PROCEEDINGS of the23rd International Congress on Acoustics
9 to 13 September 2019 in Aachen, Germany

Pertinence of a simplified plane wave model for reverberationenergy decays in rooms with a pair of parallel surfaces
Jean-Jacques EMBRECHTS1
1

University of Liege, Montefiore Institute, Acoustics Laboratory

ABSTRACTAn analytical model has been developed recently to take into account specular and diffuse reflections inrooms with a pair of parallel surfaces. Analytical models can help to better understand the influence ofscattering coefficients’ values in those rooms in which unusually long reverberation times can be created bylocked-in rays. Also, analytical models can be used complementary to ray-tracing methods to quickly
estimate the late part of reverberation decays. The developed model is based on the formalism of soundparticles emitted in the room by a point isotropic source. Some approximations lead to an intuitive analyticalmodel including the influence of absorption and scattering coefficients on reverberation. The purpose of thispaper is to develop and analyse the pertinence of an additional simplification to the model, based on theassumption of plane waves propagating between the two parallel walls. In this case, the model predicts areverberation in the room composed of two exponential decays. The pertinence will be analysed bycomparing this simplified plane wave model with reverberation decays computed by a sound ray program.
Keywords: Reverberation decays, scattering

1. INTRODUCTION
The influence of surface scattering on the reverberation in rooms has been deeply investigated in
the last decades. In particular, the author of this paper has developed a geometrical acoustics approachlinking surface scattering and reverberation decays (1). More recently, a simple analytical model ofthis approach was proposed for rooms with one pair of paralle l surfaces (2). In these rooms, long andperiodic propagation paths or ‘locked-in rays’ can create non-exponential reverberation decays whichstrongly depend on the surfaces’ scattering properties.
This analytical model will be summarized in the next section. It is shown that the reverberationenergy decay can be expressed as the sum of two exponential decays, one of them being weighted by aଵfunction ቀ మ ቁ, where ‘t’ is the time after the source’s cut-off. The purpose of this study is to analyze the௧pertinence of a further simplification to this model consisting in assuming a plane wave propagation ofthe remaining energy between the two parallel surfaces.

2. THE MPW MODEL (MIXED REFLECTIONS BY PARALLEL WALLS)
2.1 Sound particles
The model first considers a sound source emitting an impulse of energy E s at time t=0. This energy
is represented by sound particles. Some particles are emitted in the direction of the two parallelsurfaces A1 or A2 (see Figure 1) and a fraction of them may enter a back-and-forth mode of propagation
between both surfaces, depending on their scattering coefficients. The energy transported by all thoseparticles is called the ‘specular flux’ Ȱ௦௣௘௖ ሺ‫ݐ‬ሻ. All the other particles (reflected by the other surfaces,
either specularly or diffusely) are supposed to follow a ‘diffuse’ mode of propagation characterized by
an exponential energy decay ݁ ିఊ௧ : this is an assumption of the model partly justified by the diffuse
reflections in the room and partly by the quasi-isotropic distribution of the image sources not createdby the pair of parallel surfaces (1,2). The value of the decay rate J only depends on the absorption
properties of all surfaces in the room (1,2) and is given by:
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jjembrechts@uliege.beߛ ൌ ݉ܿ ൅ ܿߙሺ݀ܵሻන݀ܵͶܸ ௌ ͳ െ ͲǤͷߙሺ݀ܵሻ

(1)

In this formula, m is the air attenuation constant, c is the sound propagation speed, V and S are thevolume and total surface of the room respectively and D(dS) is the absorption coefficient of the surface
element dS.

