Handbook for Sound Engineers

224 Chapter 9

One problem associated with the ray-tracing tech-
nique is that the accuracy of the detection is strongly
influenced by size of the detector. A large spherical
detector will record a larger number of hits from the
rays than another spherical detector of smaller diameter,
even if the respective centers of the spheres are located
at the exact same point in space. Furthermore, the
ray-tracing method may lead to an underestimation of
the energy reaching the detector (even if its size is
considered adequate) unless large numbers of rays are
used since the energy is sampled via rays that diverge as
they spread from the source thus increasing the possi-
bility that low-order reflections may miss the detector.

Techniques combining the image methodology and
the ray-tracing approach have been developed.^5 The
algorithms aim at reducing the number of images to be
considered by using the more computationally efficient
ray-tracing technique to conduct the visibility test
required by the image method.

Beam-Tracing Models. The triangular area that is
defined between two adjacent rays emanating from a
source is called a 2D ray; more than two rays can also
be used to define a 3D pyramidal or conical region of
space in which the acoustical energy is traveling away
from the source. In these instances, the source is viewed
at emitting beams of energy, and the associated model-
ing techniques are known as beam-tracing methods.
Figure 9-14 shows an example of a beam and its reflec-
tion path from a surface.

The beam-tracing technique offers the advantage of
guaranteeing that the entire space defining the model
will receive energy since the directions of propagations
are not sampled as in the case of the traditional
ray-tracing approach. Virtual source techniques are used
to locate the points that define the reflection zones
across the boundaries of the room. On the other hand,

the technique requires very complex computations to

determine the reflection patterns from the surfaces since

the reflection cannot be viewed as a single point as in

the case of the ray-tracing technique: when 2D beams

are used, the reflections from the surfaces must be

considered as lines, while 3D beams define their reflec-

tions as areas. Care must also be taken to account for

overlapping of the beams by each other or truncation of

the beams by obstacles in the room.

Although the computational complexity of the model

is substantially increased when it comes to assessing the

direction of the reflections, the departure from the

single point reflection model presents numerous advan-

tages over the traditional image and/or ray-tracing tech-

nique. The issues associated with the divergence of the

reflections as a function of increased distance from the

source are naturally handled by the beam-tracing

approach. Furthermore, the effects of acoustical diffu-

sion can be modeled—at least in an estimated

fashion—since the energy contained in the beams can

be defined as having a certain distribution over either

the length of the intersecting lines (for 2D beams) or

areas (for 3D beams). For example, an adaptive

beam-tracing model^6 that controls the cross-sectional

shape of the reflecting beam as a function of the shape

of the reflecting surfaces also allows for an evaluation

of the diffuse and specular energy contained inside a

reflecting beam. If the energy contained inside the inci-

dent beam is EB and the energy reflected from a surface

is ER, then one can write

(9-11)

where,

Dis the surface’s absorption coefficient,

Gis the surface’s diffusion coefficient.

The energy ED that is diffused by the surface is

found to be proportional to the area of illumination A

and inversely proportional to the square of an equivalent

distance L between the source and the reflection area

(9-12)

The adaptive algorithm allows for a separate assess-

ment of the specular and of the diffuse reflections from

the same geometrical data set that represents the travel

map of the beams inside the space. In this instance the

diffused energy from a given surface is redirected to

other surfaces in a recursive fashion via radiant

exchange, a technique also used in light rendering appli-

cations. The diffuse and the specular portions of the

Figure 9-14. A 3D beam is emitted by a source S and
reflects at a surface.

S (source)

Incident beam

Reflected beam

Reflection area

Surface

S 1 (image of

the source)

ER=EB 1 – D 1 – G

ED

EBAG 1 – D

4 SL^2

v-------------------------------