User:Pwicky7/sandbox

Second-order Stokes wave on arbitrary depth
Stokes3_amplitude_double_frequency.svg 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 $\mathcal{S}$ divided by the wave steepness ka (with $\mathcal{S} = a_{2} / a$).

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

Stokes and Ursell parameters
The ratio $\mathcal{S}$ 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 $\mathcal{S}$ has the asymptote$$\lim_{kh \to \infty} \mathcal{S} = \frac{1}{2}\, ka.$$For long waves, i.e. small kh, the ratio $\mathcal{S}$ behaves as$$\lim_{kh \downarrow 0} \mathcal{S} = \frac{3}{4}\, \frac{ka}{(kh)^3},$$or, in terms of the wave height $k = 2π / λ$ and wavelength $H = 2a$:$$\lim_{kh \downarrow 0} \mathcal{S} = \frac{3}{32\,\pi^2}\, \frac{H\, \lambda^2}{h^3} = \frac{3}{32\,\pi^2}\, \mathcal{U},$$with$$ \mathcal{U} \equiv \frac{H\, \lambda^2}{h^3}.$$