Acoustics 1 SGN-14006 / A.K.

Sound, acoustics Slides based on: Rossing, ”The science of sound,” 1990.

Contents: 1.  Introduction 2.  Vibrating systems 3.  Waves 4.  Resonance 5.  Room acoustics

Acoustics 2 SGN-14006 / A.K. 1 Introduction

!  The word acoustics refers to the science of sound and is a subcategory of physics

!  Room acoustics (confusingly, sometimes just acoustics) studies sound propagation indoors (esp. concert halls)

!  The goal in this lecture is to learn the principles, not so much the equations

Acoustics 3 SGN-14006 / A.K. 2 Vibrating systems

!  Common to vibrating systems –  Motion repeats in each regular time interval (the period) –  Some force restores the system toward equilibrium

2.1 Simple harmonic motion !  Spring-mass system

–  force (spring constant x stretch) acts as the restoring force –  In equilibrium, earth gravity mg = F

!  In Simple harmonic motion, the restoring force is directly proportional to the distance from equilibrium –  In that case frequency f does not depend on amplitude –  Mass + spring:

xKF ⋅−=

mK

tAxxmK

x

xmmaKx

=⇒+=⇒=+

≡=−

ωϕω )cos(,0

,

!!

!!

Acoustics 4 SGN-14006 / A.K. 2.2 Energy and damping (1)

!  In a vibrating system, kinetic energy and potential energy are alternating

!  In the mass-spring case –  Kinetic energy EK = ½mv2

–  Potential energy EP = Kx2

!  Figure: –  top: displacement vs. time,

bottom: speed vs. time –  At times t1 and t3

potential energy is at its maximum,

–  At times t2 and t4 kinetic energy is at its maximum

x

Acoustics 5 SGN-14006 / A.K. Energy and damping (2)

!  In all realistic vibrating systems, there are energy losses due to friction etc.

!  Unless energy is brought to the system from outside, the amplitude of the vibration will decay (see figure) –  Typically a certain fraction of the mechanical energy is lost during

each vibration. In that case, the amplitude envelope is exponentially decaying (see figure)

Amplitude envelope

Acoustics 6 SGN-14006 / A.K. 2.3 Simple vibrating systems (1)

!  These are simple harmonic vibrating systems (in addition to the mass-spring system)

1.  Pendulum (small angle) –  Mass attached to a string –  Gravity as the restoring force

2.  ”Spring” of air –  A piston of mass m moves freely

in a cylinder, of area A and length l

Acoustics 7 SGN-14006 / A.K. Simple vibrating systems (2)

3.  Helmholz resonator –  Air in the ”neck” acts as the mass –  Air in the cavity acts as the spring

–  Examples of Helmholz resonators •  Blowing air across and ampty bottle •  Bass reflex tube in loudspeakers (figure) •  Sound hole in the guitar

Acoustics 8 SGN-14006 / A.K. 2.4 Systems with two masses

!  In the above examples, one coordinate sufficed to describe the motion " only one degree of freedom

!  In the following, we consider systems with two or three degrees of freedom –  Then there are also more than one mode of vibration –  Typically each mode has a different frequency of vibration

!  Figure: system with two masses and three springs –  Two modes: masses moving

(a) in the same direction (b) in opposite directions

–  Modes are independent of each other and have different mode frequencies

–  In realistic cases the movement is usually a combination of modes

Acoustics 9 SGN-14006 / A.K. Systems with two masses (2)

!  Figure: above-descrived two-mass system has two transverse vibration modes in addition to the longitudinal ones –  Vibration is perpendicular to the spring –  Transverse vibration: for example membrane of a drum –  Longitudinal vibration: for example air column in a wind instrument

Acoustics 10 SGN-14006 / A.K. 2.5 Systems with many modes of vibration

!  Figure: in the general case a system with N masses has N longitudinal and N transverse vibration modes –  2N modes, but only N frequencies, since corresponding

longitudinal and transverse modes have the same frequency

!  More masses " ”wavelike” shape emerges –  Vibrating

string can be considered as a mass-spring system where N is very large

Acoustics 11 SGN-14006 / A.K. 2.6 Vibrations in musical instruments (1)

1.  Vibrating string –  Can be viewed as a mass-spring system: string mass and elasticity –  Many vibration modes that are typically nearly exact integer multiples of

a fundamental frequency "  Harmonic modes (see bottom row of the figure on the previous slide)

2.  Vibrating membrane –  Can be thought of as a two-dimensional ”string”, tension of the

membrane acts as a restoring force (membrane attached to a rim) –  figure: four vibration modes are illustrated

– 

Acoustics 12 SGN-14006 / A.K. Vibrations in musical instruments (2)

3.  Vibrating bar –  For example marimba, xylophone, glockenspiel –  Stiffness of the bar provides a restoring force –  Vibration modes are not harmonic, but the frequencies in glockenspiel for

example are 1 : 2.76 : 5.40 : 8.93 :... (harmonic would be 1:2:3:...)

