2 acoustic waves, standing waves, room modes

2

Acoustic Waves, Standing Waves, Room Modes

The acoustic waves represent vibrations in fluid media that are progressingboth with time and over space. The simplest form of acoustic waves in fluidmedia is a single harmonic wave, governed by the wave equation in Helmholtzform [17]

∇2p+ β2 p = 0. (2.1)

More complex waves can often be decomposed into harmonic waves withFourier analysis. This chapter briefly introduces single-harmonic waves.

2.1 Forward- and Backward-Propagating Waves

The wave equation provides solutions that are of temporal and spatial na-ture. The most fundamental wave solution expressed in Cartesian coordinatesrepresents a single-harmonic (single-tone) pressure wave

p(t) = p cos(ω t− β x) = pRe[e j(ω t−β x)

]= p cos

[2π

(t

T− x

λ

)], (2.2)

where p is the amplitude of the sinusoidal sound pressure. Angular frequency,ω = 2π f , contains frequency, f , in Hertz [Hz]. And quantity, T , is period ofthe sinusoidal pressure wave in second [s]. β is the propagation coefficient,β = ω/c = 2π/λ with c being speed of sound in fluid media. λ is wavelength

λ =c

f= cT =

2π

β. (2.3)

24 2 Acoustic Waves, Standing Waves, Room Modes

Equation (2.2) represents a forward-propagating wave component, denoted as

p→(t) = pRe[e j(ω t−β x)

]= p cos

[2π

(t

T− x

λ

)], (2.4)

with corresponding particle velocity

v→(t) =p

ρ cRe[e j(ω t−β x)

]=

p

ρ ccos

[2π

(t

T− x

λ

)], (2.5)

where ρ c is the characteristic resistance of the fluid media, and ρ is themedium density.

The backward-propagating wave component can be written as

p←(t) = pRe[e j(ω t+β x)

]= p cos

[2π

(t

T+x

λ

)], (2.6)

with corresponding particle velocity

v←(t) =−pρ c

Re[e j(ω t+β x)

]=

−pρ c

cos

[2π

(t

T+x

λ

)], (2.7)

2.2 Plane Waves Reflection and Standing Waves

The forward-propagating wave component represents a plane wave propaga-tion. When it perpendicularly impinges on a plane wall of infinite size with acomplex-valued normal-incident reflection coefficient

R = |R | e jϕR , (2.8)

the reflected wave component becomes

pr←(t) = p |R | cos[2π

(t

T+x

λ

)+ ϕR

]= pRe

[(|R | e jϕR

)e j(ω t+β x)

], (2.9)

The linear superposition of the incident plane wave in Equation (2.4) andreflected wave component in Equation (2.9) yields the standing wave pressure

ps= p→(t) + p

r←(t)

= p[e j(ω t−β x) + |R | e j(ω t+β x+ϕR)

](2.10)

in front of the (infinitely) extended plane wall. The magnitude of this pressurestanding wave represents the standing wave envelop

2.2 Plane Waves Reflection and Standing Waves 25

| ps| = p

√1 + |R |2 + 2|R | cos

(4πx

λ+ ϕR

)(2.11)

Note that the envelop of the standing wave in Equation (2.11) is a functionof x only. The ’cosine’ function inside the square-root contains a factor of 4π.Comparison between Equation (2.11) with Equation (2.5) and Equation (2.9)indicates that the envelop has a spatial frequency being twice of the incidentand reflected waves [see also Figure (2.1)]

Fig. 2.1. Flow diagram of pseudo-script for the standing waves.

