the beam patterns for the other sound radiators, the array factor (6.7.1.8) may result in acoustic pressure nodes and lobes at the far field points. Yet, the array factor in (6.7.1.8) may also have nodes that

coexist with lobes. For instance, when 1 sin2

kd mθ π= both the numerator and denominator vanish.

Consequently, the development and analysis of arrays of acoustic radiators are not as straightforward to undertake as they are for more elementary radiators.

While shading tailors the amplitudes of acoustic pressure provided for radiators in an array, steering is the method of controlling the phase of radiators, typically by defined phase delays in the signals sent to the

radiators. If a signal delay iτ is provided for the thi element in the array, it may be shown that the new array factor is

( )( )

( )

0

0

1sin sin sin1 2

1sin sin sin2

N kdD

N kd

θ θθ

θ θ

− = −

(6.7.1.9)

such that the major lobe is no longer directed at broadside to the array, 0θ = , but is now directed to

0sin cdτθ = (6.7.1.10)

Another common term used to denote the outcome of phase delay based manipulations of acoustic arrays is beamsteering. Beamforming is conventionally used to denote either or both the use of shading and steering.

6.7.2 Arrays of directional sources

If the radiators that compose the array are themselves directive, then the product theorem provides a straightforward means to determine the overall angular directional influences for pressure delivery to far

field points. Namely, if each of the elements in the array possesses beam pattern ( ),eD θ φ , such that the

central source in the array is

( ) ( )0 , j t kr

eA D er

ωθ φ −=p (6.7.2.1)

then substitution will plainly show that the ultimate beam pattern for the system is

( ) ( ) ( ), , ,e aD D Dθ φ θ φ θ φ= (6.7.2.2)


143

The ( ),aD θ φ is the array factor that corresponds to the geometric arrangement of sources in the array as

if they are point sources located at the geometric centers of the actual, directional sources. For instance, the array factor for the line array is given by (6.7.1.8).

The product theorem is a statement that the overall capabilities of the array to deliver acoustic pressure to far field points of different angles is the product of the beam pattern of each sound radiator element and the array factor of the configuration of the radiators as if they are only point sources.

6.8 Spatial Fourier transform

The integrals that enable the determination of the beam patterns for the line source and baffled piston are remarkably similar to Fourier transforms. Indeed, the established mathematical conveniences of the Fourier transform pair (86) and (87) may be used to great ends in the study of directional acoustic sources.

6.8.1 Line source far field acoustic pressure response by spatial Fourier transform method

As an example, consider the line source where we now employ

( )2d aU x dxπ=Q g (6.8.1.1)

The differential source strength dQ is generally complex due to the complex aperture function ( )xg . The

limits of ( )xg are −∞ to +∞ , while for convenience, the function is often prescribed to have amplitude

according to the range ( )0 1x≤ ≤g .

Then, (6.8.1.1) is substituted into the expression for the differential pressure of the line source element

( )0 4

j t krdd j er

ωρ ωπ

′−=′

Qp (6.8.1.2)

The total pressure emitted by all elements to a common field point, distances r′ away from each differential line element, is found by integrating (6.8.1.2) over the length of the line source. In the far field, the integral evaluation is

( ) ( ) ( )/2

0 /2

1, ,2

Lj t kr jux

L

ar t j cU ke x e dxr

ωθ ρ −

−= ∫p g (6.8.1.3)

where sinu k θ= . Because of the support of the aperture function ( )xg , we alternatively express (6.8.1.3)

using

( ) ( ) ( )01, ,2

j t kr juxar t j cU ke x e dxr

ωθ ρ+∞−

−∞= ∫p g (6.8.1.4)

It is evident that the integral of (6.8.1.4) is one of the Fourier transform pairs, according to

( ) ( ) juxu x e dx+∞

−∞= ∫f g (6.8.1.5)


144

( ) ( )12

juxx u e duπ

+∞ −

−∞= ∫g f (6.8.1.6)

Due to its complex composition, ( )xg weights the (i) existence, (ii) amplitude, and (iii) phase of the

acoustic source integral, that all together determine the beam pattern. Note that the ( )uf is the beam pattern

of the radiator prior to its normalization to a unit value at the aperture function maximum.

Because the transform pair of (6.8.1.5) and (6.8.1.6) convert between the spatial description and wavenumber description of the acoustic source characteristics, the pair is referred to as the spatial Fourier transform pair. Formally, (6.8.1.5) is the transform while (6.8.1.6) is the inverse transform.

An example is useful to understand the relation between the spatial Fourier transform and computation of beam patterns. For an unshaded and unsteered line source, we have that

( )1; / 2 / 2

0; / 2, / 2L x L

xx L x L− ≤ ≤ +

= < − > +g (6.8.1.7)

Thus, evaluating (6.8.1.4) leads to

( ) ( )0

sin1 2, ,2

2

j t kr

uLar t j cU ke ur

ωθ ρ −

=

p (6.8.1.8)

To meet convention, the beam pattern result of the integration, which is the bracketed term of (6.8.1.8), should be normalized so that at most it has unit amplitude. Assuming, by intuition and past experience, that the beam pattern is maximized at the broadside location, 0θ = , we may simplify the bracketed beam pattern term in (6.8.1.8) using

sin sin2 2

sin2 2

kL kL

Lk k

θ

θ

≈ = (6.8.1.9)

Thus, without normalization, the magnitude of the beam pattern is the line source length L . Therefore, we divide the beam pattern by this value and pre-multiply the leading terms of the (6.8.1.8) by L in order to maintain the same integral evaluation.

( ) ( ) ( )0 0

1sin sin sin1 12 2, , 12 2 sin

2 2

j t kr j t kr

uL kLa ar t j cU kLe j cU kLeuLr r kL

ω ωθ

θ ρ ρθ

− −

= =

p (6.8.1.10)

where the result of (6.8.1.10) is plainly equal to (6.3.5).


145

The aperture function ( )xg may include amplitude- and phase-shifting components so that the acoustic

source differential elements do not contribute equally to the far field point, or do not emit sound at the same phase. For example, beamsteering may be achieved by a complex exponentialThe evaluation limits of the integral for the aperture function (6.8.1.4) are analogous to the spatial distribution of the source, i.e. the existence of the vibrating surface.

