Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 335 / 2 / i / Solution

Let $V=k_0^2(n^2-1)$ be the scattering potential supported in $D$. The background Helmholtz equation and its outgoing kernel give the scalar Lippmann-Schwinger equationReplacing the unknown field in the integral by the incident field gives the Born approximation for scalar wave scattering:For the Rytov approximation, work where ${\psi }_i\ne 0$ and set $\psi ={\psi }_i{e}^{\varphi }$. Dividing the wave equation by the total field and subtracting the incident equation givesNeglect the quadratic gradient term. If ${\varphi }_1={\psi }_{s,B}/{\psi }_i$, direct differentiation shows $\Delta {\varphi }_1+2\nabla \log {\psi }_i\cdot \nabla {\varphi }_1=-V$, soExpanding the exponential in the potential gives ${\psi }_R={\psi }_i+{\psi }_{s,B}+O(V^2)={\psi }_B+O(V^2)$. Thus both approximations have the same first-order field, although exponentiation retains a particular family of higher powers.