INTRODUCTORY
OCEANOGRAPHYINTRODUCTORY
OCEANOGRAPHY
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To understand why we say that c a L (or T) for deep water waves and c a d for shallow water waves, it is necessary to write the characteristic equation (obtained by solving the small amplitude wave equations) for celerity. Shown below is full expression of that equation.
where k = 2p/L is the wavenumber, g is acceleration of gravity and p = 3.14159 -- note that in this equation, c2 is proportional to both L and d (and represents celerity for the intermediate depth wave).
DEEP WATER. When water is deep, kd is large and tanh(kd) ~ 1, so the equation can be written as:
or writing as a product of celerities,
and substituting c = L/T, we get
Note that the L divide out and you are left with
which shows that c a T (our goal).
SHALLOW WATER. When water is shallow, tanh(kd) ~ kd, and k's divide out to give
Taking the square root of both sides gives the expression for shallow water, c a d (our goal).
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