Obtain an expression for the compressive energy $W(\rho)$ per unit volume for adiabatic motion of a perfect gas, for which the pressure $p$ is given in terms of the density $\rho$ by a relation of the form

$$p=p_{0}\left(\rho / \rho_{0}\right)^{\gamma},$$

where $p_{0}, \rho_{0}$ and $\gamma$ are positive constants.

For one-dimensional motion with speed $u$ write down expressions for the mass flux and the momentum flux. Deduce from the energy flux $u\left(p+W+\frac{1}{2} \rho u^{2}\right)$ together with the mass flux that if the motion is steady then

$$\frac{\gamma}{\gamma-1} \frac{p}{\rho}+\frac{1}{2} u^{2}=\text { constant }$$

A one-dimensional shock wave propagates at constant speed along a tube containing the gas. Ahead of the shock the gas is at rest with pressure $p_{0}$ and density $\rho_{0}$. Behind the shock the pressure is maintained at the constant value $(1+\beta) p_{0}$ with $\beta>0$. Determine the density $\rho_{1}$ behind the shock, assuming that $(\dagger)$ holds throughout the flow.

For small $\beta$ show that the changes in pressure and density across the shock satisfy the adiabatic relation $(*)$ approximately, correect to order $\beta^{2}$.