User:Pwicky7/sandbox

Second-order Stokes wave on arbitrary depth

The ratio S = a₂ / a of the amplitude a₂ of the harmonic with twice the wavenumber (2 k), to the amplitude a of the fundamental, according to Stokes's second-order theory for surface gravity waves. On the horizontal axis is the relative water depth h / λ, with h the mean depth and λ the wavelength, while the vertical axis is the Stokes parameter S divided by the wave steepness ka (with k = 2π / λ). Description: * the blue line is valid for arbitrary water depth, while * the dashed red line is the shallow-water limit (water depth small compared to the wavelength), and * the dash-dot green line is the asymptotic limit for deep water waves.

The surface elevation η and the velocity potential Φ are, according to Stokes's second-order theory of surface gravity waves on a fluid layer of mean depth h:

$${\begin{aligned}\eta (x,t)=&a\left\{\cos \,\theta +ka\,{\frac {3-\sigma ^{2}}{4\,\sigma ^{3}}}\,\cos \,2\theta \right\}+{\mathcal {O}}\left((ka)^{3}\right),\\\Phi (x,z,t)=&a\,{\frac {\omega }{k}}\,{\frac {1}{\sinh \,kh}}\\&\times \left\{\cosh \,k(z+h)\sin \,\theta +ka\,{\frac {3\cosh \,2k(z+h)}{8\,\sinh ^{3}\,kh}}\,\sin \,2\theta \right\}\\&-(ka)^{2}\,{\frac {1}{2\,\sinh \,2kh}}\,{\frac {g\,t}{k}}+{\mathcal {O}}\left((ka)^{3}\right),\\c=&{\frac {\omega }{k}}={\sqrt {{\frac {g}{k}}\,\sigma }}+{\mathcal {O}}\left((ka)^{2}\right),\\\sigma =&\tanh \,kh\quad {\text{and}}\quad \theta (x,t)=kx-\omega t.\end{aligned}}$$

Observe that for finite depth the velocity potential Φ contains a linear drift in time, independent of position (x and z). Both this temporal drift and the double-frequency term (containing sin 2θ) in Φ vanish for deep-water waves.

Second Order Stokes Wave as amplitude grows.

Stokes and Ursell parameters

Further information: Ursell parameter

The ratio S of the free-surface amplitudes at second order and first order – according to Stokes's second-order theory – is:

$${\mathcal {S}}=ka\,{\frac {3-\tanh ^{2}\,kh}{4\,\tanh ^{3}\,kh}}.$$

In deep water, for large kh the ratio S has the asymptote

$$\lim _{kh\to \infty }{\mathcal {S}}={\frac {1}{2}}\,ka.$$

For long waves, i.e. small kh, the ratio S behaves as

$$\lim _{kh\downarrow 0}{\mathcal {S}}={\frac {3}{4}}\,{\frac {ka}{(kh)^{3}}},$$

or, in terms of the wave height H = 2a and wavelength λ = 2π / k:

$$\lim _{kh\downarrow 0}{\mathcal {S}}={\frac {3}{32\,\pi ^{2}}}\,{\frac {H\,\lambda ^{2}}{h^{3}}}={\frac {3}{32\,\pi ^{2}}}\,{\mathcal {U}},$$

with

$${\mathcal {U}}\equiv {\frac {H\,\lambda ^{2}}{h^{3}}}.$$