4.  Vibrating plate –  As in a vibrating bar, the stiffness of the plate

itself acts as a restoring force (different from a stretched membrane of a drum)

Acoustics 13 SGN-14006 / A.K. Vibrations in musical instruments (3)

5.  Air-filled pipes –  Vibrating air column –  For example organ pipe, trumpet –  Comparable to a vibrating string –  Many vibration modes

Acoustics 14 SGN-14006 / A.K. 2.7 Vibration spectra

!  When a vibrating system is excited, it usually starts to vibrate in several modes at the same time –  Each mode has its specific frequency and amplitude " Spectrum of the vibration

!  Figure: spectrum of a plucked string " Mode frequencies of an instrument (for example) can be studied

by recording its sound and then looking at its Fourier transform

Acoustics 15 SGN-14006 / A.K. 3 Waves

!  Waves transport energy and information through a medium so that the medium itself is not transported

!  In the case of sound, the medium is usually air !  Sounds may reflect, refract, or diffract !  Speed of sound in air ≈ 340 m/s (20 ºC)

–  cf. speed of light ≈ 3·108 m/s

Acoustics 16 SGN-14006 / A.K. 3.1 Progressing waves

!  For a progressing wave where v is velocity, f is frequency, and λ is wavelength –  figure: wave in a rope

λfv =

Acoustics 17 SGN-14006 / A.K. 3.2 Properties of waves

!  Figure: reflection

!  Figure: linear superposition –  Waves may travel ”through” each

other without changing their properties

!  Standing wave –  Found e.g. on a stretched string –  Wave traveling in the string reflects from both ends so that the

sum of waves traveling in opposite directions appears not to move –  ”Nodes” and ”antinodes” can be observed on the string

Acoustics 18 SGN-14006 / A.K. Standing waves on a string

!  Standing wave is formed when a wave travels in opposite directions on a string, reflecting at both ends

!  Figure: resonance frequencies of a vibrating string –  Wavelength of the lowest

resonance λ = 2 x string length " Fundamental mode

frequency f1 = v / 2L –  Higher modes:

fn = n · v / 2L = n · f1 where propagation speed T is string tension and µ is mass per unit length (speed and frequency get lower by reducing tension or increasing string mass)

µTv =

Acoustics 19 SGN-14006 / A.K. 3.3 Sound waves

!  Sound waves are longitudinal waves that travel in gas, liquid, or solid material –  Speed of sound is lowest in gas –  Hearing works also underwater, although due to the changed

speed of sound, the direction of arrival of sounds is unclear !  Figure: reflection of a

sound pulse in a pipe (a) Sent positive pressure

pulse (b) Reflection at open

end (negated) (c) Reflection at closed

end (d) Absorption

(no reflection)

Acoustics 20 SGN-14006 / A.K. 3.4 Propagation in two or three dimensions

!  Usually sound waves propagate in two or three dimensions !  Sources with different geometries radiate different kinds of

patterns –  Point source radiates spherical waves (left figure) –  Line source radiates cylindrical waves (right figure) –  Large flat source radiates plane waves –  Real-life sources can only approximate these geometries

Acoustics 21 SGN-14006 / A.K. 3.5 Doppler-effect

!  Normally the frequency of the waves arriving to an observer is the same as the frequency of vibration at the sound source

!  The situation changes if either the source of the observer is in motion –  Observer ”meets” the waves more frequently when moving

towards the wavefront (" frequency increases) –  When moving apart from each other, observed frequency

decreases = Doppler-effect

Acoustics 22 SGN-14006 / A.K. 3.6 Reflection

!  Reflection of sound waves can be experienced by clapping hands at some distance from a large wall

!  Figure: reflected waves appear to come from an imaginary source behind the reflecting surface –  Think of a mirror

Acoustics 23 SGN-14006 / A.K. 3.7 Refraction

!  Refraction occurs when the speed of waves changes –  Direction of the waves changes

!  Figure: propagation speed changes abruptly as wave passes from one medium to another

!  Speed can also change gradually –  Figure: wind does not ”blow the sound back” (speed of wind is

small compared to sound), but because higher wind speed at higher altitude tends to refract the sound to the sky

Acoustics 24 SGN-14006 / A.K. 3.8 Diffraction

!  Sound waves tend to bend around an obstacle !  Figures:

–  left: sound bends behind a wall (see arrows) –  right: sound waves traveling through a narrow opening appear as

a ”new” point source

Acoustics 25 SGN-14006 / A.K. 4 Resonance

!  Idea of resonance illustrated by a child in a swing: giving the swing a small push at a suitable frequency, its amplitude gradually increases

4.1 Mass-spring vibrator resonance –  figure: mass-spring systems attached

to a crank –  Natural vibration frequency of the

mass-spring system is f0 –  Crank is revolved at frequency f,

which is slowly varied " Vibration amplitude A changes and

reaches its maximum Amax when f = f0 !  Curve: amplitude A as a function of

frequency f

Acoustics 26 SGN-14006 / A.K. 4.2 Standing waves on a string

!  Standing wave is formed when a wave travels in opposite directions on a string, reflecting at both ends

