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Acoustics in a Closed Space

Chapter 5

- Yang-Hann Kim

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

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.

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.

Acoustically small space The fluid particles in the space can be regarded as if they are all moving with the same phase.

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

5.3 Theory for Acoustically Large Space (Sabines

theory)

- The spatial distribution of the acoustic waves is

not well dependent upon the location 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 likely if more

randomly distributed wall impedance exists. - A diffuse field implies a space in which the

sound is likely to be equally distributed

irrespective of the position.

5.3 Theory for Acoustically Large Space (Sabines

theory)

- 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 .

5.3 Theory for Acoustically Large Space (Sabines

theory)

- 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 field can be

determined using an equation that expresses the

conservation of sound energy (Equation 2.36)

5.3 Theory for Acoustically Large Space (Sabines

theory)

- 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

5.3 Theory for Acoustically Large Space (Sabines

theory)

- 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

loss induced by the direct sound

loss induced by the reverberant sounds

5.3 Theory for Acoustically Large Space (Sabines

theory)

- 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 t. Equation 5.7

and 5.8 then lead to

5.3 Theory for Acoustically Large Space (Sabines

theory)

- The concept of energy decay as expressed by 1/t

or the characteristic decay time (t) 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

area of the open window

5.3 Theory for Acoustically Large Space (Sabines

theory)

- 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 (strictly speaking, the sound in a

diffuse field) can be represented by only one

parameter the characteristic decay time t.

5.3 Theory for Acoustically Large Space (Sabines

theory)

- T60 (the reverberation time or the reverberation

period) is defined as the time 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 window As can be

rewritten as - where N is the number of elements that comprise

the room of interest, an is the absorption

coefficient (which is the ratio of the absorbed

sound power to the incident sound power), and n

is an index that represents each material.

5.3 Theory for Acoustically Large Space (Sabines

theory)

- An expression that relates the reverberation

period to the open window area and the volume of

the closed space is found to be

5.3 Theory for Acoustically Large Space (Sabines

theory)

5.4 Direct and Reverberant Field

- 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 r1 or r2 has to be

conserved provided that there are no energy loss

in the medium.

5.4 Direct and Reverberant Field

- If r2 2r1, then the intensity ratio is
- This indicates that the sound intensity will be

reduced by 6dB. - For a diffuse field, the acoustic properties are

independent with respect to the location. The

solution of Equation 5.21 is therefore 0dB.

5.4 Direct and Reverberant Field

- 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.

5.4 Direct and Reverberant Field

- 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 the sound 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

At total area of the closed space As

equivalent area of an open window

5.4 Direct and Reverberant Field

- ?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 r has to be identical to

what the sound source generates. This physical

balance can be mathematically written as

5.4 Direct and Reverberant Field

- If a monopole source and the radius are widely

spaced relative to the wavelength of interest,

the acoustic intensity and power can be written

as - Similarly, the acoustic energy density of the

direct sound can be obtained as - Equations 5.28 and 5.29 provide the following

relationship, that is,

5.4 Direct and Reverberant Field

- 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

5.4 Direct and Reverberant Field

- 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.

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.

5.5 Analysis Methods for a Closed Space

- 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.)

5.5 Analysis Methods for a Closed Space

- A sound field that falls into a given frequency

within the closed space can be 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 that correspond to individual

coordinate directions of the Cartesian coordinate

system. - Let us consider a cube-shaped space in which

sound can potentially be generated. Under the

rigid wall boundary condition, - where Lx, Ly, Lz represent the lengths in each

direction.

5.5 Analysis Methods for a Closed Space

- 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 a monopole sound source placed in a

three-dimensional space. If the excitation is

generated using a monopole sound source at the

location of ,

(S monopole amplitude)

5.5 Analysis Methods for a Closed Space

- 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.

5.5 Analysis Methods for a Closed Space

- Consider hlmn(k), a function that represents the

frequency characteristics of a space. If the

walls of a cubic room have the rigid body

condition, k2lmn has a real-number value ( )

and is expressed as - If the excitation frequency (f kc/2p) of

Equation 5.40 is the same as or similar 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.

5.5 Analysis Methods for a Closed Space

- 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 becomes higher.

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 Helmholtz resonator.

5.6 Characteristics of Sound in a Small Space

- 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.

(where p pin).

5.6 Characteristics of Sound in a Small Space

- 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 A is 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

(l length of the neck or effective length of

the neck, to be more precise)

5.6 Characteristics of Sound in a Small Space

- 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

5.6 Characteristics of Sound in a Small Space

Neglecting higher order terms

By the state equation, dp/d?sc2

(mA acoustic inertance)

- Equation 5.52 ( ) can be written

as

5.6 Characteristics of Sound in a Small Space

- The resonance frequency increases as the area of

the neck becomes larger, but falls as the volume

of the cavity becomes larger. - This is because the wavelength associated with

the resonance frequency is very long relative to

the size of resonator. This causes the entire

fluid at the 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

expressed depending on whether it has a flange or

not - To design the resonance frequency of a resonator

precisely, the end correction factor should be

taken into account.

l length of the neck a radius of cross-section

5.6 Characteristics of Sound in a Small Space

- 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 mainly determines resonance

frequency

can be obtained by open end correction

can be derived under the assumption that the

pressure in the cavity is maintained uniformly.

