 Solution

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

For

k > 0

, phase velocity is the speed of a fixed phase,

c_{p} = ω / k

c_{g} = d ω / d k

transports a narrow-band wave envelope. With

q = k H

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

H \to 0

with

k

fixed,

tanh (k H) / (k H) \to 1

. Therefore

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

Both velocities tend to the shallow-water speed

g H

at leading order.

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