MATLAB note: Implementing the incident, reflected and standing wavesin terms of sound pressure requires basic understanding of the concept intro-duced in Sec. 1.1. One needs to discretize both time and spatial domain, withthe sampling theorem, the time resolution [see Equation (1.5)], and spatialstep size coming to use, included in the initial step in Fig. 2.1. The standingwave envelop in Equation (2.11) is independent from time variable. It is alsoprepared outside the repetitive calculations (loop) with the time variable. Therepetitive calculation and graphical presentation of superposition of the inci-dent and reflected wave components [in Equations (2.6), (2.9)] along with thepositive and negative envelop within a looped operation animates the dynamicpropagation and changes of the standing wave. A proper implementation ofanimations would allow readers to understand how harmonic waves propa-gate and comparing the standing wave pressure with its envelop. Vary bothmagnitude and phase of the reflection coefficient to examine the influence ofthe wall with varied reflection coefficients. One can also view each of the in-cident, the reflected and their superposition, the standing wave components,separately for ease of comparison to enhance coherent understanding of thestanding wave concept.


Fig. 2.2. Standing waves at two different instances in front of an infinite plane wallwith a magnitude of the reflection coefficient 0.8. Solid line represents the standingwave, while dotted lines are its envelop.

2.3 Standing Wave Ratio

The envelop of standing wave in Equation (2.11) contains the complex-valuedreflection coefficient in Equation (2.8). This envelop reaches its maxima andminima when the cosine function inside Equation (2.11) becomes ±1 at

4πx

λ+ ϕR = nπ, n = 0,±1,±2, . . . , (2.12)

with n being even for its maxima, and odd for its minima. This makes Equa-tion (2.11) become

|ps,extreme

| = p√

1 + |R |2 ± 2|R |, (2.13)

leading to

|ps,max

| = p (1 + |R |), |ps,min

| = p (1− |R |). (2.14)

2.3 Standing Wave Ratio 27

Fig. 2.3. Example of MATLAB-script for animating/viewing the standing wavealong with its envelop, where symbol ’...’ in Line 13 and 17 connects the followingcode line (Line 14, Line 18).

The standing wave ratio is determined by

S =| p

s,max|

| ps,min

|=

1 + |R |1− |R|

. (2.15)

Solving for |R | leads to the normal-incident magnitude of the reflection coef-ficient (reflectance)

|R | = S − 1

S + 1, (2.16)

and the normal-incident absorbtion coefficient, α n,

α n = 1− |R |2 =4S

(S + 1)2. (2.17)

The standing wave phenomena as expressed in Equation (2.11) and inFig. 2.1 are dependent on the acoustic properties (R ) of the wall (termina-tion). This can be used for experimental determination of absorption coeffi-cient for materials of interest. Figure 2.4 conceptually illustrates a standing


Fig. 2.4. Measurement tube using the standing wave ratio method. The wavelengthof a sinusoidal signal driving the loudspeaker has to be much larger than the cross-sectional size of the tube so as to ensure a plane wave propagation.

wave tube, where a single-frequency plane wave can be generated within thetube, impinging upon a piece of material under test that properly fits the innertube area at the opposite end from the sound source. The movable microphonescans the standing wave field in order to measure the standing wave ratio, S,via adjacent maximum | p

s,max| and minimum | p

s,min| values of the standing

wave envelop, leading to the absorption coefficient, α n using Equation (2.17).

2.4 Normal Modes in Rectangular Rooms

The standing waves exist in front of a plane wall (of infinite size). But if thereis another plane wall stands in parallel with the first wall at distance lx, thesuperposition of the standing waves will sustain only under certain conditions,more so for an enclosed room. In order to showcase the conditions for which thestanding waves will sustain (in form of room modes), consider a rectangularroom with all rigid walls as shown in Fig. 2.5. For this room geometry, theHelmholtz wave equation in Cartesian coordinates in Equation (2.1) can beexpressed as

∂2p

∂x2+∂2p

∂y2+∂2p

∂z2+ β2p = 0, (2.18)

where the propagation coefficient follows

β2 = β2x + β2

y + β2z , (2.19)

with βx, βy, βz being the propagation coefficients along x-, y-, and z- directions.A general solution of the sound pressure will be

p(x, y, z) = px(x) p

y(y) p

z(z). (2.20)

Substituting the trial solution in Equation (2.20) and the propagation coeffi-cient in Equation (2.19) into the Helmholtz equation in Equation (2.18) leadsto separation of three wave equations in three orthogonal components

2.4 Normal Modes in Rectangular Rooms 29

Fig. 2.5. Rectangular room with rigid walls.

