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 $r$.^2.3

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

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

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. (1.1) on page  becomes

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

as depicted in Fig. 1.8.

More generally, frequency-dependent air absorption can be modeled using the substitution

$$z^{-1}\leftarrow G(z)z^{-1}$$

where $G(z)$ denotes the filtering per sample in the propagation medium. Since air absorption cannot amplify a wave at any frequency, we have $\left \vert G(e^{j\omega T})\right \vert\leq 1$. A lossy delay line for plane-wave simulation is thus described by

$$Y(z) = G^M(z) z^{-M}X(z)$$

in the frequency domain, and

$$y(n) = \underbrace{g\ast g\ast \dots \ast g \, \ast }_{\hbox{$M$\ times}} x(n-M)$$

in the time domain, where `$\ast$' denotes convolution, and $g(n)$ is the impulse response of the per-sample loss filter $G(z)$.

For spherical waves, the loss due to spherical spreading is of the form

$$Y(z) \propto \frac{G^M(z) z^{-M}}{r}X(z)$$

where $r$ is the distance from $X$ to $Y$. We see that the spherical spreading loss factor is ``hyperbolic'' in the propagation distance $r$, while air absorption is exponential in $r$.