Figure 1 – The two parallel surfaces in the room are L meters apart. The sections A1 and A2 are included inthese surfaces and they define the largest prism connecting them perpendicularly. The sound source must be
situated inside this prism.
2.2 Specular and diffuse fluxes
After n reflections (n even) on A 1 and A 2, it is shown in (2) that the specular flux is expressed by:ȳ௡(2)ሺܴ ܴ ሻ௡Ȁଶʹߨ ଵ ଶ
where tn is the time between specular reflections of orders (n) and (n+1) on A 1 and A 2 , which can be௡௅
approximated by ቀ ቁ if the number of reflections n is ‘great enough’, : n is the solid angle subtended௖by A 1 or A2 from the corresponding image sources of order n and R1 and R2 are the specular reflection
coefficients of these two surfaces. In (2), the decrease of the solid angle : n is taken into account as஺ȳ௡ ؆  ሺ௡௅ሻమ if n is ‘great enough’ (A being the common area of A 1 and A 2 ). Introducing the decayȰ௦௣௘௖ ሺ‫ݐ‬௡ ሻ ൌ  ‫ܧ‬௦constant ߚ ൌ  െ௖ଶ௅ሺܴଵ ܴଶ ሻ finally leads to:݁ ିఉ௧೙(3)Ȱ௦௣௘௖ ሺ‫ݐ‬௡ ሻ ൌ  ‫ܧ‬௦ ൬൰ʹߨܿ ଶ‫ݐ‬௡ଶEquation 3 gives the remaining energy of the specular flux at time tn after the impulse emission by
the source. This function of time decreases more or less by discrete steps, but an approximatedcontinuous decrease is simply obtained by replacing tn by t in eq.3.
The diffuse flow of sound particles contains all particles not travelling between A 1 and A 2 throughspecular reflections. By assumption, the corresponding energy decreases proportionally to ݁ ିఊ௧ .
Moreover, at each new reflection on A 1 and A 2 , additional sound particles (and thus energy) join thisଶ௅
flow, mostly those diffusely reflected by A 1 and A 2. It is shown in (2) that, if ቀο‫ ݐ‬ൌ ቁ is the time௖interval including two successive specular reflections on A 1 and A 2, the energy remaining in thediffuse flux at time (tn +' t) is:ഥ ൅ ߥܴ‫ܦ‬തതതത ሻȰௗ ሺ‫ݐ‬௡ ൅ ο‫ݐ‬ሻ ൌ  Ȱௗ ሺ‫ݐ‬௡ ሻ݁ ିఊο௧ ൅ Ȱ௦௣௘௖ ሺ‫ݐ‬௡ ሻ‫ܦ‬ଵଶ ‫ܦ‬ଵଶ ൌ ߥ ିଷȀଶ ሺ‫ܦ‬(4)ഥ ൌ ͲǤͷ ‫ כ‬ሺ‫ܦ‬ଵ ൅ ‫ܦ‬ଶ ሻ ,D1 and D2 are the diffuse reflection coefficients of A 1 and A 2 respectively, ‫ܦ‬6029

തതതതܴ‫ ܦ‬ൌ ͲǤͷ ‫ כ‬ሺܴଶ ‫ܦ‬ଵ ൅ ܴଵ ‫ܦ‬ଶ ሻ and ߥ ൌ ݁ ఊ௅Ȁ௖ .
2.3 Solution for the total remaining energy and the reverberation decay
In (2), a continuous expression is found for Ȱௗ ሺ‫ݐ‬ሻ by solving a differential equation based on eq.4.
This expression is combined with eq.3 to give the total remaining energy in the sound particle flow attime t after the impulse emission by the source, provided t is ‘great enough’. Indeed, someapproximations and assumptions have been used, such that the early part of the decay cannot bepredicted by the resulting analytical expression.
The model, called in (2) the MPW model, is rather expressed not as an impulse response, but as theusual energy decay curve or reverberation decay. Applying the Schroeder backward integration to theprevious ‘impulse response’ leads to:݁ ିఊ௧݁ ିఉ௧(5)൅  ܹ௦ ܳଵଶߛߚ‫ ݐ‬ଶIn this expression, Ws is the acoustic power of the source and Q12 is a parameter depending on theacoustic properties (absorption and scattering) of the parallel surfaces (A 1 and A2 ) and on the decay
constants J and E . The constant Cd in eq.5 could be fixed if the decaying energy could be evaluated atsome particular time of the reverberation decay, for example at t=0. However, as the MPW modelcannot predict the early part of the decay, another approximation is used in (2).
We must here recall the main purpose of this model. We are not searching for a detailed model of theearly part of the decay, but rather for an analytical model of the total energy decay in the room, fromwhich we could extract an ‘average’ reverberation time or a ‘global’ (‘non-local’) description of thereverberation decay including the influence of absorption and scattering coefficients.
With this aim in mind, the early part of the reverberation decay is si mply approximated by an
exponential decay with J as the decay rate. This approximation is extensively motivated in (2). Also,
the initial time ti from which the eq.5 can be applied has been chosen as the time corresponding to thefirst 10dB decrease in the energy decay curve ሺ‫ݐ‬௜ ൌ ሺͳͲሻ Ȁߛሻ. This has been considered as a goodcompromise between the ‘asymptotic’ assumption of the model ( t great enough) and the fact that tishould also belong to the early part of the decay (see (2) for more details). With this choice, theconstant C d in eq.5 could be fixed.
‫ܦ‬௦ ሺ‫ݐ‬ሻ ൌ  ‫ܥ‬ௗ 

