Normal surface water waves, as generated by wind, do not have sine form but wave peak is higher and shorter than wave trough with different wave steepness. What parameters characterize such a surface water wave and how can one predict amplitude of water for given waves as function of time?


Deep water waves are often described as "cnoidal", with a mathematical description involving the Jacobian elliptic function cn(). This is an exact solution to the nonlinear Korteweg–de Vries differential equation. A more accurate equation is the Boussinesq. These are the basis for describing all water waves, whether stirred up by wind or otherwise. The basic parameters for a particular solution are wave height, period (or wavelength), depth of the water, and acceleration of gravity.

I hate to cite Wikipedia due to its propensity to change, but the best explanations I could find, including math, are there.

As for the details of wind pushing on the wave peaks, and the peaks disturbing the air flow, and big waves breaking over in ways that excite surfers, there are no nice mathematical forms I know of, but then I'm not expert on this. Numerical modeling is king in the area. Some original research was done for the movie "The Perfect Storm" on how to do better simulations and crunch the numbers faster.

  • $\begingroup$ Deep water waves are not described by the KdV equation, rather the KdV equation governs particular scenarios in shallow water. In fact, the evolution of deep water waves, to second order in wave steepness and bandwidth is governed by the nonlinear Schrodinger equation. The KdV equation is also not the basis for describing all water waves, rather, we leave that distinction for linear theory. $\endgroup$ – Nick P Feb 21 '14 at 5:18
  • $\begingroup$ Shallow-water theory leads to the KdV equation. First-order deep-water theory results in the Benjamin-Ono equation. The Stokes equation can be used for deep water waves but if it is too deep it will become unstable and the nonlinear Schrodinger equation is used instead. Finite depth is governed by the Joseph equation. $\endgroup$ – Isopycnal Oscillation Apr 15 '15 at 23:42

The generation of waves by wind is still an open question. Jeffrey's (1925) made a prediction based on wave shadowing, that is, he proposed that wind over waves would lead to higher pressure over the troughs and lower pressure over the crests, leading to wave growth. It turns out the theoretical growth rates for waves, based on this mechanism, are much too small to account for the wave growth observed in the ocean. Phillips (1956) predicted that random pressure fluctuations on the surface could lead to the generation of waves, but again, his predictions were too small to account for the growth observed. It was Miles (1957) that proposed that flow over water is analogous to the Kelvin-Helmholtz instability, and this mechanism could lead to wave growth in the ocean. The growth rates were much closer to what's observed. It is likely that in reality both the Miles and Phillips mechanism are in play.

This is not the entire story. In Miles' theory, the profile of the wind over the waves was logarithmic and $laminar$ as one approached the surface of the water. In reality, it has been shown in the laboratory that flow separation occurs over steep and breaking waves. This effectively attenuates the effects of the wind on the waves. It is now thought that capillarity plays a large part in this process, but the details are still open.

Next, we consider the form of deep-water surface gravity waves, which is the relevant scenario in the context of wave generation. This is a vast subject, so I'll just sketch some of the basics.

We begin by assuming the flow is irrotational, which means that there exists a velocity potential $\phi$, where $\nabla \phi = \textbf{u}$, with $\textbf{u}$ the fluid velocity, such that $\nabla^2\phi =0$ everywhere in the water. The conditions at the free surface $z=\eta$ are \begin{equation} i)\ \ \eta_t+\nabla \phi \cdot \nabla \eta= \phi_z, \end{equation} i.e. the kinematic boundary condition, and \begin{equation} ii)\ \ \phi_t+\frac{1}{2}(\nabla \phi)^2 +gz = 0, \end{equation}

which is the dynamic boundary condition, ensuring continuity of pressure across the interface.

Finally, we have the condition that there is no flow at the bottom, i.e. \begin{equation} iii)\ \ \phi_z \to 0 \quad \text{as} \quad z\to -\infty \end{equation} Now, although the governing equation is linear (it's Laplace's equation), the boundary conditions are nonlinear and are evaluated at one of the variables that we are solving for, namely $\eta$.

The waves I think you're referring to, with `peakier' peaks and flatter troughs are Stokes waves (Stokes 1847). They are solutions to the above equations when we restrict solutions to permanent progressive waves. Without grinding through the details, to second order in wave steepness, measured by $ak$ for wave amplitude $a$ and wavenumber $k$, the surface displacement becomes

$\eta(x,t) = a\cos (kx-\omega t) +\frac{1}{2} a^2 k\ \cos 2(kx -\omega t)$ where

$\omega = \sqrt{gk}(1+1/2(ak)^2)$, is the dispersion relation, and we notice that there is a Stokes correction, proportional to $(ak)^2$. These are known as finite amplitude waves, since the linear solutions are valid for infinitesimal amplitudes only.

Now, Stokes theory reigned supreme until the 1960s, when people started to see signs that these waves were not the entire story. People had been trying to generate monochromatic wave trains in the lab, but they were find that far from the paddle, the wave form did not have the permanent progressive form predicted by Stokes' theory. Lighthill (1965) showed that the governing equations for the amplitude of deep-water waves, valid to second order, was an elliptic equation, and hence potentially unstable to perturbations. This was made rigourous by Benjamin and Feir in (1967), who showed second order stokes waves were unstable to subharmonic perturbations. At the same time, Zakharov (1967) showed this result in the context of a Hamiltonian formulation.

The growth of this instability is described by the nonlinear schrodinger equation, which captures the four wave resonance that is operative in weakly nonlinear narrow banded surface gravity waves.

But this is also not the entire story, as the growth of the Benjamin-Feir instability can eventually lead to breaking, which is clearly not described by potential flow theory. So we don't really have equations that are valid for very long times when the initial conditions are steep enough to be subject to these instabilities.

This is clearly just a (rambling) taste of a very rich subject. Feel free to ask any details that you might find interesting.


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