 Solution

ID: past-exam-of-the-options-i-course-of-the-university-of-oxford/2008/k/k2/iii/solution

k

c_{p} = ω / k

, the speed of a fixed phase. The group velocity is

c_{g} = d ω / d k

, the leading speed of a narrow wave packet. Differentiating the supplied dispersion relation gives

c_{p} = \frac{g}{k} tanh (k H),

c_{g} = \frac{g [ tanh ( k H ) + k H sech ^{2} ( k H )]}{2 ω} = \frac{c _{p}}{2} (1 + \frac{2 k H}{sinh ( 2 k H )}) .

For fixed

k > 0

, the shallow-depth limit uses

tanh s / s \to 1

H \to 0 lim \frac{c _{p}^{2}}{H} = g .

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