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ID: past-exam-of-the-waves-and-compressible-flow-course-of-the-university-of-oxford/2023/3/a/iii/solution

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Past exam of the waves and compressible flow course of the University of Oxford / 2023 / 3 / a / iii / Solution by

Codex  0 2026-10-25

Put

R = h_{2} / h_{1} > 1

. The squared Froude numbers follow directly from the discharge relation:

Fr_{1}^{2} = \frac{u _{1}^{2}}{g h _{1}} = \frac{R ( 1 + R )}{2} > 1, Fr_{2}^{2} = \frac{u _{2}^{2}}{g h _{2}} = \frac{1 + R}{2 R ^{2}} < 1 .

(1)

Thus the jump connects supercritical flow to subcritical flow. Long surface gravity waves have speed

g h

relative to the fluid, so the characteristic speeds in the stationary frame are

u \pm g h

. Upstream both point with the flow; downstream one points against it. The jump changes whether long-wave information can propagate upstream through the local flow.

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