User Guide

Sound System Design Reference Manual
5-17
relationship is often indicated by the simple
expression: 4W/cSα. W represents the output of the
sound source, and the familiar expression S α
indicates the total absorption of the boundary
surfaces.
Remembering our statistical room model, we
know that sound travels outward from a point source,
following the inverse square law for a distance equal
to the mean free path, whereupon it encounters a
boundary surface having an absorption coefficient α.
This direct sound has no part in establishing the
reverberant sound field. The reverberant field
proceeds to build up only after the first reflection.
But the first reflection absorbs part of the total
energy. For example, if α is 0.2, only 80% of the
original energy is available to establish the
reverberant field. In other words, to separate out the
direct sound energy and perform calculations having
to do with the reverberant field alone, we must
multiply W by the factor (1 - α ).
This results in the equation:
E =
4W
cR
rev
*
This gives the average energy density of the
reverberant field alone. If we let R = Sα/(1 - α), the
equation becomes:
E =
4W 1-
cS
rev
α
α
()
Note that the equation has nothing to do with
the directivity of the sound source. From previous
examples, we know that the directivity of the source
affects critical distance and the contour of the
boundary zone between direct and reverberant
fields. But power is power, and it would seem to
make no difference whether one acoustic watt is
radiated in all directions from a point source or
concentrated by a highly directional horn.
Is this really true? The equation assumes that
the porportion of energy left after the first reflection is
equivalent to W(1 - α). Suppose we have a room in
which part of the absorption is supplied by an open
window. Our sound source is a highly directional
horn located near the window. According to the
equation the energy density of the reverberant field
will be exactly the same whether the horn is pointed
into the room or out of the window! This obviously is
fallacious, and is a good example of the importance
of understanding the basis for acoustical equations
instead of merely plugging in numbers.
* With room dimensions in meters and acoustic power
in watts, the reverberant field level in dB is:
L
rev
= 10 log W/R+ 126 dB. See Figure 5-21.
Figure 5-21. Steady-state reverberant field SPL vs. acoustic power and room constant