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
  and 5.8 then lead to

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
  surface. (To get more information, see Section
  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
  lets the sound radiate well.

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.