Exponentially Decaying Traveling Waves

Let $g(r,\omega)$ denote the decay factor associated with propagation of a plane wave over distance $r$ at frequency $\omega$ rad/sec. For an ideal plane wave, there is no ``spreading loss'' (attenuation by $1/r$ ). 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:

$$g(r_1+r_2,\omega) = g(r_1,\omega)g(r_2,\omega)$$

This property implies that $g$ is an exponential function of distance $r$ .^3.3

Frequency-independent air absorption is easily modeled in an acoustic simulation by making the substitution

$$z^{-1}\rightarrow gz^{-1}$$

in the transfer function of the simulating delay line, where $g$ denotes the attenuation associated with propagation during one sampling period ($T$ seconds). Thus, to simulate absorption corresponding to an $M$ -sample delay, the difference equation Eq.(2.1) on page  becomes

$$y(n) = g^Mx(n-M),$$

as depicted in Fig.2.9.