For a time-harmonic perturbation, the deviations from equilibrium may be described by the following ansatz:

The changes induced by sound occur sufficiently rapidly that the process can be treated as adiabatic. Density ρ and pressure P are related through the adiabatic compressibility κ_s:

Substituting ρ and v into Eqs. (3.2) and (3.1), and retaining only linear terms, gives the coupled equations:

Only solutions for which the velocity perturbation is parallel to the wave vector are considered; these correspond to longitudinal waves:

This yields a homogeneous linear system of two unknowns:

A nontrivial solution exists only when the determinant vanishes. The resulting dispersion relation is: