###
Frequency-Dependent Air-Absorption
Filtering

More generally,

*frequency-dependent* air
absorption can be modeled using the substitution

where

denotes the

*filtering per sample* in the

propagation medium. Since air absorption cannot amplify a wave at any
frequency, we have

. A lossy

delay line for

plane-wave simulation is thus described by

in the

frequency domain, and

in the time domain, where `

' denotes

convolution, and

is
the

impulse response of the per-sample loss filter

. The effect
of

on the

poles of the system is discussed in §

3.7.4.
For spherical waves, the loss due to spherical spreading is of the form

where

is the distance from

to

. We see that the spherical
spreading loss factor is ``hyperbolic'' in the propagation distance

, while air absorption is

exponential in

.

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