A short ripple of wavelength travels on deep water of kinematic viscosity

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

Signal Speeds in Fluids, Ch Waves of finite height: Stokes waves, shoaling and breaking

Problem

A short ripple of wavelength λ=3.0 cm\lambda = 3.0\,\mathrm{cm} travels on deep water of kinematic viscosity ν=1.0×10−6 m2 s−1\nu = 1.0\times10^{-6}\,\mathrm{m^2\,s^{-1}}, surface tension σ=0.073 N m−1\sigma=0.073\,\mathrm{N\,m^{-1}} and density ρ=1000 kg m−3\rho=1000\,\mathrm{kg\,m^{-3}}. ((a)) On a clean surface, using the amplitude decay rate 2νk22\nu k^2, find the time for the ripple's amplitude to fall by a factor of ee. ((b)) Find the ripple's angular frequency from ω2=gk+σk3/ρ\omega^2=gk+\sigma k^3/\rho, and the thickness δ=2ν/ω\delta=\sqrt{2\nu/\omega} of its Stokes layer. ((c)) The surface is now covered by an inextensible film, so the amplitude decays at the rate νk/2δ\nu k/2\delta. Find the new ee-folding time, and the factor by which the film has shortened it.

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