∂2px

∂x2 + β2x p x

= 0

∂2py

∂y2 + β2y p y

= 0

∂2py

∂y2 + β2y p y

= 0.

(2.21)

A general solution for x-component, for example, is

px= Px cos(βx x) +Qx sin(βx x), (2.22)

which is subject to the boundary condition (rigid wall),

∂px

∂x= 0. (2.23)

This boundary condition is so-called Newman boundary condition (the pres-sure gradient, namely the particle velocity, has to be zero at boundaries). Fig-ure 2.1 illustrates that (0, lx) indicate the room boundary for x-component.Substituting Equation (2.22) into Equation (2.23) for x = 0 leads to

∂px

∂x

∣∣∣∣x=0

= −βx Px sin(βx x)︸︷︷︸=0

+βxQx cos(βx x)︸︷︷︸=1

= 0. (2.24)

This forces Qx = 0 to fulfill the Neumann boundary condition, while Px = 0.Meanwhile for x = lx, substituting x = lx into the first term of Equation (2.24)leads to

∂px

∂x

∣∣∣∣x=lx

= −βx Px sin(βx lx) = 0, (2.25)

resulting in

βx lx = nxπ, → βx =nx π

lx, (2.26)


where nx, along with ny and nz in the following, are positive integers. Simi-larly, the rectangular dimensions, ly, lz, as shown in Fig. 2.1, lead to

βy =ny π

ly, βz =

nz π

lz. (2.27)

Substituting Equation (2.26) with Px = 0 and Qx = 0 into Equation (2.22),the solution of the x-component wave equation specified by the boundarycondition becomes

px= Px cos

(nxπ

lxx

). (2.28)

Substituting Equation (2.26) and Equation (2.27) into Equation (2.19) resultsin

βnx,ny,nz = π

√(nxlx

)2

+

(nyly

)2

+

(nzlz

)2

. (2.29)

Substituting β = 2π f/c into Equation (2.29) leads to discrete frequencies,

fnx,ny,nz=

c

2

√(nxlx

)2

+

(nyly

)2

+

(nzlz

)2

, (2.30)

at which the room modes occur. The frequency only at different combinationsof modal numbers nx, ny, nz is termed modal frequency.

Finally, due to Equation (2.20), the three-dimensional pressure solutionbecomes

pnx,ny,nz ( x, y, z ) =

p cos

(nxπ

lxx

)cos

(nyπ

lyy

)cos

(nzπ

lzz

). (2.31)

For a two-dimensional rectangular room with rigid walls, the time-dependentsound pressure is written as

pnx,ny(x, y, t) = p cos

(2π

t

T

)cos

(nxπ

lxx

)cos

(nyπ

lyy

). (2.32)

MATLAB note: Given a two-dimensional room with dimension lx =6.2 m and ly = 8.5 m, implement Equation (2.32) for a specific combina-tion of nx and ny. The implementation requires basic understanding of theconcept introduced in Sec. 1.1. One needs to discretize both time and spatialdomain, where the sampling theorem, the time resolution [see Equation (1.5)],and spatial step size come to use.

The script in Algorithm 2.6 animates the pressure distributions for 50frames with time, also storing the animation images into a multimedia file.


Fig. 2.6. MATLAB-script for animating/viewing normal room modes of a two-dimensional room of dimension 8.5 m × 6.2 m with nx = 4 and ny = 3. Lines 16,17, 27, and 29 are responsible for creating a multimedia file ’NormalModes.avi’.