As an example of amplitude and phase weighting, consider Figure 72 where an aperture function is illustrated according to

( )

2

2

2

2

0;2

2 21 ; ; 02 2 2

2 21 ; ;02 2 2

0;2

j

j

Lx

L Lx e x xL L

xL Lx e x x

L LLx

φ

φ

π πφ

π πφ

< − + = − + + − ≤ ≤ =

− = − − ≤ ≤ >

g (6.8.1.11)

Therefore, a general expression for an aperture function, considering existence is established over a domain, is

( ) ( ) ( )j xx x e φ=g g (6.8.1.12)

Figure 72. Example of aperture function amplitude and phase change.

6.8.2 Forced, baffled rectangular plate far field acoustic pressure response by spatial Fourier transform method

A second example involving the sound radiation from rectangular, simply supported, baffled plates demonstrates the incorporation of the spatial Fourier transform as a perspective as well as powerful tool. Consider Figure 73 which gives a schematic of a simply supported, baffled plate, excited by a point force, which therefore radiates sound to a far field point. Sec. 3.7.2 gave preliminaries regarding the standing

wave response of the plate in consequence to a point force at ( )0 0,x y . For a transverse bending

displacement response composed of an infinite sum of normal modes and modal amplitudes


146

( ) ( ) ( ) ( )1 1 1 1

, , , sin sinj t j tmn mn m n

m n m nx y t e x y e k x k yω ω

∞ ∞ ∞ ∞

= = = =

= =∑∑ ∑∑w W A (6.8.2.1)

the modal amplitudes mnA were found from

( ) ( ) ( ) ( )2 2 0 0

4 , sin sinx yL L

mn m nx y mn

P x y k x k y dxdyhL Lρ ω ω

=− ∫ ∫A (6.8.2.2)

The point force is expressed using the pressure distribution ( ) ( ) ( )0 0 0,P x y P x x y yδ δ= − − such that

(6.8.2.2) yields

( ) ( )( )

0 002 2

sin sin4 m nmn

x y mn

k x k yPhL Lρ ω ω

=−

A (6.8.2.3)

Connecting these results to the Rayleigh's integral (6.5.4), we see that the acceleration distribution of the vibrating surface is no longer uniform over the radiating surface. In other words, there is a clear relation

between the normal modes and the weighting function ( ),x yg .

Figure 73. Simply supported, baffled plate, excited by a point force, radiating sound to a far field point.

For ( ) ( ), ,d U x y x y dS=Q g , the development above suggests that

( ) ( ) ( )1 1

sin sin ,mn m nm n

d j k x k y x y dxdyω∞ ∞

= =

= ∑∑Q A g (6.8.2.4)

where


147

( ) ( ) ( )

( )

( )

1 2

1

2

,

1;00; 0,

1; 00; 0,

x

x

y

y

x y x y

x Lx

x x L

y Ly

y y L

=

≤ ≤= < >

≤ ≤= < >

g g g

g

g

(6.8.2.5)

Then, (483) becomes

( ) ( ) ( ) ( ) ( )20 1 2

1 1

1 sin sin2

j t krmn m n

m nd e k x k y x y dxdy

rωρ ω

π

∞ ∞′−

= =

= −′ ∑∑p A g g (6.8.2.6)

where the differential pressure is multiplied by 2 when compared to (6.8.1.2) because the plate is baffled so that all of the differential elements are comparably baffled.

In the far field,

sin cos sin sinr r x yθ φ θ φ′ ≈ − − (6.8.2.7)

By substitution of (6.8.2.7) into (6.8.2.6) and integration over the domain considering the far field assumptions, yields

( ) ( ) ( ) ( ) ( )2 sin cos sin sin0 1 2

1 1

1 sin sin2

j t kr jkx jkymn m n

m ne k x e x k y e y dxdy

rω θ φ θ φρ ω

π

∞ ∞ +∞ +∞−

−∞ −∞= =

= − ∑∑ ∫ ∫p A g g (6.8.2.8)

The separability of the integrals in (6.8.2.8) reveals a product of spatial Fourier transforms weighted by the normal modes

( ) ( ) ( ) ( ) ( )20 1 2

1 1

1 sin sin2

yx jk yj t kr jk xmn m n

m ne k x e x dx k y e y dy

rωρ ω

π

∞ ∞ +∞ +∞−

−∞ −∞= =

= − ∑∑ ∫ ∫p A g g (6.8.2.9)

with sin cosxk k θ φ= and sin sinyk k θ φ= .

Consequently, one may identify the spatial Fourier transforms from (6.8.2.9) as follows

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

1 ,

2 ,

sin

sin

x x

y y

jk x jk xm x m m

jk y jk yn y n n

k x e x dx x e dx x

k y e y dy y e dy y

+∞ +∞

−∞ −∞

+∞ +∞

−∞ −∞

= =

= =

∫ ∫∫ ∫

g g F

g g H

(6.8.2.10)

It is apparent from (6.8.2.10) that the transforms are weighted, by the new aperture functions ( ),x m xg and

( ),y n yg , via the normal modes of order ( ),m n .

Given time and patience to undertake the trigonometric simplifications, one obtains [17]

( ) ( )( ) ( )

20 2 2 2 2

1 1

1 1 1 11, , ,2

y yx xm n jk Ljk L

m nj t krmn

m n x m y n

k e k er t e

r k k k kωθ φ ρ ω

π

−−∞ ∞

−

= =

− − − − = −− −∑∑p A (6.8.2.11)


148

where xk and yk are the wavenumber components. Note that the baffle of the plate indicates that the domain

outside of the plate area will not contribute to the integration of (6.8.2.9).

6.8.3 Arrays and far field acoustic pressure design by spatial Fourier transform method

The spatial Fourier transform method is easily extended to determining array factors associated with certain arrangements of point sources, regardless of their individual directive properties for far field sound radiation. For arrays of point sources in the determination of the corresponding array factors, the weighting

function is a summation of delta functions ( ) ( ) ( )i i ix x y y z zδ δ δ− − − , where ( ), ,i i ix y z is the location

of an individual source center, such that the integral evaluation is straightforward.

An interesting application of the spatial Fourier transform pair is to define a beam pattern via the function

( )uf and solve for the source distribution, amplitude shading, and phase shift distributions, represented in

the weighting function ( )xg , that permit such beam pattern. In practice, this is an open-ended problem of

phased array design that has no singular solution. Others have conceived this problem as a means to weight

vibration control routines of distributed structures, like forced plates, so that the weighting function ( ),x yg

is realized in a way to culminates in the desired far field response [30]. Numerous texts in acoustics and array theory have addressed this and similar problems [31] [32] [33] [34]. Transducer and array development and optimization remain significant areas of research to this day.