2.4 Test of the MPW model
As several approximations have led to the MPW analytical model, a great number of comparisons
have been made with reverberation decays computed with a ray tracing program, in or der to validate
the model. These comparisons included tests in shoebox rooms with different dimensions andcombinations of absorption and scattering coefficients. Results are discussed in (2), some comparisons
will be illustrated in the next section.
The adequacy of the model is measured by the correspondence between the reverberation time T 30
computed by both models (MPW and ray tracing), the second one being considered as the reference.
The correlation between the corresponding values of T 30 has been shown to be excellent, except for one
particular room with a very flat geometry. In any case, the predictions of the MPW model were muchbetter than the classic diffuse field theory. Except for the very flat room, it is shown in (2) that theaverage deviation between the values of T 30 computed by both models was less than 15%.

3. SIMPLIFIED PLANE WAVE MODEL FOR THE ENERGY DECAY CURVE
3.1 Development of the model
A further simplification of the MPW model can be developed, which leads to a reverberation energy
decay expressed by the sum of two exponential decays. This kind of ‘pure’ exponential model has
already been suggested by some authors to simplify the reverberation decay analytical expression (3,4).
The relevance of a ‘pure’ exponential model will be investigated in the following, but first of all wedevelop the model.
Starting from the sound particles model described in the previous section, suppose that, after N(even integer) reflections, the flow of ‘specular’ particles travels perpendicularly to the surfaces A 1and A 2, just like plane waves would propagate between both surfaces. Another way of expressing thisis to assume that the corresponding image sources are ‘very far’ from A 1 and A 2, close to infinity. Inthis case, eq.2 expressing the energy of the specular flux becomes (for n even):

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ȳே
(6)
ሺܴ ܴ ሻ௡Ȁଶʹߨ ଵ ଶ
Regarding the diffuse flux of particles, eq.4 can be converted with the ‘plane wave’ assumption attime (tN+2't) as:Ȱ௦௣௘௖ ሺ‫ݐ‬௡வே ሻ ൌ  ‫ܧ‬௦Ȱௗ ሺ‫ݐ‬ே ൅ ʹο‫ݐ‬ሻ ൌ  ൫Ȱௗ ሺ‫ݐ‬ே ሻ݁ ିఊο௧ ൅ Ȱ௦௣௘௖ ሺ‫ݐ‬ே ሻ‫ܦ‬ଵଶ ൯݁ ିఊο௧ ൅  Ȱ௦௣௘௖ ሺ‫ݐ‬ே ሻ ሺܴଵ ܴଶ ሻ‫ܦ‬ଵଶ

(7)which can be generalized as (for any integer value m >0):௠ିଵȰௗ ሺ‫ݐ‬ே ൅ ݉ο‫ݐ‬ሻ ൌ  Ȱௗ ሺ‫ݐ‬ேሻ݁ ିఊ௠ο௧൅ Ȱ௦௣௘௖ ሺ‫ݐ‬ே ሻ‫ܦ‬ଵଶ  ෍ ሺܴଵ ܴଶ ሻ௞ ݁ ିሺ௠ିଵି௞ሻఊο௧ (8)௞ୀ଴or after evaluation of the sum by:Ȱௗ ሺ‫ݐ‬ே ൅ ݉ο‫ݐ‬ሻ ൌ  Ȱௗ ሺ‫ݐ‬ே ሻ݁ ିఊ௠ο௧ ൅ Ȱ௦௣௘௖ ሺ‫ݐ‬ே ሻ‫ܦ‬ଵଶ ݁ ିሺ௠ିଵሻఊο௧ ቆ‫ܤ‬ఔଶ௠ െ ͳቇ‫ܤ‬ఔଶ െ ͳ (9)In the previous formula:‫ܤ‬ఔଶ ൌ  ሺܴଵ ܴଶ ሻ݁ ఊο௧ ൌ  ݁ ሺఊିఉሻο௧ (10)Equations 6 and 9 can then be added to obtain the remaining energy in the flow of sound particles
at time (tN+m' t) after the impulse emission by the source. As explained in section 2, the remaining
energy decreases more or less by successive steps as a function of time, but we can approximate thisstepwise decrease by a continuous function, if (m' t) is simply replaced by any W >0 in the model.
Ȱ௧௢௧ ሺ‫ݐ‬ே ൅ ߬ሻ ൌ  Ȱௗ ሺ‫ݐ‬ே ሻ݁ ିఊఛ ൅ Ȱ௦௣௘௖ ሺ‫ݐ‬ே ሻቀ݁ ିఉఛ ൅  ‫ܭ‬ఔ ݁ ିఊఛ ൫݁ ሺఊିఉሻఛ െ ͳ൯ቁ஽ ௘ ംο೟ with the parameter ‫ܭ‬ఔ ൌ  భమ
஻మ

