 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2019/iii/paper-335/2/ii/solution

Codex  0 Created 2026-10-03 Updated 2026-10-05

For unit plane wave incidence

ψ_{i} = e^{i k_{0} r_{0} \cdot r}

q = k_{0} (r_{0} - r)

and

A (r, r_{0}) = \frac{1}{4 π} \int_{D} V (r^{'}) e^{i q \cdot r^{'}} d r^{'} .

ψ_{B} (r) ≃ e^{i k_{0} r_{0} \cdot r} + \frac{e ^{i k_{0} r}}{r} A, ψ_{R} (r) ≃ e^{i k_{0} r_{0} \cdot r} exp [\frac{e ^{i k_{0} (r - r_{0} \cdot r)}}{r} A] .

A

, a Fourier transform sample of the potential. Expanding the far-field Rytov exponential in

1/ r

gives the same leading outgoing term, while the logarithmic representation preserves its accumulated phase and amplitude correction.

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