149

7 Enclosures and waveguides

Acoustic waves in confined volumes are critical to many applications of acoustics study, for instance wave propagation in HVAC ducts, measurement of acoustical and elastic properties of materials, sound in musical pipes, resonances in rooms, development of speakers, and so on. We refer to fully enclosed volumes as enclosures, and term enclosed volumes with one or more open pathways as waveguides. This section is not a complete discourse on relevant matters of acoustics, and in particular we will omit study of diffuse field sound. Individuals interested in such physics, which pertains to numerous applications of architectural acoustics and noise control, should consider enrolling in ME 5194 "Engineering Acoustics".

7.1 Normal modes in acoustic enclosures

Unless otherwise stated, we study rigid-walled enclosures. Yet, in practice, the walls of acoustic enclosures do not need to be perfectly rigid, uniform, or continuous, in order to take advantage of the following analytical techniques for their study. On the other hand, the enclosure walls must be mostly rigid and must not contain significantly-sized openings to other acoustic volumes, lest the assumptions inherent in the analysis be too dramatically violated for their usefulness to be borne out.

7.1.1 Rectangular enclosure

Rectangular enclosures are the most common volumetric geometry encountered in civil or architectural contexts of acoustics, and play fundamental roles in the development of loudspeaker systems. Myriad other disciplines involve rectangular enclosures as well, see for instance the scope of Figure 1. Consider the rectangular enclosure shown in Figure 74. The dimensions of the enclosure in the x , y , and z axes are,

respectively, xL , yL , and zL . We assume that the surfaces of the six enclosure walls are perfectly rigid, so

that the normal component of the particle velocity vanishes at the walls. Thus, by Euler's equation and by virtue of the coordinate definition from one corner of the rectangular enclosure, we have that

0

0xx x L

p px x= =

∂ ∂ = = ∂ ∂ (7.1.1.1)

0

0yy y L

p py y= =

∂ ∂= = ∂ ∂

(7.1.1.2)

0

0zz z L

p pz z= =

∂ ∂ = = ∂ ∂ (7.1.1.3)


150

Figure 74. Rectangular enclosure. A view with perspective.

Similar to the study of the strings, rods, and other structures that are bounded on all ends, we anticipate that the acoustic response within the rigid enclosure will be composed of standing waves. To this end, we apply a separable form of assumed solution

( ) ( ) ( ) ( ), , , j tx y z t x y z e ω=p X Y Z (7.1.1.4)

to the governing equation (4.1.1.7), the acoustic wave equation, and separate the independent equations to yield

22

2 0xd kdx

+ =X X (7.1.1.5)

22

2 0yd kdy

+ =Y Y (7.1.1.6)

22

2 0zd kdz

+ =Z Z (7.1.1.7)

where it may be shown that the individual wavenumbers ik must satisfy

22 2 2 2

x y zk k k kcω = + + =

(7.1.1.8)

Based on the boundary conditions, appropriate assumed solutions to the (7.1.1.5), (7.1.1.6), and (7.1.1.7), are, respectively,

( )( ) cos ; ; 0,1,2,...l xl xllx

lx k x k lLπ

= = =X X (7.1.1.9)

( )( ) cos ; ; 0,1,2,...m ym ymmy

my k y k mLπ

= = =Y Y (7.1.1.10)


151

( )( ) cos ; ; 0,1,2,...n zn znnz

nz k z k zLπ

= = =Z Z (7.1.1.11)

leading to the result for the ( ), ,l m n normal mode

cos cos cos lmnj tlmn lmn xl ym znk x k y k ze ω=p A (7.1.1.12)

where the associated natural frequency of the ( ), ,l m n mode is

22 2

lmnx y z

l m ncL L Lπ π πω

= + +

(7.1.1.13)

It is seen that the rigid acoustic enclosure leads to boundary condition definition and associated normal modes analogous to those for the vibrations of a supported string, membrane, or rod with all free ends. In contrast, further derivation shows that an acoustic enclosure with pressure release boundaries (something of a mathematical anomaly) leads to boundary condition definition and associated normal modes analogous to those for the vibrations of a string, membrane, or rod with all fixed ends.

Using Euler's identity, such that

( )1cos2

jx jxx e e−= + (7.1.1.14)

reveals the description of these normal acoustic modes in terms of standing waves that travel from one end of the enclosure to the other, in separable ways respecting the coordinate system definition

( )18

lmn xl ym znj t k x k y k zlmn lmne

ω ± ± ±=p A (7.1.1.15)

7.1.2 Cylindrical enclosure

Cylindrical enclosures are another common geometry to encounter, although they are more often observed in waveguide applications as described in the subsequent sections. Nevertheless, as rigid enclosures, a cylindrical volume permits the following Helmholtz equation, given an assumed solution to the wave

equation of j te ω=p P

2 2 22

2 2 2 2

1 1 0kr r r r zθ

∂ ∂ ∂ ∂+ + + + =

∂ ∂ ∂ ∂P P P P P (7.1.2.1)

The boundary conditions for a rigid enclosure involve the extremities of the radial and length coordinates. Using the schematic of Figure 75 for a right cylinder, we have that

0

0z z L r az z r= = =

∂ ∂ ∂ = = = ∂ ∂ ∂

P P P (7.1.2.2)


152

Figure 75. Cylindrical enclosure. A view with perspective.

