Uncertainty quantiﬁcation of the diffuse sound ﬁeld assumption in
structure-borne sound radiation predictions
Edwin P.B. REYNDERS, Pengchao WANG, Geert LOMBAERT and Cédric VAN HOORICKXKU Leuven, Department of Civil Engineering, Belgium, edwin.reynders@kuleuven.be

Abstract
When predicting the radiation of structure-borne sound into a room, it is often assumed that the generated soundﬁeld is diffuse. A diffuse ﬁeld is by deﬁnition a random ﬁeld: it represents a conceptual ensemble of rooms
with the same volume and total absorption, but otherwise any possible arrangement of boundaries and small
objects that scatter incoming sound waves. Adopting a diffuse sound ﬁeld model therefore inherently introduces
uncertainty on the computed results. This uncertainty can be important, especially at the lower frequency end of
the spectrum. In this work, practical formulas are derived for predicting not only the mean, but also the varianceof energetic level quantities, such as the band-integrated spatially averaged sound pressure level, in a diffuse
sound ﬁeld caused by a mechanically excited structure. The formulas are ﬁrst validated in a simulation study,
and then applied for predicting the sound pressure level in a room caused by impact excitation of a plate. Boththe average sound pressure level and its standard deviation can be well predicted.
Keywords: Uncertainty quantiﬁcation, diffuse sound ﬁeld, structure-borne sound.

1

INTRODUCTION

When predicting the sound energy that is radiated by a vibrating structure into a room, the radiated sound ﬁeld
is usually taken to be diffuse, which means that it is a random ﬁeld, composed of a large number of statisticallyindependent plane waves, the spatial phase of which is uniformly distributed and independent from the amplitude[1]. The randomness is caused by uncertainty regarding wave scattering inside and at the boundaries of theroom, either because of a lack of information, or because the wave scattering is not modeled in detail. The
higher the frequency, the more sensitive the harmonic (i.e., pure tone) sound pressure at a speciﬁc point in theroom becomes to such wave scattering. Since a diffuse ﬁeld represents a state of maximum uncertainty (termed
maximum entropy) due to wave scattering [7], it can accurately represent the state of knowledge regarding thesound ﬁeld in the receiver room from a certain frequency onwards. Below this frequency, the diffuse ﬁeld
model will over-estimate the uncertainty caused by random wave scattering.
It is also known that the variance of the total sound energy across the random ensemble representing the diffuse
ﬁeld decreases with increasing modal overlap factor. Since the modal overlap factor increases with frequency,
the mean total energy of the random ensemble will be close to the total energy in any particular member ofthe ensemble at high frequencies, so at high frequencies, diffuse ﬁelds behave essentially as deterministic, andthey are often treated as such in technical acoustics. At low frequencies however, considerable variation canoccur when comparing the mean total sound energy of a diffuse room model with the total sound energy for
one particular member of the random ensemble.
It would therefore be desirable to quantify the uncertainty that is inherent in the diffuse sound energy as afunction of frequency. Based on this uncertainty information, the analyst could then estimate, for a given
vibrating structure, by how much the radiated sound energy could vary between various receiver rooms withthe same gross properties. Or when a single receiver room is of interest, the analyst could then estimate fromwhich frequency onwards taking the mean, nominal diffuse sound energy is sufﬁciently accurate.
The purpose of the present paper is precisely to deliver a method for quantifying the uncertainty of structureborne sound predictions that is inherently present in the diffuse ﬁeld assumption. This will be based on insightsfrom fundamental physics that have led to diffuse ﬁeld variance formulas which can be used in engineering

218acoustics [4, 3]. A ﬁrst contribution of the present work is that these results are complemented with the derivation of variance formulas for level quantities such as the spatially averaged band-integrated sound pressure level.
A second contribution lies in the numerical and experimental validation of the proposed method.

