Directional sound field decay analysis in performance spaces

Abstract

The analysis of the spatio-temporal features of sound fields is of great interest in the field of room acoustics, as they inevitably contribute to a listeners impression of the room. The perceived spaciousness is linked to lateral sound incidence during the early and late part of the impulse response which largely depends on the geometry of the room. In complex geometries, particularly in rooms with reverberation reservoirs or coupled spaces, the reverberation process might show distinct spatio-temporal characteristics.

In the present study, we apply the analysis of directional energy decay curves based on the decomposition of the sound field into a plane wave basis, previously proposed for reverberation room characterization, to general purpose performance spaces. A simulation study of a concert hall and two churches is presented uncovering anisotropic sound field decays in two cases and highlighting implications for the resulting temporal evolution of the sound field diffuseness.

Keywords

Sound field analysis diffuseness isotropy sound field decay directional reverberation

Introduction

Even though the theory of a diffuse sound field is most fundamental to many applications and measurement procedures, involving Sabine’s theory of sound, a robust and well accepted quantitative measure is currently lacking.¹ Joyce² presented theoretical reflections on the ergodicity of sound field decay and the relationship to the idea of a diffuse sound field as well as inter-dependencies between the distribution of boundary conditions imposed by the absorption on room surfaces and sound field diffuseness. Kuttruff³ showed however that the shape of the decay curve in non-coupled spaces is primarily related to the distribution of boundary conditions and as such may not suffice as a strict criterion to quantize sound field diffuseness. As a result multiple authors proposed modified versions of Sabine’s decay equation for non-uniform distribution of boundary conditions or non-diffuse sound fields, which are however usually only valid for a limited number of special cases.⁴ An overview is found in Kuttruff⁴ and Stephenson.⁵

In room acoustics the diffuse sound field is also often linked to the idea of a mixing time, after which a sound field is assumed to be characterized as diffuse and thus marking a temporal separation into an interval containing the direct sound together with a number of early reflections—which can be attributed to distinct directions of incidence—and an interval containing the reverberant part of the sound field with negligible directional dependence.⁶ This idea is most often exploited to reduce the computational effort when rendering the acoustics of a room in virtual acoustic environments by approximating the late reverberation as a stochastic process waiving information on the geometry of the space.^7,8 More recently, efforts were made to extend such approaches to include directional information, while at the same time not necessitating geometrical information.⁹

Several single or dual microphone methods to estimate the diffuseness of a sound field have been made, based on the smoothness of the energy decay curve (EDC),^10,11 the temporal structure of the room impulse response (RIR),¹² or statistical measures exploiting the correlation coefficient^13
–16 as well as higher order statistical moments.¹⁷ The perceived diffuseness of a sound field has been linked to the inter-aural cross correlation (IACC).¹⁸ However, especially correlation based estimators prove to be sensitive to already a small number of incoming waves due to their non-linear response. Similarly to the estimation of diffuseness, a large number of methods for the estimation of the mixing time, based on the geometry of a room, the reflection density, or again stochastic measures have been proposed by various authors. An overview over a selection of mixing time estimators, as well as a perceptual study were presented by Lindau et al.¹⁹ Aiming at the separation of the sound field into directional and diffuse components for the application in sound reproduction systems, Pulkki and co-authors proposed a diffuseness estimator hinging on the relation between the time averaged intensity vector and the total energy of the sound field.^20
–22 The approach was later used by Götz et al.²³ for the estimation of the mixing time in a room with simple geometry. Nevertheless, Epain and Jin²⁴ showed that the method suffers from a very high sensitivity in sound fields comprised of only few sound waves. Since the fundamental definition of the diffuse sound field is its isotropy condition—requiring it to be composed of an infinite number of uncorrelated waves with incidence directions distributed uniformly over the spherical domain—arrays of microphones of high order prove to be a well suited analysis tool. Early works by Thiele²⁵ were extended to microphone array based approaches from Gover et al.^26,27 and more recently Epain and Jin²⁴ and Nolan et al.²⁸ However, aforementioned studies were either limited to the steady-state sound field or in the case of Gover to a very coarse temporal resolution.

