the equation of mass continuity,

and an equation of state which we take as the gas law

where the gas constant, , and the temperature, , are taken as constants and the fluid is isothermal.

Now the basic situation is regarded as a uniform medium at rest
(
) with a uniform pressure () and uniform density
() satisfying

Note that the other equations, (6.5) and (6.6) are satisfied automatically. We assume that there is a small change in pressure, , which introduces a small velocity, , and causes small change in the density, . These perturbations

are small in magnitude and much smaller than the magnitude of the initial uniform medium values of and . Hence,

Thus, we can write the actual density as

and similarly for the pressure. Hence, (6.5), (6.6) and (6.7) are expressed as

and

Now we use the fact that and are uniform, neglect all products of small quantities to give

where is the square of the isothermal sound speed (typical values for the sound speed are 300m/s for air, 270m/s for hydrogen, 1410m/s for water and 5600m/s for glass). Notice that these equations are linear equations in the perturbed quantities and that they have constant coefficients.

We can eliminate the velocity and from the
equations to derive a single equation for the density perturbations,
. Hence,

and we have the usual wave equation. Solutions to this wave equation represent

and is the constant amplitude of the wave.

More generally, a single wave mode in three dimensions has
and we consider

with wavenumber

Substituting (6.10) into (6.9) gives the dispersion relation

Note

- Sound waves propagate equally in all directions, there is no
preferred direction.
- From (6.8)

and so is in the same direction as . This means that sound waves propagate longitudinally.