(i) Write down the shock condition associated with the equation

$$\rho_{t}+q_{x}=0$$

where $q=q(\rho)$. Discuss briefly two possible heuristic approaches to justifying this shock condition.

(ii) According to shallow water theory, waves on a uniformly sloping beach are described by the equations

$$\begin{aligned} &\frac{\partial h}{\partial t}+\frac{\partial}{\partial x}(h u)=0, \\ &\frac{\partial u}{\partial t}+u \frac{\partial u}{\partial x}+g \frac{\partial \eta}{\partial x}=0, \quad h=\alpha x+\eta, \end{aligned}$$

where $\alpha$ is the constant slope of the beach, $g$ is the gravitational acceleration, $u(x, t)$ is the fluid velocity, and $\eta(x, t)$ is the elevation of the fluid surface above the undisturbed level.

Find the characteristic velocities and the characteristic form of the equations.

What are the Riemann variables and how do they vary with $t$ on the characteristics?