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:[1][2]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.
The ratio S of the free-surface amplitudes at second order and first order – according to Stokes's second-order theory – is:[3]In deep water, for large kh the ratio S has the asymptoteFor long waves, i.e. small kh, the ratio S behaves asor, in terms of the wave height H = 2a and wavelength λ = 2π / k:with