In the deep ocean, wind-generated waves are usually well modeled by linear theory; there is no evidence that the statistics of measured wave fields are in any way different from expectations of a linear superposition of sinusoids [ Elgar, et al., 1984]. On the other hand, as waves propagate into shallower depths, a familiar change in wave shape occurs with crests steepening and troughs becoming more broad. These shape changes are just the visible manifestation of nonlinear spectral evolution and have be quantified with a number of measures including group length [ Elgar, et al., 1984], time series moments [ Herbers and Guza, 1992] and higher order spectral quantities like the bi-spectrum [ Elgar and Guza, 1985].
A substantial literature exists modeling the nonlinear evolution of shoaling, initially-monochromatic waves [e.g. Mei, 1983], including excellent work on refraction and diffraction over variable topography [e.g. Kirby and Dalrymple, 1983]. Results are applicable to conditions of extremely narrow-banded incident waves, such as in wave tank studies. On the other hand, real world wave fields are always of finite bandwidth; no matter how clean the incident waves appear to the eye, energy is always distributed over a range of frequencies. The statistics of such a wave field can only be modeled by accounting for the full spectrum of possible interactions.
Over the last decade, models have been successfully developed based on a weakly nonlinear perturbation of the shallow-water Boussinesq equations [ Freilich and Guza, 1984]. Interactions occur in triads, with two components of the incident wave field interacting to transfer energy to a third component at the sum (and difference) frequency and wavenumber. For a spectral wave field, the sum interactions of all possible frequency pairs within the spectrum must be integrated. This approach has provided excellent predictions of many aspects of shoaling waves on low-sloping beaches.
Recently this work has been extended to include directionally-spread seas [ Elgar, et al., 1993]. As expected, interactions produce sum frequency components whose direction is found from the vector sum of the interacting wavenumbers. The coupling of this type of modeling with recently-available high-resolution directional spectrum climatology [ Long, 1994] will provide insight into the expected nonlinear wave statistics in shallower water in the nearshore. The requirement for careful higher-order modeling is not solely academic. Bowen [1980] showed that third order statistics of incident waves (skewness) provides an important driving force for sediment transport in the nearshore.
Very recently, interest has re-focused on the nonlinear interaction originally proposed by Longuet-Higgins [1950] whereby two opposing wave trains generate energy at twice the incident frequency (frequency-doubling) and at long wavelengths, such that pressure fluctuations can be measured at depths much larger than the decay depth of the individual waves. This energy is detectable in ground-based seismometers and forms the basis for remote microseism measurements of ocean wave height [ Zopf, et al., 1976]. Using a large-aperture, 24-element pressure array in 13 m depth off North Carolina, Herbers and Guza [1994] showed observations of this energy to be in excellent agreement with nonlinear predictions of second order theory [ Hasselmann, 1962], although the dominant interactions resulted from opposing wave trains driven by veering winds, not from incident wave reflection from the beach (as was required by Zopf [1976] in order for microseism data to provide meaningful measurements of nearshore wave height).