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
![$ V$](http://www.dsprelated.com/josimages_new/pasp/img239.png)
in time interval
![$ \Delta t$](http://www.dsprelated.com/josimages_new/pasp/img2677.png)
must equal the acoustic
intensity
![$ \underline{I}(t,\underline{x})$](http://www.dsprelated.com/josimages_new/pasp/img3095.png)
exiting the volume,
times
![$ \Delta t$](http://www.dsprelated.com/josimages_new/pasp/img2677.png)
(approximating
![$ I$](http://www.dsprelated.com/josimages_new/pasp/img238.png)
as constant between times
![$ t$](http://www.dsprelated.com/josimages_new/pasp/img122.png)
and
![$ t+\Delta t$](http://www.dsprelated.com/josimages_new/pasp/img3096.png)
):
The term
![$ \underline{I}(t,\underline{x})\cdot\underline{n}(\underline{x})$](http://www.dsprelated.com/josimages_new/pasp/img3098.png)
is the dot-product of the (vector)
intensity
![$ \underline{I}$](http://www.dsprelated.com/josimages_new/pasp/img3099.png)
with a unit-vector
![$ \underline{n}$](http://www.dsprelated.com/josimages_new/pasp/img3100.png)
chosen to be normal to the
surface at position
![$ \underline{x}$](http://www.dsprelated.com/josimages_new/pasp/img260.png)
along the surface. Thus,
![$ \underline{I}\cdot \underline{n}$](http://www.dsprelated.com/josimages_new/pasp/img3101.png)
is
the component of the acoustic intensity
![$ \underline{I}$](http://www.dsprelated.com/josimages_new/pasp/img3099.png)
exiting the volume
normal to its surface. (The tangential component does not exit.)
Dividing through by
![$ \Delta t$](http://www.dsprelated.com/josimages_new/pasp/img2677.png)
and taking a limit as
![$ \Delta t\to 0$](http://www.dsprelated.com/josimages_new/pasp/img3102.png)
yields the following conservation law, originally published by
Kirchoff in 1867:
Thus, the rate of change of energy in an ideal acoustic volume
![$ V$](http://www.dsprelated.com/josimages_new/pasp/img239.png)
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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