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

We examine the free surface flow over a fixed bed covered by rigid sinusoidal dunes. The mean current velocity is near the critical value at which the linearized theory predicts unbounded response. By allowing transients we examine the instability of the steady and nonlinear solution of Mei (1969) and the possibility of chaos when the current has a small oscillatory component.

Nonlinear resonance of free surface waves in a current over a sinusoidal bottom: a numerical study

Since the speed of sound in water is much greater than that of the surface gravity waves, acoustic signals can be used for early warning of tsunamis. We simplify existing works by treating the sound wave alone without the much slower gravity wave, and derive a two-dimensional theory for signals emanating from a fault of finite length. Under the assumptions of a slender fault and constant sea depth, the asymptotic technique of multiple scales is applied to obtain analytical results. The modal envelopes of the two-dimensional sound waves are found to be governed by the Schrodinger equation and are solved explicitly. An approximate method is described for the inverse estimation of fault properties from the pressure record at a distant hydrophone.

Sound signals of tsunamis from a slender fault

A new technique of asymptotic expansion is applied to water wave propagation over an uneven bottom that has straight and parallel contours. Attention is focused on the establishment of a rigorous scheme of successive approximation for higher-order corrections. The bottom depth is assumed to vary slowly within a wavelength. By introducing a compressed coordinate in the direction normal to the contours and by assuming an expansion of the WKB type, the weakness of the depth variation in the normal direction is incorporated in the mathematical formalism. The conventional linearized theory of wave refraction is obtained as the first-order solution without the explicit assumption of Snell's law. In the second order, a steady-state depression of the mean water surface is found for the general case where the incident wave approaches the contours obliquely. Reflection is neglected.

An asymptotic theory for water waves on beaches of mild slope

By studying the nonlinear problem to higher order, the steady free surface flow past a wavy bed is analyzed near the resonance speed. It is shown that the free surface amplitude always remains finite when nonlinearity of the problem is taken into account. Response curves relating the amplitude-stream velocity are given.

Steady Free Surface Flow Over Wavy Bed

Employing matched asymptotics, the recent theory of Mei (1986) is extended to study the phenomenon of upstream influence by a slender ship moving near the critical speed. For a special class of channel width and ship slenderness, it is shown that the response on the sea surface is essentially one-dimensional with the wave crests perpendicular to the ship's axis. In particular, solitons are radiated upstream. The hydrodynamic pressure on the ship, as well as the total sinkage force, wave resistance and trimming moment are calculated. These forces are functions of time despite the constant speed of the ship. The sinkage and trim for a ship model fixed on an advancing carriage are computed and show rapid variations across the critical speed as in the reported experiments of Graff, kracht & Weinblum (1964). Because of the assumed slenderness of the ship, this theory does not predict two-dimensional waves in the wake. Nevertheless, there is crude agreement in the time-averaged hydrodynamic forces between theory and experiment.

Chiang C. Mei

Papers

Nonlinear resonance of free surface waves in a current over a sinusoidal bottom: a numerical study

Sound signals of tsunamis from a slender fault

An asymptotic theory for water waves on beaches of mild slope

Steady Free Surface Flow Over Wavy Bed

Forces on a slender ship advancing near the critical speed in a wide canal