= Coherent acoustic layer above a rough reflector
{title2=$D=t_{21}I/(1-r_{12}r_be^{2i\beta d})$}
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 $De^{i\alpha x}[e^{-i\beta(z-d)}+r_be^{2i\beta d}e^{i\beta(z-d)}]$. Feedback through diffuse roughness channels must be neglected in this scalar model or included in a context-dependent effective reflection coefficient.
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