For this Helmholtz equation (7.1.2.1), we assume a separable form of response solution

( ) ( ) ( ) ( ), ,r z r zθ θ= ΘP R Z (7.1.2.3)

It may then be shown that the following three equations result from substitution of (7.1.2.3) into (7.1.2.1):

( )2

2 2 2 22 0mn

d dr r k r mdr dr

+ + − =R R R (7.1.2.4)

22

2 0d mdθΘ+ Θ = (7.1.2.5)

22

2 0zld kdz

+ =Z Z (7.1.2.6)

By applying the boundary conditions (7.1.2.2), solutions to (7.1.2.4), (7.1.2.5), and (7.1.2.6), are, respectively,

( ) ( );mn m mn mn mnmn J k r k a j′= =R R (7.1.2.7)

( ) [ ]cos ; 0,1,2,...m lmnm m mθ γΘ = Θ + = (7.1.2.8)

( ) cos ; ; 0,1,2,..l zl zll k z k L l lπ= = =Z Z (7.1.2.9)

where mnj′ is the thn extremum of the thm Bessel function of the first kind. Like for the circular membrane,

the term lmnγ is associated with azimuthal dependence in the initial conditions. Like for the rectangular

enclosure, three integer values ( ), ,l m n define the specific acoustic mode in the cylindrical enclosure, now

corresponding to numbers of nodes in radial, azimuthal, and length dimensions. Therefore, the acoustic

pressure of the ( ), ,l m n normal mode is, by substitution into (7.1.2.3),

( ) [ ]cos cos lmnj tlmn lmn m mn lmn zlJ k r m k ze ωθ γ= +p A (7.1.2.10)

The associated natural frequency is


153

2 2lmn mn zlc k kω = + (7.1.2.11)

Considering (7.1.2.7) and (7.1.2.8), the normal modes of the rigid, right cylindrical enclosure are analogous to those for a fixed circular membrane although the enclosure introduces the length-dimension variation of acoustic pressure by virtue of the third spatial dimension.

7.2 Acoustics waveguides of constant cross-section

Numerous applications in science and engineering practice involve waveguides, which are enclosures that possess one significantly greater dimension than the other two and when that dimension has at least one end open. Considering acoustic waveguides, the end that is not open is often presumed to possess an acoustic source such that waves propagate along the waveguide axis. A long pipe with one end open and with a noise source at the other end is a good example. We study here waveguides that have a constant cross-section in the off-axis dimensions.

7.2.1 Rectangular waveguide

Consider the rigid-walled rectangular enclosure shown in Figure 74, but now the enclosure is terminated in the 0z = plane and continues without boundary for increasing z , making this a properly defined waveguide. For instance, a rectangular piston may exist in the 0z = plane and may drive acoustic wave propagation down the semi-infinite waveguide (often termed duct in engineering practices), for example as in an HVAC application.

Based on the results of the rectangular enclosure, we now hypothesize that the solution to the wave equation will contain the same x and y axis variation of acoustic pressure but now the z axis variation will exhibit

wave travel. Because the waveguide is semi-infinite in the z axis and because the coordinate system is defined with z =0 as the rigid plane, there is no z− traveling wave. Thus, we have that the acoustic pressure

for the ( ),l m mode is a slight modification to (7.1.1.12), such that

( ) cos coszj t k zlm lm xl yme k x k yω −=p A (7.2.1.1)

22

z lmk kcω = −

(7.2.1.2)

2 2 2lm xl ymk k k= + (7.2.1.3)

; 0,1,2,...xlx

lk lLπ

= = (7.2.1.4)

; 0,1,2,...ymy

mk mLπ

= = (7.2.1.5)

Although (7.2.1.1) is not significantly different than (7.1.1.12), the implications regarding wave propagation are great. Namely, the frequency ω can take on any value by virtue of the traveling wave term


154

in (7.2.1.1) that satisfies the corresponding wave equation and boundary conditions. Yet, based on (7.2.1.2), the traveling wavenumber zk may be either real or imaginary, based on the relative values of the two

squared terms in (7.2.1.2).

For example, when / lmc kω > , then the traveling wavenumber zk is real, which is the case that we are

familiar with: acoustic waves propagate along the waveguide in the z+ direction. For this specific value

of ( ),l m considered, this is termed a propagating mode. On the other hand, when / lmc kω < , the (7.2.1.2)

returns an imaginary result for the traveling wavenumber. Thus, when input into (7.2.1.1) the wave propagation in the z axis is no longer a complex exponential but is instead characterized by a standard

exponential form zk ze− which indicates that the wave decays exponentially as it travels in the z+ direction. This is termed an evanescent wave which we encountered previously in the study of wave propagation in dispersive structures, such as beams.

The frequency at which the transition occurs between propagating modes and evanescent waves is termed the cut-off frequency.

lm lmckω = (7.2.1.6)

Therefore, if the waveguide is excited at a frequency below the cut-off of a given ( ),l m mode, then the

( ),l m mode will only exist locally to the acoustic source as a standing wave, but will have exponentially

decaying amplitude for distances away from the source. In other words, acoustic energy from that mode will not propagate when the acoustic excitation is below the associated cut-off frequencies.

In the case when ( ) ( ), 0,0l m = , plane waves propagate along the waveguide. Since, for a rigid-walled

waveguide with lossless fluid media, there is no attenuation with increasing distance for plane waves, plane wave propagation in ducts can be a severe trouble in many engineering applications of noise control: waves (i.e. plane wave noise) can propagate extraordinarily long distances with very low attenuation. Comparatively, when the source in the waveguide is driven at frequencies above cut-off for higher-order modes, the higher frequency modes may not project as much acoustic noise by virtue of positive and negative acoustic pressure regions in a given cross-section. In other words, the total cross-sectional area that delivers positive acoustic pressure may be similar to the area that delivers negative acoustic pressure, resulting in low energy transfer.

For a rectangular cross-section waveguide with longest dimension L (e.g. in the x axis), and the mode

index for the other dimension axis being 0 (e.g. in the y axis), we have that 1,0lmk kLπ

= = and that

1,0 1,0 2cc f

L Lπω = → = . This means that only plane waves will propagate in a rectangular cross-section

waveguide, of greatest cross-section dimension L , for frequencies less than / 2c L [Hz].


155

Further derivation for the rigid-walled, cylindrical cross-section waveguide of radial dimension a reveals that the cut-off frequency below which only plane waves propagate is 1.84 / 2c a π [Hz].

7.3 Pipes

The acoustic behavior of pipes is a subset of the behavior of waveguides. For pipes, the emphasis is on plane wave propagation and the incorporation of other lumped acoustic elements in the waveguide that may manipulate the wave propagation behaviors. As a result, the assumption in our analysis of pipes is that the acoustic wavelength is very long, respecting the largest characteristic dimension of the pipe, so that only

the plane wave mode, e.g. ( ) ( ), 0,0l m = in a rectangular waveguide, is able to project propagating waves.

It is under such condition that the lumped parameter modeling of acoustic elements is valid.

Consider a pipe with constant cross-sectional area S and length L inside of which is a uniform acoustic

fluid of characteristic specific acoustic impedance 0cρ , as shown in Figure 76. The end of the pipe in the

plane 0x = is driven by a rigid piston at an angular frequency ω .

