Handbook for Sound Engineers

228 Chapter 9

Numeric Model: Finite Difference Time-Domain
Methodology. As mentioned earlier, the BEA tech-
niques are best suited to yielding data in the frequency
domain, although they can be adapted to provide
time-domain information albeit at a cost in computing
efficiency. Another numerical methodology that uses a
discrete representation of the acoustical environment is
known as Finite-Difference Time-Domain (FDTD) and
it is very efficient in terms of computational speed and
storage while also offering excellent resolution in the
time domain. It has been demonstrated^15 that the tech-
nique can be used to effectively model low-frequency
problems in room acoustics simulations and the results
are suitable for the generation of reflectograms.

In the finite difference (FD) approach instead of
describing the surface with a mesh (as with the BEA
technique), a grid is used and the algebraic equations
are solved at the points of the grid as shown in
Fig. 9-17.

In this instance the size of the grid can be made as
small as needed to provide a high degree of resolution
when needed and the grid points can be defined using
the most effective coordinate system for the application.
For example, a flat surface could be defined with a (x, y,
z) Cartesian grid system while cylinders (for pillars) and
spheres (for audience’s heads) could be expressed with
cylindrical and spherical systems respectively.

9.2.2.4 Statistical Models

The use of statistics in acoustical modeling is primarily
reserved for the study of the behavior of sound in rect-
angular and rigid rooms where the dominant phenomena
that are taking place are related to modes. The issues of

modal frequencies, modal density, and mode distribu-

tions are presented, along with the appropriate descrip-

tive equations in Chapter 5—Small Room Acoustics.

Another application of statistics in acoustical

modeling can be found in situations where resonance

effects take place at high frequencies, as opposed to the

traditionally low frequencies associated with room

modes. A technique known as Statistical Energy

Analysis^16 (SEA) can be used to accurately account for

the effect of modal resonance effects that take place in

systems such as partitions and walls, by analyzing the

kinetic energy and the strain energy associated with

vibrating structures. An SEA model will describe a

vibrating system (such as a wall) with mass and spring

equivalents and will allow for the analysis of the effect

that adding damping materials will have on the vibra-

tion spectrum. SEA techniques are optimized for

frequency-domain analysis and the output cannot be

used for time-domain applications to add information to

the impulse response of a room, or to yield a reflecto-

gram; still, the main advantage of SEA is that real mate-

rials such as composite partitions with different degrees

of stiffness, construction beams, and acoustical sprays

can be modeled in a precise manner (i.e., not only in

terms of a unique physical coefficient) over an extended

range of frequency.

9.2.2.5 Small Room Models

A room that is acoustically small can be defined as one

in which classically defined reverberation phenomena

(using the assumption of a diffuse sound field) do not

take place, but rather that the sound decays in a nonuni-

form manner that is a function of where the measure-

ment is taken. The use of diffusion algorithms in large

room models that rely on either ray-tracing, image

source, or adaptive algorithms has vastly improve the

reliability of the prediction models in a wide range of

spaces, however accurate predictions of the sound field

can be made in small rooms considering the interference

patterns that result from modal effects. Figs. 9-18 and

9-19 shows the mapping^17 of the interference patterns

resulting from modal effects in an 8 m × 6 m room

where two loudspeakers are located at B1 and B2. In the

first instance, a modal effect at 34.3 Hz creates a large

dip in the response at about 5.5 m, while the second

case shows a very different pattern at 54.4 Hz. Such

models are very useful to determine the placement of

low-frequency absorbers into a room in order to mini-

mize the impact of modal effects at a specific listening

location, and they are a good complement to the large

Figure 9-17. A grid describes a surface in the FD method.

Room

Grid around

surface

Finite difference

operator