### 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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