to be done in this area. Face recognition is a problem that scarcely clamoured for attention before the computer age but, having surfaced, has involved a wide range of techniques and has attracted the attention of some fine minds (David Mumford was a Fields Medallist in 1974). This singular application of mathematical modelling to a messy applied problem of obvious utility and importance but with no unique solution is a pretty one to share with students: perhaps, returning to the source of our opening quotation, we may invert Duncan's earlier observation, 'There is an art to find the mind's construction in the face!'.

Linear and nonlinear waves, by G. B. Whitham. Pp.636. £50. 1999. ISBN 0 471 35942 4 (Wiley).

We survey the newly developed Hilbert spectral analysis method and its applications to Stokes waves, nonlinear wave evolution processes, the spectral form of the random wave field, and turbulence. Our emphasis is on the inadequacy of presently available methods in nonlinear and nonstationary data analysis. Hilbert spectral analysis is here proposed as an alternative. This new method provides not only a more precise definition of particular events in time-frequency space than wavelet analysis, but also more physically meaningful interpretations of the underlying dynamic processes.

/pdf/a-new-view-of-nonlinear-water-waves-the-hilbert-spectrum-42def0zzxb.pdf

A new view of nonlinear water waves: the Hilbert spectrum

http://personalpages.to.infn.it/~onorato/papers/2007_Cavaleri_etal_PO.pdf

Wave Modelling - The State of the Art

Numerical methods are of great importance in the study of electromagnetic scattering from random rough surfaces. This review provides an overview of rough surface scattering and application areas of current interest, and surveys research in numerical simulation methods for both one- and two-dimensional surfaces. Approaches considered include numerical methods based on analytical scattering approximations, differential equation methods and surface integral equation methods. Emphasis is placed on recent advances such as rapidly converging iterative solvers for rough surface problems and fast methods for increasing the computational efficiency of integral equation solvers.

Numerical simulation methods for rough surface scattering

Every mariner is aware that dangerous large breaking water waves occur on the world's oceans. The scope of this review is somewhat greater. Wave breaking occurs at a large range of scales and we do not restrict ourselves to the deep ocean. "Deep water" in the context of water wave studies implies water sufficiently deep that the surface waves are unaffected by the direct effects of variations in bed topography. Thus even a small pond can support breaking deep-water waves. Shallow water breaking is reviewed in Peregrine (1983). Some comments on the visual aspect of breakers are in order, since direct observation still has a role to play in the study of this complex phenomenon. The most dramatic breakers are plunging breakers where the breaking commences by the wave overturning and forming a forward moving sheet of water which plunges down into the water in front causing splashes, air entrainment, and eddies. Although plunging breakers are common on beaches they are less common on deep water, so much so that some people have argued that they do not occur naturally. However, read Coles (1991) for a distillation of an experienced yachtsman's account of waves at sea. Most other breakers are described as spilling breakers. From their

Wave Breaking in Deep Water

This paper deals with the development of techniques for satellite synthetic aperture radar (SAR) ocean image interpretation. We derived a theoretical model of a radar image for a Korteweg-de Vries type ocean internal soliton and validated the model using ocean internal wave signals taken from ERS-1 SAR and RADARSAT SAR images. The results indicate that the model perfectly simulates ocean internal soliton signatures with double-sign variations of radar backscatter. On the basis of the model, we developed the curve fitting method and the peak-to-peak method for determining the internal soliton characteristic half widths, which then were used to calculate the internal soliton amplitudes. The test results indicate that ocean internal soliton amplitudes derived by the two methods agree with in situ data acquired on the Portuguese Continental Shelf and in the South China Sea with reasonable accuracy. The role that wind fields play in ocean radar remote sensing was also analyzed. Finally, the modulation ratio of ocean internal waves on radar images was quantitatively estimated.

https://agupubs.onlinelibrary.wiley.com/doi/pdf/10.1029/2000JC000726

Theoretical expression for an ocean internal soliton synthetic aperture radar image and determination of the soliton characteristic half width

Probability density function of the surface elevation of a nonlinear random wave field is obtained. The wave model is based on the Stokes expansion carried to the third order for both deep water waves and waves in finite depth. The amplitude and phase of the first-order component of the Stokes wave are assumed to be Rayleigh and uniformly distributed and slowly varying, respectively. The probability density function for the deep water case was found to depend on two parameters: the root-mean-square surface elevation and the significant slope. For water of finite depth, an additional parameter, the nondimensional depth, is also required. An important difference between the present result and the Gram-Charlier representation is that the present probability density functions are always nonnegative. It is also found that the 'constant' term in the Stokes expansion, usually neglected in deterministic studies, plays an important role in determining the details of the density function. The results compare well with laboratory and field experiment data.

A non-Gaussian statistical model for surface elevation of nonlinear random wave fields

A new approach using phase information to view and study the properties of frequency modulation, wave group structures, and wave breaking is presented. The method is applied to ocean wave time series data and a new type of wave group (containing the large 'rogue' waves) is identified. The method also has the capability of broad applications in the analysis of time series data in general.

The local properties of ocean surface waves by the phase-time method

The modification of the shape of the wave spectrum in the high-frequency range and the amount of energy loss, due to wave breaking are examined. The original waves are assumed to be Gaussian, stationary, and of finite bandwidth. Breaking is assumed to occur when the vertical acceleration at any point on the surface reaches g/2. Based on the wave breaking model, an approximate but accurate spectrum of breaking waves and an exact expression of the amount of energy loss due to wave breaking are derived. It is shown that the spectrum which corresponds to minimum rate of energy loss has an upper limit proportional to in the high-frequency range.

Statistical Characteristics of Breaking Waves

[1] A reappraisal of wave theory from the beginning to the present day is made here. On the surface, the great progress in both theory and applications seems to be so successful that there would be no great challenge in wave studies anymore. On deeper examination, we found problems in many aspects of wave studies starting from the definition of frequency, the governing equations, the various source functions of wave models, the directional development of wind wavefield, the wave spectral form and finally the role of waves as they affect coastal and global ocean dynamics. This is a call for action for the wave research community. For future research, we have to consider these problems seriously and also to examine the basic physics of wave motion to determine their effects on other ocean dynamic processes quantitatively, rather than relying on parameterization in oceanic and geophysical applications.

https://agupubs.onlinelibrary.wiley.com/doi/pdf/10.1029/2011JC007768

Yeli Yuan

Papers

Theoretical expression for an ocean internal soliton synthetic aperture radar image and determination of the soliton characteristic half width

A non-Gaussian statistical model for surface elevation of nonlinear random wave fields

The local properties of ocean surface waves by the phase-time method

Statistical Characteristics of Breaking Waves

A reappraisal of ocean wave studies