Accounting for delay of energy transfer between coupled rooms in statistical-acoustics models of reverberant-energy decay^a)

Systems of coupled rooms are ubiquitous in the built environment as a result of unintentional architectural decisions and also continue to find intentional use as a tool in the acoustical design of auditoria and other performance spaces. Due to increased interest over the last ten years, many numerical techniques are now available for modeling sound fields in systems of coupled rooms, particularly in the high-frequency geometrical-acoustics (GA) limit.^1,2 Yet the first such technique developed, based on diffuse-field statistical-acoustics (SA) models (as described in Refs. 3 and 4), continues to find application in design due to the speed and ease of use it offers. While it is well known that SA models of this type must make multiple assumptions and approximations, they can offer reasonable accuracy when certain conditions are met.³ Accuracy can often be further improved through semi-empirical corrections that heuristically account for violations of the underlying assumptions.³ Among the assumptions inherent in SA models of systems of coupled rooms is the implicit treatment of energy transfer between subrooms as instantaneous; a condition that cannot be met by any physical system.⁵ In the following, modifications to a previously introduced SA model³ are developed that explicitly account for the finite travel time of acoustic energy between subrooms. The effects of these modifications are illustrated for a simple two-room coupled system previously considered in Ref. 3 and the predictions of the modified SA model are then compared with predictions of a validated computational GA model.¹ 

The reverberant sound field in a system of N coupled rooms has been traditionally described by an SA model in which ɛ_i, the energy density associated with the ith subroom, is given by the solution of a system of coupled ordinary differential equations (ODEs)³ 

where η is a parameter that is determined by the decay model (i.e., Sabine, Eyring, or Kuttruff, as described in Ref. 3, $ζi$ is the decay rate of the ith room in the uncoupled state due to absorption alone (given by $cSiα¯i/8Vi$⁠), $Si$ is the surface area of the ith room, $α¯i$ is the mean absorption coefficient of the wall surfaces in the ith room, $Vi$ is the volume of the ith room, and $Sij$ is the coupling area between the ith and jth rooms. Solutions of Eq. (1) are of the form $ɛi(t)=∑ɛj0(i)exp(−2δjt)$⁠, where, in the two-room case for which $ζ1>ζ2$⁠, $2δ1>η1(2ζ1+cS12/4V1)>η2(2ζ2+cS12/4V2)>2δ2$⁠.

One can object to the shift in eigenvalues induced by coupling in so much as it implies in the two-room case with $ζ1>ζ2$ that coupling causes the effective absorption coefficient of S₁₂ in Room 1 to be greater than one.⁵ But this anomaly should not be surprising. The case of an uncoupled room with an aperture modeled as an open window is physically equivalent to being coupled by the aperture to a perfectly dead room (reverberation time $T=0$⁠). Because SA models are valid only in the small-absorption limit⁴ and can never predict $T=0$⁠, there can be no expectation of recovering the uncoupled limit using an SA model of coupled rooms.

Even so, it remains true that the model expressed by Eq. (1) implicitly assumes that the exchange of energy between the subrooms is instantaneous in so much as the rate change of the uniform energy density in the ith subroom at a particular time $dɛi(t)/dt$ is related to $ɛj(t)$⁠, the uniform energy density in adjacent subrooms at that same time. As Lyle has observed,⁵ this assumption is not physically accurate due to the finite velocity of sound. Instead, one might reason that $dɛi(t)/dt$ should depend on past values of $ɛj(t)$⁠. In Ref. 5 Lyle proposes a model in which $dɛi(t)/dt$ depends on $ɛj(t−τ¯ij)$⁠, where $τ¯ij$ is the travel time between the centroids of the ith and jth rooms. This leads to a modified form of Eq. (1)

The delay term introduced here should be significant when $τ¯ij$ is sufficiently large compared to the reverberation times of the rooms, as in coupled-room concert halls or other spaces with a similar relationship between decay rate and physical dimension. Analytical solution of the system of delay differential equations (DDEs) is not trivial⁵ but robust algorithms for solving systems of DDEs numerically are widely available (see, e.g., the Runge–Kutta approach described in Ref. 6).

