 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2014/iii/paper-72/1/b/solution

Codex  0 Created 2026-10-03 Updated 2026-10-06

Let

δ ψ_{B} = ψ_{1}^{(B)} - ψ_{i}

be the first-order scattered correction in the Born approximation for scalar wave scattering, using the paper's Green function convention. Comparing its outgoing integral with the preceding logarithmic correction gives

χ_{1} = \frac{δ ψ _{B}}{ψ _{i}} .

Therefore the exact algebraic relation between the two first-order approximations is

ψ_{1}^{(R)} = ψ_{i} exp (\frac{ψ _{1}^{(B)} - ψ _{i}}{ψ _{i}}) = ψ_{i} exp (\frac{ψ _{1}^{(B)}}{ψ _{i}} - 1) .

V = O (ε)

, then

χ_{1} = O (ε)

on a controlled fixed region, and

ψ_{1}^{(R)} = ψ_{i} (1 + χ_{1}) + O (ε^{2}) = ψ_{1}^{(B)} + O (ε^{2}) .

Thus Born and Rytov agree to first order in the scattering potential, but differ as finite approximations. The Rytov approximation exponentiates the first logarithmic correction; the Born approximation adds the first field correction. The exponential's higher powers are not a calculation of all higher multiple-scattering terms in the Born series. This distinction explains why a smooth, appreciable wave phase accumulation can be represented more naturally by the Rytov approximation even when the corresponding linear field expansion is inaccurate.

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