2

PREDICTING THE VIBRATION FIELD OF THE RADIATING STRUCTURE

For simplicity, it will be assumed that the vibration ﬁeld of the radiating structure depends on the direct mechanical excitation and possibly also on radiation damping, but not on diffuse sound pressure loading. The
prediction of the radiated sound energy then starts with the prediction of the displacement ﬁeld of the soundradiating structure, such as a ﬂoor in a building. If excitation of the structure is harmonic
ω, then at frequency
the displacement ﬁeld u(x,t) at position x and time t can be expressed as u(x,t) = Re u(x, ω)eiωt , where thecomplex vector u(x, ω) is termed the phasor. This phasor notation is used throughout the present and the next
section. It can be easily generalized towards stationary excitation. It is assumed that the radiating structure has
ﬁnite size, and that u(x, ω) can be accurately represented with a ﬁnite number N of modal (or generalized)
displacements, which are gathered in a vector q ∈ CN . If additionally the mechanical excitation of the structure
is known, the modal displacements can be computed from
D(ω)q(ω) = f(ω),(1)

where D(ω) ∈ CN×N represents the dynamic stiffness matrix of the structure in modal coordinates, and f(ω) ∈CN the vector of modal forces. The three quantities depend on frequency, but this dependency will be dropped
in what follows in order not to overload the notation.
The concern in noise control is usually with energetic quantities, such as the vibrational cross-spectrum
Sqq := qqH ,(2)

where the superscript H denotes Hermitian (or complex conjugate) transpose. From Eq. (1), the response crossspectrum relates to the force cross-spectrum via
Sqq = DSff D−H .

3(3)

PREDICTING THE RADIATED SOUND ENERGY

In stationary conditions, the time-averaged harmonic net acoustic power that is injected into the receiver roomby the vibrating structure Win , equals the time-averaged power that is dissipated in the room, Wdiss . Adopting ahysteretic damping model for the receiver room then results in
Win = Wdiss = ωηE,

(4)where η denotes the (frequency-dependent) loss factor of the receiver room, and E the total time-averaged
acoustic energy in the receiver room, which is the quantity of interest. In order to compute the input power
Win , the reaction forces that are applied onto the structure by the acoustic pressure resulting from the structure’s
movement q, are determined. These forces are split into forces caused by the direct sound ﬁeld, fdir , and theforces caused by the reverberant sound ﬁeld, frev . The input power can then be expressed as the sum of the
input powers to the direct and reverberant ﬁelds:
1 
Win = Win,dir +Win,rev = Re iωqH (fdir + frev ) .
2(5)

The direct ﬁeld is the sound ﬁeld that would occur in the absence of any diffuse reﬂections or scattering, i.e.,the sound ﬁeld that would occur when the structure would be embedded in an inﬁnite planar bafﬂe and radiate
into an acoustic halfspace. For planar structures, this sound ﬁeld can be obtained by evaluating the Rayleigh219

integral [2]. The result is a linear relationship between the degrees of freedom of the structure and the loading
onto the structure caused by the sound pressure of the direct ﬁeld, fdir :− Ddir q = fdir ,(6)where Ddir is the dynamic stiffness matrix of the direct ﬁeld of the receiver room at the interface between theroom and the vibrating structure. Once the direct ﬁeld dynamic stiffness matrix Ddir has been determined, theinput power to the direct ﬁeld can, by application of (6), be expressed as

 ω

1 
(7)
Win,dir = − Re iωqH Ddir q = ∑ Im Ddir, jk Sqq, jk .
2
2 j,kIn order to proceed, the reverberant sound ﬁeld is now taken to be diffuse, which means that it is a random
ﬁeld, composed of a large number of statistically independent plane waves, the spatial phase of which is uniformly distributed and independent from the amplitude. The central limit theorem can then be invoked, fromwhich it follows that the complex phasor pressure amplitudes have zero mean [1, Sec. 2.2], hence f̂rev = 0,
where the hat denotes the ensemble average. The structural vibration q was assumed to depend only on the
structure-borne noise source f (and possibly also fdir , if radiation damping is accounted for, i.e., if Ddir is included in D) but not on the diffuse reaction forces f̂rev . It then follows from (5) that Ŵin,rev = 0 and Ŵin = Ŵin,dir .
Combining Eqs. (4) and (7) and taking the expected value then results in11
Ŵin,dir =Im Ddir, jk Ŝqq, jk .
(8)
Ê =
ωη
2η ∑
j,k
From this expression, the ensemble mean of the total acoustic energy in the receiver room can be determined.
When the dissipation takes place primarily in the diffuse ﬁeld, the damping loss factor of the receiver room canbe determined from the reverberation time T or the equivalent total absorption area A. Finally, in a sound ﬁeldthat is composed of plane waves, such as a diffuse sound ﬁeld, the spatially averaged squared sound pressurelevel p2av relates to the total time-averaged acoustic energy in the room via
E=



V

e(x)dx =

p2avV
.
ρa c2

(9)

