Past exam of the mathematics course of the University of Cambridge / 2009 / iii / Paper 78 / 4

Fix time dependence $e^{-i\omega t}$ and use pressure amplitude for $\psi$. Write the incident acoustic plane wave as ${\psi }_i=A{e}^{i\alpha x-i\beta z}$, where $\alpha =k\cos \theta$, $\beta =k\sin \theta >0$. “Perfectly reflecting” specifies zero absorbed energy but does not uniquely select a boundary condition: a rigid wall has a Neumann boundary condition, whereas a pressure-release surface has a Dirichlet boundary condition. We first give the rigid-wall result, and then the alternative pressure-release result.

For the rigid surface the normal-velocity condition, without an immaterial normalizing factor, isExpand $\psi ={\psi }^{[0]}+{\psi }^{[1]}+{\psi }^{[2]}+\cdots$ in the height amplitude, scaling a fixed differentiable profile. The flat total field is ${\psi }^{[0]}=A{e}^{i\alpha x}(e^{-i\beta z}+e^{i\beta z})$. Taylor expansion givesLet $\hat{h}(q)=\int h(x)e^{-iqx}dx$ and $\beta (q)=\sqrt{k^2-q^2}$ on the outgoing sheet: it is positive real for propagating modes and positive imaginary for evanescent modes. Fourier transforming this boundary derivative gives $2A(k^2-\alpha q)\hat{h}(q-\alpha )$. The outgoing angular spectrum therefore yields the small-height Neumann scattering fieldFor an infinite stationary surface the Fourier quantities may be interpreted as random distributions or through a finite illumination window and its limiting mean. The correction is linear in $h$ and $h^{\prime }$, so zero mean height and stationarity giveOnly the rough correction has zero mean at first order; the flat reflected wave remains.

The first nontrivial coherent correction depends on correlation, not just the r.m.s. height. Define $C_h(s)=⟨h(x+s)h(x)⟩$, its power spectrum of surface height $S_h(q)=\int C_h(s)e^{-iqs}ds$, and ${\sigma }^2=(2\pi {)}^{-1}\int S_h(q)dq$. At second order the flat-field third normal derivative and mixed derivative vanish at zero, leavingFor each first-order outgoing Fourier component, its derivative data are $g_1(q)=2A(k^2-\alpha q)\hat{h}(q-\alpha )$, its second normal derivative is $i\beta (q)g_1(q)$, and its tangential derivative is $q{g}_1(q)/\beta (q)$. Averaging their products with $h,h^{\prime }$ gives the mean derivativeDividing by the specular outgoing derivative $i\beta$ therefore gives oblique coherent reflection from a stationary rigid rough surface:This second-order expression requires the relevant spectral moments to exist and the boundary expansion to remain controlled. Small $|kh|$ alone does not control arbitrarily large slopes or high-frequency roughness. The given stationarity and $\sigma$ determine the first-order mean above, but do not determine this spectrum-weighted correction numerically.

For a pressure-release surface, instead impose $\psi (x,h)=0$. The flat reflected coefficient is $-1$, and Taylor expansion gives ${\psi }^{[1]}(x,0)=2i\beta Ah{e}^{i\alpha x}$. Thus small-height Dirichlet scattering givesThe zero-mean first correction again vanishes on averaging. At second order ${\psi }^{[2]}(x,0)=-h{\partial }_z{\psi }^{[1]}(x,0)$, givingIf the roughness spectrum is concentrated near $q=\alpha$ after the incident tangential shift, the additional long-correlation approximation yields $r_N\simeq 1-2{\beta }^2{\sigma }^2$ and $r_D\simeq -1+2{\beta }^2{\sigma }^2$. A local Gaussian random-height phase model would then give $r_N=e^{-2{\beta }^2{\sigma }^2}$ and $r_D=-e^{-2{\beta }^2{\sigma }^2}$, but Gaussian statistics and that local approximation were not supplied. They must not be inferred from stationarity and r.m.s. height alone.

Translation invariance of the stationary surface statistics implies that a single incident tangential wavenumber retains that wavenumber in its coherent mean. Hence, away from the surface, writeThe unknown effective reflection coefficient $r_b$ is referenced to the mean plane $z=0$. It may be the rigid or pressure-release coefficient just derived, or a more general rough-boundary response. Lossless microscopic reflection does not require $|r_b|=1$: diffuse scattering removes power from the coherent specular field.

For the added planar upper interface, keep the same tangential wavenumber $\alpha$. Let ${\beta }_2=\sqrt{k_2^2-{\alpha }^2}$ on its outgoing sheet. The normal acoustic impedances in the layer and upper medium areContinuity of pressure and normal velocity gives the reflection coefficient for upward incidence from the layer and transmission coefficient for downward incidence from above:If a separate impedance sheet rather than an ordinary fluid interface is intended, its specified effective impedance replaces the upper termination; a sheet impedance has not otherwise been given. These formulas correspond to the stated two-fluid interface.

Let the upper incident pressure amplitude at $z=d$ be $I$. If $D$ and $U$ denote downward and upward layer amplitudes at that plane, their interface and lower-reflection relations areSolving, or summing the geometric series of repeated round trips, gives the coherent acoustic layer above a rough reflector:The expression describes the field at heights where the common homogeneous layer exists; perturbative mean fields near the moving rough boundary need their physical trace evaluated separately. For an upper incident wave normalized as $e^{i\alpha x-i{\beta }_{2}z}$, set $I=e^{-i{\beta }_{2}d}$. With a matched upper medium, $r_{12}=0$ and $t_{21}=1$, the formula reduces to the single lower-reflection field. Zeros of the denominator give ideal layer resonances, interpreted with outgoing continuation or small loss.

This is a coherent specular description: a single effective $r_b$ closes the downward/upward mean amplitudes. Diffuse paths reflected back from the upper interface can modify the mean response; they must either be neglected in this approximation or included in the effective lower reflection coefficient for the actual layered environment. Without a full roughness spectrum and boundary type, a unique numerical mean field cannot be specified beyond this general coefficient form.