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Let
denote the decay factor associated with
propagation of a plane wave over distance at frequency
rad/sec. For an ideal plane wave, there is no ``spreading
loss'' (attenuation by ). Under uniform conditions, the
amount of attenuation (in dB) is proportional to the distance
traveled; in other words, the attenuation factors for two successive
segments of a propagation path are multiplicative:
This property implies that is an exponential function of
distance .2.3
Frequency-independent air
absorption is easily modeled in an acoustic simulation by making
the substitution
in the transfer function of the simulating delay line, where
denotes the attenuation associated with propagation during one
sampling period ( seconds). Thus, to simulate absorption
corresponding to an -sample delay, the difference equation
Eq. (1.1) on page becomes
as depicted in Fig. 1.8.
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 .
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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