The end of the pipe at x L= is terminated in a layer that possesses specific acoustic impedance mLZ (units

[N.s/m3]) and hypothetically no thickness. It is often desired to compute what this impedance is, as a function of frequency. In fact, it may be determined according to standing wave behavior within the pipe. The following approach provides the mathematical framework to achieve this empirical determination of the specific acoustic impedance.

To facilitate more straightforward mathematics, we define a new distance, d , being the difference from the impedance at x L= to the current location x . In other words, d L x= − . As a result, we write that plane wave propagation in the pipe is

( ) ( )( ) ( )( )

( ) ( ) ( )

j t k L x j t k L x

j t kd j t kd

x e e

d e e

ω ω

ω ω

+ − − −

+ −

= +

= +

p A B

p A B (7.3.1)

The latter notation indicates that the acoustic pressure p is a function of the coordinate d , as opposed to

an interpretation that it is evaluated at d .


156

Figure 76. Schematic of pipe. A view with perspective.

The pressure reflection coefficient is

=BRA

(7.3.2)

The spatial coordinate transformation retains the familiar form as from studies in Sec. 5.1 with new interpretation regarding the location under consideration for different values of x by virtue of d L x= − . In other words, for a known L , a 0d = implies that x L= .

Using the definition of the pressure reflection coefficient, we rewrite (7.3.1) as

( ) jkd jkdd e e− + p = A R (7.3.3)

The particle velocity in the pipe is

( )0 0

jkd jkdd e ec cρ ρ

−= −A Bu (7.3.4)

which is defined in terms of the pressure reflection coefficient to be

0

jkd jkde ecρ

− = − Au R (7.3.5)

Then the specific acoustic impedance at any point in the pipe is [3]

( ) 0

jkd jkd

jkd jkd

e ed ce e

ρ−

−

+= =

−p RZu R

(7.3.6)

The boundary condition at 0d = is that the impedance (7.3.6) must equal the impedance of the termination. Thus,


157

0 011mL c cρ ρ+ +

= =− −

R A BZR A B

(7.3.7)

If the driver has complex mechanical input impedance ( ) ( )0 0 / 0m x x= = =Z F u , where

( ) ( )0 0x x S= = =F p is the acoustic force provided by the piston, using (7.3.6) reveals that the mechanical

input impedance is defined in terms of the reflection coefficient and traveling pressure wave amplitudes as

0 0 0

jkL jkL jkL jkL

m jkL jkL jkL jkL

e e e ecS cSe e e e

ρ ρ− −

− −

+ += =

− −R A BZR A B

(7.3.8)

As described in analogous studies of waves in elementary structures of finite duration, Sec. 3, standing waves occur in acoustic volumes of finite extent. The alternative perspective is of forward and backward propagating waves. The standing wave ratio is defined as the ratio of the maximum absolute value of pressure in the pipe to the minimum absolute value

max

min

SWR =pp

(7.3.9)

Consider the case that the incident plane wave to the impedance termination has a purely real amplitude

A=A (7.3.10)

while we invoke the general complex reflection coefficient to define a reflected wave that is shifted in phase by θ relative to the incident wave

jBe θ=B (7.3.11)

Thus we have that (7.3.7) is

0

1

1

j

mLj

B eAc B eA

θ

θρ

+=

−Z (7.3.12)

Substitution of (7.3.10) and (7.3.11) into (7.3.1) ultimately yields after simplifications [1]

( ) ( ) ( ) ( ){ }1/22 22 2cos / 2 sin / 2A B k L x A B k L xθ θ= + − − + − − − p (7.3.13)

It may be shown that max

A B= +p , while min

A B= −p . Consequently, the standing wave ratio is

SWR A BA B+

=−

(7.3.14)

In terms of the pressure reflection coefficient, (7.3.14) is

1SWR

1+

=−

RR

(7.3.15)


158

Rearranging (7.3.15) yields

SWR 1SWR 1

BA

−= =

+R (7.3.16)

The first local minima from the general impedance termination at x L= occurs when a node appears in (7.3.13). This computed by selecting one of the sinusoids of (7.3.13) to take on an argument that results in a function evaluation of zero, and then by subsequent substitution of that argument into the other sinusoid. One finds that the first minima from x L= when

( )min

1 12 2

kd θ π= + (7.3.17)

Alternatively, (7.3.17) is expressed in terms of acoustic wavelengths

( )min1

4d λ θ

π = +

(7.3.18)

Bringing these pieces back together, one may empirically determine the frequency dependent specific acoustic impedance of a material layer placed at one end of a pipe, as driven by a source of single frequency plane waves on the opposing end, Figure 77. A microphone traverses linearly along the central axis of the pipe. The center is selected so as to minimize inconsistencies of developed plane wave fields near the pipe walls. The microphone traverses forward and backward to identify the standing wave ratio according to maximia and minima of the standing waves. This enables computation of SWR (7.3.15) and thus the

pressure reflection coefficient amplitude /B A=R (7.3.16). From the traversing data, the location of the

first local minima is identified as a number of wavelengths of the known sound source frequency. Then, by (7.3.18), one computes the phase difference from incident to reflected waves θ for that frequency and

wavelength considered. Then, using (7.3.12) with the /B A=R and θ , the specific acoustic impedance

of the material layer mLZ is known.

This experimental procedure was standardized as ASTM C384 for the purpose of measuring normal incidence acoustical absorption-relevant properties of materials. Yet, because it requires a new test at each frequency, and due to the inconvenience of moving measurement apparatus that can lead to errors over time, an alternative method ASTM E1050 ultimately established precedence as the more commonly employed test method for acoustical property measurement of materials' absorption-relevant properties. The standard ASTM E1050 uses fixed microphone positions, and determines the entire spectral-dependence on the properties in one test by virtue of a white noise source and the mathematics behind the approach.


159

Figure 77. Standing wave tube experiment illustration for measuring impedance of unknown material. A view in perspective.

