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

Comment on “A Perturbation Analysis of an Interaction between Long and Short Surface Waves” by Woodruff and Messiter

Many papers have been devoted to nonlinear waves on a thin layer of viscous fluid flowing down an incline at low to moderate Reynolds numbers (see Chang 1994 for a survey). Motivated by interests in chemical engineering, surface tension is emphasized in past studies where the Weber number W e is ususally assumed to be large W e = O(∈—2) where ∈ = is a small parameter denoting the depth-to-wavelength ratio. Among the few papers on high Reynolds numbers, the boundary layer approximation to O(∈2) accuracy and the momentum integral method are used for analytical convenience. Due to the complexity of these nonlinear evolution equations, most reported studies concentrate on permanent (or stationary) waves which propagate at a constant speed without changing form. However in these papers there exist inconsistencies since pressure is taken to be only hydrostatic which implies omission of 0(e 2) terms in the transverse momentum equation. A consistent second order theory has been worked out for large Reynolds numbers and small-to-moderate surface tension (Lee, 1995, Lee & Mei, 1995).

Continuous Bores on a Viscous Fluid Down an Incline

We describe the results of a theoretical study of pumping of groundwater from a layered soil. The model soil system consists of three horizontal layers where a very soft and highly impervious aquitard is sandwiched by two hard and highly porous aquifers. Water is pumped from the bottom aquifer through a vertical well. This type of problem is usu­ ally solved in hydrology by assuming constant total stress in the soil, and that the drawdown can be calculated in advance of the soil stress. By a perturbation theory and the use of Lagrangian coordinates, these common assumptions are found to be inadequate if the aquitard is very soft or thick. Specifi­ cally the drawdown should be nonlinearly coupled to the soil deformation and that the total stress is not quite uniform in depth. By a model constitutive law representative of clays, two patterns of cyclic pumping or recharging are compared.

Subsidence Due to Intensive Pumping from a Layered Soil

An approximate theory is developed for the two-dimensional propagation of tsunami emanated from a slender fault of fine length. Assuming significant contrasts between the sea depth, fault width, fault length and the bathymetric length scales, we invoke parabolic approximation to deduce a linear Kademtsev-Petviashivili (K-P) equation governing the two-dimensional propagation of dispersive long waves over great distances. Analytical techniques are employed to explore the far-field radiation in the forward and spanwise directions in a sea of constant depth. The solution can be used as a convenient input for predicting local variations of wave scattering and possibly breaking along a coast.

Asymptotic analysis of dispersive tsunami from a slender fault

We present a survey of nonlinear effects on water waves interacting with a structure or topography with nontrivial geometry. Examples will be discussed for narrow-banded incident waves whose average wavelength is comparable to the sea depth. In particular the role played by the diffraction of short waves on the excitation of long-period oscillations will be highlighted.

Chiang C. Mei

Papers

Comment on “A Perturbation Analysis of an Interaction between Long and Short Surface Waves” by Woodruff and Messiter

Continuous Bores on a Viscous Fluid Down an Incline

Subsidence Due to Intensive Pumping from a Layered Soil

Asymptotic analysis of dispersive tsunami from a slender fault

Nonlinear Effects in Water Wave Diffraction