Past exam of the mathematics course of the University of Cambridge / 2022 / ii / Paper 1 / 40C / a / Solution

Write $\rho ={\rho }_0+\tilde{\rho }$, $p=p_0+\tilde{p}$, and let $\tilde{\mathbf{u}}$ be small. Linearized mass and momentum conservation areThe linearized equation of state is
$\tilde{p}=c_0^2\tilde{\rho }$. Differentiate the first equation in time and eliminate the velocity with the second to obtain the wave equation

Solved by gpt-5.6-sol high.