7.3.1 Rigid termination of pipes

If the termination at x L= is rigid, mL = ∞Z . As a result, the pressure reflection coefficient is R =1. Using

(7.3.6), we find the specific acoustic impedance anywhere in the duct ( )dZ normalized respecting the

specific acoustic impedance of the fluid 0cρ

( )0

cotd

j kdcρ

= −Z

(7.3.1.1)

In terms of the mechanical input impedance at the driven end of the pipe, we have


160

0

0

cotm j kLcSρ

= −Z (7.3.1.2)

As stated in Sec. 3.1.4.1, the general case of resonance is when the reactance of the mechanical input impedance vanishes. Consequently, (7.3.1.2) vanishes when cot 0kL = , which occurs for

2 12n

nk L π−= ; 1,2,3,...n = (7.3.1.3)

The criteria for resonances (7.3.1.3) are identical to the resonances of a forced-fixed string, Sec. 3.1.4.1. Yet, the analogous state variables are string transverse displacement and particle velocity (not acoustic pressure). Indeed, using the boundary condition for a harmonically oscillation piston with velocity

j tu Ue ω= to compute the complex amplitude A of acoustic pressure in (7.3.5), it can be shown with (7.3.6)

that the acoustic pressure in the driven-closed pipe for the thn mode is

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-1

-0.5

0

0.5

1

normalized pipe lengthmod

e sh

ape

acou

stic

pre

ssur

e [d

imen

sion

less

]mode number 1


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-1

-0.5

0

0.5

1


e sh

ape

acou

stic

pre

ssur

e [d

imen

sion

less

]

mode number 2


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-1

-0.5

0

0.5

1


e sh

ape

acou

stic

pre

ssur

e [d

imen

sion

less

]

mode number 3


Figure 78. Examples of the normal acoustic modes of a driven-closed pipe. Here the driven end is at .


161

0cossin

j t nn

n

k dj cUek L

ωρ= −p (7.3.1.4)

Figure 78 presents several examples of the lowest order normal acoustic modes of driven-closed pipes. It is evident that an acoustic pressure node always occurs at the location of the driving source, while a location of zero normal particle velocity, by Euler's equation, occurs at the closed termination end. These trends of acoustic pressure for the normal modes are the opposite of those trends for transverse displacement for the forced-fixed string.

7.3.2 Short, closed volume

When the pipe is very short and the frequency considered is not too high, then 1kL << . In this case, the mechanical input impedance is

0

0

1 1tan

m

cS j kL jkLρ= ≈

Z (7.3.2.1)

Substituting terms and rearranging yields

20

0mc S

j Lρω

=Z (7.3.2.2)

The acoustic impedance 0aZ , recalling Sec. 4.2.1, is the mechanical impedance divided by 2S . Thus,

2 20 0

0ac c

j LS j Vρ ρω ω

= =Z (7.3.2.3)

For the simple harmonic oscillator, we saw that the mechanical impedance was msR j mj

ωω

= + +

Z .

Thus, the short, closed volume of acoustic fluid behaves analogously as a spring of stiffness 2 2

0 /effs c S Vρ= . This is not surprising given our experiences of, for example, closing a lid on a container

and detecting the resistance to the effort. The result also shows that for smaller volumes, there is greater stiffness. We may not be able to speak of experiences that demonstrate this phenomenon, but in the packaging and design of microphones, the stiffness of entrapped air can be disastrous towards measurement quality because it can result in a resonant-like property at low frequencies. This is one reason that microphones have vent holes [35].

7.3.3 Open-ended pipes

While a rigid termination to a pipe results in mL = ∞Z , the case of an open end termination is not 0mL =Z

. Although the acoustic pressure amplitude does drop significantly when the wall of the pipe is absent so as to confine and maintain the pressure, the acoustic pressure does not vanish at the open end of a pipe.

In fact, the impedance of the open termination to a pipe is more comparable to the radiation impedance of a baffled piston into the free field above the baffled plane. Therefore, using (6.6.4) to (6.6.6) and taking


162

into account our focus on low frequencies respecting the characteristic dimension of the pipe (here, we assume the pipe is cylindrical and the pipe radius is a ) and that the pipe has a flange at its open termination (effectively a baffle), Figure 79, we have that the acoustic impedance is

( )200

1 82 3a

c ka j kaSρ

π = +

Z (7.3.3.1)

Noting that 2S aπ= , we have

02

0 0 2

81 3

2a

aSck j

S

ωρπρ

π

= +Z (7.3.3.2)

The resistance of this acoustic impedance is 20 0

12aR ckρπ

= . Assessment of the values in (7.3.3.2) shows

that it is dominated by the reactance for 1ka << . By considering the mechanical impedance of the simple harmonic oscillator, the reactance term of (7.3.3.2) is analogous to a lumped "acoustic" mass equal to

( )0 8 / 3S aρ π . (An entirely different way to arrive the determination of this effective mass contribution is

to compute the resonances of the associated mechanical impedance according to vanishing reactance [1]). Therefore, in the assessment of the impedance of an open-ended pipe, the wave does not "see" an open end

but instead encounters an effective extension of the pipe of length 8 0.853

aL aπ

∆ = ≈ where in the associated

mass ( )0 8 / 3S aρ π exists. Thus, when conducting analyses of the resonance frequencies of open-ended

pipes, like organ pipes, in the way carried out in Sec. 7.3.1 for the forced-closed pipe, the length of the open-ended and flanged pipe must be considered as the effective length effL L L= + ∆ . If the pipe does not

have a flange at its end, in other words the pipe is similar to a bare pipe open to the environment, then the effective length correction term is 0.6L a∆ = . If the pipe is unflanged at the open end, it is found that the

resistance of this acoustic impedance is 20 0

14aR ckρπ

= .


163

Figure 79. Comparison of driven-open pipes with the open ends with or without flanges.

7.3.4 Helmholtz resonator

The effective mass encountered at the open end of a pipe, due to an effective extension of the pipe, and the stiffness encountered for a small enclosed volume, leads one to ask if a short length of pipe that opens into a small volume can act in a way analogous to a mass-spring oscillator.

Indeed, this is the case and the device is termed a Helmholtz resonator. The Helmholtz resonator is a common device employed to suppress noise in pipes/ducts. Although, similar to an undamped mass-spring oscillator, the resonances associated with Helmholtz resonators are inherently narrowband and thus valuable for noise control purposes only when plane waves travel along the duct at targeted frequencies near resonance of the device.

Consider the schematic shown in Figure 80 at left. A small acoustic volume of V is attached to a pipe by a short neck with length L , cross-sectional area S , and radius a . The use of a cylindrical neck connecting the main pipe to the Helmholtz resonator is not in practice, but it simplifies the mathematics involved towards computing the resonance effects.

Figure 80. Flanged and unflanged Helmholtz resonators in pipes.


