Building Acoustics

Room acoustics 135

4.6.2 Image-source models

Image-source (or mirror-source) modelling is based on regarding all reflections from the
boundary surfaces as sound contributions from images of the real source(s). The strength
of this type of modelling, when carried out rigorously, is that it covers all transmission
paths between source and receiver. It may give the impulse responses correct inside the
framework of geometrical acoustics.
It is relatively simple mathematically to find all these mirror sources. The main
problem is that except for rooms of very simple shapes, most of these sources are either
not visible in a given receiver position or may be invisible in any part of, for example,
the audience area. This means that a number of reflections are not physically valid. To
separate out the “valid” image sources is a time-consuming task when coming to the
higher order reflections. We may illustrate this by calculating the number of image
sources of the order N in a room having M surfaces, which is given by M(M – 1)N–1. In a
room having e.g. M equal to 12, we get approximately 16 000 image sources of the forth
order, approximately 175 000 of the fifth order and so on. Except for rooms having a
very simple shape, e.g. rectangular ones, maybe only a few hundred of these sources are
valid. As in the case of ray tracing the question arises on when to stop the calculations.
“Adding on” to the results using statistical arguments are common having carried out
calculations correctly up to a given order of reflections.

Source

Receiver

S 1

S 12 S 123

Figure 4.14 Example on trajectory between a receiver and a third-order image source.

Sketches which illustrate some of the aspects discussed above are shown in Figures
4.14 and 4.15. The first one shows, in a cross section (horizontally or vertically) through
a room of rectangular shape (parallelepiped), an example of the trajectory between a
receiver and a third-order image source. Figure 4.15 gives an example on a first-order
image source S 1 (mirrored in wall W), which is not visible in any of the possible receiver
positions R within the indicated sector, a sector given by the solid angle defined by the
wall surface as seen from the image source.
Finding the image-source positions is in many cases quite easy where regular room
shapes are concerned and one may also find analytical expressions as to the sound
propagation. An example that we shall also use later on (see section 4.8) is sketched in
Figure 4.16, which shows a vertical section of a long “flat” room. Here we shall assume
that the ceiling height is much smaller than the other dimensions of the room; i.e. we
shall neglect the influence of the sidewalls.