Recently, Berzborn et al.^29,30 presented studies of the angular distribution of incident energy as well as the sound field isotropy during the decay process in a reverberation room. The analysis method is based on energy decay curves including directional information calculated by applying the Schroeder integral to a decomposition of the sound field into a plane wave basis. A clearly anisotropic decay with additionally decreasing isotropy during the decay was identified in some reverberation room configurations.³⁰ A similar study of the isotropy condition in the decaying sound field was presented by Nolan et al.³¹ Alary et al.³² used a slightly modified version of the method presented in Berzborn and Vorländer²⁹ for the analysis of sound fields in a narrow corridor as well as a church captured with a 32-channel spherical microphone array (SMA).

In the this paper, we apply the analysis of the directional energy decay curves (DEDCs) to rooms with more general purpose and arbitrary use-cases, aiming at the detection anisotropic features in the decay process such as reverberation reservoirs with distinct directions. We further investigate their influence on the isotopy condition of the sound field which we calculate from the DEDCs. The following section introduces the decomposition of a sound field captured in the spherical harmonic domain into plane waves and the subsequent calculation of the DEDC as well as the estimation of the sound field isotopy. A simulation study of three rooms, a rectangular-like concert hall, a domed baroque church, and a gothic art church, is introduced. Finally, the results are presented and discussed, followed by the conclusions given in the last section.

Directional sound field decay analysis

SMAs allow for capturing directional room impulse responses (DRIRs) comprising spatial information on sound field in rooms, by spatially sampling the sound field in a spherical domain or volume. They further allow for an elegant representation of the sound field as a set of Fourier coefficients in the spherical harmonic (SH) domain independent of the array geometry used for capturing.³³

Directional room impulse responses

Assuming a sound field composed of a sum of $L$ plane waves, their resulting density function in a single receiver point can be expressed as the SH coefficients³³

h_{n m} (t) = \sum_{l = 1}^{L} s_{l} (t) Y_{n}^{m} (θ_{l}, ϕ_{l}),

(1)

where $Y_{n}^{m} (θ_{l}, ϕ_{l})$ are the real-valued spherical harmonic basis functions of order $n$ and degree $m$ evaluated at for the angle of incidence of the sound wave onto the receiver array—described by the co-latitude and azimuth angles $(θ_{l}, ϕ_{l})$ —and the signal $s_{l} (t)$ carried by the wave. Note that the angular properties of the sound field described by equation (1) are preserved by the encoding into the respective SH series expansion, allowing to omit the angular dependency of $h_{n m} (t)$ . Equation (2) is also referred to as plane wave composition or SH expansion of the sound field³³ and is measured in Pascals. In the context of DRIRs captured with arrays of microphones, $s_{l} (t)$ is the impulse response of the impinging wave, describing the corresponding propagation path from an arbitrary source point to the receiver point including interactions with the room surfaces, that is, the impulse response of the respective reflection path. In the following we will also refer to equation (1) as the spherical harmonic domain directional room impulse response (SH-DRIR). Note that equation (1) does not include a physically extended array geometry but is purely valid for a single point while still retaining the angular information on the sound field. It is the time domain solution to the interior problem for a spherical domain, cf. Williams,³⁴ and can be calculated from array measurements³³ or simulated numerically.³⁵ For the remainder of this paper we will assume that the sound field is readily available in the form of equation (1), consequently assuming an ideal equalization of the array geometry and therefore omitting the limitations to the operating frequency range of SMAs,³⁶ while at the same time simulating the limitations to the achievable angular resolution due to the discrete angular sampling of the sound field. Consequently, the analysis method presented in the following can be applied to measured SH-DRIRs without modification when a suitable array geometry for the frequency range of interest is chosen.³⁶ For additional information on solving for the interior problem from physical array measurements, the reader is referred to additional literature.^33,34

Plane wave decomposition

By applying plane wave decomposition beamforming we decompose the captured sound field into a discrete set of $Q$ plane waves, yielding the spatial domain plane wave density function evaluated at the $q' th$ steering direction $(θ_{q}, ϕ_{q})$ ³⁷

a (t, θ_{q}, ϕ_{q}) = \frac{4 π}{{(N + 1)}^{2}} y_{n m}^{T} (θ_{q}, ϕ_{q}) h_{n m} (t),

(2)

where

y_{n m} (θ_{q}, ϕ_{q}) = {[Y_{0}^{0} (θ_{q}, ϕ_{q}), \dots, Y_{N}^{N} (θ_{q}, ϕ_{q})]}^{T},

