Handbook for Sound Engineers

Acoustical Modeling and Auralization 223

process since virtual sources located beyond a certain
distance from the receiver location can be eliminated
while not compromising the fact that all of the reflec-
tions within a specific time frame are being recorded,
and the image method can lead to very accurate results
in the modeling of the arrival time of reflections at a
specific location.
Efficient computer implementations of the image
methodology have been developed^4 to allow for a fast
output of the reflections while also checking for the
validity of the images and for the presence of obstacles.
Still the method is best suited to the generation of very
accurate reflectograms of short durations (500 ms or
less) and limited number of reflections (fifth order
maximum for typical applications). These factors do not
negatively affect the application of the image method in
acoustical modeling since in a typical large space—like
an auditorium or a theater—the sound field will become
substantially diffuse after only a few reflections and
some of the most relevant perceived attributes of the
acoustics of the space are correlated to information
contained in the first 200 ms of the reflectogram.

Ray-Tracing Models. The ray-tracing methodology fol-
lows the assumptions of geometrical acoustics presented
at the onset of this section, but in this instance the source
is modeled to emit a finite number of rays representing
the sound waves in either an omnidirectional pattern for
the most general case of a point source, or in a specific
pattern if the directivity of the source is known. Fig. 9-12
shows an example of a source S generating rays inside a
space and how some of the rays are reflected and reach
the receiver location R.

In this instance, the goal is not to compute all of the
reflection paths reaching the receiver within a given
time frame but to yield a high probability that a speci-

fied density of reflections will reach the receiver (or

detector usually modeled as a sphere with a diameter

selected by the user) over a specific time window.

In the case of the image method, the boundaries of

the room are replaced by virtual sources that dictate the

angle of the reflections of the sound waves. In a similar

fashion, the ray-tracing technique creates a virtual envi-

ronment in which the rays emitted by the source can be

viewed as traveling in straight paths across virtual

rooms until they reach a virtual listener as presented in

Fig. 9-13.

The time of travel and location of the ray are then

recorded and can yield a reflectogram similar to that

presented earlier in Fig. 9-9.

The main advantage of the ray-tracing technique is

that since the model is not trying to find all of the reflec-

tion paths between source and receiver, the computa-

tional time is greatly reduced when compared to an

image technique; for a standard ray-tracing algorithm,

the computational time is found to be proportional to

the number of rays and to the desired order of the reflec-

tions. Another advantage inherent to the technique is

that multiple receiver locations may be investigated

simultaneously since the source is emitting energy in all

directions and the model is returning the number and

the directions of rays that are being detected without

trying to complete a specific path between source and

receiver. On the other hand, since the source is emitting

energy in many directions and one cannot dictate what

the frequency content of a specific ray is versus that of

another, the simulations pertaining to the assessment of

frequency-dependent absorption must be performed

independently and in their entirety for each frequency of

interest.

Figure 9-12. Rays are generated by a source, S. Some of
the rays reach the receiver, R.

Source Receiver

Figure 9-13. The rays can be seen as traveling in straight

paths across virtual images of the room until they intercept

a receiver. Adapted from Reference 4.

This ray found a

virtual receiver

but missed the

three neighbors

that were closer

to the source

Source

This ray

found a virtual

receiver in the

virtual room

nearest to

the source

This ray did

not find any

receiver in

any of the

virtual rooms

Receiver