Acústica em ambientes Fechados Chapter 5

8/10/2019 Acstica em ambientes Fechados Chapter 5

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Sound Propagation: An Impedance Based Approach Yang-Hann Kim 2010 John Wiley & Sons (Asia) Pte Ltd

Sound Propagation

An Impedance Based Approach

Acoustics in a Closed Space

Yang-Hann Kim

Chapter 5


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Outline

5.1 Introduction/Study Objectives

5.2 Acoustic Characteristics of a Closed Space

5.3 Theory for Acoustically Large Space (Sabines Theory)

5.4 Direct and Reverberant Field

5.5 Analysis Methods for a Closed Space 5.6 Characteristics of Sound in a Small Space

5.7 Duct Acoustics

5.8 Chapter Summary

5.9 Essentials of Acoustics in a Closed Space

2


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5.1 Introduction/Study Objectives

Depending on the distribution of the impedance, the sound propagation

differs significantly.

Sound propagation will be determined by the overall volume of the space

and the wall impedances which characterize the space.

The volume of space has to be considered with regard to the wavelength

of interest.

If the volume is fairly large, the waves would behave as if in a large space, and

would reach all possible places.

If the volume is small compared to the wavelength, then the wave would

appear to be everywhere in the space instantly.

3


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5.2 Acoustic Characteristics of a Closed Space

It is usually not plausible to express the sound that is likely to propagate in

a space of interest mathematically. The volume of a space of interest determines the major acoustical

characteristics of sound propagation in the space. Intuitively, a measure

has to be scaled with respect to the wavelength of interest.

For an acoustically large space, Sabine found that the reverberation period

represents the acoustic characteristics of the space well.

4

Acoustically small space The fluid particles in the space can be regarded

as if they are all moving with the same phase.

Acoustically large spaceThe acoustic wave travels in the space as a ray.

3

( )Vl>>

3( )Vl


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5.3 Theory for Acoustically Large Space (Sabines theory)

The spatial distribution of the acoustic

waves is not well dependent upon thelocation of the space. In other words, if

the pressure is measured at any position

in the space, it would be almost identical

to the mean value.

This phenomenon would be more likelyif more randomly distributed wall

impedance exists.

A diffuse field implies a space in which

the sound is likely to be equallydistributed irrespective of the position.

5

Figure 5.1 Illustration of sound propagation patterns in rooms. (These are the cases in which the size of room is acoustically

large; in other words, the typical dimension of the room is much larger than the typical wavelength)


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We first define acoustic energy density as

The sound energy at an arbitrary location is not expected to be perfectly

uniform. If considering an averaged sound energy density with respect to a

certain time and a small volume, expressed as

If a diffuse field is expressed using this measure, then the sound field

would satisfy the equality .

6

22

0 2

0

1 1( , ) .2 2

pr t uc

e rr

= +r

(5.1)

kinetic energy per unit volume potential energy induced by the

expansion and contraction of the unit

volume of the medium

1 1( , ) ( , ') '.

t T

t Vr t r t dVdt

T Ve e

+D

D=

D D r r

(5.2)

( , ) ( )r t te e=r


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7

The sound field before the sound wave is

reflected from the walls (direct sound field)is quite different compared to after the

wave has been reflected as sound from

the walls (reverberant sound field or

reflected sound field).

The sound energy of a reverberant fieldcan be determined using an equation that

expresses the conservation of sound

energy (Equation 2.36):

Figure 5.1 Illustration of sound propagation patterns in rooms. (These are the cases in which the size of room is acoustically

large; in other words, the typical dimension of the room is much larger than the typical wavelength)

0.d

I

dt

e+ =

v

(5.3)


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8

inP

outP

V

inP

outP

V

Figure 5.2 The energy balance by the power through the boundary (Vis volume, in is the incoming power, and outis the

outgoing power)

With the assumption that the volume does not include any sound source

bounded by the surface of the room as well as by the sound source, if

Equation 5.3 is integrated with regard to the volume, we have

.in out

V

ddV

dte = - (5.4)

( : power)


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It is possible to regard the sound in a closed space as being composed by

two sound fields: the first is direct and the second is reverberant.

