220 Chapter 9
absorption and associated construction techniques is
difficult to model accurately.
9.2.2 Computational Models
This section presents models that create a mathematical
representation of an acoustical environment by using
assumptions that are based on either a geometrical, ana-
lytical, numerical, or statistical description of the physi-
cal phenomena (or parameters) to be considered, or on
any combination of the afore- mentioned techniques, In
all instances, the final output of the modeling phase is
the result of extensive mathematical operations that are
usually performed by computers. With the development
of powerful and affordable computers and of graphical
interfaces these modeling tools have become increas-
ingly popular with acoustical designers. To various
degrees, the aim of computational models is to ulti-
mately yield a form of the impulse response of the room
at a specific receiver location from which data pertain-
ing to time, frequency, and direction of the sound
energy reaching the receiver can be derived. This infor-
mation can then be used to yield specific quantifiers
such as reverberation time, lateral reflection ratios,
intelligibility, and so on.
The inherent advantage of computational models is
flexibility: changes to variables can be made very
rapidly and the effects of the changes are available at no
hard cost, save for that of computer time. Issues related
to source or receiver placement, changes in materials
and/or in room geometry can be analyzed to an infinite
extent. Another advantage of computational models is
that scaling is not an issue since the models exist in a
virtual world as opposed to a physical one.
Computational models are themselves divided into
subgroups that are fundamentally based on issues of
adequacy, accuracy, and efficiency. An adequate model
uses a set of assumptions based on a valid (true)
description of the physical reality that is to be modeled.
An accurate model will further the cause of adequacy by
providing data that is eminently useful because of the
high confidence associated with it. An efficient model
will aim at providing fast and adequate results but
maybe to a lesser—yet justified—extent in accuracy.
Although issues of accuracy and of efficiency will be
considered in this portion of the chapter, the discussion
of the various classes of computational models will be
primarily based on their adequacy.
9.2.2.1 Geometrical ModelsThe primary assumption that is being made in all geo-
metrical models^2 applied to acoustics is that the wave
can be seen as propagating in one or more specific
directions, and that its reflection(s) as it strikes a surface
is (are) also predictable in terms of direction; this is a
very valid assumption when the wavelength can be con-
sidered small compared to the size of the surface and
the condition ka >5 presented in Table 9-1 quantifies
the limit above which the assumption is valid. Under
this condition, the propagating sound waves can be rep-
resented as straight lines emanating from the sources
and striking the surfaces (or objects) in the room at spe-
cific points. The laws of optics involving angles of inci-
dence and angles of reflection will apply and the term
geometrical acoustics is used to describe the modeling
technique.
The second assumption of relevance to geometrical
acoustics models is that the wavelength of the sound
waves impinging on the surfaces must be large
compared to the irregularities in the surface, in other
words the surface has to appear smooth to the wave and
in this instance the irregularities in the surface will
become invisible since the wave will diffract around
them. If the characteristic dimension of the irregularities
is denoted by b, then the condition kb <1 is required
using the criteria outlined in Table 9-1. This is a neces-
sary condition to assume that the reflection is specular,
that is that all of its energy is concentrated in the new
direction of propagation. Unless this condition is met in
the actual room the energy of the reflected wave will be
spread out in a diffuse fashion and the geometrical
acoustics assumption of the model will rapidly become
invalid, especially if many reflections are to be
considered.Image Models. In this class of geometrical acoustics
models, the assumption that is being made is that the
only sound reflections that the model should be con-
cerned about are those reaching the receiver, so the
methodology aims at computing such reflections within
time and order constraints selected by the user of the
model while ignoring the reflections that will not reach
the receiver. To find the path of a first-order reflection a
source of sound S 0 is assumed to have an image—a vir-
tual source S 1 —located across the surface upon which
the sound waves are impinging as presented in Fig. 9-5.
As long as the surface can be considered to be rigid,
the image method allows for the prediction of the angles
of reflections from the surface and can find all of the
paths that may exist between a source and a receiver.^3 It