6.5: Deepwater Waves

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Page ID: 47256

Franz S. Hover & Michael S. Triantafyllou
Massachusetts Institute of Technology via MIT OpenCourseWare

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In the case that \(H \longrightarrow \infty\), the above equations simplify because

\[ \phi(x, \, z, \, t) \longrightarrow -\dfrac{a \omega}{k} e^{kz} \sin (\omega t - kx + \psi). \]

We find that

\begin{align} \omega^2 \, &= \, kg \textrm{ (dispersion)} \\[4pt] p \, &= \, \rho ga e^{kz} \cos (\omega t - kx + \psi) - \rho gz; \\[4pt] u \, &= \, a \omega e^{kz} \cos (\omega t - kx + \psi); \\[4pt] w \, &= \, -a \omega e^{kz} \sin (\omega t - kx + \psi); \\[4pt] \xi_p \, &= \, a e^{kz} \sin (\omega t - kx + \psi); \\[4pt] \eta_p \, &= \, a e^{kz} \cos (\omega t - kx + \psi). \end{align}

The dynamic part of the pressure undergoes an exponential decay in amplitude with depth. This is governed by the wave number \(k\), so that the dynamic pressure is quite low below even one-half wavelength in depth: the factor is \(e^{- \pi} \approx 0.05\). Particle motions become circular for the deepwater case. The radii of the circles also decay exponentially with depth.