(3)

is the SH steering vector containing the ${(N + 1)}^{2}$ coefficients for a respective maximum order $N$ . Analogously, the vector $h_{n m} (t)$ contains the Fourier coefficients of the SH-DRIR defined in equation (1). An exact solution may be found if the directions of incidence $(θ_{l}, ϕ_{l})$ and the steering directions $(θ_{q}, ϕ_{q})$ coincide. Since the directions of incidence are generally not known a priori, a discrete, but dense grid of $Q$ steering directions uniformly covering the unit sphere with sufficiently high angular resolution needs to be chosen.

Directional energy decay curves

The EDC measured with an omnidirectional receiver is one of the most fundamental tools in room acoustics, yielding information about the transient behavior of the sound field decay subsequent to reaching a steady-state. It serves as the foundation for the calculation of the energy based room acoustic parameters.⁴ With the purpose of analyzing the decay process while retaining directional information we calculate the DEDC^29,30

d (t, θ_{q}, ϕ_{q}) = \int_{t}^{\infty} | a (τ, θ_{q}, ϕ_{q} {) |}^{2} d τ,

(4)

as the Schroeder integral³⁸ of the plane wave density function (cf. equation (2)). The DEDC may provide insights into the angular distribution of energy remaining in the decay process over time, uncovering directions of non-uniform energy incidence causing anisotropic decays or directionally dependent reverberation such as flutter echoes or reverberation reservoirs found in complex structures or coupled volumes. Analogously to EDCs calculated from omnidirectional receivers, equation (4) is proportional to the squared sound pressure, that is, ${Pa}^{2}$ . Normalization however, should only be a applied with a joint normalization constant over all directions.

The infinite integration in equation (4) has to be limited in practical applications due to the finite length of the SH-DRIR and the presence of measurement noise, consequently resulting in a truncation error. Multiple solutions when working with EDC measured with omnidirectional receivers have been proposed in the past³⁹ or even standardized⁴⁰ and can be adapted to DEDCs. In order to make no limiting assumptions on the shape of the decay function here, the integration is truncated at the intersection time between the RIR and the noise floor without further compensation of the neglected energy during integration (cf. Method B listed by Guski and Vorländer³⁹). It has to be noted that this method requires the truncation time to be sufficiently high for the resulting error to be negligible in the desired evaluation range.^39,41 Lundeby et al.⁴¹ state this criterion to be fulfilled for times before decay levels of $20 dB$ above the noise level at the intersection time. Considering directionally varying decay rates, a joint valid evaluation range for all directions has to be chosen which is prescribed by the shortest intersection time over all directions.

Isotropy estimation

Using a directional receiver Thiele²⁵ and Gover et al.^26,27 suggested to estimate the isotropy of a sound field using the normalized absolute difference of incident energy onto the receiver for $Q$ steering directions from the mean energy incident over all directions. Adapted to the DEDC this estimation can be written as^27,30

σ_{d} (t) = \frac{1}{{〈 d (t) 〉}_{Ω}} \sum_{q = 1}^{Q} | d (t, θ_{q}, ϕ_{q}) - {〈 d (t) 〉}_{Ω} |,

(5)

where ${〈 d (t) 〉}_{Ω}$ is the directional mean over the DEDC spanning the entire spherical domain $Ω = (θ, ϕ)$ at time instance $t$ ,

{〈 d (t) 〉}_{Ω} = \frac{1}{Q} \sum_{q = 1}^{Q} d (t, θ_{q}, ϕ_{q}) .

(6)

For $t = 0$ this equals the variation of incident energy of the steady-state sound field. To compensate for the non-ideal beamformer directivity pattern we normalize by the variation over the latter for a single plane wave incidence, here referred to as $σ_{e, 0}$ .²⁷ Re-normalization by subtracting from one, yields the estimated isotropy of the sound field,²⁷

μ_{d} (t) = 1 - \frac{σ_{d} (t)}{σ_{e, 0}} .

(7)

Equation (7) is a function in the interval $μ_{d} (t) \in [0, 1]$ , which will be zero for a single plane wave incident and one for a perfectly isotropic sound field.

