On water of finite depth

PHYSICS · Subjective · Fermi University Physics · Fluids & Continuum Physics · Waves, acoustics and compressible flow · Waves, acoustics and compressible flow

Signal Speeds in Fluids, Ch Waves of finite height: Stokes waves, shoaling and breaking

Problem

On water of finite depth hh, the linear wave η=acos⁡(kx−ωt)\eta=a\cos(kx-\omega t), with ω2=gktanh⁡kh\omega^2=gk\tanh kh, has the velocity field u=aωcosh⁡k(z+h)sinh⁡khcos⁡(kx−ωt),w=aωsinh⁡k(z+h)sinh⁡khsin⁡(kx−ωt).\begin{aligned}u&=a\omega\frac{\cosh k(z+h)}{\sinh kh}\cos(kx-\omega t),\\ w&=a\omega\frac{\sinh k(z+h)}{\sinh kh}\sin(kx-\omega t).\end{aligned} (a) Following the deep-water argument, find the parcel displacements ξ\xi and ζ\zeta, and show that the Stokes drift is uS(z)=ωka2cosh⁡2k(z+h)/(2sinh⁡2kh)u_{\rm S}(z)=\omega ka^2\cosh 2k(z+h)/(2\sinh^2kh). (b) Show that this reduces to ωka2e2kz\omega ka^2e^{2kz} when kh→∞kh\to\infty, and explain why it does not vanish at the sea bed, where the vertical motion does. (c) For a swell of period T=8.0 sT=8.0\,\mathrm{s} and amplitude a=0.50 ma=0.50\,\mathrm{m} in water h=15 mh=15\,\mathrm{m} deep, find kk and the drift at the surface and at the bed, and compare the surface value with the deep-water formula at the same period.

Browse the problem library