A tank of homogeneous fluid of depth rotates about a vertical axis at

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

Signal Speeds in Fluids, Ch Waves in a rotating fluid: inertial, Poincare and Kelvin waves

Problem

A tank of homogeneous fluid of depth H=0.30 mH = 0.30\,\mathrm m rotates about a vertical axis at Ω=2.0 rad s−1\Omega = 2.0\,\mathrm{rad\,s^{-1}}. An obstacle of width L=0.08 mL = 0.08\,\mathrm m on the floor starts to move slowly at t=0t = 0. ((a)) Taking the inertial waves it excites to have ∣K∣≈2π/L|\mathbf K| \approx 2\pi/L, estimate the speed at which their energy travels up the rotation axis, and the time tformt_{\rm form} the column above the obstacle takes to reach the surface, also in rotation periods Trot=2π/ΩT_{\rm rot} = 2\pi/\Omega. ((b)) Find how high the column has grown at t=Trott = T_{\rm rot}. Sketch side views of the tank, obstacle on the floor and free surface at the top, at t=0t = 0, t=Trott = T_{\rm rot} and t=tformt = t_{\rm form}, shading the fluid that moves with the obstacle.

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