If Equation 5.5 is applied when only a reverberant sound field exists, the

energy conservation equation for the reverberant sound is:

9

( ) ( ), , , .direct rev in direct out direct out revV V

ddV dV

dte e+ = - + (5.5)

, .rev out revV

ddV

dt

e = - (5.6)

loss induced by the

direct sound

loss induced by the

reverberant sounds


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Sabine found that the reverberant sound field created by the reflection

from the walls can be regarded as a diffuse sound field. Equation 5.6 can

be rewritten as

Sabine also noted that

To convert Equation 5.8 into a formula, a coefficient that has a time scale

must be used. Here, time scale is denoted as .Equation 5.7 and 5.8 then

lead to

10

, .rev

out rev

dV

dt

e= - (5.7)

, .out rev revVe (5.8)

,rev revd

dt

e e

t= - (5.9)

/

0( ) .t

rev t e te e -= (5.10)


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The concept of energy decay as expressed by 1/or the characteristic

decay time () is strongly related to the walls that form a closed space as

well as the items located in the space, as these items act as sound

absorbing elements.

They can be regarded as an open window that dissipates sound energy

from the closed space to outside.

concept of the area of an open window

11

1.

sAt (5.11)

area of the open window


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Intuitively, it is natural to postulate that a greater size would lead to a

longer time required to dissipate the acoustic energy in the room. Equation

5.11 could be rewritten in the proportional form:

Sabine successfully found a coefficient that can convert the proportional

form of Equation 5.12 into the following equality:

This equation essentially states that the sound in the room (strictlyspeaking, the sound in a diffuse field) can be represented by only one

parameter: the characteristic decay time .

12

.s

V

At (5.12)

4.

s

V

c At= (5.13)

( c: speed of sound)


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T60(the reverberation time or the reverberation period) is defined as thetime required to reduce the sound by 60dB. Applying this definition to

Equation 5.10 yields

Rearranging Equation 5.14 provides

Equations 5.13 and 5.15 result in

where the area of the open windowAscan be rewritten as

whereNis the number of elements that comprise the room of interest, nisthe absorption coefficient (which is the ratio of the absorbed sound powerto the incident sound power), and n is an index that represents eachmaterial.

13

60 /006

.10

Te

te e -= (5.14)

6

60 ln10 .T t= (5.15)

60

55.30.161 ,

s s

V VT

c A A= = (5.16)

1

,N

s n n

n

A Aa=

= (5.17)


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An expression that relates the reverberation period to the open window

area and the volume of the closed space is found to be

14

60

1

0.161 .N

n n

n

VT

Aa=

=

(5.18)

Table 5.1 Reverberation time of famous concert halls (RTocand RTunocare reverberation time when occupied and unoccupied)

(Adapted from L.L. Beranek, Concert Halls and Opera Houses: Music, Acoustics, and Architecture. Springer-Verlag, New York Inc.,

2004.)


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15

Figure 5.3Several famous concert halls: (a) Vienna Grosser Musikvereinsaal (Concert Halls and Opera Houses,2nd edition, 2004,

pp.174, Vienna Grosser Musikvereinssaal, L. Beranek, Springer-Verlag New York, Inc.: With kind permission of Springer

Science + Business Media); (b) Berlin Philharmonie (Concert Halls and Opera Houses, 2nd edition, 2004, pp. 298, Berlin

Philharmonie, L. Beranek, Springer-Verlag New York, Inc.: With kind permission of Springer Science + Business Media.); (c)

Tokyo Suntory Hall (Concert Halls and Opera Houses, 2nd edition, 2004, pp.408, Tokyo Suntory Hall, L. Beranek, Springer-

Verlag New York, Inc.: With kind permission of Springer Science + Business Media.); and (d) Boston Symphony Hall (Concert

Halls and Opera Houses, 2nd edition, 2004, pp.48, Boston Symphony Hall, L.L. Beranek, Springer-Verlag New York, Inc.: With

kind permission of Springer Science + Business Media.)