From the above analysis, the mean radiated sound energy (or spatially averaged squared sound pressure) acrossthe considered random ensemble of rooms, can be computed. For an accurate computation of the ensemble
variance, the standard notion of a diffuse ﬁeld should be generalized. When random wave scattering is present
within the room, the natural frequencies and mode shapes are also random. Weaver [8] found that, when there
is sufﬁcient random wave scattering such that a state of maximum information entropy is reached, the statisticsof the eigenvalues of the room conform to those of the eigenvalues of a Gaussian Orthogonal Ensemble (GOE)random matrix, while the mode shapes are independent Gaussian random ﬁelds. Based on this GOE deﬁnitionof a diffuse ﬁeld, Langley and Brown [4] were able to derive the relative variance of the input power (or,equivalently, the relative variance of the total energy or the spatially-averaged squared sound pressure) of adiffuse ﬁeld under harmonic loading with a deterministic amplitude. The result of this derivation reads
 
1
α −1
Var [Win ] Var [E] Var p2av
+
=
= 
≈
,
(10)
2
2
2
πm
Ŵin
Ê
(πm)2
2
p
av
where m denotes the modal overlap factor of the room and α a loading factor. The approximation in Eq. (10)is accurate when m ≥ 0.2. In the present setting, the diffuse sound ﬁeld originates from a vibrating structure
via an area coupling, in which case α ≈ 2.
The harmonic mean and variance of the acoustic energy due to radiation of structure-borne sound into a diffuse
ﬁeld can be computed from Eqs. (8) and (10), respectively. This diffuse sound energy has a lognormal probability distribution at moderate and high modal overlap (typically for m ≥ 2) [5]. This implies that the energy,
when represented on a logarithmic scale, is normally distributed.

220

4

BAND INTEGRATION AND LEVEL QUANTITIESIn technical acoustics, it is customary to integrate energetic quantities over frequency bands. Since ensemble
averaging is an additive operation, the mean band-averaged energetic quantities can be immediately obtained byintegration, e.g., for the band-integrated energy EB , one has
ÊB := ωuωlÊ(ω)dω.

(11)

For the relative variance, the derivation that has led to (10) has been extended to yield [3]:Var [EB ] α − 1 
1
2B arctan (B) − ln 1 + B2 +≈ln 1 + B2 ,
2
2
2
πmB
(πmB)
ÊB

where

B :=

ω u − ωl
,
ωη

(12)

in which ω represents the nominal (or center) frequency of the band. For 1/3-octave bands, B ≈ 0.23/η. In
the derivation of Eq. (12), it has been assumed that damping and the load spectrum (which in the present
application, boils down to the vibration spectrum Sqq of the radiating structure) be constant over the consideredfrequency band [3]. This implies that, in order for Eq. (12) to hold with good accuracy, the frequency band of
integration should be sufﬁciently narrow.
The prediction of the variance of the band-integrated energy over wide frequency bands is an open problem. In
this work, a practical solution is investigated, which consists of dividing the large band into several narrowerbands that are (i) sufﬁciently narrow such that (12) holds with good accuracy, and (ii) sufﬁciently wide suchthat the band-integrated energies belonging to different bands are approximately statistically independent. Whenboth conditions are satisﬁed, the mean and variance of the band-integrated energy for the narrower bands canbe estimated from (11) and (12), thanks to condition (i). Thanks to condition (ii), the mean and variance of theintegrated energy over the wide band are then obtained asÊB = ∑ ÊB, j(13)
andVar [EB ] = ∑ Var EB, j .
jj

The input power and spatially averaged squared sound pressure are usually represented on a logarithmic scale,as the sound power level LW and the sound pressure level Lp , respectively:
LW := 10 log

WB
W0

Lp := 10 log

and

p2av,B
p20

,

(14)

where log denotes the logarithm with base 10, and the reference values are chosen as W0 = 10−12 W and p20 =ρa cW0 . The subscript B is employed to emphasize that it usually concerns band-averaged quantities. From
Sec. 3, it is clear that in a diffuse ﬁeld, in input power, total energy and spatially averaged squared soundpressures all have the same relative variance, and that they are all approximately lognormally distributed. If thisalso holds for the band-averaged quantities, then the levels are normally distributed, with mean values
L̂W = 10 log

ŴB C
−
W0 2

and

L̂p = 10 log


p2av,B
p20

−

C
,
2

(15)

and variances  10C
Var (LW ) = Var Lp =,ln 10

and where

C := 10 log 1 +Var [EB ]
.
ÊB2

(16)

From this analysis, two remarkable conclusions can be drawn: (i) the variances of all energetic levels (soundpower level, sound pressure level, etc.) are the same, and (ii) the mean of an energetic level is not simply thelevel of the mean energetic quantity, but the error by neglecting the additional term will be smaller than 1 dB
as long as the relative variance of the band-averaged energy will be smaller than 0.6.

