(11.84) | ||

(11.85) |

where and are the small velocity and pressure perturbations, respectively, due to the wave. To first order in small quantities, the fluid equations of motion, (11.1) and (11.2), reduce to

respectively. We can also define the displacement,

The curl of Equation (11.87) implies that . Hence, we can write , and Equation (11.87) yields

Finally, Equation (11.86) gives

The most general traveling wave solution to Equation (11.90), with wave vector , and angular frequency , is

It follows from Equation (11.89) that

and from Equation (11.88) that

Here, is the phase velocity of the wave.