 Solution

ID: past-exam-of-the-options-i-course-of-the-university-of-oxford/2010/k/k2/solution

For

k > 0

, try

η = A cos (k x - ω t), ϕ = B cosh (k (y + h)) sin (k x - ω t) .

The potential satisfies Laplace's equation, and its derivative with respect to

y

vanishes at

y = - h

. The surface kinematic condition gives

B k sinh (kh) = A ω .

The dynamic condition gives

- B ω cosh (kh) + g A = 0

. Eliminating the nonzero amplitudes yields

ω^{2} = g k tanh (kh) .

Thus choosing either frequency sign satisfying this relation and

B = A ω / (k sinh (kh))

produces the required harmonic wave.

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