A dish of clean water ( , , ) is shaken vertically at

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

Modes, Packets and Media, Ch Water waves and de Broglie waves

Problem

A dish of clean water (

\sigma = 0.0728\,\mathrm{N\,m^{-1}}

\rho = 998\,\mathrm{kg\,m^{-3}}

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

) is shaken vertically at

80\,\mathrm{Hz}

. The Faraday ripples that form are a surface mode responding parametrically at exactly half the shaking frequency; that mode has quality factor

Q = 45

. ((a)) Find the wavelength of the ripples. ((b)) The modulation depth of the mode's squared frequency is

h_m = 4ak

(for a mode driven at exactly half the shaking frequency, drive amplitude

a

, ripple wavenumber

k

). Using the threshold modulation depth for parametric resonance,

h_{m,\rm c} = 2/Q

, find the shaking amplitude at which the ripples first appear.

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