Figure 2.7 illustrates one snapshot in two different presentations. Figure 2.7along with Equation (2.32) shows there are discrete values of x and y thatcosine functions of x and y co-incidently become ±1, namely

x = 0, or even integers oflx2nx

, y = 0, or even integers ofly2ny

. (2.33)

These are ±peak values of the pressure distributions, termed normal roommodes or modal distributions. Numbers nx, ny (also nz) represent exactlythe number of the modes along the respective directions (axes), termed modalnumbers. For an even modal number (in this case, nx = 4 along x-axis) the halfmodes of the same polarity exist on each room boundaries (along x-direction),while for an odd modal number (in this case, ny = 3 along y-axis) the halfmodes of opposite polarity exist on each room boundaries (along y-direction).

In addition, the cosine functions of x and y become zero for all values ofx and y when


Fig. 2.7. Modal distribution of sound pressure in a two-dimensional rectangularroom enclosed by rigid walls with nx = 4 and ny = 3. (a) Modal distributionin three-dimensional presentation together with modal contours showing clearly themodal lines. (b) Top view of the modal distribution in two-dimensional presentation.(c) Plane traveling wavefronts creating the modal behavior in the rectangular room.


x = odd integers oflx2nx

, y = odd integers ofly2ny

. (2.34)

These are the straight lines, termed nodal lines, being perpendicular to indi-vidual axes as shown in Figure 2.7 (a,b). Note that numbers nx and ny (aswell as nz) also represent exactly the number of the nodal lines, also termednodal numbers. Figure 2.7 (a,b) show nx = 4 nodal lines perpendicular tox-axis, while ny = 3 nodal lines perpendicular to y-axis.

The modal distribution illustrated in Fig. 2.7 also conveys that high soundpressure in forms of modes can always be found at rigid boundaries, more soat the room corners. The room modes are due to the plane wave propagationacross the reflecting interior surface. At each modal frequency, the propagationdirections are determined by the angles αx, αy, and αz, at which the wavefrontnormal is oriented with respect to the respective axes,

cosαx = ±nxlx, cosαy = ±ny

ly, cosαz = ±nz

lz. (2.35)

In order to clarify this, Equation (2.31) is rewritten using Euler formulacosx = (e jx + e−jx)/2 [5]

pnx,ny,nz

(x, y, z ) =p

8

∑exp

[jπ

(±nxlxx ± ny

lyy ± nz

lzz

)], (2.36)

where the sum includes all 23 combinations of ± signs.

Fig. 2.8. Number of normal modes in a rectangular room with room dimension of(L = 4.88 m, W = 3.6 m, H = 2.44 m ) [2]. Curve labeled by NC is predicated basedon the model frequency - modal number relations in Equation (2.30). Points markedby ’*’ are predicated by Nf in Equation (2.36). Curve labeled by NB is predictedby Bolt [2]. Curve by NA is calculated using only the first term in Equation (2.36)


According to Maa [6], the number of normal modes below an upper limitfrequency, fu, is determined by

Nf =4π V f3u3 c 3

(1 +

3S c

16V fu+

3L c2

8π V f2u

), (2.37)

where S is the total interior surface area, L = lx+ly+lz is the sum of the threedimensions, V is the volume of the room, and c is the sound speed. Significanceof this equation is that the number of normal modes is solely determined usingthe room dimensions (V , S, L) and the upper limit frequency, fu. The firstterm becomes dominate only for geometrical-acoustics-valid frequency rangewhen the wavelength under consideration is orders of magnitude smaller thanthe room dimension [2, 8], which is the case when fu ≫ c/V 3 [8]. Figure 2.8illustrates the prediction results as function of the upper limit frequency, fu.A historical review can be found in a recent article [16].

