A swell of wavelength and surface amplitude travels over water of uniform depth

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

Fluids and Continuum Physics, Ch Surface gravity and capillary waves

Problem

A swell of wavelength λ=60 m\lambda = 60\ \mathrm{m} and surface amplitude a=0.80 ma = 0.80\ \mathrm{m} travels over water of uniform depth H=200 mH = 200\ \mathrm{m}. The water is treated as inviscid and incompressible with irrotational flow, surface tension is neglected, the amplitude is small enough for linear theory, and g=9.8 m s−2g = 9.8\ \mathrm{m\ s^{-2}}. ((a)) Find the period of the wave. ((b)) Find its phase speed. ((c)) Find its group velocity. ((d)) Using the linearised kinematic condition ∂η/∂t=∂ϕ/∂z\partial\eta/\partial t = \partial\phi/\partial z at z=0z = 0, find the largest magnitude of the vertical fluid velocity at the mean surface level.

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