(a) (b)

(c) (d)


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A direct sound field refers to a field that does not have any reflected sound

waves.

If there is no reflection, then the total sound power through the surface at

r1orr2has to be conserved provided that there are no energy loss in the

medium.

16

Figure 5.4 Sound propagation in an open space (direct sound field)


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The sound that we hear is generally the sum of the direct and the

reverberant sound.

The direct sound would be dominant if a listener is close to the source;

however, reverberant sound would be more likely to dominate when the

listener is further away from the sources and close to the wall or walls.

18

Figure 5.5 Spatial variation of sound field with respect to the distance from source


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It is necessary to derive a certain measure or scale that can determine the

degree of participation of the direct and reverberant fields, or the direct and

reflected sound waves in a room.

For a steady state condition, Equation 5.5 can be rewritten as

The sound power generated by the sound sources is balanced by thesound power reflected due to the direct sound and due to what is induced

by the reverberant sound on the surface that we select.

How much is reflected is directly related to the absorption coefficient of the

walls. The average absorption coefficient of the walls is denoted

19

( ), , ,0 .in direct out direct out rev= P - P + P (5.22)

.st

AA

a= (5.23)

At : total area of the closed space

As: equivalent area of an open window


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out, direct, which is the power reflected from the walls by the incident sound

power (in, direct), are related as

Equations 5.22 and 5.24, the time rate change of the reverberant sound

energy, are related to the direct sound power, that is:

The sound power passing through the surface of a sphere with a radius of

rhas to be identical to what the sound source generates. This physical

balance can be mathematically written as

20

, , .out direct in direct aP = P (5.24)

, ,

(1 ) .out rev in direct

aP = - P (5.25)

2

,4 .

in direct r I rpP = (5.26)

(Ir: intensity at distance rfrom the source)


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A similar relationship can be obtained for the reverberant sound. The

reverberant sound energy can be regarded to be distributed in a closed

space, which can be envisaged as the space surrounded by the surfaces

of discontinuities that have various wall impedances.

The total energy density comprising the direct and reverberant sound can

therefore be written as

22

, ,rev

out rev

Ve

tP = (5.31)

( ) ,4

1 .rev in direct

scA

e a= - P (5.32)

, 2

2

2

0

( )

161 (1 )

4

1 .

direct rev

in direct

s

direct

r

r

r c A

r

r

e e e

pa

p

e

= +

P = + -

= +

(5.33)


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The new parameter used in Equation 5.33 is

This expresses the radius at which it is likely that the direct and reverberant

sound participate equally.

23

016 (1 )

.16 (1 )

s

t

Ar

A

p a

a

p a

= -

=-

(5.34, 35)

Figure 5.6 Total energy density (the sum of direct and reverberant sound energy density; r0, an intersection point, is the radius

of reverberation)


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5.5 Analysis Methods for a Closed Space

Sound waves in a closed space can be regarded as the solutions that

satisfy the boundary conditions of the closed space and the governing

equation. There are two distinct approaches to acquire these solutions.The first is to obtain the solutions in the time domain, and the second is to

acquire them in the frequency domain.

In the frequency domain, it describes the sound waves in terms of the

superposition of mode shapes. These approaches can be implemented by

the following three methods.

The first regards the sound field of interest as the superposition of natural or

normal modes that satisfy the boundary condition and the governing equation.

The second method describes the sound field using singular functions that

satisfy the governing equation.

The latter method describes the sound field using acoustic rays, and is often

referred to as ray acoustics. It assumes that the wavelength of interest is very

much smaller than the characteristic length of the surface of reflection.

24


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The latter method cannot be applied if the walls are no longer considered

as locally reacting surfaces, or if the acoustic wavelength fails to meet the

basic assumption of a locally reacting surface. (To get more information,see Section 3.9.1 from textbook.)

25

Figure 5.7 Conceptual example of ray acoustics


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A sound field that falls into a given frequency within the closed space canbe expressed by superposition of unique modes that meet the boundary

condition and the governing equation, as

where subscripts l, m, n refer to the respective orders of modes thatcorrespond to individual coordinate directions of the Cartesian coordinate

system.