5.6 Characteristics of Sound in a Small Space

- 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.

5.6 Characteristics of Sound in a Small Space

- 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 sound radiate well.

5.6 Characteristics of Sound in a Small Space

- 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 - where B p(mxri)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 a1 and a2 are the magnitude of the

incident and reflected wave, respectively.

Particle velocity can be obtained by linerarized

Eulers equation, that is

5.6 Characteristics of Sound in a Small Space

- Then the impedance (Z0) at xl can be written as
- Writing Equation 5.80 with respect to a2/a1

yields - In addition, the impedance (Zi) at x 0 also can

be obtained as

5.6 Characteristics of Sound in a Small Space

- 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 kl

tends 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 d, the

pressure change in the cavity can be expressed as

5.6 Characteristics of Sound in a Small Space

- 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 l denotes neck length that generally

includes end correction. - Therefore, the resonance frequency that sets

reactance to zero can be obtained as

5.6 Characteristics of Sound in a Small Space

- 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 noise transmission 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.

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,

5.7 Duct Acoustics

- As noted on Figure 5.13,
- In terms of wavelength instead of wave number,

Equation 5.93 can be rewritten as - Equations 5.93 and 5.94 delineate the dispersion

properties of a sound wave being propagated

within a duct.

5.7 Duct Acoustics

- kz, the propagation constant in the z direction,

can be a real or imaginary number. - If it is a real number, it is propagated in the

positive z direction. - 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?/c can propagate without being

attenuated, that is - The duct serves as a sort of low pass filter with

the cut-off wave number of k.

5.7 Duct Acoustics

- If the cross-sectional area of a duct changes

dramatically, this also significantly alters the

way that a wave is propagated.

wave blocking

wave tunneling

5.7 Duct Acoustics

- 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), will continue 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 z

direction 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.

5.7 Duct Acoustics

- Even in the absence of higher-order modes,

massive changes takes place in the propagation of

waves when the section experiences dramatic

change.

5.7 Duct Acoustics

- These waves should meet the continuity condition

at the planes whose sections are expanded (z0)

and contracted (zL), respectively.

5.7 Duct Acoustics

- At z0, the pressure and the velocity need to be

continuous, - where S1 and S2 refer to the cross-sectional

areas of the two tubes before and after

expansion. - The continuity condition at zL can also be

written as

5.7 Duct Acoustics

- 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.

5.7 Duct Acoustics

- In Equation 5.105, transmission loss reaches its

peak when sin kL has the highest value of 1

transmission losses becomes zero, which is the

minimum value, when sin kL is zero.

5.7 Duct Acoustics

5.7 Duct Acoustics

- 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.

5.7 Duct Acoustics

- 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.

5.7 Duct Acoustics

- As another special case, lets look at an

acoustic horn. Websters horn equation governs

fluid particles in the acoustic horn. The

acoustic horn can reduced reflection waves at the

right end by slowly changing cross-section.

5.7 Duct Acoustics

- The forces acting on the fluid between x and x?x

and its motion will obey the conservation of

momentum, that is - where S represents the cross-sectional area of

the horn, u is the particle velocity in the x

direction, ? is volume density of the fluid, and

?S is the projected area of the area at x?x to

the area at x. - By neglecting higher-order terms and using the

assumptions in Section 2.2, - where p is the sound pressure, ?0 is the static

volume density.

5.7 Duct Acoustics

- The conservation of mass can be expressed as
- As ?x?0,
- By differentiating the right-hand side with

respect to ?u and S,

5.7 Duct Acoustics

- With Equations 5.107 and 5.110,
- the state equation can, finally, provide us with

Websters horn equation, that is, - As a solution,
- where S0 is the area at the left end of the

horn, and a is a flare constant which expresses

exponential increase as x becomes larger.

5.7 Duct Acoustics

- 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.

(Details can be found in the book.)

5.7 Duct Acoustics

- 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

5.7 Duct Acoustics

- Suppose that we have a velocity source at the

throat (x 0) with u0e-j?t 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 (-j?0c). This means that the waves in the

horn cannot propagate well.

5.7 Duct Acoustics

- 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.

5.8 Chapter Summary

- The characteristics of sound generated in a

relatively large space compared to the wavelength

differ significantly from those of sound created

in a smaller space. The sound generated in the

former case can be considered to have direct and

reverberant sound field. - Reverberation time, suggested by Sabine,

represents the properties of acoustically large

space. - In contrast to a larger space, the resonator

properties of a small space are more similar to

those of a 1-DOF vibratory system than its

propagation properties, and this phenomenon can

be utilized in controlling a variety of noises. - If , as in the case of a duct, the characteristic

length of its section is shorter than a

wavelength and the length of its propagation

direction is considerably longer than that of a

wavelength, unique phenomena such as wave

blocking and tunneling can be observed.