!  Figure: resonance frequencies of a vibrating string –  Wavelength of the lowest

resonance λ = 2 x string length " Fundamental mode

frequency f1 = v / 2L –  Higher modes:

fn = n · v / 2L = n · f1 as mentioned in §3.2 above

Acoustics 27 SGN-14006 / A.K. 4.3 Partials, harmonics, overtones

!  Terminology for discussing vibration spectra, for example the spectra of musical instruments: –  Partial : any mode of a vibrating system (any component of sound) –  Harmonic : if partials are (nearly) integer multiples of the

fundamental (as in a vibrating string for example), the partials are called harmonics (fundamental is the first harmonic)

–  Harmonic sound : sound where partials are nearly integer multiples of the fundamental

Acoustics 28 SGN-14006 / A.K. Partials, harmonics, overtones (2)

!  Relative strengths of the partials largely determine the timbre (tone

colour) of the sound –  Temporal evolution of the partial amplitudes is another important factor of

timbre (and there are also others like tonal vs. noisy quality etc.)

frequency

ampl

itude

Fundamental mode (frequency = fundamental frequency)

harmonic overtones (in a harmonic sound, frequencies are integer multiples of the fundamental)

Acoustics 29 SGN-14006 / A.K. 4.4 Open and closed pipes

!  Reflection of a positive sound pulse at the ends of a pipe –  Reflected as negative at open end and as positive at closed end

!  Human vocal tract can be modeled as an acoustic tube –  When varying the shape of the vocal tract, resonance frequencies

(formants) move " phonemes !  Left: vibration modes in a pipe open at both ends fn = n · f1 , n = 1,2,3,... !  Right: vibration modes in a pipe with one end closed

fn = n · f1 , n = 1, 3, 5,... " only odd harmonics

! 

Acoustics 30 SGN-14006 / A.K. 4.5 Sympathetic vibration

!  The amount of sound radiated by a source is proportional to the amount of air it displaces as it moves –  Thin vibrating string displaces very little air and therefore radiates only a

small amount of sound –  Membrane of a drum or the element of a loudspeaker displaces more air

!  Radiation can be increased by attaching the vibrating system to a wood plate or sounding box –  Vibrating system makes the plate move –  Due to its large area, sympathetic vibrations of the plate increase the

amount of radiated sound, even though its resonance frequency would not be exactly right

!  In musical instruments, two or more vibrators often work together –  Piano strings and soundboard, guitar strings and body –  Clarinet reed and air column

!  String instruments are based on the sympathetic vibrations of a sounding box or soundboard –  Resonance frequencies of the sounding box largely determine the timbre

of the instrument

Acoustics 31 SGN-14006 / A.K. 5 Room acoustics

5.1 Sound propagation outdoors and indoors !  Free field

–  Source is small enough to be considered a point source –  Source is outdoors and far away from reflecting objects " Sound waves propagate from the source in shape of sphreres and sound

pressure is proportional to 1/r [Pa] (r: distance from source) " Average sound intensity I ∝ 1/r2 [W/m2] –  Indoors, free field can be found only in an unechoic room

!  Indoors sound waves encounter walls and other obstacles –  Figures: obstacles reflect and absorp sound in ways that determine the

acoustic properties of the room

Acoustics 32 SGN-14006 / A.K. 5.2 Direct, early, and reverberant sound

!  In auditorium, direct sound reaches listeners in 20-200 ms –  Depends on the distance from the source to the listener

!  Soon after, the same sound reaches the listener from reflecting surfaces (walls, ceiling) –  These are called early reflections –  Time difference to direct sound usually < 50 ms

!  Last group of reflections is called reverberant sound –  Sound reflected several times from various surfaces –  Weaker, many reflections, close to each other in time –  When the source is turned off, reverberation decays in an

approximately exponential manner (dB level is a straight line)

Acoustics 33 SGN-14006 / A.K. Direct, early, and reverberant sound (2)

!  Figure: –  Room impulse response h(t) –  direct sound –  early reflections –  reverberation

Acoustics 34 SGN-14006 / A.K. 5.4 Reverberation time

!  Reverberation time is among the most familiar characteristics of auditoriums –  Reverberation reinforces direct sound –  Too much reverberation results

in a loss of clarity –  Suitable reverberation time depends

on the auditorium size and purpose (speech: short, organ music: long)

!  Figure: studying reverberation –  Switch on a steady source for time T –  Record the sound at another

location in the auditorium " Level of the reverberant sound

first increases in steps when early reflections arrive

Acoustics 35 SGN-14006 / A.K. Reverberation time

!  Reverberation time is usually denoted by T60 –  Time where the sound level decreases by 60 dB from its maximum –  Figure: sound level decreases approximately exponentially, thus

dB level decay is a straight line as a function of time

–  Curve R(t) (see the figure) describing the sound level decrease can be also obtained from room impulse response h(t) by

p [lin.] Lp [dB]

∑∞

=t

thtR )()( 2

R(t) 10 log10(R(t))