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

Approximations for shallow-water ship waves are sought which are valid near the critical speed U = (gh)1/2. For sufficiently thin bodies (struts or ships) the governing equation is dispersive. Simple analytic solutions are given which are valid for all F2 ≤ O(1). As the thickness increases, nonlinearity also enters. A soliton solution is discussed which applies to a sharp-nosed half-body at slightly supercritical speed.

Flow around a thin body moving in shallow water.

An approximate, analytical solution is found for the gravity-induced stresses in the neighbourhood of an axisymmetric topographic feature on an elastic half-space. The solution is in the form of a perturbation expansion in powers of the characteristic slope, ϵ. The leading order problem, at O(1), is for a distributed normal load on a plane half-space. The O(ϵ) correction is due to a distributed shear traction. The vertical variation of the near-surface stress perturbation and the rotation of the principal stress directions are O(ϵ) effects, and thus include contribution s of like order from both the normal and shear loads. The method requires general solutions for axisymmetric, normal and shear tractions of arbitrary radial distribution, and these are found in terms of Hankel transforms.

Gravity‐induced stresses near axisymmetric topography of small slope

The effects of randomly irregular bathymetry on the propagation of interfacial gravity waves are studied. Modelling sea water by a two-layer fluid of different densities, weakly nonlinear waves much longer than the sea depth but comparable to the horizontal scale of bathymetry are treated by Boussinesq approximation and multiplescale analysis. For transient wave pulses, the governing equation for the pulse profile is shown to be an integro-differential equation combining KdV and Burgers terms. Quantitative and qualitative effect of disorder on the attenuation of wave amplitude, reduction of wave speed and change of wave profile are examined numerically and analytically based on the asymptotic approximation. For time-harmonic waves, modecoupling equations are derived and examined for the competition between diffusion by random scattering, steepening by nonlinearity and frequency dispersion for a broad range of depth ratios.

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Attenuation of long interfacial waves over a randomly rough seabed

By assuming a head loss across the harbor entrance the wave-induced response in a rectangular model harbor is studied theoretically. The loss is assumed to be quadratic in local velocity with a constant friction coefficient. Bottom and side-wall dissipation are not considered. It is shown that odd higher harmonics are present. The resonance characteristics of the fundamental harmonic response in the harbor are significantly influenced by the additional damping due to friction, especially for the lowest peaks. While in the perfect fluid theory these peaks heighten with diminishing entrance width, in the presence of friction the opposite is true for sufficiently narrow mouth.

Effects of Entrance Loss on Harbor Oscillations

A harbor with two coupled rectangular basins is subjected to periodic incident waves. Ignoring friction the scattering problem is solved by the method of matched asymptotics for narrow junctions. The example of two identical basins is analyzed in detail for the resonant spectrum and response. It is shown that for certain modes the inner basin is less shielded.

Chiang C. Mei

Papers

Flow around a thin body moving in shallow water.

Gravity‐induced stresses near axisymmetric topography of small slope

Attenuation of long interfacial waves over a randomly rough seabed

Effects of Entrance Loss on Harbor Oscillations

Resonant scattering by a harbor with two coupled basins