Let us consider a cube-shaped space in which sound can potentially begenerated. Under the rigid wall boundary condition,

whereLx,Ly,Lzrepresent the lengths in each direction.

26

(5.36), , 0

( ) ( ),lmn lmn

l m n

r r

=

= r r

P a

(5.37)( , , ) cos cos cos ,lmnx y z

l x m y n z x y z

L L L

p p p=


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In the case of relatively simple single dimension (i.e., a square tube with of length

L),

A constant that represents the level of contribution that each unique mode makes

to the entire sound field is called modal coefficient. To look at the behavior of

modal coefficients in detail, let us observe sound fields that are radiated from amonopole sound source placed in a three-dimensional space. If the excitation is

generated using a monopole sound source at the location of ,

27

(5.38)( ) cos ,ll x

xL

p=

(5.39)0

( ) ( ).l ll

x x

=

= P a

(5.40)

*

0

2 2

*

0

( )4

( )

4 ( ) ( ),

lmnlmn

lmn lmn

lmn lmn

r

V k k

k r

p

p

-=

L -

= -

r

r

Sa

Sh

(5.41)*

0

2 2, , 0

( ) ( )4( ) .

( )

lmn lmn

l m n lmn lmn

r rr

V k

p

=

-=

L -

r r

r SP

k

0rr

(S: monopole amplitude)


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Figure 5.8 depicts some individual modes contributing to the entire sound field,

with each extent in a cubic room described by a given volume.

28

Figure 5.8 Sound pressure generated in a cubic space (0.8 0.6 0.1m3); contributions by individual unique modes


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Considerhlmn(k), a function that represents the frequency characteristics of

a space. If the walls of a cubic room have the rigid body condition, k2lmnhas a real-number value ( ) and is expressed as

If the excitation frequency (f = kc/2) of Equation 5.40 is the same as orsimilar to

then the particular mode contribution (almn) will be infinite or significantly

amplified. This frequency characteristic function, like the transfer function

of a 1-DOF vibratory system, serves to adjust the extent of amplification for

each mode depending on excitation frequency.

29

(5.42)

22 2

2 2 2 2( ) ( ) ( ) .lmn x y z

x y z

l m nk k k k

L L L

p p p = + + = + +

(5.43),2

lmn lmn

cf k

p=

2lmnk


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The total number of participating modes and modal density increase

dramatically as the frequency increases. In other words, a larger number

of modes are needed to express sound fields as the frequency becomeshigher.

30

Figure 5.9 Number of modes and modal density in a closed rectangular space (L=5.8m; W=4.5m;H=2.5m) with the walls being

rigid bodies


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5.6 Characteristics of Sound in a Small Space

An acoustically small space is one whose representative length or size is

small relative to wavelength. An acoustically small space can generally be

regarded as a vibratory system. A prime example of this is the Helmholtzresonator.

31

Figure 5.10 (a) Shape of simple resonator and equivalent vibratory system; (b) various types of resonator and conceptual

samples; and (c) meaning of acoustic compliance


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If the wavelength is considerably longer than the size of the resonators

body and neck, the movements of fluid in the neck or the body will have

almost identical phase.

From Figure 5.10(a), the pressure change (pin) per unit time will reduce the

volume change in the cavity of the resonator. If the pressure changes and

volume are small enough to be linearized,

Using acoustic compliance CA (which represents the volume change

induced by unit sound pressure) as a proportional constant,

When we have a large CA, the resonator undergoes a massive volume

change.

Equation 5.45 only highlights the correlation between pressure and volume.

32

(5.44)dp dV

dt dt

-

(5.45).Adp dV

Cdt dt

= -

(wherep = pin).