(11)

. If we group the terms proportional to each exponential decay andഌ ିଵ஺஺approximate ȳே ؆  ሺே௅ሻమ ൌ మ మ , the following expression is obtained (for t > tN):௖ ௧ಿȰ௧௢௧ ሺ‫ݐ‬ሻ ൌ  ൫Ȱௗ ሺ‫ݐ‬ே ሻ െ Ȱ௦௣௘௖ ሺ‫ݐ‬ே ሻ‫ܭ‬ఔ ൯݁ ఊ௧ಿ ݁ ିఊ௧ ൅  ‫ܧ‬௦‫ܣ‬ሺͳ ൅ ‫ܭ‬ఔ ሻ݁ ିఉ௧ ʹߨܿ ଶ ‫ݐ‬ேଶ(12)Finally, the reverberation energy decay is obtained by backward integration of eq.12 between t > tNand infinity. If the symbol Cd is used for the product of all constant factors multiplying ݁ ିఊ௧ , then the final
expression for the energy decay is very similar to the MPW model (see eq. 5):‫ܦ‬௦ ሺ‫ݐ‬ሻ ൌ  ‫ܥ‬ௗ ‫ܣ‬ሺͳ ൅ ‫ܭ‬ఔ ሻ ݁ ିఉ௧݁ ିఊ௧൅  ܹ௦ߛʹߨܿ ଶ ߚ‫ݐ‬ேଶ(13)The significant difference with the MPW model is that we have now the sum of two ‘pure’
exponential reverberation decays (tN instead of t in the denominator of the second term).
3.2 Model for the early decay
The constant Cd in eq.13 will be fixed as explained in section 2.3 for the MPW model. The same
assumption of an exponential early energy decay is used between times 0 and tN, with J as the decay rate. In
other words, the early reverberation decay is supposed to be fixed by the average absorption in the room only.
We recall that this assumption is not intended to accurately model the early reverberation decay, but the goalis only to give an initial condition to eq.13. The justification is the same as in (2).
‫ܣ‬ሺͳ ൅ ‫ܭ‬ఔ ሻ ߛ݁ ሺఊିఉሻ௧ಿ‫ܥ‬ௗ  ൌ  ܹ௦ ቆͳ െ ቇʹߨܿ ଶߚ‫ݐ‬ேଶ

(14)

Regarding the initial time tN, a first proposal is to adopt the same value as in (2), i.e.ሺ‫ݐ‬ே ൌ ሺͳͲሻ Ȁߛሻcorresponding to the first 10dB of the decay. Double this value will also be tested (corresponding to the6031

first 20 dB of the reverberation decay).
An additional procedure is needed if Cd <0 in eq.14. This situation must be avoided as it can lead to
unrealistic energy decays, as shown in (2). In that case, the same initial value of the decay tN is used, but
the constant Cd is set to zero and another decay rate is selected for the exponential early decay (݁ ିఓ௧ withPJ).
3.3 Test of the simplified model
The simplified model expressed by eqs. 13 and 14 is now tested by comparison with a ray tracing
program and the MPW model.
A shoebox room similar to the one used in (2) is selected: the length L in Figure 1 is set to 30m,
while the two other dimensions (width and height) are 10m. All the lateral surfaces are diffusely
reflecting, with a variable absorption coefficients. Only the surfaces A 1 and A 2 can reflect specularly,
with a scattering coefficient value being equal to either 0, 0.3 or 1. The sound source is situated in the
middle of the room in this study and several receptors are distributed in the volume for the ray tracing
program.
Regarding the distribution of absorption in the room, eight tests cases are defined in Table 1: A1 andA2 share the same value D12 while all other surfaces have the same absorption coefficient Dothers.
Table 1 – The different combinations of absorption coefficients
Test case

