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 Incoming links: Coherent acoustic layer above a rough reflector

Past exam of the mathematics course of the University of Cambridge / 2009 / iii / Paper 78 / 4 / b / Solution Created 2026-10-03 Updated 2026-10-14

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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, write

⟨ ψ_{s} (x, z)⟩ = A r_{b} (α, ω) e^{i αx + i β z}, ⟨ ψ_{tot} ⟩ = A e^{i αx} (e^{- i β z} + r_{b} e^{i β z}) .

(1)

The 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

α

. Let

β_{2} = k_{2}^{2} - α^{2}

on its outgoing sheet. The normal acoustic impedances in the layer and upper medium are

Z_{1} = \frac{ρ ω}{β}, Z_{2} = \frac{ρ _{2} ω}{β _{2}} .

(2)

Continuity of pressure and normal velocity gives the reflection coefficient for upward incidence from the layer and transmission coefficient for downward incidence from above:

r_{12} = \frac{Z _{2} - Z _{1}}{Z _{2} + Z _{1}}, t_{21} = \frac{2 Z _{1}}{Z _{2} + Z _{1}} .

(3)

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

I

. If

D

and

U

denote downward and upward layer amplitudes at that plane, their interface and lower-reflection relations are

D = t_{21} I + r_{12} U, U = r_{b} e^{2 i β d} D .

(4)

Solving, or summing the geometric series of repeated round trips, gives the coherent acoustic layer above a rough reflector:

⟨ ψ_{tot} (x, z)⟩ = \frac{t _{21} I e ^{i αx}}{1 - r _{12} r _{b} e ^{2 i β d}} [e^{- i β (z - d)} + r_{b} e^{2 i β d} e^{i β (z - d)}], h (x) \leq z \leq d .

(5)

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 αx - i β_{2} z}

, set

I = e^{- i β_{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.

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