Handbook for Sound Engineers

Acoustical Modeling and Auralization 225

response can then be recombined to yield a reflectogram

that presents a high degree of accuracy, especially when

compared to traditional ray-tracing techniques. Fig.

9-15 shows a comparative set of impulse response

reflectograms obtained by the adaptive beam tracing,

the image, the ray-tracing, and the nonadaptive

beam-tracing techniques in a model of a simple perfor-

mance space containing a stage and a balcony.

The adaptive model is able to yield a reflectogram
that is extremely close to that obtained with an image
method—i.e., it is able to generate all of the possible
reflections paths at a single point in space. From the
perspective of computing efficiency, the adaptive-beam
tracing methodology compares favorably with the
image methodology especially when the complexity of
the room and/or the order of the reflections is increased.
Other variants of the beam-tracing approach have
been developed. In the priority-driven approach,^7 the
algorithms are optimized to generate a series of the

most relevant beams from a psychoacoustics perspec-

tive so that the reflectogram can be generated very

rapidly, ideally in real time and the model can be used in

an interactive fashion. The beams are ranked in order of

importance based on a priority function that aims at

accurately reproducing the early portion of the reflecto-

gram since it is by far the most relevant to the percep-

tion of the space from the perspective of

psychoacoustics. The late portion of the reflectogram

(the late reverberation) is then modeled by using a

smooth and dense energy profile that emulates the

statistical decay of the energy in a large space.

A note on diffusion: The issue of diffusion has been of

prime interest to the developers of computer models

since commercially available or custom-made diffusers

are often integrated into room designs, sometimes at a

high cost. Although diffusion is an essential qualitative

part of the definition of a sound field, the quantitative

question of “how much diffusion is needed?” is often

answered using considerations that have little founda-

tion in scatter theory and/or general acoustics. A

concept as elementary as reverberation finds its clas-

sical quantitative representation (the Sabine/Eyring

equation and associated variants) rooted into the notion

that the sound field is assumed to be diffuse, and unless

this condition is met in reality, one will encounter

substantial errors in predicting the reverberation time.

Today’s advanced large room computer acoustic simula-

tion software products incorporate the ability to model

diffused reflections using either a frequency dependence

function or an adaptive geometry that spreads out the

incident energy of the sound ray over a finite area. This

allows for a much more accurate correlation between

predicted and test data especially in rooms that have

geometry involving shapes and aspect ratios that are out

of the ordinary, noneven distribution of absorptive

surfaces, and/or coupled volumes.^8 Under these condi-

tions the incorporation of diffusion parameters into the

model is necessary and a specular-only treatment of the

reflections (even when using an efficient ray-tracing

technique) will lead to errors.

9.2.2.2 Wave Equation Models

Wave equation models are based on an evaluation of the

fundamental wave equation, which in its simplest form

relates the pressure p of a wave at any point in space to

its environment via the use of the 3D Laplacian operator

(^) and the wave number k:
(9-13)
Figure 9-15. Comparative reflectograms for a simple room
model. From Reference 6.
Adaptive Beam Tracing
Time—s
Image Method
5
0
–5
–10
0 0.1 0.2 0.3
Nonadaptive Beam Tracing
Ray-Tracing
SPL(dB)
SPL(dB)
5
0
–5
–10
5
0
–5
–10
5
0
–5
–10
SPL(dB)
SPL(dB)
0 0.1 0.2 0.3
0 0.1 0.2 0.3
0 0.1 0.2 0.3
Time—s
Time—s
Time—s
’^2
’
2
pk
2
+0p=