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Energy Decay through Lossy Boundaries

Since the acoustic energy density

is the energy per unit
volume in a 3D sound field, it follows that the total energy of the
field is given by integrating over the volume:

In reverberant rooms and other acoustic systems, the field energy
decays over time due to losses. Assuming the losses occur only at the
boundary of the volume, we can equate the rate of total-energy change
to the rate at which energy exits through the boundaries. In other
words, the energy lost by the volume

in time interval

must equal the acoustic

intensity
exiting the volume,
times

(approximating

as constant between times

and

):

The term

is the dot-product of the (vector)
intensity

with a unit-vector

chosen to be normal to the
surface at position

along the surface. Thus,

is
the component of the acoustic intensity

exiting the volume
normal to its surface. (The tangential component does not exit.)
Dividing through by

and taking a limit as

yields the following conservation law, originally published by
Kirchoff in 1867:

Thus, the rate of change of energy in an ideal acoustic volume

is
equal to the surface integral of the power crossing its boundary. A
more detailed derivation appears in [

349, p. 37].
Sabine's theory of acoustic energy decay in reverberant room

impulse
responses can be derived using this conservation relation as a
starting point.

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