Simulation setup

The DEDCs were investigated in a simulation study of three different rooms: a shoe-box type concert hall, a baroque-style church with a large dome structure, and a large gothic art church. The simulated SH-DRIRs, CAD models, and boundary properties are available as a separate data publication.⁴² It has to be noted that even though the geometries of each room are inspired by existing buildings, this work presents a specific case study leading to the omission of venue names. Figure 1 shows side and top views of the room geometries. Distinct geometrical features such as the stage, balconies, or in the case of the gothic church the columns supporting the arcades, are indicated in different shades of grey or white. Combinations of three receiver positions as well as one source position were investigated. For the concert hall, a position close to the center of the stage was chosen, while for both churches, the source was placed at the position of the organ. The receiver positions were chosen with varying distances from the source in potential listener areas. The room volumes, surface areas, and average polygon sizes, as well as reverberation times calculated as the mean over all receiver positions in the respective rooms are given in Table 1. All rooms were assumed to be empty. The boundary properties separated into average values for the walls, the ceiling, and the floor are given in Figure 2. The properties are averaged using the arithmetic mean weighted by the surface area of the respective materials covering the respective surface type, that is, walls, ceiling, and floor. This allows for a compact visualization of non-uniform distributions of boundary conditions. For detailed information on the exact distribution and quantities of boundary parameters, the reader is referred to the accompanying data publication.⁴² Note that the seating in the concert hall and the baroque church are not modeled as geometric features; the scattering of sound waves is therefore represented in the scattering coefficient of the floor materials. The same approach is used for geometric details and ornaments. Similarly, the lacking geometric detail of the rounded columns in the gothic church is counteracted by use of a large scattering coefficient covering a large part of the sidewalls. In the baroque church geometric features such as columns and arches are well captured with good precision.

Figure 1.

Top and side views of the concert hall (top), the baroque church (center), and the gothic church (bottom) including the source position as well as the three receiver positions. The coordinate system represents the coordinate axes of the receiver array and refers to the top view.

Table 1.

Volumes $V$ , room surfaces $S$ , average polygon size ${\bar{S}}_{poly}$ , and reverberation times for the concert hall (CH), the baroque church (BC), and the gothic church (GC). The reverberation times are given as the mean over all three receiver positions with an omnidirectional directivity and are calculated using linear regression in the $T_{60}$ interval.

Room	$V$	$S$	${\bar{S}}_{poly}$	$T_{60}$
Room	$V$	$S$	${\bar{S}}_{poly}$	$250 Hz$	$500 Hz$	$1 kHz$	$2 kHz$
Concert hall	$20786 m^{3}$	$5429.8 m^{2}$	$8.4 m^{2}$	$2.2 s$	$2.1 s$	$2.0 s$	$1.8 s$
Baroque church	$29890 m^{3}$	$12603.8 m^{2}$	$0.8 m^{2}$	$11.2 s$	$8.3 s$	$7.0 s$	$4.8 s$
Gothic church	$160457 m^{3}$	$48455.1 m^{2}$	$8.0 m^{2}$	$10.1 s$	$8.9 s$	$7.8 s$	$5.8 s$

Figure 2.

Absorption and scattering coefficients averaged for the ceiling, the sidewalls, and the floor. For averaging, the arithmetic mean weighted by the corresponding surface area of the materials covering the respective surface is used.

The simulations of the SH-DRIRs were performed using the hybrid simulation framework RAVEN^43,44 utilizing a combination of the image sources method up to second order and a ray-tracing approach for specular reflections of higher order as well as scattered reflections; diffraction is not considered. For scattered reflections a diffuse rain algorithm based on Lambert’s cosine law is applied.^44,45 The amount of scattered energy is proportional to the respective scattering coefficient. Implementation details are outlined in Schröder.⁴⁴ The hybrid simulation approach is justified by Kuttruff’s findings that on average scattered reflections are largely dominant compared to specular reflections above second order.^45,46 The synthesis of SH-DRIRs based on ray-tracing results is performed using three-dimensional histograms representing the angular distribution of sound particles and their respective energy over time as well as direction of incidence of the respective particles.⁴⁵ In order to ensure adequate statistical power to achieve sufficient spatio-temporal precision throughout the decay process, a number of $1 \cdot 10^{8}$ particles was used for the ray-tracing algorithm. The angular resolution of the detection sphere representing the receiver was chosen as $1^{\circ}$ in azimuth an elevation angles. An energy loss of $160 dB$ , before an individual particle is discarded, was chosen as stop criterion, providing a sufficient signal-to-noise ratio and a joint evaluation range for directions with short decays and directions decaying at a slower rate. For detailed information on the simulation algorithms as well as the generation of the spatio-temporal histograms, the reader is referred to the literature by Lentz et al.⁴⁵ and Schröder.⁴⁴ The encoding into the spherical harmonic domain up to order $N = 10$ was performed analogously to the plane wave series expansion of the sound field—detailed in the previous section, cf. equation (1)—as part of the simulation framework, that is, by weighting the impulse of each respective reflection with the SH basis function evaluated for the corresponding direction of incidence. For further information on the encoding, the reader is referred to the literature by Pelzer et al.^35,47

