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Acoustics (VTAN01) HT2 (2019/2020)

01log110

)log()log(

logloglog

loglog)log(

0

=

=

⋅=

−=⎟⎟⎠

⎞⎜⎜⎝

⎛

+=⋅

xnx

yxyx

yxyx

n

Formulae – Acoustics VTAN01

TABLE OF CONTENTS

I) Some logarithmic rules: 1

II) Fundamental acoustic definitions 1 II.1) Frequency bandwidths, weighted sound pressure level 3 II.2) One dimensional wave propagation 4

III) Room acoustics 5 III.1) Inside a room : reflection and transmission 5 III.2) Between two rooms: sound isolation and absorption 7

I) Some logarithmic rules:

yxy

x

yxyx

x

x

y

xx

xxy

−

+

=

⋅=

=

=

=⇔=

101010

10101010

)10log(10log

log

II) Fundamental acoustic definitions Sound pressure - One-dimensional harmonic plane sound field:

- Effective value (RMS, Root Mean Square) for sound pressure in a point: NOTE: For a harmonic wave,

Sound pressure level SPL - Sound pressure level: , where pref = 2·10-5 Pa (pref : approximately the quietest sound a young undamaged human hearing can detect at 1000 Hz)

- Equivalent sound level:

⎟⎠

⎞⎜⎝

⎛ +−= ϕω

ω xc

tptxp cosˆ),(

∫Δ+

Δ=

tt

t

dttxpt

p0

0

),(1~ 2

⎟⎟⎠

⎞⎜⎜⎝

⎛=⎟

⎟⎠

⎞⎜⎜⎝

⎛= ∫∫

TtL

T

refTeq dt

Tdt

ptp

TL p

0

10/)(

02

2

, 101log10)(1log10

⎟⎟⎠

⎞⎜⎜⎝

⎛= 2

2~log10

refp p

pL

2/ˆ~ pp =

;2 fπω =λπω /)2(/ == ck;/ fc=λ

;/1 Tf =


- Summing of two sound sources (if uncorrelated the last term vanishes):

SPL difference (dB) between 2 sources 0 1 2 3 4 5 6 7 8 9 10 Added dB to the highest SPL 3 2,54 2,12 1,76 1,46 1,19 0,97 0,79 0,64 0,51 0,41 - Summing of N uncorrelated sources:

- Phon curves:

Figure 1 – Phon curves

Sound intensity

- The sound energy Π and sound intensity I is )()()()()(

)()()(tvtptI

SttI

tvtFt⋅=⇒

⎪⎭

⎪⎬

⎫

Π=

⋅=Π

- The sound power level and sound intensity level are calculated (in decibels) according to

⎟⎟⎠

⎞⎜⎜⎝

⎛

Π

Π=Π

ref

L log10 and ⎟⎟⎠

⎞⎜⎜⎝

⎛=

refI I

IL log10 , where Π ref = 10-12 W and Iref = 10-12 W/m2.

Relation (function of the directivity factor Q) between Lp and L∏ : .4,3,2,1;4

log10 2 =⎟⎠

⎞⎜⎝

⎛+=Π Q

rQLL p π

∫Δ+

Δ++=

tt

ttot dttptp

tppp

0

0

)()(2~~~21

22

21

2

⎟⎠

⎞⎜⎝

⎛= ∑

=

N

n

Ltotp

npL1

10/,

,10log10


The time mean values are: ∫ ⋅=ΠT

dttvtpTS

0

)()( and ∫ ⋅=T

dttvtpT

I0

)()(1

- For a wave propagating in the positive x-direction, the previous integral yields: cpI ρ2~= ,

with c being the speed of sound in m/s and cZ ρ= the specific acoustic impedance.

cair =γP0

ρair (T = 0!)1+ Tair[

!C]2 ⋅273

"

#$

%

&'= 331.4 ⋅ 1+

Tair[!C]

2 ⋅273"

#$

%

&'

II.1) Frequency bandwidths, weighted sound pressure level II.1.a) Octave bands and third octave bands:

II.1.b) Weighed sound level:

Figure 2 - A-, B-, C- and D-weightings (10 Hz – 20 kHz).