221

5

NUMERICAL VERIFICATIONIn this section, the foregoing theory is veriﬁed with a numerical experiment. The aim is to compute the soundpressure level that is radiated into a room by a concrete ﬂoor, which is in its turn excited by a point load. The
room has a width of lx = 4.15 m, a length of ly = 5.10 m, and a height of lz = 4.12 m. The concrete ﬂoor on
top of the room has a width of Lx = 4.15 m, a length of Ly = 5.10 m, and a thickness of t = 0.10 m. The point
load is applied in the vertical direction, at a distance of 1.15 m from the nearest long edge and 2.10 m fromthe nearest short edge of the ﬂoor. Its Fourier transform is constant, with an amplitude of 1 N/Hz and zero
phase. The concrete has a density of ρ = 2350 kg/m3 , a Young’s modulus of 40 GPa, and a Poisson’s ratio of
0.2. The sound speed in the room is c = 343 m/s, the air density is ρa = 1.2 kg/m3 and the reverberation time
is T = 1.5 s.
The rooms have a random acoustic mass distribution in the sense that a total of Nm = 30 point air pockets(or acoustic point masses) are distributed at random locations within each room. The air pockets are highly
idealized models for small wave scatterers in the room, in the same way as point masses would be highly
idealized models for small wave scatterers on a plate. Each air pocket has 0.4 % of the total acoustic mass
V /c2 of the room. The probability distribution of the location of each air pocket is uniform throughout theentire room, and the locations of the air pockets are statistically independent. The total number of air pockets
and their individual acoustic mass are chosen arbitrarily; the important point is that, from a certain frequency
onwards, the randomness introduced by the wave scatterers reaches a state of maximum information entropy,
which conforms to a diffuse ﬁeld [6].
The ﬂoor is modeled deterministically, in line with Sec. 2. In particular, the ﬂoor is taken to be a homogeneous,
linear elastic, simply supported thin plate with hysteretic damping. Its in-vacuo natural frequencies and modeshapes are then known in closed form, so the quantities D and f which appear in Eq. (1) can be computed
analytically.
Two stochastic models for the room are constructed. The ﬁrst room model is a detailed model, in which boththe point air pockets and the modal behavior of the room are modeled in detail. This is achieved with anassumed-modes model, in which the hard-walled room modes are taken as basis functions. Realizations of therandom room are computed by ﬁxing the positions of the point air pockets with a random number generator.
This process is repeated to generate a Monte Carlo ensemble of 1000 realizations of the sound ﬁeld. The second
room model is a diffuse ﬁeld model that is constructed according to Sec. 3. In this model, the random wave
scattering caused by placing the point air pockets at random locations within the room, is supposed to result ina diffuse sound ﬁeld. Both room models do not only differ in the level of detail in which the room is modeled.
There is also a very important difference in computation cost, since the diffuse ﬁeld model is computationallymuch more efﬁcient than the detailed model.
The mean and relative variance of the harmonic time-averaged total energy in the receiver room, computed withboth the detailed room model and the diffuse room model, are plotted in Fig. 1a and Fig. 1b, respectively.
Results from individual Monte Carlo realizations, as obtained with the detailed room model, are also displayed.
At frequencies lower than about 100 Hz, there are clear differences between the results obtained with the detailed
and the diffuse room models. This is because in that low-frequency range, the sensitivity of the local harmonicsound ﬁeld to the presence of the small random wave scatterers is relatively low, and a diffuse ﬁeld model
will overestimate the uncertainty caused by these random wave scatterers. However, once this sensitivity is
sufﬁciently large, the diffuse ﬁeld model is adequate: above about 100 Hz, the natural frequencies of the room
start to mix well across the random ensemble, resulting in a good match between the ensemble mean andvariance of the energies as predicted from the detailed room model and the diffuse room model.
The mean and standard deviation of the spatially averaged sound pressure levels in 1/3-octave bands, are plottedin Fig. 1c and Fig. 1d, respectively. Not only the unweighted results are plotted, but also the A-weighted results.
Since the sound pressure levels are normally distributed, conﬁdence intervals can be directly computed from thestandard deviations that are plotted in Fig. 1d; for example, a 95 % conﬁdence interval corresponds with 4 times
the standard deviation. The 95 % conﬁdence interval corresponding to the uncertainty due to the random wave