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First of all, the volume change with respect to time in the cavity can be

written as

where u(t)is the velocity of fluid at the neck and Ais the cross-sectional

area of the neck. We can rewrite Equation 5.44 as

Now consider the fluid motion at the neck. The balance between sound

pressure acting on the fluid at the neck and the momentum of the fluid can

be formulated as

33

(5.46)( ),dV Au tdt

= -

(5.47)1

( ).A

dpAu t

dt C

=

(5.48)0( ) ,out indu

p p A Aldt

r- =

(5.49)0 .out indup p ldt

r- =

(l : length of the neck or effective

length of the neck, to be more precise)


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Substituting Equation 5.49 into Equation 5.47, we can obtain

As noted before,pin=p; Equation 5.50 can be rewritten as

From Equation 5.51, the resonance radial frequency (n) can be obtained

as

34

(5.50)

2

0 2.Aout in

C d pp p l

A dtr

= +

(5.51)

2

02 .A outl d pp C pA dt

r

+ =

(5.52)2

0

1.n

AlCA

w

r

=


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35

(5.53)0 0( )( ),V V Vr r r= + D + D

(5.54)0 0,V Vr rD + D =

(5.55)0

,dV V d

dt dt

r

r= -

(5.56)20

.dV V dp

dt c dt r= -

(5.57)20

,AV

Ccr

\ =

Neglecting higher

order terms

By the state

equation,

dp/d]s=c2

(5.58)0 .Al

mA

r=

Equation 5.52 ( ) can be written as

2

0

1

n

AlC

Aw r

=

(5.59)2 1 .

n

A Am C

w =

(mA: acoustic inertance)

5 6 Ch i i f S d i S ll S


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The resonance frequency increases as the area of the neck becomeslarger, but falls as the volume of the cavity becomes larger.

This is because the wavelength associated with the resonance frequencyis very long relative to the size of resonator. This causes the entire fluid atthe neck to move in the same phase and the volume in the cavity to

sustain the entire fluid at the neck as a kind of spring element. If a diameter of the neck is considerably smaller than the wavelength, the

effective length (including end correction) of the neck can be expresseddepending on whether it has a flange or not :

To design the resonance frequency of a resonator precisely, the endcorrection factor should be taken into account.

36

(5.60).nA

clV

w =

(5.61)0.85 (with a flange),

0.6 (without a flange).

l l a

l l a

= +

= +

l : length of the neck

a: radius of cross-section

5 6 Ch t i ti f S d i S ll S


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Neck and cavity are also basic components that consist of the geometrical

shape of a resonator.

In particular, impedance of a resonator can also be expressed as

where Zr represents radiation impedance, and Zneck and Zcavity are

impedances for the neck and the cavity, respectively.

In particular, the reactance (imaginary part) of the impedance mainlydetermines resonance frequency:

37

(5.62),HR r neck cavity= + +Z Z Z Z

(5.63){ }2

2

0 0 0

8Im ' .

3HR

a al c

j V

pr w r w r

p w= + +Z

can be obtained by open endcorrection

can be derived under theassumption that the pressure in

the cavity is maintained uniformly.

5 6 Ch t i ti f S d i S ll S


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From observations in the book, which is omitted in this presentation, we

can find that the geometry (the shape, location, and size of the neck and

cavity) mainly affects the performance of a resonator.

In addition, the shape of the neck is one of the main attributes which

changes the absorption characteristics of resonator impedance. By

changing the shape of the neck, we can therefore improve the absorption

performance of a resonator.

The necks can be any shape depending on practical requirements other

than acoustical requirements. The shape is not very important if its spatial

variation is considerably smaller than the wavelength of interest, such as

the case of the neck of the Helmholtz resonator.

38

5 6 Characteristics of Sound in a Small Space


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39

Figure 5.12 A schematic model of a round resonator that has a gradually changing neck.Lis the axial length of the cavity, lis

the length of the neck,Ris the radius of the cavity, riis the inlet radius of the neck, and rois the outlet radius of the neck

We therefore consider a horn-shaped neck. The horn causes the

impedance of propagating sound from a small source to gradually change

to that of the impedance at the end of the horn, which lets the soundradiate well.



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40

Suppose that we have a plane wave propagating in the neck, then the

wave is governed by Websters horn equation (see Section 5.7 for details).

