 Solution

ID: past-exam-of-the-probability-statistics-and-options-i-course-of-the-university-of-oxford/2008/k/2/ii/solution

Take the undisturbed surface to be

y = 0

and treat

η

ϕ

, and their induced velocities as first order in wave amplitude. Products of first-order quantities are second order and are discarded. Evaluating surface conditions at

y = η

differs from evaluation at zero only by second-order terms. Thus the linearized surface gravity wave problem is

ϕ_{xx} + ϕ_{yy} = 0 (- H < y < 0),

η_{t} = ϕ_{y}, ϕ_{t} + g η = 0 (y = 0), ϕ_{y} = 0 (y = - H) .

Here

u = ϕ_{x}

v = ϕ_{y}

. The linear free-surface conditions can also be combined into

ϕ_{tt} + g ϕ_{y} = 0

at zero.

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