222

100

10-2

10-4
40 50 63(a)80 100 125 160 200 250 315 400 500

(b)
4

3

2

1

0

(c)

40 50 63 80 100 125 160 200 250 315 400 500(d)Figure 1. Response of a room with random wave scatterers, loaded by a point-excited vibrating concrete ﬂoor:(a) ensemble mean of the harmonic energy; (b) ensemble relative variance of the harmonic energy; (c) ensemble
mean of the 1/3 octave band-integrated sound pressure level; (d) ensemble standard deviation of the 1/3 octave
band-integrated sound pressure level. Thick solid blue lines: detailed room model statistics; thick dashed redlines: diffuse room model statistics. In plots (a,c), the results from individual Monte Carlo realizations are
additionally displayed by means of thin gray lines. All results are presented without frequency weighting,
except in plots (c,d), where results for A weighting are also included; the detailed room model statistics with
A weighting are then represented with thick dash-dotted black lines.
scattering (and causing a diffuse ﬁeld) is larger than 1 dB for all 1/3-octave bands below 400 Hz.
Finally, Table 1 contains the overall band-integrated unweighted (or Z-weighted) and A-weighted mean soundpressure levels and their standard deviations, as obtained from the Monte Carlo data for the detailed roommodel, and from Eqs. (13), (15) and (16) for the diffuse ﬁeld model. On overall, the mean and the standard
deviation as predicted by both models agree very well. There is a slight difference between both models for the
standard deviation without weighting, because the sound pressure levels at the lowest frequencies then contribute
importantly to the global response, and their uncertainty is overestimated with the diffuse ﬁeld model. This
increased uncertainty is the price that is paid when employing diffuse ﬁeld models at low frequencies.

6

EXPERIMENTAL CASE STUDY

In this section, the foregoing methodology is demonstrated and validated by predicting and measuring the soundthat is generated by a polymethyl metacrylate (PMMA) panel when impacted by a known hammer force. The

223

Table 1. Response of a room with random wave scatterers, loaded by a vibrating concrete ﬂoor: comparison
between the mean and standard deviation of the band-integrated sound pressure levels over the entire frequency
range of interest (35-560 Hz), as obtained from the detailed room model and the diffuse room model.
weighting
Z [dB]
A [dB(A)]L̂p,diffuse
75.8
67.6

σ (Lp,detailed )
0.24
0.11

70

70

60

60

50

50

40

40

30

30

20

20

10

10

0

(a)

L̂p,detailed
75.5
67.3

63

125

250

500

1000

2000

0

4000

63

σ (Lp,diffuse )
0.37
0.11

125

250

500

1000

2000

4000

(b)

Figure 2. Sound pressure level due to sound radiation by a PMMA panel that is excited by repeated hammer
impacts at a single point: (a) harmonic values and (b) 1/3-octave band values. Solid blue lines: measurements.
Dashed red lines: mean predictions; solid red lines: 95% conﬁdence interval of the predictions.