164

A plane wave arrives at the location of the Helmholtz resonator, it encounters an acoustic impedance that is the sum of the acoustic impedance of a short open-ended pipe and of a small, closed acoustic volume [3]. Thus, the plane wave encounters the total acoustic impedance

22 0 0

0 01

2aL cck j

S Vρ ρ

ρ ωπ ω

′ = + −

Z (7.3.4.1)

where 0.85effL L a′ = + by virtue of the fact that, as seen in the schematics of Figure 80, the short neck has

two open terminations: one into the Helmholtz resonator and the other into the pipe. The latter open termination is almost always flanged, so that L′ modifies effL with another 0.85a contribution. As a result,

the neck length must account for both contributions, thus 0.85 0.85L L a a′ = + + for the flanged Helmholtz resonator, at left in Figure 80.

As described elsewhere, resonance most generally occurs when the reactance vanishes. Thus, by (7.3.4.1), the Helmholtz resonator natural frequency is computed to be

2nc Sf

L Vπ=

′ (7.3.4.2)

7.4 Filtering properties of lumped acoustic impedances

Low frequency acoustic wave propagation in pipes is amenable to employment of lumped acoustic impedance concepts described in Sec. 7.3. Several other cases of acoustic impedance change are needed to be introduced in order to thereafter examine the full filtering properties of various lumped acoustic impedances on plane wave propagation.

Figure 81. At left, change in cross-section of pipe. At middle, branching of pipe. At right, side-branching of pipe.

Consider the cross-section change in the pipe as shown at left in Figure 81. We first examine this as an arbitrary change in acoustic impedance from 0 /c Sρ on the incident wave side to 0 0 0R jX= +Z at the

junction of the two layers at 0x = . The incident, reflected, and transmitted acoustic pressure waves are

( )j t kxi ie

ω −=p P (7.4.1)

( )j t kxr re

ω +=p P (7.4.2)


165

( )j t kxt te

ω −=p P (7.4.3)

To conserve mass at the cross-section change in the pipe, the continuity of particle velocity is no longer sufficient for this situation. Now, the continuity of volume velocity is required. Thus, the corresponding volume velocities of the waves are

( )j t kxi ie

ω −=q U (7.4.4)

( )j t kxr re

ω +=q U (7.4.5)

( )j t kxt te

ω −=q U (7.4.6)

where the units of the complex amplitudes are iU =[m3/s]. Continuity of acoustic pressure and volume

velocity then yield

i r t

i r t

+=

+p p pq q q

(7.4.7)

By virtue of the general expression for the acoustic impedance /=Z p q , this leads to the expression for

the acoustic impedance for values of 0x < .

0jkx jkx

i r i rjkx jkx

i r i r

c e e R jXS e eρ −

−

+ += = = +

+ −p p P PZq q P P

(7.4.8)

At the cross-section change 0x = , we have the input acoustic impedance, abbreviated using 0 0 0R jX= +Z

. Then, we solve for the pressure reflection coefficient and find

0 0

0 0

//

r

i

c Sc S

ρρ

−= =

+ZPR

P Z (7.4.9)

The power reflection coefficient, recalling the definition in Sec. 5, is therefore

( )( )

2 22 0 0 0

2 20 0 0

/

/

R c S XR

R c S Xρ

ρΠ

− += =

+ +R (7.4.10)

while the power transmission coefficient is

( )0 0

2 20 0 0

4 //

R c STR c S X

ρρ

Π =+ +

(7.4.11)

Now applying these derivations to the specific case of an acoustic impedance change associated with the cross-section change Figure 81, we have that 0 0 2/c Sρ=Z assuming that the fluid 0x < is the same as the

fluid 0x ≥ . Consequently, the S indicated in the (7.4.8) to (7.4.11) is 1S . By substitution, we have


166

( )( )

21 2

21 2

S SR

S SΠ

−=

+ (7.4.12)

( )1 2

21 2

4S STS S

Π =+

(7.4.13)

Figure 82 shows results of (7.4.12) and (7.4.13) plotted as a function of the area ratio 2 1/S S . It should be

emphasized that the reduction of transmitted acoustic power is not in consequence to absorption phenomena but strictly to the reflection of the power back upstream towards its origin, presumably an acoustic source.

Figure 82. Power reflection and transmission coefficients at cross-section change in pipe.

Then, we examine the influence of side branches upon the wave propagation in the pipe. In this case, we consider the schematic of the middle in Figure 81 and define the transmitted pressures in the 1 and 2 side branches according to the acoustic impedances and volume velocities:

( )1 1 1

j t kxe ω −=p Z q (7.4.14)

( )2 2 2

j t kxe ω −=p Z q (7.4.15)

At the junction 0x = , the continuity of acoustic pressure is

1 2i r+ = =p p p p (7.4.16)

while continuity of volume velocity is

1 2i r+ = +q q q q (7.4.17)

Together, (7.4.16) and (7.4.17) yield

1 2

1 2

i r

i r

+= +

+q q q qp p p p

(7.4.18)

which is an expression for the acoustic admittance, 01 / Z

10-2

10-1

100

101

102

0

0.2

0.4

0.6

0.8

1

S2 / S1

coef

ficie

nt

power reflection coefficientpower transmission coefficient


167

0 1 2

1 1 1= +

Z Z Z (7.4.19)

It is seen from (7.4.19) that the combination of side branches has an impact upon the plane acoustic wave propagation in the same way as combinations of electrical impedances in parallel. Thus, the input acoustic impedance 0Z observed by the incident wave at the junction 0x = is a reciprocal sum of the impedances

of the downstream pipe segments.

Because we focus on the long wavelength limit, the side branches need not be at angles relative to each as shown in the middle of Figure 81. One common example is a side branch which is a length of pipe with an impedance 1 b=Z Z that is oriented normal to the axis of the pipe connecting up and downstream, at right

in Figure 81. Consider that the up and downstream sections of pipe share the same cross-sectional area

2S S= so that the downstream acoustic impedance is the same as the upstream 2 0 /c Sρ=Z . Then, the

pressure transmission coefficient is

2

0 / 2t b

i i bc Sρ= =

+P ZPP P Z

(7.4.20)

In general, the side branch acoustic impedance is b b bR jX= +Z . By substitution into the power reflection

and transmission coefficients, we find

( )( )

20

2 20

/ 2

/ 2 b b

c SR

c S R Xρ

ρΠ =

+ + (7.4.21)

( )

2 2

2 20 / 2

b b

b b

R XTc S R Xρ

Π

+=

+ + (7.4.22)

The (7.4.21) and (7.4.22) provide a means to assess the filtering properties of lumped acoustic impedances that are placed along pipes. A significant proportion of the acoustic power is reflected back upstream when the reactance of (7.4.21) vanishes and the resistance of the side branch component is small. By conservation of energy, and in agreement with the outcome of (7.4.22), the transmitted power is very low. Several common acoustic filters used in applications of noise and wave control acoustics are worth closer assessment.

