### 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 exponential function of distance .3.3 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.(2.1) on page  becomes

as depicted in Fig.2.9.
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