Someone argues: “Take internal waves in the deep ocean,

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

Signal Speeds in Fluids, Ch Internal gravity waves in a stratified fluid

Problem

Someone argues: “Take internal waves in the deep ocean, N=1.0×10−3 s−1N = 1.0\times10^{-3}\,\mathrm{s^{-1}}, travelling at 45∘45^\circ to the vertical, so k=mk = m. With horizontal wavelength λx=2π/k=20 m\lambda_x = 2\pi/k = 20\,\mathrm{m}, the dispersion relation ω2=N2k2/(k2+m2)\omega^2 = N^2k^2/(k^2+m^2) gives ω2=N2/2\omega^2 = N^2/2, a period of about 2.5 hours2.5\,\mathrm{hours}. Now keep the same direction (45∘45^\circ) but take a horizontal wavelength ten times longer, λx=200 m\lambda_x = 200\,\mathrm{m}. Then kk is ten times smaller, and since ω=Nk/k2+m2\omega = Nk/\sqrt{k^2+m^2} has kk only in the numerator, the frequency must drop by the same factor of ten, so the new period should be about 25 hours25\,\mathrm{hours}.” ((a)) Find the error in this reasoning. ((b)) State the correct period for the 200 m200\,\mathrm{m} wave.

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