7.4.1 Low pass filter

Consider that the side branch is an enlarged section of pipe of total cross-sectional area 1S when compared

to the up and downstream pipe area of S , as shown at left in Figure 83. At low frequencies, this side branch

acts like an acoustic spring. As we derived in Sec. 7.3.2, the spring stiffness is 20 /effs c Vρ= . Consequently,

the acoustic impedance of this section of pipe, spanning length L , is


168

( )2

0

1b

cjS S Lρ

ω≈ −

−Z (7.4.1.1)

Substitution of (7.4.1.1) into (7.4.22) yields the power transmission coefficient

21

1

12

TS S kL

S

Π ≈− +

(7.4.1.2)

At low frequencies kcω = , power transmission is complete while it decreases logarithmically for logarithmic increase in the frequency, like a standard first-order low pass filter.

As shown in the illustrations of Figure 83, the enlarged volume may be an axi-symmetric volume around the pipe that passes from up to downstream.

Figure 83. Acoustic filters for plane wave propagation in pipes.

7.4.2 High pass filter

We observed in Sec. 7.3.3 that open-ended pipes exhibit mass-like acoustic impedances. Such characteristics are similar to the roles of high pass filters. Thus, considering the middle schematic of Figure 83, the pipe containing a short length of unflanged pipe as a side branch is likely to exhibit high pass characteristics to the plane wave propagation. If the side branch radius is a and length is L , as shown previously the acoustic impedance is

2 00 2

14b

Lck ja

ρρ ω

π π′

= +Z (7.4.2.1)

Here the corrected length L′ is often taken as 0.85 0.6 1.4L L a a L a′ = + + ≈ + even though the math on the right-hand side of the equation does not work out exactly. In fact the result is closer to measured outcomes, which is the reason for the alteration.

At low frequencies, the resistance is negligible from (7.4.2.1). Thus, substituting only the reactance of (7.4.2.1) into (7.4.22), we find


169

22

1

12

Ta

SL kπ

Π =

+ ′

(7.4.2.2)

Indeed, (7.4.2.2) exhibits high pass filter characteristics and similar logarithmic reduction in the power transmission coefficient for logarithmic reduction in the frequency below a cut off value.

7.4.3 Band stop filter

A device termed the quarter wave resonator is used to attenuate a target frequency by resonance phenomenon. Recall from Sec. 7.3.1 derivation regarding the closed termination pipe, and see the schematic in the top right of Figure 83. The acoustic impedance of the short length L closed termination side branch of cross-sectional area S is

0 cotbcj kL

Sρ ′= −Z (7.4.3.1)

where 0.85L L a′ = + . Considering that the acoustic resistances involved are small, 0bR = and the power

transmission coefficient becomes

( )

20

2

222 0

0

cotcot

1 cot/ 2 cot 4

c kLkLST

c kLc S kLS

ρ

ρρΠ

′ ′ = = ′+′+

(7.4.3.2)

The numerator of (7.4.3.2) vanishes when ( )2 1 / 2kL n π= − for 0,1,2,...n = . Considering the lowest index

and expressing the result in terms of acoustic wavelength, we find that the length of the side branch is / 4L λ= . In other words, if the side branch length is one-quarter of the acoustic wavelength, it is capable

of completely suppressing the transmitted acoustic power down the pipe at the frequency / / 4f c c Lλ= =

. Since there are no resistive impedance components in this lumped acoustic element, this means that the acoustic power is reflected back upstream at the resonant frequency. But such perfect attenuation of the acoustic energy downstream occurs only at the resonant frequency according to such analysis. To provide a more robust attenuation of the acoustic energy from propagating downstream via a broader attenuation effect, porous material liners are often used in such quarter wave resonators in order to introduce more significant resistive impedance components. The trade-off then exists between perfect attenuation at one frequency and an ability to attenuate a range of frequencies using the same quarter wave resonator. Ultimately, the role of this lumped acoustic impedance is analogous to a band stop filter for the range of frequencies that it suppresses.

Helmholtz resonators offer another band stop filter effect by virtue of the associated mass-spring phenomena it realizes, as described in Sec. 7.3.4. The lumped acoustic impedance of the Helmholtz resonator was there found to be


170

22 0 0

0 01

2b aL cck j

S Vρ ρ

ρ ωπ ω

′ = = + −

Z Z (7.4.3.3)

With (7.4.3.3) substituted into (7.4.22) to determine the acoustic power transmission coefficient, it is seen that the resonance of the Helmholtz resonator gives rise to small values in the numerator of the power transmission term. The remaining numerator term is the contribution from the acoustic resistance of (7.4.3.3), in addition to potential porous linings or absorptive materials placed within the volume of the resonator. This in general leads to the fact that Helmholtz resonators tend to have greater inherent damping effects than quarter wave resonators, when the two types of lumped acoustic impedances are fabricated for the same frequency of wave attenuation.

In the presence of mean flow, the necks of Helmholtz resonators and the quarter wave resonators are angled into the flow direction in order to minimize vortex shedding that gives rise to adverse downstream noise and to reduce the potential for particulates in the flow to build up within the resonator volumes [36].


171

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[3] D.T. Blackstock, Fundamentals of Physical Acoustics (Wiley, New York, 2000).

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[16] M.C. Remillieux. Development of a model for predicting the transmission of sonic booms into buildings at low frequency. PhD dissertation. Blacksburg, Virginia, USA: Virginia Polytechnic Institute and State University; 2010.

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[18] M. Postema, Fundamentals of Medical Ultrasonics (Spon Press, New York, 2011).

[19] G. D'Aguanno, K.Q. Le, R. Trimm, A. Alù, N. Mattiucci, A.D. Mathias, N. Aközbek, M.J. Bloemer, Broadband metamaterial for nonresonant matching of acoustic waves, Scientific Reports 2, 340 (2012).

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173

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course notes for osu me 8260 advanced …...rl harne me 8260, adv. eng. acoust. 2019 the ohio state...

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