Linear Analysis

Consider a uniform gas with density $\rho_0$ and pressure $p_0$ without motion ${\bf u}_0=0$. 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

$$\rho=\rho_0+\delta \rho,$$ (2.27)

$$p=p_0+\delta p,$$ (2.28)

and

$${\bf u}={\bf u}_0+\delta {\bf u}=\delta {\bf u},$$ (2.29)

where the amplitudes of perturbations are assumed much small, that is, $\vert\delta \rho\vert/\rho_0 \ll 1$, $\vert\delta p\vert/p_0 \ll 1$ and $\vert\delta {\bf u}\vert/c_s \ll 1$. We assume the variables changes only in the $x$-direction. In this case the basic equations for isothermal gas are

$$\frac{\partial \rho}{\partial t}+\frac{\partial \rho u}{\partial x}=0,$$ (2.30)

$$\rho \left( \frac{\partial u}{\partial t}+u\frac{\partial u}{\partial x}\right)=-\frac{\partial p}{\partial x} +\rho g_x,$$ (2.31)

and

$$p=c_{is}^2\rho,$$ (2.32)

where $u$ and $g_x$ represent the $x$-component of the velocity and that of the gravity, respectively.

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

$$\frac{\partial \rho_0+\delta \rho}{\partial t}+\frac{\partial (\rho_0+\delta \rho)(u_0+\delta u)}{\partial x}=0.$$ (2.33)

Noticing that the amplitudes of variables with and without $\delta$ are completely different, two equations are obtained from equation(2.33) as

$$\frac{\partial \rho_0}{\partial t}+\frac{\partial \rho_0 u_0}{\partial x} = 0,$$ (2.34)

$$\frac{\partial \delta \rho}{\partial t}+\frac{\partial \rho_0 \delta u + \delta \rho u_0}{\partial x} = 0,$$ (2.35)

where the above is the equation for unperturbed state and the lower describes the relation between the quantities with $\delta$. 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

$$(\rho_0+\delta \rho) \left( \frac{\partial u_0+\delta u}{\pa... ..._0+\delta\rho) \frac{\partial \phi_0+\delta \phi}{\partial x},$$ (2.36)

gives the relationship between the terms containing only one variable with $\delta$ as follows:

$$\rho_0\frac{\partial \delta u}{\partial t}= -\frac{\partial ... ...{\partial x} - \rho_0 \frac{\partial \delta \phi}{\partial x}.$$ (2.37)

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

$$\frac{\delta p}{\delta \rho}=\left(\frac{\partial p}{\partial \rho}\right)_T=\frac{p_0}{\rho_0}=c_{is}^2,$$ (2.38)

and for isentropic gas

$$\frac{\delta p}{\delta \rho}=\left(\frac{\partial p}{\partial \rho}\right)_{\rm ad}=\gamma \frac{p_0}{\rho_0}=c_{s}^2.$$ (2.39)