Sound Wave

If the self-gravity is ignorable, equation (2.35)

$$\frac{\partial \delta \rho}{\partial t}+\rho_0 \frac{\partial \delta u}{\partial x}=0,$$ (2.40)

and equation (2.37)

$$\rho_0 \frac{\partial \delta u}{\partial t}= -c_{is}^2\frac{\partial \delta \rho}{\partial x},$$ (2.41)

where we assumed the gas is isothermal, these two equations describe the propagation and growth of perturbations. If the gas acts adiabatically, replace $c_{is}$ with $c_s$.

Making $\partial /\partial x\times$(2.40) and $\partial /\partial t\times$(2.41) vanishes $\delta \rho$ and we obtain

$$\frac{\partial^2 \delta u}{\partial t^2}-c_{is}^2\frac{\partial^2 \delta u}{\partial x^2}=0.$$ (2.42)

Since this leads to

$$\frac{\partial \delta u}{\partial t}-c_{is}\frac{\partial \delta u}{\partial x} = 0,$$ (2.43)

$$\frac{\partial \delta u}{\partial t}+c_{is}\frac{\partial \delta u}{\partial x} = 0,$$ (2.44)

equation (2.42) has a solution that a wave propagates with a phase velocity of $\pm c_s$. Since the displacement ( $\propto \delta u$) is parallel to the propagation direction $x$, and the restoring force comes from the pressure, this seems the sound wave.