Though Eq. (2) better models the dependence on the past required by the finite speed of sound, use of a single delay value is an approximation. A more physically realistic model should account for a continuous distribution of delay values corresponding to the distribution of possible paths between points in adjacent rooms. (This is analogous to the distribution of free paths in a single-volume room.⁷) Taking $fij$ to be the probability density function of delay values between the ith and jth rooms, the energy density can be modeled by a system of coupled functional differential equations (FDEs)

in which the delay is bounded on the left by zero, as causality demands, and continuous from the right from a maximum value $τ^ij$ given by the greatest distance between any two points in ith and jth rooms. This system of distributed-delay integro-differential equations must be solved numerically, using techniques such as the Runge–Kutta approach described in Ref. 8. In the limit that all $fij$ approach delta functions centered at $τ¯ij$⁠, Eq. (3) reduces to Eq. (2). However, in the general case, $fij$ will have a more complicated form and solutions to Eq. (3) may be sensitive to small perturbations in the distribution.

To illustrate the effects of incorporating travel time between subrooms in the SA model, predictions of Eqs. (2) and (3) are compared with those of Eq. (1) for a simple two-room coupled system, shown in Fig. 1, previously considered in Ref. 3. These predictions are compared with predictions of a validated computational GA model based on beam-axis tracing with late part ray tracing.¹ In all cases the GA predictions shown represent the ensemble average (as described in Ref. 3) of decay curves for all possible source-receiver combinations in Room 2 (source positions B0–B3 and receiver positions 1–9) with the number of beam axes/rays chosen to ensure negligible systematic and random error.^1,3 To ensure ergodicity and mixing, a Lambert scattering coefficient of 0.99 was assigned to all surfaces.³ 

Figure 2 compares predictions of Eqs. (1) and (2) for four different aperture conditions (S₁₂ = 25, 50, 100, and 200 m²) and two values of η, corresponding to Eyring and Kuttruff decay models within the subrooms. Solutions to Eq. (2) are computed by the method described in Ref. 6.

Figure 3 compares predictions of Eqs. (1) and (3) for two different aperture conditions (S₁₂ = 25 and 200 m²) and two values of η, corresponding to Eyring and Kuttruff decay models within the subrooms. Solutions to Eq. (3) are computed by the method described in Ref. 8. The distribution of $fij$ was calculated by a Monte Carlo approach that generated a large number of uniformly distributed random positions in each subroom and computed the distance between all pairs of points in adjacent rooms that were visible to one another given the particular configuration of the coupling aperture (single aperture, centered in the wall, square in shape).

In all cases considered here, incorporating delay between subrooms slows the decay from steady state. This is expected because the decay curves considered are in the less-reverberant subroom such that contributions from Room 1 have the effect of extending the late decay and producing a shallower second rate in the double-slope decay curve. The time lags in Eqs. (2) and (3) effectively increase the energy of these contributions, lengthening the overall decay. For smaller apertures this improves the agreement with GA predictions but only shifts the sign of the error for large apertures. The DDE model given by Eq. (2) may yield predictions that better match GA than those of Eq. (1), particularly if the Eyring decay model is used⁵ (as it tends to overpredict decay rate³). The distributed-delay model given by Eq. (3) appears to offer no advantage relative to Eq. (2) for this geometry, particularly for large apertures, for which the error is somewhat increased.

Despite incorporating greater physical realism relative to prior models, this work disagrees with the previous finding that accounting for travel time between subrooms in SA models of coupled rooms leads to a general improvement in the accuracy of predictions.⁵ In large part this is because accounting for travel time introduces additional approximations and is, in some sense, inconsistent with the underlying approximations of SA models.

As the aperture size increases, SA models fail because underlying assumptions of the model are violated.³ The limited results shown here, along with additional evaluations, suggest that the magnitude of such errors is even greater for SA models that account for finite travel time. Though Lyle was correct in citing failure of diffuse-field assumptions in subrooms for large apertures,⁵ the source of the error in DDE models for large-aperture conditions is more fundamental.

