A long wave of wavelength and surface amplitude runs along a straight channel of…

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 long wave of wavelength λ=900 m\lambda = 900\ \mathrm{m} and surface amplitude a=0.15 ma = 0.15\ \mathrm{m} runs along a straight channel of uniform depth h=12 mh = 12\ \mathrm{m}. The water is inviscid and incompressible, the flow irrotational, surface tension is neglected, linear theory applies, and g=9.8 m s−2g = 9.8\ \mathrm{m\ s^{-2}}. ((a)) Evaluate khkh and say which limit of the dispersion relation ω2=gktanh⁡kh\omega^{2} = gk\tanh kh applies. ((b)) Find the phase speed. ((c)) Find the period. ((d)) Find the group velocity. ((e)) 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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