The plane wave decomposition was directly performed on the simulated SH-DRIRs for 1681 steering directions uniformly distributed over the sphere following the equal-area partitioning algorithm proposed by Leopardi.⁴⁸ The DEDCs were calculated using Schroeder integration as detailed in equation (4) with an integration limit corresponding to the intersection time and noise power determined using Lundeby’s method.⁴¹ The resulting DEDCs were subsequently truncated to the minimum out of the time corresponding to an energy decay of $65 dB$ in the omnidirectional EDC, calculated from the zero-order SH-DRIR, and the time according to the joint valid decay range criterion specified in the previous section.

Results

The top part of Figure 3 shows the DEDCs as a contour plot evaluated for the $1 kHz$ octave frequency band at the first receiver position. The DEDCs are normalized by their respective mean over all steering directions and thus are representative of the variation from the mean incident energy for each time instance. The plots are depicted for time instances corresponding to the steady-state as well as decays of $- 5 dB$ to $- 35 dB$ in steps of $10 dB$ . The corresponding time instances were extracted from the omnidirectional EDC calculated using the zero-order SH-DRIR. The bottom part of Figure 3 correspondingly visualizes the estimated isotropy as a function of time, the markers represent the time instances for which the contour levels are shown.

Figure 3.

The DEDCs normalized by their directional mean (top) and estimated isotropy (bottom) for the $1 kHz$ octave frequency band at the first receiver position. The contour lines represent the levels marked in the color bar, dashed lines indicate negative contour levels. The diamond markers in isotopy plots correspond to the contour plots depicted above in the order from top to bottom.

Steady-state analysis

The largest variations in local amplitude changes can be observed for the steady-state sound field. Distinct global maxima representing variations of more than $9 dB$ from the mean are found in the directions of the sound source, cf. Figure 1. Particularly in the concert hall and the baroque church clear additional local maxima are found. In the concert hall, these can be attributed to first order reflections from the sidewalls ( $ϕ = \pm 60^{\circ}$ ), the stage floor, as well the ceiling. In the baroque church, causes for local maxima are first order reflections from the ceiling at $(θ, ϕ {) = (50}^{\circ} {, 180}^{\circ})$ , and the floor, and second order reflections from the balconies at $(θ, ϕ {) = (35}^{\circ}, \pm 60^{\circ})$ . In contrast, in the gothic church local energy variations are distributed more uniformly over the sphere and assigning local maxima to geometrical features is not as straight forward, indicating a higher degree of mixing. These observations are well reflected in the estimated isotropy in the steady state, which is lowest for the concert hall, and highest for the gothic church.

