Handbook for Sound Engineers

Acoustical Modeling and Auralization 221

also satisfies the boundary conditions that must take
place at the surface, that is, the acoustical pressures
have to be equal on both sides of the surface at the
reflection point, and the velocity of the wave has to be
zero at the interface. The image from the virtual source
S 1 can also be used to determine where the second-order
reflections from a second surface will be directed to
since as far as the second surface is concerned, the wave
that is impinging upon it emanated from S 1. A second
order source S 2 can thus be created as shown in Fig. 9-6
and the process can be repeated as needed to investigate
any order of reflections that constitutes a path between
source and receiver.

It is thus possible to collect the amplitude and the
direction of all of the reflections at a specific location as
well as a map of where the reflections emanate from.
Even the reflection paths from curved surfaces can be
modeled by using tangential planes as shown in Fig. 9-7
and Fig. 9-8.

Since the speed of sound can be assumed to be a
constant inside the room, the distance information
pertaining to the travel of the reflections can be trans-
lated into time-domain information; the result is called a
reflectogram (or sometimes an echogram) and it
provides for a very detailed investigation of the reflec-

tions inside a space that reach a receiver at a specific

location. A sample of a reflectogram is shown in

Fig. 9-9..

Although it was originally developed solely for the

determination of the low-order specular reflections taking

place in rectangular rooms due to the geometric increase

in the complexity of the computations required, the tech-

nique was expanded to predict the directions of the spec-

ular reflections from a wide range of shapes.^4 In the

image method, the boundaries of the room are effectively

Figure 9-5. A source and its virtual image located across a

boundary define the direction of the first-order reflection.

Figure 9-6. Higher-order reflections can be created by
adding virtual images of the source.

S 1 (image of the source)

Reflection

point

S 0 (source)

Receiver

Surface (boundary)

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)

(^2) Receiver
nd order
reflection
Boundary 2
1 st order
reflection
Figure 9-7. Image construction from a convex plane.
Adapted from Reference 1.
Figure 9-8. Image construction from a concave plane
Adapted from Reference 1.
Figure 9-9. A reflectogram (or echogram) display of reflec-
tions at a specific point inside a room.
S 1 (image of the
source)
Reflection
point
Convex surface
Tangential plane
Receiver
S 0 (source)
Reflection
point Tangential plane
Concave surface
Receiver
S 0 (source)
S 1 (image of the
source)
90 dB
70 dB
50 dB
30 dB
10 dB
0 ms 100 ms 200 ms
Direct sound