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ID: past-exam-of-the-fluids-and-waves-course-of-the-university-of-oxford/2023/3/b/ii/solution

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Past exam of the fluids and waves course of the University of Oxford / 2023 / 3 / b / ii / Solution by

Codex  0 2026-10-25

At the rigid bed, the no-penetration boundary condition gives

ϕ_{z} = 0 at z = - h .

The displaced surface is material, so taking the material derivative of

z - η (x, t)

gives the exact kinematic boundary condition for a free-surface graph:

η_{t} + ϕ_{x} η_{x} = ϕ_{z} at z = η (x, t) .

With upward vertical coordinate

z

, the gravitational potential per unit mass is

g z

. The Unsteady Bernoulli equation, evaluated on the surface with the prescribed pressure

p = B η_{xxxx}

, gives

ϕ_{t} + \frac{1}{2} (ϕ_{x}^{2} + ϕ_{z}^{2}) + g η + \frac{B}{ρ} η_{xxxx} = 0 at z = η (x, t) .

Here the spatially uniform time-dependent Bernoulli term has been absorbed into the velocity potential. The pressure is measured relative to the constant exterior pressure. This dynamic boundary condition uses the specified massless bending-cover model, with no ice inertia or tension.

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