Small-height Dirichlet scattering ($\psi (x,h(x))=0,h=\epsilon \eta$)

Small-height Dirichlet scattering expands the acoustic boundary condition about a mean plane for a fixed regular surface profile. The zeroth-order field includes the flat reflection; higher orders solve outgoing Helmholtz equation problems with induced boundary data. Small $|kh|$ controls the incident-field Taylor expansion, but rapidly varying roughness can require additional control of normal derivatives and spectral moments.

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.

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.

For normal incidence on a Dirichlet rough surface, the scattered trace on the moving boundary is exactly $-e^{-ikh(x)}$. Its trace on the mean plane is instead $-1+2ikh+2kh\mathcal{B}h+\cdots$. The two traces are related by a normal Taylor expansion and must not be identified. Their ensemble averages can therefore have different quadratic coefficients without inconsistency.

For normal plane-wave incidence, the second-order Dirichlet boundary condition gives ${\psi }_s^{[2]}=-h{\partial }_z{\psi }_s^{[1]}$ at the mean plane, since ${\partial }_z^2{\psi }^{[0]}$ vanishes there. The Dirichlet-to-Neumann map for a Helmholtz half-space then yields $2kh\mathcal{B}h$. This nonlocal expression is not generally $2k^2{h}^2$; that local limit needs roughness varying slowly compared with the wavelength.

Taylor expansion of the Dirichlet boundary condition gives the first-order boundary trace $-h{\partial }_z{\psi }^{[0]}$ at the mean plane. The outgoing angular spectrum then propagates it. A zero-mean surface gives zero mean for this linear rough correction, while the complete scattered field still contains its nonzero flat reflection.

For incident amplitude one, ${\psi }_i=e^{i{p}_{i}x-i{q}_{i}z}$ with $q_i>0$, Taylor expansion of a small-height Dirichlet boundary condition gives ${\psi }_{\mathrm{sc}}^{[1]}(x,0)=2i{q}_{i}f(x)e^{i{p}_{i}x}$. If the mean height is constant, its mean transform is supported only at tangential wavenumber $p_i$, giving the displayed specular mean through first order. Small height alone is insufficient: deterministic mean height $a\cos Kx$ produces mean sidebands $p_i\pm K$. Stationary roughness has constant mean and therefore satisfies the first-order requirement.

First-order Dirichlet roughness has mean-plane trace $2i{q}_i{A}_{i}h(x)e^{i{p}_{i}x}$, where $p_i^2+q_i^2=k^2$ and $q_i>0$. Stationary phase applied to the outgoing angular spectrum evaluates the far-field amplitude at tangential wavenumber $k\sin {\theta }_s$. For a stationary surface illuminated over length $D$, the mean squared height transform divided by $D$ tends to its power spectrum of surface height. Thus the diffuse intensity samples the spectrum at the tangential momentum transfer. An infinite stationary surface has no finite total far-field amplitude without a window or an intensity-density convention.

The outgoing angular spectrum propagates prescribed scalar boundary data into a half-space. For $e^{-i\omega t}$ time dependence, $\beta (q)=\sqrt{k^2-q^2}$ has nonnegative real part on propagating modes and nonnegative imaginary part on evanescent modes. Its trace at the boundary is $g$, and each mode solves the Helmholtz equation. The spectral representation can be interpreted through distributions for nondecaying plane-wave data.

Use $\hat{\psi }(\nu )=\int \psi (x,0)e^{-i\nu x}dx$ and $q(\nu )=\sqrt{k^2-{\nu }^2}$ on the upward branch. If the uncentered field correlation is translationally homogeneous and $⟨\hat{\psi }(\nu )\hat{\psi }({\nu }^{\prime }{)}^{\ast }⟩=2\pi S(\nu )\delta (\nu -{\nu }^{\prime })$, its correlation at height $z$ is the displayed integral. The propagating spectrum is unchanged; evanescent contributions decrease as $e^{-2\sqrt{{\nu }^2-k^2}z}$. A spectral measure of a stationary random field permits discrete coherent components. Without correlation homogeneity a double spectral covariance integral depends on both observation coordinates rather than only their separation.

An upward angular spectrum obtained from a horizontal field trace is exact in the homogeneous half-space above that reference plane. For a rough boundary $z=f(x)$, choose the plane above $\sup f$. A trace at $z=0$ is not physically defined in the medium if any part of the surface lies above it. Using such a mean-plane trace requires an analytic continuation convention; convergence of the continued spectrum down to the rough boundary is an additional hypothesis. Small-height boundary perturbation uses the reference plane formally order by order and is distinct from an exact expansion throughout an arbitrary rough domain.