Very short ripples on clean water (capillary waves) obey the dispersion relation = k^ 3

PHYSICS · Multi-part · Fermi University Physics · Oscillations, Waves & Acoustics · Waves in two and three dimensions, dispersion and wave packets · Waves in two and three dimensions, dispersion and wave packets

Optics and Photonics, Ch Harmonic waves, phasors and group velocity

Problem

Very short ripples on clean water (capillary waves) obey the dispersion relation ω=σk3ρ,\omega = \sqrt{\frac{\sigma k^{3}}{\rho}}, where σ=0.0728 N m−1\sigma = 0.0728\ \mathrm{N\,m^{-1}} is the surface tension of water, ρ=1000 kg m−3\rho = 1000\ \mathrm{kg\,m^{-3}} its density, kk the wavenumber and ω\omega the angular frequency. Consider ripples of wavelength λ=5.00 mm\lambda = 5.00\ \mathrm{mm}. ((a)) Find kk, in m−1\mathrm{m^{-1}}. ((b)) Find the frequency ν\nu, in Hz. ((c)) Find the phase velocity vv, in m s−1\mathrm{m\,s^{-1}}. ((d)) Show that vg=32vv_g = \tfrac{3}{2}v for this dispersion relation, and give vgv_g in m s−1\mathrm{m\,s^{-1}}.

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