Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2020 / Paper 1 / 8 / b / i / Solution

Transform in time, leaving $r$ as a parameter. Applying the derivative identity twice gives $\widetilde{{\partial }_t^2\sigma }=-{\omega }^2\tilde{\sigma }$. ThereforeThese manipulations can be justified for sufficiently decaying smooth fields, or with distributional Fourier transforms when ordinary integrability is unavailable.