A raindrop strikes a still pond and launches capillary ripples obeying

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

Modes, Packets and Media, Ch Dispersion and the propagation of wave packets

Problem

\omega^2=(\sigma/\rho)k^3

, with surface tension

\sigma=0.072\,\mathrm{N\,m^{-1}}

and density

\rho=1.00\times10^3\,\mathrm{kg\,m^{-3}}

(gravity is negligible at these wavelengths). ((a)) Write

\omega=Ck^{3/2}

with

C=\sqrt{\sigma/\rho}

, find the stationary-phase wavenumber

k_{\rm s}(x,t)

seen at distance

x

and time

t

after impact, and explain why, unlike gravity waves on deep water, the shortest wavelengths arrive first. ((b)) A float sits at

x_1=3.0\,\mathrm{cm}

from the impact point. Find the local wavelength it observes at

t_1=20\,\mathrm{ms}

after impact. ((c)) A second float sits at

x_2=6.0\,\mathrm{cm}

, on the same ray from the impact point (that is,

x_2/t_2=x_1/t_1

). Find the time

t_2

at which it sees the same local wavelength as the first float, and, using the one-dimensional stationary-phase amplitude (

\propto t^{-1/2}

) together with the geometric spreading factor

(x_1/x_2)^{1/2}

for a point source (as for the stone dropped in a lake), find the ratio of the amplitude seen by the second float to that seen by the first.

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