Exponentially Decaying Traveling Waves

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

*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.(2.1) on page becomes

as depicted in Fig.2.9.

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Center for Computer Research in Music and Acoustics (CCRMA), Stanford University