Handbook for Sound Engineers

222 Chapter 9

being replaced by sources and there is no theoretical limit
to the order of reflection that can be handled by the
image methodology. From a practical standpoint, the
number of the computations that are required tends to
grow exponentially with the order of the reflections and
the number of surfaces as shown in Eq. 9-10, where NIS
represents the number of images, NW is the number of
surfaces that define the room, and i is the order of the
reflections.^5

(9-10)

Furthermore reflections must be analyzed properly in
terms of their visibility to the receiver in the case of
complicated room shapes where some elements may
block the reflections from the receiver as shown in
Fig. 9-10.

The model must also constantly check for the
validity of the virtual sources to insure that they actually
contribute to the real reflectogram by emulating reflec-
tions taking place inside the room and not outside its
physical boundaries. Fig. 9-11 illustrates such a
situation.
In Fig. 9-11 a real source S 0 creates a first-order
image S 1 across boundary 1. This is a valid virtual
source that can be used to determine the magnitude and
the direction of first-order specular reflections on the
boundary surface 1. If one attempts to create a
second-order virtual source S 2 from S 1 with respect to
boundary surface 2 to find the second order reflection,
the image of this virtual source S 2 with respect to
boundary 1 is called S 3 but it is contained outside the
boundary used to create it and it cannot represent a
physical reflection.
Once the map of all the images corresponding to the
reflection paths has been stored, the intensity of each
individual reflection can be computed by applying
Eq. 9-9 introduced earlier. Since the virtual sources do

represent the effect of the boundaries on the sound

waves, the frequency dependence of the absorption

coefficients of the surfaces is modeled by changing the

power radiated by the virtual sources; thus, once the

image map is obtained the model can be used to rapidly

simulate an unlimited number of “what if” simulations

pertaining to material changes as long as the locations

of the sources and the receiver remain unchanged. A

further correction for the air absorption resulting from

the wave traveling over extended distances can also be

incorporated at this time in the simulation. The same

reasoning applies to the frequency distribution of the

source: since the image map (and the resulting location

of the reflections in the time domain) is a sole function

of source and receiver position, the image model can

rapidly perform “what if ” simulations to yield reflecto-

grams at various frequencies.

The image methodology does not readily account for

variations in the absorption coefficient of the surfaces as

a function of the angle of incidence of the wave. When

taking into account all of the properties in the transmis-

sion medium, it can be shown that many materials will

exhibit a substantial dependence of their absorption

coefficient on the incidence angle of the wave, and in its

most basic implementation the image method can

misestimate the intensity of the reflections. It is

however, possible to incorporate the relationship

between angle of incidence and absorption coefficient

into a suitable image algorithm in order to yield more

accurate results, although at the expense of computa-

tional time.

In an image model the user can control the length of

the reflection path as well as the number of segments

(i.e., the order of the reflections) that comprise it. This

allows for a reduction in the computational time of the

Figure 9-10. The reflection is not visible to the listener due
to the balcony obstruction.

NIS

NW

NW 2–

=---------------->@ NW 1– i 1–

Reflection

Balcony

Listener

Figure 9-11. Invalid images can be created when a virtual

source is reflected across the boundary used to create it.

Adapted from Reference 4.

S 2 (2nd order image of the

source across boundary 2)

S 1 (image of

the source

across

boundary 1)

Boundary 1

S 0 (source)

Boundary 2

S 3 (invalid image

of the source)