Note that individual, well-separated room modes are also distinct in en-closures with arbitrary room geometries in modal frequency ranges. Equa-tion (2.37) is also valid for estimating the number of normal modes. Withthe increasing frequency distinct room modes will transit to many overlap-ping normal modes. For airborne sound in a reverberant enclosure, there is across-over frequency, termed Schroeder frequency [12],1 given by

fc = 2000

√T

V, (2.38)

with V being the volume, and T the reverberation time of the enclosurearound fc (see Chapter 3 for detailed discussion on the reverberation time).Schroeder [11] derived Equation (2.38) by equating the half-power bandwidthof the room modes with threefold average spacing between modal frequencies.When applying Sabine’s formula [9, 15] for determining the reverberation timeof the enclosure,

T = 13.84V

cA, (2.39)

Equation (2.38) becomes [12]

λc ≈√A

6, (2.40)

with A being the equivalent absorption area of the room. In other words, threetimes or more modal overlaps lead to the wavelength shorter than λc, whichcan be termed Schroeder wavelength or cross-over wavelength.

1 In Schroeder’s original paper in German authored by Schroder [10], he applieda more conservative modal overlapping factor, 4000, corresponding to ten modaloverlaps. This German publication was translated later into English [11].

2.5 Vortex Modes in Rectangular Rooms 35

Fig. 2.9. Sound intensity vectors in a square two-dimensional room of dimensionlx = ly = λ/2.

2.5 Vortex Modes in Rectangular Rooms

In modal frequency range under certain conditions, sound energy in the steadystate flows in circular patterns in enclosure spaces, they are termed vortexroom modes. With recent advent of particle velocity sensors [3], sound inten-sity vector-fields and sound energy flow-fields can be experimentally quanti-fied with high accuracy, the vortexial modes (circular energy flows in closedpaths) become more and more noticeable and relevant in practice. For clarity,this section discusses these vortex modes in rectangular enclosures with rigidboundaries. For simplicity, we take a two-dimensional square room (while thefield value along the third-dimension is considered to be uniform) for study-ing the sound intensity vector field, the velocity potential field consists of twoaxial modes with explicit time-dependence [14] as

Φ x(t) = Φ cos(βx x)ejω t,

Φ y(t) = Φ cos(βy y)ej(ω t+π/2) = j Φ cos(βy y)e

jω t.

(2.41)

Superposition of the two axial modes leads to

Φ(x, y, t) = [cos(βx x) + j cos(βy y)] ejω t, (2.42)

where the amplitude Φ is dropped for simplicity. The sound pressure becomes

p(x, y, t) = −ϱ ∂Φ∂t

= −j ϱω Φ = ϱω [cos(βy y)− j cos(βx x)]ejω t. (2.43)

and the sound particle velocity components

36 2 Acoustic Waves, Standing Waves, Room Modesv x(t) =

∂Φx

∂x = −βx sin(βx x)ejω t,

v y(t) =∂Φ y

∂y = −jβy sin(βy y)ejω t.

(2.44)

Using Equations (2.43) and (2.44) we obtain the sound intensity componentsIx = 1

2Re( p v∗x) = −ϱω βx sin(βx x) cos(βy y),

Iy = 12Re( p v

∗y) = ϱω βy cos(βx x) sin(βy y),

(2.45)

where the asterisk denotes the complex conjugate. Figure 2.9 illustrates thesteady-state intensity vector field in the square room of dimension lx = ly =λ/2, or βxlx = βxly = π.

Fig. 2.10. Streamlines of sound energy flows in a square two-dimensional room. (a)Streamlines when lx = ly = λ/2. (b) Streamlines when lx = ly = λ, overlapped withthe amplitude distribution of the streamline function.

Figure 2.9 indicates that the steady-state energy flows along vortexialpaths. These paths are termed streamlines. We determine the streamline func-tion of the sound energy through its infinitesimal relationship [13]

dψ =∂ψ

∂xdx+

∂ψ

∂ydy, (2.46)

so that

ψ(x, y) =

∫Ix dy −

∫Iy dx

= −1

2ϱω sin(βx x) sin(βy y). (2.47)

Figure 2.9 illustrates energy streamlines in the square room with two differentwavenumbers. Note that Figure 2.9 (a) contains the same streamlines as thefirst quadrant in Figure 2.9 (b).

