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).

Sea wave energy is being increasingly regarded in many countries as a major and promising resource. The paper deals with the development of wave energy utilization since the 1970s. Several topics are addressed: the characterization of the wave energy resource; theoretical background, with especial relevance to hydrodynamics of wave energy absorption and control; how a large range of devices kept being proposed and studied, and how such devices can be organized into classes; the conception, design, model-testing, construction and deployment into real sea of prototypes; and the development of specific equipment (air and water turbines, high-pressure hydraulics, linear electrical generators) and mooring systems.

Wave energy utilization: A review of the technologies

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.

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A new view of nonlinear water waves: the Hilbert spectrum

Dynamics of structures

Equations of motion are derived for long waves in water of varying depth. The equations are for small amplitude waves, but do include non-linear terms. They correspond to the Boussinesq equations for water of constant depth. Solutions have been calculated numerically for a solitary wave on a beach of uniform slope. These solutions include a reflected wave, which is also derived analytically by using the linearized long-wave equations.

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0022112067002605

Long waves on a beach

A quantitative theory is described for the formation mechanism of sand bars under surface water waves. By assuming that the slopes of waves and bars are comparably gentle and sediment motion is dominated by the bedload, an approximate evolution equation for bar height is derived. The wave field and the boundary layer structure above the wavy bed are worked out to the accuracy needed for solving this evolution equation. It is shown that the evolution of sand bars is a process of forced diffusion. This is unlike that for sand ripples which is governed by an instability. The forcing is directly caused by the non-uniformity of the wave envelope, hence of the wave-induced bottom shear stress associated with wave reflection, while the effective diffusivity is the consequence of gravity and modified by the local bed stress. During the slow formation, bars and waves affect each other through the Bragg scattering mechanism, which consists of two concurrent processes: energy transfer between waves propagating in opposite directions and change of their wavelengths. Both effects are found to be controlled locally by the position of bar crests relative to wave nodes. Comparison with available laboratory experiments is discussed and theoretical examples are studied to help understand the coupled evolution of bars and waves in the field.

Formation of sand bars under surface waves

In most past theories on Bragg reflection of waves by a finite patch of rigid bars, only outgoing waves are allowed on the transmission side, simulating the effect of an idealized shoreline where all the incident wave energy is consumed by breaking. In these theories the amplitudes of both the incident and reflected waves are found to decrease monotonically over the bar patch in the shoreward direction. This result has motivated the idea of artificially constructing bars to protect a beach from incident waves. However, some numerical calculations have suggested that this tendency does not always hold when there is some reflection from the shore. We show here that with finite reflection by the shoreline the spatial distribution of wave energy over the patch can indeed be reversed, indicating that the mechanism can increase the hazards to the beach. The phase relation between the bars and the shoreline reflection is found to be the key to this qualitative change of wave response.

Do longshore bars shelter the shore

Periodic water waves through an aquatic forest

SUMMARY A hybrid elcment method developed recently for two-dimensional problems of water waves in an infinite fluid is extended to three dimensions. In this method only a limited fluid domain close to irregular bodies is discretized into conventional finite elements, while the remaining infinite domain is treated as one element with analytical representations of high accuracy. Continuity at the junction surface is treated as natural boundary conditions in a variational principle. Computation experience and numerical results for several ocean structures are presented.

A hybrid element method for diffraction of water waves by three‐dimensional bodies

In the study of long gravity waves of finite amplitude, the
main body of the existing theories has been built upon the simplifying assumption that the viscosity is either totally negligible, or adequately described by an
empirical law. To date very little systematic account of the viscosity effect has appeared that is based on the Navier-Stokes' equations of motion. Thus, in the
important problems of flood waves in rivers, Chezy's formula and a variety of empirical laws of hydraulics have been used to replace the viscous stress terms,
and this is the approach taken in most of the hydraulic studies on open channel flows. Among theoretical contributions along this line, one may mention the
book by Stoker (1957), the works of Dressler (1949), and of Lighthill and Whitham (1955, I). Dressler developed a rigorous theory of roll waves. In particular
he obtained a discontinuous periodic solution in the case of relatively large amplitudes and a continuous periodic (cnoidal) solution in the case of small
amplitudes. General flood movement in long rivers has been masterfully investigated by Lighthill and Whitham (1955, I), as a type of kinematic waves. Their
method of predicting the transient motion of large amplitude waves with discontinuities
(or shocks) is especially noteworthy.

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Chiang C. Mei

Papers

Formation of sand bars under surface waves

Do longshore bars shelter the shore

Periodic water waves through an aquatic forest

A hybrid element method for diffraction of water waves by three‐dimensional bodies

Nonlinear Gravity Waves in a Thin Sheet of Viscous Fluid