Using the Laplace transform, either Eq. (1) or Eq. (2) can be rewritten as a series of power-energy transfer functions $ɛ̃(σ)/Π̃(σ)$⁠, which express the ratio of energy density to source power in the Laplace domain for each subroom.⁹ In the limit that $Sij→0 for∀i,j$⁠, the power-energy transfer function reduces to the same expression for either Eq. (1) or Eq. (2). For Eq. (2), however, the poles of the power-energy transfer functions are notably more complicated and give different limiting behavior than Eq. (1) as S₁₂ becomes large and adjacent subrooms approach single-room behavior. The DDE model given by Eq. (2) becomes inaccurate in this limit at relatively smaller aperture sizes than the model that does not account for time delay given by Eq. (1) due to the additional approximations made in Eq. (2). As the coupling aperture becomes larger, it is increasingly inaccurate to assume that the travel time of energy between the subrooms can be estimated from the distance between their centroids (e.g., a larger number of points satisfy the visibility requirement).

While Lyle correctly observed that the transfer of energy between diffuse fields cannot be instantaneous in a physical system, the model expressed in Eq. (1) is more consistent with the physical assumptions implicit in SA, viz., nominally homogeneous energy density associated with each subroom (arrived at by a balance between geometric mixing and demixing due to absorption).⁴ Just as consistent SA models represent a discrete loss of energy by a phonon at a wall surface as a loss from the nominally homogeneous energy within the room (that represents the sum of all energies of the ensemble of classical phonons), so also are energy gains or losses from apertures represented as gains or losses to or from the homogeneous energy (i.e., a phonon is simply labeled as belonging to a new subroom)—both correspond to the assumption of an irradiation strength on the boundary consistent with nominally homogeneous energy.⁴ Corrections such as accounting for finite travel time are therefore heuristic and semi-empirical.⁴ For this reason, accounting for propagation delay does not ensure a more accurate prediction, though, just as other heuristic semi-empirical corrections introduced in Ref. 3, it may significantly improve predictions in certain cases.^3,10

A DDE model first introduced by Lyle⁵ has been incorporated into the general framework introduced by Summers³ and extended to account for distributed decay in a new FDE model. Distribution of delays between rooms in coupled systems is physically realistic but violates underlying assumptions of SA. Therefore it is an heuristic correction that can improve predictions in some cases but cannot be more accurate consistently.

Modern computational methods for GA, such as those described in Refs. 11 and 12, are capable of explicitly investigating the distribution of phonon delays between subrooms. These methods can therefore indicate those systems of coupled rooms for which delays between subrooms (and the shape of delay distributions) are a significant factor. Because FDE models are substantially more sensitive to model parameters than conventional ODE models of coupled rooms, these relationships merit further study, particularly in systems of three or more rooms including rooms of irregular shape (such as corridors).

This suggests classes of coupled-room systems for which these models may be particularly relevant and experimental investigation is warranted: (1) Auditoria having irregularly shaped auxiliary coupled rooms with coupling located remotely from the audience area, and (2) office, schools, and other buildings comprised of multiple rooms connected by corridors. Also likely to show effects of delay between subrooms are enclosures such as cathedrals in which nondiffuse energy transfer between nonadjacent subrooms³ has a significant effect.¹⁰ In such cases the distribution of delay values will have a minimum greater than zero, set by the physical distance between the coupling apertures of the subrooms.

Experimental investigations of the effects of delay between subrooms on decay curves will need to incorporate new data models, as current Bayesian methods for parameter estimation and model-order selection^2,13 rely on analytic solutions to systems of ODEs for the data model. An extension of these techniques to data models based on a numerical solution to systems of FDEs will be necessary to investigate the effects of delay.

Portions of this work were supported by the Bass Foundation and the Rensselaer Polytechnic Institute School of Architecture.