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Through second order in height, a normally incident acoustic plane wave has the displayed mean reflected amplitude for a Dirichlet boundary. The propagating part of the power spectrum of surface height contributes a positive real correction to the negative flat reflection coefficient, while the evanescent part contributes an imaginary correction. The mean linear rough correction is zero.

The quadratic coherent correction weights the power spectrum of surface height by $\beta (q)$, so equal root mean square heights can give different mean fields. Only an additional long-spatial-scale approximation, with spectrum concentrated at $|q|\ll k$, reduces it to $2k^2{\sigma }^2$. Second-order surface scattering therefore contains information absent from first-order coherent reflection.

Let the stationary height have covariance function $C_h(\xi )=⟨h(x)h(x+\xi )⟩$, with $C_h(0)={\sigma }^2$. Define the power spectrum of surface height byFor real stationary heights this spectrum is even and nonnegative. The Fourier multiplier identity in part (c) givesThus coherent reflection from a stationary rough surface at normal incidence isRequire the corresponding weighted spectral moment to exist. If the stationary process has a spectral measure rather than a density, the same formula uses that measure with the matching normalization.

Splitting the propagating and evanescent parts makes the effect clear:through second order. The positive real correction reduces the magnitude of the initially negative unit coherent reflection at this order, as some reflection becomes diffuse. The evanescent part produces a coherent wave phase correction. In contrast, the first-order mean was exactly the flat reflected wave.

The height-correlation dependence of coherent reflection cannot generally be determined from $\sigma$ alone: the quadratic term weights the whole spectrum by $\beta (q)$, while ${\sigma }^2=(2\pi {)}^{-1}\int S_h(q)dq$ is unweighted. If the roughness varies only on scales much longer than the wavelength, so its spectrum is concentrated at $|q|\ll k$, then $\beta \simeq k$ andThis is a useful limiting formula, not the general second-order answer under only small-height assumptions. Also the mean field sampled at the moving physical boundary is $-1+k^2{\sigma }^2/2$ through second order, from part (c), and is a different observable.

For a real zero-mean stationary height process, its power spectrum of surface height is the Fourier transform of the height covariance function. It is even and nonnegative, and ${\sigma }^2=(2\pi {)}^{-1}\int S_h(q)dq$. More general stationary processes have a spectral measure rather than a density. Weighted spectral moments control the existence of boundary derivatives and second-order mean scattering corrections.