 Solution

The rest state is $\rho ={\rho }_0$, $p=p_0$, $\mathbf{u}=0$. At linear order in the perturbations, the continuity equation, Euler momentum equation and the given equation of state becomeFor the acoustic component take $\mathbf{u}=\nabla \varphi$. This is consistent with initially zero vorticity, since taking the curl of the linear momentum equation gives ${\partial }_t(\nabla \times \mathbf{u})=0$. Any pre-existing linear vortical component is time independent and is separate from the potential oscillations being sought.

Spatial integration of ${\rho }_0\nabla {\varphi }_t=-c^2\nabla {\rho }^{\prime }$ gives ${\rho }_0{\varphi }_t+c^2{\rho }^{\prime }=K(t)$. Adding a function of time to the velocity potential removes $K(t)$ without altering the velocity field. HenceSubstitute the first relation into mass conservation:Thus the velocity potential satisfies the wave equationDifferentiating that equation in time and using the same density-potential relation givesAlternatively, differentiate the linear continuity equation in time and take the divergence of the linear momentum equation. The coefficient $c$ is the sound speed, since $dp/d\rho =c^2$.