Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 335 / 1 / ii

After separating the conserved horizontal factor as

ψ = e^{i p x} f (z)

f^{''} + q_{0}^{2} f = - α q_{0}^{2} H (- z) f,

where

H

d^{2} / d z^{2} + q_{0}^{2}

G (z, z^{'}) = \frac{e ^{i q_{0} ∣ z - z^{'} ∣}}{2 i q _{0}} .

The Born approximation at first order replaces

f

on the right-hand side by the incident profile

f_{i} (z^{'}) = e^{- i q_{0} z^{'}}

. For

z > 0

this gives

f_{s, B} (z) = - α q_{0}^{2} \int_{- \infty}^{0} \frac{e ^{i q_{0} (z - z^{'})}}{2 i q _{0}} e^{- i q_{0} z^{'}} d z^{'} .

The integral is understood with the usual outgoing-wave convergence factor. Since

\int_{- \infty}^{0} e^{- 2 i q_{0} z^{'}} d z^{'} = - \frac{1}{2 i q _{0}},

we obtain

f_{s, B} (z) = - \frac{α}{4} e^{i q_{0} z} .

Therefore the Born reflected field is

ψ_{r, B} (x, z) = - \frac{α}{4} e^{i (p x + q_{0} z)} .

b

The Rytov approximation at first order writes the total profile as

f_{R} = f_{i} e^{χ_{1}}

and identifies the first logarithmic perturbation with the Born relative field:

χ_{1} = \frac{f _{s, B}}{f _{i}} = - \frac{α}{4} e^{2 i q_{0} z} .

Hence, in the upper half-space,

ψ_{R} (x, z) = e^{i (p x - q_{0} z)} exp (- \frac{α}{4} e^{2 i q_{0} z}) .

Expanding the exponential function to first order shows that its reflected component is

ψ_{r, R}^{(1)} (x, z) = - \frac{α}{4} e^{i (p x + q_{0} z)} .

The exponentiated expression is the Rytov approximation to the total field; only its term linear in

α

is a single specular reflected plane wave.

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