Starting with the above given equations of motion for a medium at rest:
Let us now take  to all be small quantities.
In the case that we keep terms to first order, for the continuity equation, we have the  term going to 0. This similarly applies for the density perturbation times the time derivative of the velocity. Moreover, the spatial components of the material derivative go to 0. We thus have, upon rearranging the equilibrium density:
Next, given that our sound wave occurs in an ideal fluid, the motion is adiabatic, and then we can relate the small change in the pressure to the small change in the density by
Under this condition, we see that we now have
Defining the speed of sound of the system:
Everything becomes