Directional decay analysis

During the decay, cf. Figure 3 starting at $t_{- 5 dB}$ , the global maxima found in the steady state vanish and local variations in incident energy are reduced and distributed more uniformly over the sphere, indicating an increased mixing of the sound field after the direct sound and first order reflections have decayed. The improved mixing is further reflected by an increase in the estimated isotropy. The highest local variations at $t_{- 5 dB}$ are still observed for the concert hall with amplitudes variations of more than $\pm 6 dB$ , while variations are reduced to approximately $\pm 3 dB$ in both churches. In contrast to the initially increased mixing of the sound field in the concert hall, an increasing concentration of energy in the equator region is found at $t_{- 15 dB}$ and later, with local maxima in the axial directions in the $x y$ -plane of the room, cf. azimuth angles $ϕ = (- 90^{\circ}, 0^{\circ} {, 90}^{\circ} {, 180}^{\circ})$ . This effect is well explained by the symmetry of the room and the increased absorption of the ceiling compared to the remaining room surfaces. Even though a lateral distribution of energy is still achieved, the estimated isotropy decreases over time, as all higher order reflections with grazing incidence on the ceiling are subject to increased damping. Similarly, the sound field in the baroque church exhibits a distinctively anisotropic sound field decay after $t_{- 15 dB}$ , reflected by an increasing concentration of energy incident from the poles. The global maximum, which varies more than $6 dB$ from the mean in the late decay is observed in the direction of the north pole, indicates that the dome-roof structure acts as a reverberation reservoir with a longer reverberation time than the main volume of the church. The second maximum toward the south pole is caused due to the reflection from the floor and is less pronounced compared to the maximum of incident energy in the north pole. After $t_{- 15 dB}$ , an additional increase in incident energy compared to the mean is detected behind the receiver at $θ {= 50}^{\circ}$ elevation which is either due to a reflection from the balcony or resembles a reverberation reservoir in one of the side aisles. Again, these trends are reflected in the estimated isotropy over time, which evidently reaches its maximum between $t_{- 5 dB}$ and $t_{- 15 dB}$ , followed by a decrease. In contrast, Figure 3 does not show similarly striking anisotropic features in the sound field in the gothic church. The overall variations of the incident energy decrease over time, indicating a steady increase in the mixing of the sound field, also observed estimated isotropy. Small persistent concentrations corresponding to the direction of the arcades are however found in the right hemisphere, preventing the isotropy to increase above a value of $0.91$ . Nonetheless, a corresponding significant temporal effect is not evident.

Frequency band comparison

Figure 4 shows the isotropy evaluated for the octave bands from $250 Hz$ to $2 kHz$ . Evidently, the temporal evolution of the isotropy proves to be almost independent from the analyzed frequency band. Small differences in the absolute level of estimated isotropy are primarily found in the steady state as well as the early part of the decay process. The differing lengths of the isotropy curves are due to the truncation of the DEDCs outlined in the previous section. Figure 5 shows the omnidirectional EDCs for reference. It has to be noted that the air attenuation in both churches introduces losses large compared to the losses introduced at the room surfaces and therefore dominates the omnidirectional EDCs.

Figure 4.

Estimated isotropy for the octave frequency bands from $250 Hz$ to $2 kHz$ at receiver position 1.

Figure 5.

Energy decay curves calculated for a receiver with an omnidirectional directivity at position 1.

Receiver position comparison

Figure 6 shows the estimated isotropy for the three receiver positions evaluated in the $1 kHz$ octave band. Again, it may readily be seen that the overall progression of the estimated isotropy is very similar for all receiver positions in the concert hall and the gothic church. The largest quantitative variations in the isotropy over receiver positions for all rooms are found in the steady state. In the concert hall the estimated isotopy of the sound field seems to be linked to the distance to the sound source, showing the largest isotropy estimate for receiver 3 and the smallest for receiver 1. This relationship however is no longer observed during the sound field decay. A clear relationship between the distance from the source to the receiver and the estimated isotropy is not found for both the steady-state and the decaying sound fields in both churches.

Figure 6.

Estimated isotropy in the $1 kHz$ octave frequency band for the three receiver positions.

Discussion

The simulation study presented in the previous section is expected to provide valid results above the Schroeder frequency. The used framework provides the possibility to simulate spherical array measurements including their limitations in angular resolution while at the same time omitting limitations to their usable operating frequency range, cf. Rafaely.³⁶

