Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 1 / 40D / a / Solution

Let $\tilde{\rho }$, $\tilde{p}$, and $u$ be first-order perturbations. Linearized mass and momentum conservation and the homentropic relation areDifferentiate the first equation in time and use the divergence of the second to obtainThe perturbation is irrotational, so write $u=\nabla \varphi$. Integrating the momentum equation spatially, with a purely time-dependent constant absorbed into $\varphi$, gives