226 Chapter 9
Solving the fundamental wave equation allows for
an exact definition of the acoustic pressure at any
specific point since appropriate boundary conditions
defining the physical properties of the environment
(surfaces, medium) can be used whenever required. As
an example, in a model based on the wave equation the
materials that comprise the environment (like the room)
can be defined in terms of their acoustical impedance z
given by
(9-14)where,
p refers to the pressure of the wave,
U to its velocity in the medium.
When using the wave equation, issues having to do
with diffraction, diffusion, and reflections are automati-
cally handled since the phenomena are assessed from a
fundamental perspective without using geometrical
simplifications. The main difficulty associated with the
method is found in the fact that the environment
(surfaces and materials) must be described accurately in
order for the wave equation to be applied: either an
analytical or a numerical approach can be used to
achieve this goal.
9.2.2.2.1 Analytical Model: Full-Wave Methodology.
An analytical model aims at providing a mathematical
expression that describes a specific phenomenon in an
accurate fashion based on underlying principles and/or
physical laws that the phenomenon must obey. Because
of this requirement the analytical expression governing
the behavior of a model must be free of correction terms
obtained from experiments and of parameters that can-
not be rigorously derived from—or encountered
in—other analytical expressions.
The complexity of the issues associated with sound
propagation has prevented the development of a single
and unified model that can be applied over the entire
range of frequencies and surfaces that one may
encounter in acoustics; most of the difficulties are found
in trying to obtain a complete analytical description of
the scattering effects that take place when sound waves
impinge on a surface. In the words of J.S. Bradley,^9 one
of the seminal researchers in the field of architectural
acoustics:
Without the inclusion of the effects of diffrac-
tion and scattering, it is not possible to accu-
rately predict values of conventional room
acoustics parameters [...]. Ideally, approxima-
tions to the scattering effects of surfaces, or of
diffraction from finite size wall elements should
be derived from more complete theoretical anal-
yses. Much work is needed to develop room
acoustics models in this area.In this section, we present the full-wave method-
ology,^10 one analytical technique that can be used for
the modeling of the behavior of sound waves as they
interact with nonidealized surfaces, resulting in some of
the energy being scattered, reflected, and/or absorbed as
in the case of a real space. Due to the complexity of the
mathematical foundation associated with this analytical
technique, only the general approach is introduced here
and the reader is referred to the bibliography and refer-
ence section for more details.
The full-wave approach (originally developed for
electromagnetic scattering problems) meets the impor-
tant condition of acoustic reciprocity requiring that the
position of the source and of the receiver can be inter-
changed without affecting the physical parameters of
the environment like transmission and reflection coeffi-
cients of the room’s surfaces. In other words if the envi-
ronment remains the same, interchanging the position of
a source and of a receiver inside a room will result in
the same sound fields being recorded at the receiver
positions. The full-wave method also allows for exact
boundary conditions to be applied at any point on the
surfaces that define the environment, and it accounts for
all scattering phenomena in a consistent and unified
manner, regardless of the relative size of the wavelength
of the sound wave with that of the objects in its path.
Thus the surfaces do not have to be defined by general
(and less than accurate) coefficients to represent absorp-
tion or diffusion, but they can be represented in terms of
their inherent physical properties like density, bulk
modulus, and internal sound velocity.
The full-wave methodology computes the pressure at
every point in the surface upon which the waves are
impinging, and follows the shape of the surface. The
coupled equations involving pressure and velocity are
then converted into a set of equations that separate the
forward (in the direction of the wave) and the backward
components of the wave from each other, thus allowing
for a detailed analysis of the sound field in every direc-
tion. Since the full-wave approach uses the fundamental
wave equation for the derivation of the sound field, the
model can return variables such as sound pressure or
sound intensity as needed.
The main difficulty associated with the full-wave
method is that the surfaces must also be defined in an
analytical fashion. This is possible for simple (i.e.,z p
U=--- -