Past exam of the astrophysical fluid dynamics course of the University of Cambridge / 2022 / Section B / 5 / b

Table of contents
- Solution b

Solution

The pressure-free equation has the scalar conservation law form

\partial_{t} u + \partial_{x} f (u) = 0, f (u) = \frac{1}{2} u^{2} .

(1)

A smooth travelling ansatz

u = U (x - V t)

obeys

(U - V) U^{'} = 0

, so it cannot continuously connect two distinct constant states. The inviscid shock wave must be interpreted as a discontinuous weak solution. Its Rankine-Hugoniot condition is

V (u_{2} - u_{1}) = f (u_{2}) - f (u_{1}) = \frac{1}{2} (u_{2}^{2} - u_{1}^{2}) .

(2)

For

u_{1} \neq = u_{2}

V = \frac{u _{1} + u _{2}}{2} .

(3)

The result is independent of which state is called upstream. For a physical compressive entropy shock of the Inviscid Burgers equation, the left and right states satisfy