panel was mounted in the small vertical transmission opening in the KU Leuven Laboratory of Acoustics, whichmeasures 1.25 m by 1.5 m. The receiver room has a volume of 87 m3 . In order to quantify the impact force, an
instrumented impact hammer is employed as excitation device. Repetitive impacts were given at a ﬁxed locationon the plate. The squared sound pressure amplitudes are measured at 8 locations in the room and subsequently
averaged in order to arrive at an estimate of the average sound pressure level. In the predictions, the sound ﬁeld
in the room is modeled as diffuse, with nominal values for the air density and sound speed at ρa = 1.20 kg/m3
and c = 343 m/s, respectively. For the damping loss factor of the plate and the reverberation time of the room,measured values are employed.
A ﬁnite element model of the plate was constructed using 4-node Reissner-Mindlin shell elements. The densityof the PMMA material was determined at 1260 kg/m3 . For the Young’s modulus and the Poisson’s ratio, valuesof 2.4 GPa, and 0.35 were assumed, respectively. The out-of-plane movement at the plate edges was prohibitedsince the plate was mounted against the transmission opening of the laboratory. Additionally, since the bottomedge of the plate was resting on the bottom part of the transmission opening, the horizontal movement of thisplate edge was also prohibited in the direction parallel to the edge.
The excitation consisted of repetitive hammer impacts at a single location on the plate and for a total durationof 20 s. The impact location that is considered in the present paper, was at a distance of one-third of both thehorizontal and vertical edge lengths of the plate, when measured with respect to one of its corners.
Fig. 2 contains the predicted and measured sound pressure levels. At low frequencies, the harmonic results
contain sharp peaks that sometimes also appear in the predicted mean sound pressure level, but sometimes
not. The peaks that are predicted represent modal resonances of the PMMA panel, since these are included in224the model. Modal resonances of the particular room in which the test has been performed are not visible in
the predictions since a diffuse sound ﬁeld model is adopted; they are instead captured by the 95% conﬁdence
interval that is associated with the diffuse ﬁeld assumption. Furthermore, it can be observed that the uncertainty
that is due to the diffuse ﬁeld assumption is very large at the lowest frequencies, but it rapidly decreases withincreasing frequency: in 1/3-octave bands, the width of the related 95% conﬁdence interval drops below 5 dB at
125 Hz, and below 2 dB at 250 Hz. At higher frequencies, the uncertainties due to the diffuse ﬁeld assumption
are so small that the diffuse ﬁeld model becomes essentially deterministic. At such frequencies, other sources of
uncertainty will be more important than the diffuse ﬁeld assumption which is investigated in the present work.

7

CONCLUSIONS

Practical, closed-form formulas have been derived for predicting not only the mean, but also the variance ofenergetic level quantities, such as the band-integrated spatially averaged sound pressure level, in a diffuse soundﬁeld caused by a mechanically excited structure. The uncertainty that is accounted for is the one that is inherent
in the diffuse ﬁeld model. The main theoretical results are Eqs. (15-16), which make the connection between
the mean and variance of the level quantities and the existing results for the relative variance of the harmonic orband-averaged total energy. For simplicity, it has been assumed that the vibration ﬁeld of the radiating structure
depends on the direct mechanical excitation and possibly also on energy that is radiated into the direct soundﬁeld. In this case, the uncertainty of the sound pressure level which is due to the diffuse ﬁeld assumption
depends only on the room volume, the equivalent total absorption area of the room, the considered frequency
and the integration bandwidth: the larger these quantities, the smaller the uncertainty. The formulas that arepresented in the present work allow the analyst to quantify this uncertainty in a computationally cheap way,
and therefore to estimate the error that is made when approximating the sound ﬁeld in a speciﬁc room andat a particular frequency (band) with the mean diffuse ﬁeld value. Special attention was given to the detailed
numerical veriﬁcation and experimental validation of the newly derived formulas.

ACKNOWLEDGMENTS
This research was funded by the European Research Council (ERC) Executive Agency, in the form of an
ERC Starting Grant provided to Edwin Reynders under the Horizon 2020 framework program, project 714591
VirBAcous. The ﬁnancial support from the European Commission is gratefully acknowledged.

REFERENCES[1] K. Ebeling. Statistical properties of random wave ﬁelds. In W. Mason and R. Thurston, editors, Physical acoustics Vol. XVII,
pages 233–310. Academic Press, Orlando, FL, 1984.
[2] R. Langley. Numerical evaluation of the acoustic radiation from planar structures with general bafﬂe conditions using wavelets.
Journal of the Acoustical Society of America, 121(2):766–777, 2007.
[3] R. Langley and A. Brown. The ensemble statistics of the band-averaged energy of a random system. J. Sound Vib., 275(3–5):847–857, 2004.
[4] R. Langley and A. Brown. The ensemble statistics of the energy of a random system subjected to harmonic excitation. J. Sound
Vib., 275(3–5):823–846, 2004.
[5] R. Langley, J. Legault, J. Woodhouse, and E. Reynders. On the applicability of the lognormal distribution in random dynamicalsystems. J. Sound Vib., 332(13):3289–3302, 2013.
[6] R. Lyon. Needed: a new deﬁnition of diffusion. Journal of the Acoustical Society of America, 56(4):1300–1302, 1974.
[7] P. Shorter and R. Langley. On the reciprocity relationship between direct ﬁeld radiation and diffuse reverberant loading.
Journal of the Acoustical Society of America, 117(1):85–95, 2005.
[8] R. Weaver. On the ensemble variance of reverberation room transmission functions, the effect of spectral rigidity. J. Sound
Vib., 130(3):487–491, 1989.

225