This can be written as

whereB = (mx+ri)2, and m (= (ro ri)/l) is the slope of the neck. ri, ro, and l

are depicted in Figure 5.12. The solution is then

where a1and a2are the magnitude of the incident and reflected wave,

respectively. Particle velocity can be obtained by linerarized Eulers

equation, that is

(5.77)21 0,

d dB k

B dx dx

+ =

pp

(5.78)1 2 ,( )

i imx r mx r jk jkm m

i

m a e a ek mx r

+ +-

= + + p

(5.79)

2

1 220

1 2

( )

( ) ( ) .( )

i i

i i

mx r mx r jk jk

m m

i

mx r mx r jk jk

m m

i

j m

a e a eck k mx r

ma jk e a jk e

k mx r

r

+ +-

+ +-

= - + +

+ + -

+

u


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42/66



We can rewrite the impedance at the inlet of the neck (Zi) as

If we assume that the wavelength of interest is much larger than the length

of the neck, tan kltends to kl. Equation 5.83 can then be simplified as

We now examine the impedance at x = l(Zo). If fluid around the neck is

moved about , the pressure change in the cavity can be expressed as

42

(5.83){ }

{ }

00

0

2

0

( ) tan.

( )

( ) tan

o o o oi i

o i o i o

o o o

kr m j c kr kl c kr

c kr kr jmk r r

c kr m j kr kr m kl

rr r

r

- - = - -

+ + +

Z ZZ

Z

Z

(5.84)2

00 2

0

.i o i oio i o o

r j ckr r l c

cr jkr r l

rr

r

+=

+

ZZ

Z

(5.85)2 20 0 .x l oc B c r V jkV

r d r p== = up



43/66



The impedance without regard to energy dissipation (resistance) at x = l

can be written as

Substituting Equation 5.86 into Equation 5.84, the impedance at the inlet

of the neck (Zi) can be rewritten as

where ldenotes neck length that generally includes end correction.

Therefore, the resonance frequency that sets reactance to zero can be

obtained as

43

(5.86)

2

0 .o

o

x l

rcjkVpr

=

= =pZu

(5.87)

22

0 22

,

oi i o

i

oo i o

rr jkr r l

jkVcr

r r r l V

p

rp

+

=+

Z

(5.88).2

i on

rrcf

lV

p

p=



44/66



A very common misunderstanding of a resonator is that it reduces sound

by absorption. In reality, an abrupt impedance mismatch takes place at the

resonance frequency of a resonator when installed on a noisetransmission path (e.g. automotive engine suction/exhaustion units).

This impedance mismatch reflects incident waves, and transmitted noise is

finally reduced. In other words, it acts like an invisible wall.

On the other hand, the amount of sound absorbed by a resonator is

governed by its dissipation properties. The energy dissipation occurs

primarily around the neck of the resonator, which is induced by friction

between the fluid moving around the resonators neck and the confronting

surface of the neck. The amount of dissipated energy, however, is

generally much smaller than what is reflected by an impedance mismatch.

44

5.7 Duct Acoustics


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5.7 Duct Acoustics

A duct is a space where the length of one direction is significantly greater

than the cross-sectional direction. The sound propagation within a duct

can be primarily expressed with respect to a single direction or coordinate. In the case of an infinite square duct as in Figure 5.13,

45

(5.89)( )

, 0

( , , , ) ( , ) ,zj t k z

mn

m n

x y z t x y e w

- -

=

=p P

(5.90)( )

, 0( , , , ) cos( )cos( ) .

zj t k z

mn x y

m nx y z t k x k y e

w

- -

== p a

Figure 5.13 Three directions in which compressive fluids can move within an infinite duct


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5.7 Duct Acoustics


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5.7 Duct Acoustics

kz, the propagation constant in the zdirection, can be a real or imaginary

number.

