 Solution

ID: past-exam-of-the-theoretical-mechanics-course-of-the-university-of-oxford/2010/b/6/a/solution

For positive depth put

c = g h

. The equations become

u_{t} + u u_{x} + 2 c c_{x} = 0, c_{t} + u c_{x} + \frac{c}{2} u_{x} = 0.

Adding or subtracting twice the second equation to the first gives

(\partial_{t} + (u \pm c) \partial_{x}) (u \pm 2 c) = 0.

d x / d t = u \pm c

and the corresponding conserved Riemann invariants are

Q_{\pm} = u \pm 2 c

. At a dry front

c = 0

the two speeds coincide; the positive-depth formulas are applied up to that limit.

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