( )∑ += 10/)(10log10 weighingLweighed

nL


II.2) One dimensional wave propagation

The general form of the equation of motion in fluids and solid media (applying Newton’s 2nd law) is

tv

xp

∂

∂−=

∂

∂ρ ; p: pressure, ρ: specific mass, ν: fluid particle velocity

For fluids, the relation between pressure and particle velocity is

xvP

tp

∂∂

−=∂∂

0γ ; γ = adiabatic index (γ air =1.4)

For solid media we have the corresponding relation between force and displacement

xuESF∂

∂−= ; E: modulus of elasticity, S: surface

II.2.a) Waves in fluid media

- For a longitudinal wave propagating in air (one dimensional propagation in positive x-direction) expressed with sound pressure and particle velocity, respectively, as the field variables are:

012

2

22

2

=∂

∂⋅−

∂

∂

tp

cxp

2

22

2

2

xvc

tv

∂

∂⋅=

∂

∂

where ργ 0Pc = is the propagation speed for the pressure wave in the air, with P0 being the atmospheric pressure. The general solution to the wave equation (with p being the field variable) is

)/()/(),( cxtpcxtptxp ++−= −+

The harmonic solution to the wave equation in complex form is

)()( ˆˆ),( kxtikxti epeptxp +−

−+ += ωω

For physical interpretation, take the real part of the result, where

+p̂ and −p̂ are the pressure amplitudes for

the waves propagating in positive and in negative direction, respectively. ω is the angular frequency and ck ωλπ == 2 is the wave number. This yields the equality λfc =

- Specific acoustic impedance is defined as vpZ ≡

- For a wave propagating in air (1-D propagation in positive x-direction) the acoustic impedance is

cvpZ ρ==+

+


II.2.b) Waves in solid media

-For longitudinal and waves (infinite medium) and quasi-longitudinal waves (finite medium), their wave equations read, respectively:

02

2

2

2

=∂

∂⋅−

∂

∂

tv

xv

E xx ρ

0' 2

2

2

2

=∂

∂⋅−

∂

∂

tv

xvE xx ρ

The propagation speed for longitudinal waves is ρEcL = , whereas for quasi-longitudinal waves it is

)1(' 2υρρ −== EEcqL , υ being the Poisson’s ratio of the material where the wave is propagating through. Particle displacement, particle velocity, strain or force can be used instead of pressure as a field variable in the wave equation.

- For shear waves the wave equation can be expressed using the transversal displacement w as:

02

2

2

2

=∂

∂⋅−

∂

∂

tw

Gxw ρ where

ρGcsh = , G being the shear modulus of the material.

)1(2 υ+=

EG

- For bending waves in beams and plates the wave equation in one dimension is

02

2

4

4

=∂

∂+

∂

∂

twS

xwB ρ

where

12

3bhEB = for a rectangular cross section.

The propagation speed depends on the frequency (dispersive waves), i.e.

4)(SB

kc f ρ

ωω

ω ⋅==

III) Room acoustics

III.1) Inside a room: reflection and transmission

III.1.a) Standing waves, eigenfrequencies

At normal incidence on a hard surface the pressure function is

tiekxptxp ω⋅= + )cos(ˆ2),(

and the particle velocity function

( ) )2/(sinˆ2),( πω −+ ⋅= tiekxvtxv

The Helmholtz equation: 0)()( 22

2

=+∂

∂ xpkxxp has, in the one dimensional case with two hard boundary

surfaces at x = 0 and x = L the solution

tfi nexLnBtxp ππ 2cos),( ⋅⎟

⎠

⎞⎜⎝

⎛=


where fn are the resonance frequencies Lnccf

nn 2

==λ

For the three dimensional case with six hard boundary surfaces the eigenfrequencies are

222

,, 2⎟⎠

⎞⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛+⎟

⎠

⎞⎜⎝

⎛=

Hn

Bn

Lncf zyx

nnn zyx

nx, ny and nz being indexes indicating the order mode in each direction of the room (L, B and H).

