 Solution

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

Let

H

be the depth behind the bore. The wall's kinematic boundary condition makes the velocity there

U

; ahead of the bore the depth and velocity are

h_{0}

and zero. Across a rightward hydraulic bore moving at

V

V (H - h_{0}) = H U, V H U = H U^{2} + \frac{g}{2} (H^{2} - h_{0}^{2}) .

The first gives

R = H / h_{0} = V / (V - U) = v / (v - 1) > 1

. Substituting into the second gives

F^{2} v = (R^{2} - 1) /2

, where

v = V / U

and

F = U / g h_{0}

F^{2} v^{2} = \frac{v ( 2 v - 1 )}{2 ( v - 1 ) ^{2}} = 1 + \frac{3}{2 ( v - 1 )} + \frac{1}{2 ( v - 1 ) ^{2}}, v > 1 .

The positive depth increase is consistent with the energy-loss condition in ii, now viewed in the bore's own shock frame.

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