Handbook for Sound Engineers

Acoustical Modeling and Auralization 227

planar, or curved) surfaces for which equations are

readily available, but for more complicated surfaces—

such as those found in certain shapes of diffusers—an

analytical description is more difficult to achieve, and

the methodology becomes restricted to receiver loca-

tions that are located at a minimum distance from the

surfaces upon which the sound waves are impinging.

Still for many problems the full-wave methodology is a

very accurate and efficient way to model complicated

scattering phenomena.

9.2.2.3 Numerical Model: Boundary Element
Methodology

The boundary element analysis (BEA) techniques are
numerical methods that yield a quantitative value of the
solution to the problem under investigation. BEA
techniques11,12,13,14 can be used in solving a wide range
of problems dealing with the interaction of energy (in
various forms) with media such as air and complex
physical surfaces, and they are well suited to the investi-
gation of sound propagation in a room. Although the
method is based on solving the fundamental differential
wave equation presented earlier, the BEA methodology
makes use of an equivalent set of much simpler alge-
braic equations valid over a small part of the geometry,
and then expands the solution to the entire geometry by
solving the resulting set of algebraic equations simulta-
neously. In essence, the BEA technique replaces the task
of solving one very complex equation over a single com-
plicated surface by that of solving a large quantity of
very simple equations over a large quantity of very sim-
ple surfaces. In a BEA implementation the surface is
described using a meshing approach as shown in Fig.
9-16.

In the BEA method the analytical form of the solu-
tion over the small domain (area) of investigation is not
directly accessible for modification. The use exercises
control over the solution by properly specifying the
domain (geometry) of the problem, its class (radiation
or scattering), the parameters of the source (power,
directivity, location), and, of course, the set of boundary
conditions that must be applied at each area defined by
the mesh. It is thus possible to assign individual mate-
rial properties at every location in the mesh of the
model in order to handle complex scattering and absorp-
tion scenarios, if needed. Although it can be adapted to
solving acoustical problems in the time domain the
BEA technique is better suited to providing solutions in
the frequency domain since the characteristics of the

materials are considered to be time-invariant but

frequency dependent.

The main issue that is associated with the use of

BEA methodology for the investigation of acoustical

spaces is that the size of the elements comprising the

mesh representing the surfaces dictates the accuracy of

the solution. A small mesh size will, of course, allow for

a very accurate description of the surfaces, both

geometrically and in terms of its materials, but it will

also drastically affect the computational time required

to yield a solution. On the other hand, a large mesh size

will yield very fast results that may be inaccurate

because the algebraic equations that are used in lieu of

the fundamental wave equation improperly being

applied over large surfaces. A comparison of the accu-

racy yielded by BEA techniques over very simple

geometries indicates that a minimum ratio of seven to

one (7:1) must exist between the wavelength of the

sound and the element size in order to bind the depen-

dence of the BEA analysis on the size of its element to

less than a ±0.5 dB resolution. In other words, the wave-

lengths considered for analysis must be at least seven

times larger than the largest mesh element in order for

the methodology to be accurate. For this reason the

BEA methodology is very efficient and accurate to

model sound propagation at low frequencies (below

1000 Hz), but it becomes tedious and cumbersome at

higher frequencies since in this instance the mesh must

be modeled with better than a 50 mm resolution. Still

the technique can be shown to yield excellent results

when correlating modeled projection and actual test

data from complicated surfaces such as diffusers.^13

Figure 9-16. A mesh describes a boundary in the BEA

method.

Room

Mesh around

surface

(boundary)

Boundary element

operator