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ID: past-exam-of-the-applied-partial-differential-equations-course-of-the-university-of-oxford/2023/3/b/solution

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Past exam of the applied partial differential equations course of the University of Oxford / 2023 / 3 / b / Solution by

Codex  0 2026-10-25

Apply the Rankine-Hugoniot conditions to the conserved variables

(h, h u)

and flux

(h u, h u^{2} + h^{2} /2)

. They give

s (h_{1} - 1) = h_{1} u_{1}, s h_{1} u_{1} = h_{1} u_{1}^{2} + \frac{h _{1}^{2} - 1}{2} .

For a genuine shock,

h_{1} \neq = 1

. Eliminating

u_{1}

yields

s^{2} = h_{1} (h_{1} + 1) /2

. Taking the positive branch requested,

s = \frac{h _{1} ( h _{1} + 1 )}{2}, u_{1} = (h_{1} - 1) \frac{h _{1} + 1}{2 h _{1}} .

When

h_{1} = 1

, the jump conditions instead force

u_{1} = 0

and describe a constant state; the labelled interface can then move at any speed and is not a shock. The nontrivial formulas describe a shallow-water shock into stationary unit-depth water.

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