Nonlinearity

How much nonlinearity would one expect to be present in large waves? One source of guidance is provided by studies of nonlinear unidirectional waves of translation having permanent form. A compendium of 40 numerical solutions covering the full range of oceanic depths and wave heights was prepared by Dean [8] and published as a two-volume special report by the U.S. Army Engineer Coastal Engineering Research Center. The 40 waves included all combinations of

d/L₀ = .002, .005, .01, .02, .05, .1, .2, .5, 1, 2

and

H/H_b = .25, .50, .75, 1.0.

Here,

$$d = \text{depth}, \qquad H = \text{wave height}, \qquad T = \text{wave period},$$

$$L_0 = \frac{gT^2}{2\pi} = \text{deep wave wave length}, \qquad H_b = \text{height of breaking wave}.$$

The wave conditions are close enough together so that reasonable interpolations can be made between the solutions.

Unfortunately, the solutions are published as numerical tables, and manual interpolation for a wave with arbitrary parameters entails a great deal of labor. To overcome this difficulty, a set of subroutines in the software language, MATLAB, were prepared [4] under the sponsorship of the Offshore Technology Research Center of Texas A&M University, College Station.

MATLAB is an ``open code'' language with the source readily available to users for use and customization. The Borgman and Petrakos software for spline interpolation with the Dean wave solutions are listed for downloading as the ``DSFWAV Toolbox'' on Borgman's web page ( http://gallatin.gg.uwyo.edu/~borgman ). A document describing the toolbox is also provided. (Other MATLAB toolboxes developed by Borgman and various research associates are also freely available on the web page.)

The skewness and kurtosis are commonly-used measured of nonlinearity. Let $[\eta (\theta), -\pi < \theta < \pi]$ denote the profile where

$$\theta = 2 \pi \left( \frac{x}{L} - \frac{t}{T} + \phi \right), \qquad x = \text{horizontal position},$$

$$L = \text{wavelength}, \qquad t = \text{time},$$

$$T = \text{wave period}, \qquad \phi = \text{arbitrary phase}.$$

The r.m.s., skewness, and kurtosis are defined as

$$\text{r.m.s.} = \sqrt{\frac1{2\pi} \int^\pi_{-\pi} \eta^2 (\theta) d \theta},$$

$$\text{skewness} = \frac{\frac1{2\pi} \int^\pi_{-\pi} \eta^3 (\theta) d \theta}{\text{(r.m.s.)}^3},$$

$$\text{kurtosis} = \left[ \frac{\frac1{2\pi} \int^\pi_{-\pi} \eta^4 (\theta) d\theta}{\text{(r.m.s.)}^4} \right] - 3.$$

Graphs for these quantities, within the Dean stream function solutions, are given as Fig. 1.

The skewness can be quite large in very shallow water, ranging up to +4.0 or more. It is much more modest in deep water, rising to 0.6 or so even for breaking waves. The kurtosis can be large positively in shallow water, but goes slightly negative in deeper water, hovering around -3.0 for all wave heights. This means that high waves in shallow water have a water level elevation probability density with tails heavier than the Gaussian or normal probability law. The reverse appears to be true in deep water where the negative kurtosis indicates that the probability law tails are lighter than the normal density, or perhaps bounded and truncated.

  

Figure 1: Relations for Numerical Solutions by R.G. Dean