Past exam of the mathematics course of the University of Cambridge / 2012 / iii / Paper 78 / 3 / a / Solution

Use the scattering potential convention $V=k_0^2(n^2-1)$. With time dependence $e^{-i\omega t}$ the outgoing Green function is $G_0(r,r^{\prime })=e^{i{k}_0|r-r^{\prime }|}/(4\pi |r-r^{\prime }|)$ and satisfies $(\Delta +k_0^2)G_0=-\delta$. The total-field Lippmann-Schwinger equation isReplacing the field in the integral by the incident field gives the first Born approximation for scalar wave scattering:For the first Rytov approximation, write $\psi ={\psi }_0{e}^{\chi }$ on a region where the incident field is nonzero and a continuous complex logarithm can be chosen. The logarithmic wave perturbation equation isThe last dot product is bilinear, not a squared modulus. Dropping it leaves a linear equation for ${\chi }_1$. Multiplication by ${\psi }_0$ shows that ${\psi }_0{\chi }_1$ satisfies the same outgoing inhomogeneous Helmholtz equation as ${\psi }_1$. HenceIntroduce a small multiplier $\lambda$ on $V$. Then ${\psi }_1=O(\lambda )$ andThe two total-field approximations agree to first order. Exponentiating the first logarithmic wave perturbation does not supply the generally missing second-order Born series contribution.