If it is a real number, it is propagated in the positivezdirection. If it is an imaginary number, the magnitude of sound waves attenuates

exponentially as it progresses toward the propagation direction. (evanescent

wave)

In the wave number domain, only those modes whose wave numbers in

the cross-sectional direction are lower than k=/ccan propagate withoutbeing attenuated, that is

The duct serves as a sort of low pass filter with the cut-off wave number ofk.

47

(5.95)2 2 .x yxyk k k= +

5.7 Duct Acoustics


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If the cross-sectional area of a duct changes dramatically, this also

significantly alters the way that a wave is propagated.

48

Figure 5.14 Propagation of waves in a wave guide with square section

wave blocking

wave tunneling

5.7 Duct Acoustics


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We now examine Equation 5.94 for a special case: the length in each

sectional direction being shorter than half a wavelength. In this case, all

modes in the sectional direction, excluding one where (m,n)=(0,0), willcontinue to be attenuated exponentially while being propagated.

The only mode that is propagated without attenuation, (0,0), is a plane

wave whose sound pressure remains constant in the sectional direction

and whose wave number in the zdirection is k. The wave in this case can

be expressed as

This implies that, if the characteristic length of a section is considerably

smaller than the wavelength, the wave of a duct may be considered a one-

dimensional problem.

49

(5.96)( )00( , ) .

j t kzz t a e w- -=p

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Even in the absence of higher-order modes, massive changes takes place

in the propagation of waves when the section experiences dramatic

change.

50

Figure 5.15 Reflection and transmission of waves in simple divergent tube (piis an incident wave; pra reflected wave; pstand

psrwaves transmitted into and reflected by a silencer; andpta transmitted wave)

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These waves should meet the continuity condition at the planes whose

sections are expanded (z=0) and contracted (z=L), respectively.

51

(5.97)( ) ( )( , ) , ( , ) ,j t kz j t kzi rz t e z t ew w- - - += =i rp P p P

(5.98)( ) ( )( , ) , ( , ) ,j t kz j t kzsi srz t e z t ew w- - - += =

si sr p P p P

(5.99)( )( , ) .j t kztz t e

w- -=tp P

Figure 5.15(b) Reflection and transmission of waves in simple divergent tube (piis an incident wave; pra reflected wave; pstandpsrwaves transmitted into and reflected by a silencer; andpta transmitted wave)

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Sound Propagation: An Impedance Based Approach Yang-Hann Kim 2010 John Wiley & Sons (Asia) Pte Ltd52

Atz=0, the pressure and the velocity need to be continuous,

where S1and S2refer to the cross-sectional areas of the two tubes before

and after expansion.

The continuity condition atz=Lcan also be written as

(5.100),i r si sr

+ = +P P P P

(5.101)10 0 20 0 0 0

.i si sr rP P

S Sc c c c

r rr r r r

- = -

P P

(5.102),jkL jkL jkL

si sr te e e-+ =P P P

(5.103)0 2 0 10 0 0

.jkL jkL jkLsi sr tS e e S e

c c cr r

r r r- - =

P P P

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On this basis, the magnitude ratio of transmitted waves against incident

waves and transmission loss (TL), which indicates the power of incident

waves being lost while passing through a silencer, are derived:

The amounts of transmission and reflection are related to the sectional

area and frequency of the two tubes.

53

(5.104)1 2

2 1

,

cos sin2

jkL

t

i

e

S SjkL kL

S S

t-

= =

- +

P

P

(5.105)

10 2

2

21 210

2 1

1

10log

110 log 1 sin .

4

TL

S SkL

S S

t=

= + -

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In Equation 5.105, transmission loss reaches its peak when sin kLhas the

highest value of 1; transmission losses becomes zero, which is the

minimum value, when sin kLis zero.

Figure 5.16 Comparison of transmission losses in simple divergent tube by wavelength (a) maximum transmission loss (L=/4);

and (b) minimum transmission loss (L=/2)

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Figure 5.17 Transmission loss by frequency and cross-sectional area ratio of simple divergent tube

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A similar phenomenon occurs in a pipe of shape is illustrated in Figure

5.18. In this case, the length of the tube needs to be understood as the

length of an effective tube, as described in Equation 5.60.

