Starting from the equations of conservation of mass and momentum for an inviscid compressible fluid, show that for small perturbations about a state of rest and uniform density the velocity is irrotational and the velocity potential satisfies the wave equation. Identify the sound speed $c_{0}$.

Define the acoustic energy density and acoustic energy flux, and derive the equation for conservation of acoustic energy.

Show that in any (not necessarily harmonic) acoustic plane wave of wavenumber $\mathbf{k}$ the kinetic and potential energy densities are equal and that the acoustic energy is transported with velocity $c_{0} \widehat{\mathbf{k}}$.

Calculate the kinetic and potential energy densities for a spherically symmetric outgoing wave. Are they equal?