Swell of period and amplitude crosses deep open ocean

PHYSICS · Multi-part · Fermi University Physics · Oscillations, Waves & Acoustics · Fluids & Continuum Physics · Waves, acoustics and compressible flow

Fluids and Continuum Physics, Ch Surface gravity and capillary waves

Problem

Swell of period

12\ \mathrm{s}

and amplitude

1.1\ \mathrm{m}

crosses deep open ocean. The water is inviscid and incompressible seawater of density

\rho = 1025\ \mathrm{kg\,m^{-3}}

and surface tension

\gamma = 0.074\ \mathrm{N\,m^{-1}}

;

g = 9.8\ \mathrm{m\,s^{-2}}

; the waves are of small amplitude and irrotational. ((a)) Assume the phase speed of a deep-water gravity wave can depend only on

g

and the wavenumber

k

. Use dimensional analysis to establish that

c \propto \sqrt{g/k}

, then use the deep-water dispersion relation to find the phase speed and the wavelength of this swell. ((b)) Find the power carried per metre of crest. ((c)) Find the slowest phase speed any wave on this surface can have and the wavelength at which it occurs, and state the ratio of the swell's phase speed to it.

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