A storm sends out deep-water swell in a narrow band of frequencies near
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
Modes, Packets and Media, Ch Water waves and de Broglie waves
Problem
A storm sends out deep-water swell in a narrow band of frequencies near f, for which the group velocity is vg(f)=g/(4πf).
((a)) Show that, following this band as it travels a distance D from the storm, the length of ocean it occupies grows in direct proportion to D, and that, because the wave energy carried by the band (per unit length of crest) is unchanged along the way, its wave height falls as D−1/2.
((b)) Spreading over the curved sea surface from a compact storm multiplies this by a further factor [Rsin(D/R)]−1/2, where R=6371km is the Earth's radius. This geometric factor is itself close to D−1/2 when D≪R, but it is a separate effect from the stretching of part (a). Two recording stations lie on the same great circle from a storm, at distances D1=6.0×103km and D2=9.0×103km. Find the ratio h1/h2 of the heights of this frequency band at the two stations.