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Acoustics in a Closed Space

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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
    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.
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