 Coherent acoustic layer above a rough reflector

In a coherent specular model, let $r_b$ be the effective lower pressure reflection coefficient referenced to $z=0$, and let the planar upper interface lie at $z=d$. With layer and upper normal impedances $Z_1,Z_2$, $r_{12}=(Z_2-Z_1)/(Z_2+Z_1)$ and $t_{21}=2Z_1/(Z_1+Z_2)$. Summing repeated round trips gives the displayed downward amplitude at the upper interface. The mean layer field is $D{e}^{i\alpha x}[e^{-i\beta (z-d)}+r_b{e}^{2i\beta d}{e}^{i\beta (z-d)}]$. Feedback through diffuse roughness channels must be neglected in this scalar model or included in a context-dependent effective reflection coefficient.