2.5 Vortex Modes in Rectangular Rooms 37

Figure 2.10 illustrates the pressure wavefronts causing the vortex mode inthe square room overlapped with the energy flow streamlines. The pressurewavefront is determined by

cos(βy y) = − cos(βx x) · tan θ(t), (2.48)

where θ(t) represents the phase angle of the advancing wavefront. Takingphase angles in an angular step π/16 with uniform time steps, Figure 2.11illustrates the energy wavefronts, indicating that the wavefronts are normalto the streamlines. In other word, at all points the pressure wavefronts areorthogonal to the direction of sound energy flows.

Fig. 2.11. Sound energy wavefronts overlapped with the streamlines of sound energyflows in a square room as shown in Figure 2.10 (a). Arrows indicate the wavefrontpropagating direction, starting at t = 0, propagating with equitemporal interval.

In a rectangular room of dimension (lx, ly, lz) with rigid boundaries, themodel frequency fnx,ny,nz

of tangential vortex modes for modal numbersnx, ny, nz is determined [14] by Equation (2.30).

In similar fashion, the streamline function in Equation (2.47) can be gen-eralized to a two-dimensional room with a modal number of nx, ny as

ψnx,ny(x, y, t) = −ψ sin

(nxπ

lxx

)sin

(nyπ

lyy

). (2.49)

Figure 2.12 conceptually illustrates the streamlines overlapped with the func-tion amplitude.

The normal mode theory discussed in this Chapter is of practical sig-nificance for acoustics of small rooms in general, for acoustics of recordingstudios, in particular [4]. Recent work based on experimental determinationof an impulse response at one room corner [1] facilitates estimates of the num-ber (N) of room modes, model frequencies, fn, modal decay times, Tn, modal


Fig. 2.12. Sound energy flows along streamlines of a rectangular two-dimensionalroom with modal number of nx = 4, ny = 3.

amplitudes, An and their phases, ϕn, below the Schroeder frequency based onProny model of the room impulse response,

h(t) =

N∑n=1

Ane−6.9 t/Tn cos(2π fnt+ ϕn), (2.50)

where Bayesian inference has been successfully applied [1].

2.6 Exercises 39

2.6 Exercises

Exercise 2.1.

Use the incident and reflected wave components in Equation (2.4) and Equa-tion (2.9) to show that the pressure standing wave can be expressed in Equa-tion (2.11)

Exercise 2.2.

Use the incident and reflected particle velocity components in Equation (2.5)and Equation (2.7) in consideration of the reflection coefficient in Equa-tion (2.8), determine the standing wave velocity, and its envelop.

Exercise 2.3.

Write a MATLAB-script for animating (visually inspecting) the standing wavein front of plane wall of infinite size with a reflection coefficient R = |R| e jϕ r ,similar to MATLAB-script in Fig. 2.3 (or use a language of reader’s con-venience). Explore changes when the amplitude, the acoustic frequency, themagnitude and the phase of the wall reflection coefficient.

Exercise 2.4.

[Hands-on Lab] Use a standing-wave tube to measure two pieces of absorp-tive materials at 250 Hz, 315 Hz, 400 Hz, 500 Hz, 630 Hz, 800 Hz, 1 kHz, and1.25 kHz. (Note other than relying on a commercially available standing-wavetube with a moving microphone plunger, students can build the standing-wavetube with a PVC pipe of certain length /diameter, a small loudspeaker, a low-cost electret microphone of 4 mm in diameter. This experimental test helpsreaders experience the standing waves and the standing-wave ratio method forexperimentally determine the absorption coefficient of materials of interest).

Exercise 2.5.

Find an empty rectangular room, and measure the dimension of the roomlx, ly, and lz, establish a table of columns of nx, ny, and nz to determine themodel frequencies for combinations of the modal numbers from 0 to 3.

