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Title: Acoustics in a Closed Space

1
Acoustics in a Closed Space
Chapter 5
• Yang-Hann Kim

2
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

3
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.

4
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
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.

6
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 .

7
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)

8
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

9
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
10
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

11
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
12
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.

13
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.

14
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

15
5.3 Theory for Acoustically Large Space (Sabines
theory)
16
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.

17
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.

18
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.

19
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
20
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

21
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,

22
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

23
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.

24
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.

25
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
3.9.1 from textbook.)

26
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.

27
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)
28
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.

29
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.

30
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.

31
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.

32
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).
33
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)
34
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

35
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

36
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
37
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.
38
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.

39
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

40
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

41
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

42
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

43
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

44
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.

45
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,

46
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.

47
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.

48
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
49
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.

50
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.

51
5.7 Duct Acoustics
• These waves should meet the continuity condition
at the planes whose sections are expanded (z0)
and contracted (zL), respectively.

52
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

53
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.

54
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.

55
5.7 Duct Acoustics
56
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.

57
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.

58
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.

59
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.

60
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,

61
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.

62
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.)
63
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

64
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.

65
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.

66
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.