 Solution

ID: past-exam-of-the-mathematics-and-the-environment-and-mathematical-physiology-course-of-the-university-of-oxford/2008/a/3/iii/solution

h u = 1

after normalizing the reference discharge. The reduced momentum equation integrates to

η + F^{2} u^{2} /2 = 1 + F^{2} /2

with the reference free-surface level one. Since

h = 1/ u

s = 1 - \frac{1}{u} + \frac{F ^{2}}{2} (1 - u^{2}) .

Also

h u = 1

c_{x} = E^{*} (u) - c = - s_{t} .

With

E^{*} (1) = 1

, a steady uniform reference state is

u = h = c = η = 1, s = 0

. The integration constants above specify this reference normalization; a different imposed discharge or head changes them.

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