Exercise 2.6.

Write a MATLAB-script for animating the room modes of a two-dimensionalrectangular room with rigid walls, similar to MATLAB-code in Fig. 2.6 oruse a language of reader’s convenience) to explore changes when modal num-bers nx and ny change, see how the modal frequencies and modal pressuredistribution in the room change.


Exercise 2.7.

[Hands-on Lab] Use an empty fish tank (aquarium) to explore the soundpressure distribution over the interior bottom plane of the aquarium. Deter-mine modal frequency needed to excite the aquarium in accordance with theaquarium’s dimension. Discuss where should a small loudspeaker be placedwhen exciting the empty aquarium. And use the MATLAB code in Exer-cise 2.6. to guide the spatial scanning near the interior bottom using a low-costelectret microphone of 4 mm in diameter to experience the room modes (asimilar lab kite is required as in Exercise 2.4, but an empty aquarium insteadof standing-wave tube).

References

1. Beaton, D., Xiang, N.: Room acoustic modal analysis using Bayesian inference.J. Acoust. Soc. Amer. 141(6), 4480–4493 (2017). DOI 10.1121/1.4983301

2. Bolt, R. H.: Frequency Distribution of Eigentones in a Three-Dimensional Con-tinuum. J. Acoust. Soc. Amer. 10, 228–234 (1939). DOI 10.1121/1.1915980

3. de Bree, H.-E., Leussink, P., Korthorst, T., Jansen, H., Lammerink, T. S. J.,Elwenspoek, M.: The µ-flown: A novel device for measuring acoustic flows. Sens.Actuators, A 54(1–3), 552–557 (1996).

4. Kleiner, M., Tichy, J.: Acoustics of Small Rooms. CRC Press, Taylor and FrancisGroup, Boca Raton, London, New York (2014).

5. Kuttruff, H.: Room Acoustics, 6th Ed. CRC Press, Taylor and Francis Group,Boca Raton, London, New York (2017).

6. Maa, D.Y.: Distribution of eigentones in a rectangular chamber at low frequencyrange. J. Acoust. Soc. Am. 10, 235—238 (1939). DOI 10.1121/1.1915981

7. Meissner, M.: Acoustic energy density distribution and sound intensity vectorfield inside coupled spaces. J. Acoust. Soc. Am. 132, 228—238 (2012).

8. Morse, P., M.: Vibration and Sound. McGraw-Hill, New York, London (1936),pp. 295.

9. Sabine, W. C.: Collected Papers on Acoustics. Harvard University Press, Cam-bridge, MA, (1922).

10. Schroder, M.: Die statistischen Parameter der Frequenzkurve von großenRaumen. Acustica 4, 594–600 (1954).

11. Schroeder, M. R.: Statistical Parameters of the Frequency Response Curves ofLarge Rooms. J. Audio Eng. Soc. 35, 299–306 (1987).

12. Schroeder, M. R.: The ’Schroeder frequency’ revisited. J. Acoust. Soc. Am. 99,3240–3241 (1996).

13. Waterhouse, R. V., Crighton, D. G., Ffowcs-Williams, J.E.: A criterion for anenergy vortex in a sound field, J. Acoust. Soc. Am. 81(5), 1323–1326 (1987).

14. Waterhouse, R. V.: Vortex modes in rooms, J. Acoust. Soc. Am. 82(5), 1782–1791 (1987).

15. Xiang, N.: Generalization of Sabine’s reverberation theory, J. Acoust. Soc. Am.148(3), R5–R6 (2020).

16. Xiang, N.: Maa’s equation on the number of normal modes of sound waves inrectangular rooms, J. Acoust. Soc. Am. 150(3), R9–R10 (2021)

17. Xiang, N., Blauert, J.: Acoustics for Engineers – Troy Lectures. 3rd ed. Springer-Verlag, Berlin Heidelberg (2021). Chap 9.

2 acoustic waves, standing waves, room modes

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