Using an expansion chamber-based silencer, certain frequency elements

in the noise of your choice can be reduced dramatically by adjusting the

length of the expansion chamber.

56

Figure 5.18 Impedance mismatch due to duct resonance in 1/4 wavelength tube

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A silencer that reduces noise using an impedance mismatch generated by

a sharp change in shape is referred to as a reactive silencer or reactive

muffler.

One that tries to reduce noise using a perforated tube or sound-absorbing

material is referred to as a dissipative silencer.

In general, a dissipative silencer is known to be more effective for

controlling high-frequency noise and absorbing noise better at relatively

wider bandwidths.

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The forces acting on the fluid betweenxandx+xand its motion will obey

the conservation of momentum, that is

where Srepresents the cross-sectional area of the horn, uis the particle

velocity in the xdirection, is volume density of the fluid, and Sis the

projected area of the area atx+xto the area atx.

By neglecting higher-order terms and using the assumptions in Section 2.2,

wherepis the sound pressure,0is the static volume density.

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(5.106)( ) ( ) ( )2

,xx x x x

dupS pS p S S xdt

rD+D +

- + D = D

(5.107)0 ,x t

r

- =

p u

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The conservation of mass can be expressed as

As x0,

By differentiating the right-hand side with respect touand S,

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(5.108)( ) ( ) ( ) .x x xS x uS uS t

r r r+D

D = -

(5.109)( ) .S uSt x

rr

= -

(5.110)0 01

.u S

u

t x S x

rr r

= - -

Figure 5.20Conversation of mass in an infinitesimal fluid element in an acoustic horn. Sdenotes the cross-sectional area of the

horn, uis the particle velocity in thexdirection, andis volume density of the fluid

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With Equations 5.107 and 5.110,

the state equation can, finally, provide us with Websters horn equation,

that is,

As a solution,

where S0is the area at the left end of the horn, and is a flare constantwhich expresses exponential increase asxbecomes larger.

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(5.107)0

,

x t

r

- =

p u

(5.110)0 01

,u S

ut x S x

rr r

= - -

(5.111)

2 2

2 2 2

1 10.

p S p p

x S x x c t

+ - =

(5.112)0 ,

xS S ea=

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Websters horn equation can then be written as

The solution can be given by

We can see the right-going waves are amplified as the waves propagate to

the mouth.

We can also obtain the phase velocity ( ) for the horn as

which varies with frequency.

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(5.113)2 2

2 2 2

1

0.

p p p

x x c ta

+ - =

(5.118)

2

2( , ) exp 1 .2

x

c

c

x t Pe j t k xk

aa

w-

= - -

p

(Details can be found in the book.)

(5.119)2,

12

cc

c

f

aw

=

-

cf

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There is a certain frequency which causes the phase velocity to be infinite,

referred to as the cut-off frequency in the case of wave guides, that is

63

(5.120).4

cutoff cf ap=

Figure 5.21Phase velocity (c) and cutoff frequency(fcutoff) for an exponential acoustic horn, where cis the speed of sound in air(343 m/sat 20C)

f

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Suppose that we have a velocity source at the throat (x = 0) with u0e-jt; we

can easily obtain the pressure from the principle of momentum as

By inserting the positive values of Equation 5.117 and Equation 5.120, we

can obtain the radiation impedance of the exponential horn as

When we have determined the cut-off frequency, the radiation impedance

has a purely imaginary value (-j0c). This means that the waves in the horn

cannot propagate well.

64

(5.121)00.p u

k

wr=

(5.122)0

2.

1

r

cutoff cutoff

cf f

jf f

r=

- +

Z

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Figure 5.22Driving point impedance of an exponential acoustic horn. The solid line denotes the real part of impedance

(resistance), and the dashed line represents the imaginary part of the impedance (reactance)

As frequency increases, the resistance term approaches the characteristic

impedance of the medium (0c) but the reactance term tends to 0. Note

that the resistance term is 0 below the cut-off frequency.


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Acústica em ambientes Fechados Chapter 5

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