Ocean Waves

Development of a Swell from a Localised Storm

Use buttons below to load and run a simple animation of the development of a swell from a localised storm.

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The initial storm is modelled as a circular region of incoherent random fluctuations. The disturbance is then evolved according to the dispersion relation for small amplitude waves in a deep ocean.

w(k) =   __
g k
 
(1)
where w is the frequency, k = 2 p/ l is the wavenumber, and g is the gravitational acceleration.

To compute the evolved distubance f(r,t) we fourier transform the initial disturbance f(r, t = 0) to obtain

~
f
 

k 
(t = 0) =
d2 r f(r, 0) exp(i k ·r)
(2)
assuming the wave starts at rest, the modes evolve as
~
f
 

k 
(t) = ~
f
 

k 
(0) cos(w(k) t)
(3)
so we simply multiply each mode by the appropriate time evolution factor and then inverse transform.

The model is somewhat unrealistic in that the waves are freely propagating after the initial disturbance, but captures the essential features that the long wavelength components outrun the high frequency modes, resulting, at late times, in a locally quasi-monochromatic wave with a characteristic patters of relatively slowly modulated wave-trains or 'sets'.


File translated from TEX by TTH, version 1.67.