Consider a uniform gas with density and pressure without motion .
In this uniform gas distribution, we assume small perturbations.
As a result, the distributions of the density, the pressure and the velocity are perturbed from the uniform ones as

and

where the amplitudes of perturbations are assumed much small, that is, , and . We assume the variables changes only in the -direction. In this case the basic equations for isothermal gas are

and

where and represent the -component of the velocity and that of the gravity, respectively.

Using equations (2.27), (2.28), and (2.29),
equation (2.30) becomes

where the above is the equation for unperturbed state and the lower describes the relation between the quantities with . Equation (2.34) is automatically satisfied by the assumption of uniform distribution. In equation (2.35) the last term is equal to zero. Equation of motion

(2.36) |

The perturbations of pressure and density are related with each other as follows: for the isothermal gas

(2.38) |

(2.39) |