1234

56

78

Dothers

0.2

0.2

0.2

0.5

0.5

0.8

0.8

0.8

D120.1

0.2

0.4

0.1

0.4

0.1

0.2

0.4

If the scattering coefficients of A 1 and A 2 are both equal to 1, then all reflections in the shoebox
room are of the diffuse type. In that case, the energy decays computed by the sound ray program arevery similar from one receptor to another and also quite exponential (linear decay in dB), whatever thecombination of absorption coefficients. Also, both the MPW and the simplified model lead to the same
results: since R 1=R 2=0, the decay rate ߚ ՜ λ and the decay is simply exponential in both cases, withthe decay rate J . The resulting reverberation times T 30 are close to those predicted by the ray tracing
program (averaged over all receiving positions).
More interesting is the test case in which specular reflections are defined on A 1 and A 2 (scatteringcoefficients equal to 0). Now, the shape of the reverberation energy decays predicted by the ray tracing
program strongly depends on the combination of absorption coefficients, as shown in Figure 2 (greencurves). It also significantly depends on the receiving position in this shoebox room, so only theaverage decay is compared with the MPW and the simplified models. Indeed, those models do notpredict local variations of the reverberation, but only the decay of the global energy.
It is first shown in Figure 2 that the MPW model (purple curve) developed in (2) correctly predicts
the reverberation decays (compared with ray tracing), for all combinations of absorption coefficients.
This is just an illustration of the discussion made in section 2.4 about the validity of this model. The
reverberation times T 30 have also been computed from these energy decays. The average relative
difference between the T 30 values issued from ray tracing and the MPW model is 8% for the cases
illustrated in Figure 2, the maximum being 20% for test case number 4.
However, the predictions of the simplified model developed in sections 3.1 and 3.2 are far fromaccurate. If the deviations with the ray tracing results are quite limited if the absorption of the lateralwalls is weak (cases 1 to 3), they are highly significant for all other cases. Not only the energy decay,
but also the reverberation times are badly predicted. Another point is the clear discontinuity of theslope between the early decay and eq.13 in most cases, when the decay rate E is much less J. This
discontinuity can also exist in the energy decays computed by the MPW model, but it is much lessଵvisible, due to the smoothing effect of the factor ቀ௧ మ ቁ.
In Figure 2, the initial time has been defined as ሺ‫ݐ‬ே ൌ ʹሺͳͲሻ Ȁߛሻ, i.e. corresponding to the first
20dB of the early decay. Changing this value to ሺ‫ݐ‬ே ൌ ሺͳͲሻ Ȁߛሻ does not modify the conclusions ofthe comparison.

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Figure 2 – Comparisons of the reverberation energy decays computed in the shoebox room by the ray tracing
program (at a given receptor, close to average results), the MPW model and the simplified “plane wave”(PW) model with ሺ‫ݐ‬ே ൌ ʹሺͳͲሻ Ȁߛሻ. The eight tests cases (depending on absorption coefficients) arepresented in order (see table 1), from left to right and up to down. The surfaces A1 and A2 are specularlyreflecting.

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4. CONCLUSIONSThe question asked at the beginning of this study was if it was relevant to still simplify the MPW
analytical model of the reverberation energy decay in a room with a pair of parallel surfaces. Thismodel was developed by the author of this paper in (2) and led to an expression of the decay as the sumଵof two exponential decays, the second one being multiplied by ቀ మ ቁ. The simplification consists in௧assuming plane wave propagation between the two parallel surfaces, from a certain time of the decay.
This results in a modified expression of the decay as the sum of two ‘pure’ exponential decays , a modelthat has already been proposed by other authors.
Several comparisons have been done with the decays predicted by a sound ray program which isconsidered as the reference method. In particular, it has been shown in section 3 that, for a particularshoebox room, the simplified model gives significantly worse predictions than the MPW model. This
example is the only one illustrated in this paper, but bad predictions have also been observed in manyother cases.
It appears that no clear adaptation of the model’s parameters could improve the accuracy of thesimplified model. The main problem is well illustrated in Figure 2, i.e. the (reference) energy decays
predicted by the ray tracing program have a continuously decreasing slope (in dB), something thatଵ
cannot be accounted for by the ‘pure’ exponential decay, but well by the factor ቀ మ ቁ.
௧This confirms the necessity of taking this into account in the MPW model.

REFERENCES
1. Embrechts JJ. A geometrical acoustics approach linking surface scattering and reverberation in room
acoustics. Acta Acustica united with Acustica 2014;100:864-879.
2. Embrechts JJ. An analytical model for reverberation energy decays in rooms with specular and diffuse
reflections. J Acoust Soc Am. 2019;145(4):2724-2732.
3. Sakuma T. Approximate theory of reverberation in rectangular rooms with specular and diffuse
reflections. J Acoust Soc Am. 2012;132(4):2325-2336.
4. Hanyu T. A theoretical framework for quantitatively characterizing sound field diffusion based on
scattering coefficient and absorption coefficient of walls. J Acoust Soc Am. 2010;128(3):1140-1148.

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