In transmission from one medium with wave impedance Z1 = ρ1c1 to another medium with wave impedance Z2 = ρ 2c2 the transmission factor t and reflection factor r are given, respectively, as:

12

2

1122

22 22ˆˆ

ZZZ

ccc

pp

ti

t

+=

+==

ρρρ

12

12

1122

1122

ˆˆ

ZZZZ

cccc

ppri

r

+

−=

+

−==

ρρρρ

while the transmission coefficient τ and reflection coefficient ρ is

( ) ( )212

122

1122

1122 44ZZZZ

cccc

II

i

t

+=

+

⋅==

ρρ

ρρτ

( ) ( )212

212

21122

21122

ZZZZ

cccc

II

i

r

+

−=

+

−==

ρρ

ρρρ

III.1.b) Eigenfrequencies types:

Axial modes 1D Tangential modes 2D Oblique modes 3D Figure 3 – Related geometry of eigenfrequencies

Source: Sengpielaudio

III.1.c) Reverberation time

Reverberation time can be estimated by using Sabine’s formula. Sabine based his empirical formula on the sound decay of 60 dB (1/1000 SPL) after the abruptly end of a test tone (shot, shooting, etc.).

𝑇"#(𝑓) = 0,16 𝑉

𝐴(𝑓)

with: V: volume of the room in cubic meters A: effective absorption area of the room in square meter: 𝐴(𝑓) = ∝0(1). 𝑆04

056 with Si : surface of every absorption element; αi(f): absorption coefficient of each element


III.2) Between two rooms: sound isolation and absorption

III.2.a) Airborne sound insulation

- Sound reduction index R:

- Measurement of sound reduction index R of a wall of surface S:

- Spectrum adaptation term: Li given in Fig.6

-Traffic spectrum adaptation term: Li given in Fig.6

- Combined R of a wall of surface S made by different elements (Si):

- Slot leakage:

- The mass law for a single leaf wall:

- The mass law for a double leaf wall:

- Coincidence frequency (critical frequency):

III.2.b) Impact sound insulation

- Measurement of step sound level L:

- Spectrum adaptation term:

NOTE: A prime (´) in the parameters R/L and its related ones means measurements performed in situ, whereas the same letter without the prime refers to measurements performed in the laboratory.

⎟⎟⎠

⎞⎜⎜⎝

⎛+−=

)(log10)()()(

fASfLfLfR recsend

⎟⎠

⎞⎜⎝

⎛+=10)(log10)()( fAfLfL recn

⎟⎠

⎞⎜⎝

⎛=⎟⎟⎠

⎞⎜⎜⎝

⎛

Π

Π=

τ1log10log10

t

iR

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎠

⎞⎜⎝

⎛−= ∑ −

i

fRi

iSS

fR 10/)(1101log10)(

⎟⎠

⎞⎜⎝

⎛+⋅−= −

SSfR sfR 10/)(10log10)(

hKBmcfc /

2

20 =

ʹ′ʹ′=

π

002log40)(

cmffRρπ ʹ′ʹ′

≈

002log20)(

cmffRρπ ʹ′ʹ′

≈

wnfiLn

l LC ,

2500

50

10/)(,250050, 1510log10 −−⎟

⎠

⎞⎜⎝

⎛= ∑−

wi

RiLi RC −⎟⎠

⎞⎜⎝

⎛−= ∑

=

−−

19

1

10/)(315050 10log10

wi

RiLitr RC −⎟⎟

⎠

⎞⎜⎜⎝

⎛−= ∑ − 10/)(10log10

hm ρ=''

)1(121 2

3

2 υυ −=

−=

EhIEB


III.2.c) ISO-REFERENCE CURVES Airborne sound isolation

Measurement of sound reduction index:

Figure 4 – Reference curve for air borne sound insulation (ISO 717-1)

Impact sound isolation

Measurement of step sound level Ln:

Figure 5 – Reference curve for step sound insulation (ISO 717-2)

⎟⎠

⎞⎜⎝

⎛+=10)(log10)()( fAfLfL recn

⎟⎟⎠

⎞⎜⎜⎝

⎛+−=

)(log10)()()(

fASfLfLfR recsend


Figure 6 – Spectra for calculating the spectrum adaptation terms for different frequency spectra (ISO 717-1)

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