The presented results show that the sound field in a room possesses distinct anisotropic features in the steady-state, primarily caused by the direct sound from the sound source, as well as incident waves attributed to early reflections. An improved mixing is generally observed during the transition from steady-state to the early decay, where the estimated isotropy increases. In the concert hall and the baroque church however, this initial increase is followed by a steady decrease, which is most prominent in the baroque church and indicates a de-mixing of the sound field instead of an improved mixing. In the concert hall this effect is well explained by the non-uniform distribution of boundary conditions in combination with the symmetry of the room which results in a multi-exponential sound field decay, cf. Nilsson⁴⁹ and Jacobsen and Møller Juhl,⁵⁰ with directionally dependent decay constants, cf. Figures 2 and 3. This phenomenon was also found in a similar study of a rectangular reverberation room,³⁰ where it is even more pronounced due to the non-uniform boundary conditions when an absorber sample occupies the floor, causing a separation of the sound field into waves with grazing incidence into the absorber and waves traveling in parallel. The de-mixing in the baroque church can not be be attributed to the distribution of boundary conditions, but the church geometry. The dome-roof represents a coupled volume with a larger decay time compared to the main volume of the church, therefore yielding a source for directionally dependent reverberation, which is observed in the increase of incident energy over time from above the receiver in Figure 3. At the same time the dome roof has a focusing effect on the sound field. This is supported by the variations of the estimated isotropy for the different receiver positions in Figure 6. The multi-exponential nature of the EDCs in Figure 5 are typical for rooms with coupled volumes,⁵¹ supporting the observation of a directionally dependent reverberation reservoir made in Figure 3. It does however have to be noted that due to the additional damping introduced by the attenuation in air—which is generally independent from the angle of incidence onto the receiver—the observation of multi-exponential decay curves alone does not suffice for the discrimination of the spatio-temporal phenomena outlined above.

Interestingly, a clear relation between the distance from the source to the receiver and the estimated isotropy is not found for both the steady-state and the decaying sound fields. Instead, the geometrical features as well as boundary conditions of the individual rooms have a larger influence on the sound field isotropy. In the concert hall, receiver 3 shows an overall reduced isotropy estimate compared to the remaining receiver positions due an increased energy incident from directions corresponding to the side and back walls. Contour plots are not shown here for brevity. Analogously, differences between the receiver positions in the baroque church are mainly caused by their position relative to the dome roof and the circular balcony rather than their distance to the source position, cf. Figure 1. The gothic church mainly shows an overall quantitative increased estimated isotropy for receiver 2 due to the reduced coupling to the arcades in close proximity to receivers 1 and 2, cf. Figure 6. The coupling to the arcades for the third receiver position is especially evident in the late part of the decay. A contour plot is omitted for brevity.

Conclusion

We presented an analysis method aiming at the characterization of the directional properties of steady-state and decaying sound field in rooms. A simulation study of three rooms of varying geometric complexity revealed an anisotropic sound field decay for most receiver positions in each room. Reasons are a non uniform distribution of boundary conditions and a strong symmetry in the room geometry in the case of the shoebox-like concert hall as well as coupled volumes in the case of the baroque and gothic churches. For the concert hall and the baroque church an increase in anisotropy during the decay process was detected, caused by the existence of multiple-exponential decay processes with distinct directional properties. This effect was further reflected in an decrease of the estimated sound field isotropy over decay time.

This phenomenon clearly contradicts the idea of a mixing time in these rooms, which assumes a point in time after which the sound field is diffuse and therefore would require the sound field to at least converge to a constant degree of sound field isotropy, instead of decreasing over time. While this phenomenon was most prominent for the sound field in the baroque church due to the coupling with the large reverberation reservoir introduced by the dome roof, it is also observed in the simple geometry of the almost rectangular concert hall, yet occurring later in the decay process. This further demonstrates the advantage of geometry based simulations of room acoustics over simplified modeling based on Sabine’s or Eyring’s equation which inherently assume a diffuse sound field, an assumption which was clearly not satisfied in the present study. Even thought the present study is limited to simulated SH-DRIRs, the method is applicable to measurements with spherical microphone arrays without modification. Similar results were also found in a measurement based study conducted in a reverberation room.³⁰

While a subjective study on the audibility of directionally dependent reverberation should be subject to future work, the presented method for the calculation and analysis of directional energy decay curves can be used to gain a quantitative description of the sound field decay with respect to the angular distribution of incident energy and decay constants.

Footnotes

Acknowledgements

The authors would like to thank Lukas Aspöck for many fruitful discussions as well as Johannes Imort for his help with the simulations, as well as Gottfried Behler and Wolfgang Ahnert for providing the CAD models of the gothic and baroque churches, respectively.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: Work presented here was funded by the German Research Foundation (Deutsche Forschungsgemeinschaft) under the Grant No. DFG VO 600 41-1.

ORCID iD

Marco Berzborn

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