Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 105 / 3 / c / Solution

Let $\mathcal{R}g(s,\omega )$ be the Radon transformTaking the large-radius limit in the Kirchhoff formula, with $\xi =t-r$ fixed for the outgoing limit and $\eta =t+r$ fixed for the incoming limit, gives the radiation fieldsIndeed, the expanding spheres converge after multiplication by $r/t$ to the planes $y\cdot \omega =-\xi$ and $y\cdot \omega =\eta$, respectively.

The Radon transforms of smooth compactly supported functions are smooth. If the data are supported in a ball of radius $R$, these transforms vanish for $|s|>R$. Hence both radiation fields are well-defined smooth functions of compact support in the null-time variable.