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).

Second-order wavemaker theory for irregular waves

[1] In this paper the energy budget of wave group-induced subharmonic gravity waves in the nearshore region is examined on the basis of the energy equation for long waves in conjunction with analyses of a high-resolution laboratory data set of one-dimensional random wave propagation over a barred beach. The emphasis is on the growth of forced subharmonics and the deshoaling of the reflected free waves in the shoaling zone. The incident lower-frequency subharmonics are nearly fully reflected at the shoreline, but the higher-frequency components appear to be subject to a significant dissipation in a narrow inshore zone including the swash zone. The previously reported phase lag of the incident forced waves behind the short-wave groups is confirmed, and its key role in the transfer of energy between the grouped short waves and the shoaling bound waves is highlighted. The cross-shore variation of the local mean rate of this energy transfer is determined. Using this as a source function in the wave energy balance allows a very accurate prediction of the enhancement of the forced waves in the shoaling zone, where dissipation is insignificant. The phase lag appears to increase with increasing frequency, which is reflected in a frequency-dependent growth rate, varying very nearly from the free-wave variation ∼ h -1/4 (Green's law) for the lower frequencies to the shallow-water equilibrium limit for forced subharmonics ∼h -5/2 for the higher frequencies. This observed frequency dependence is tentatively generalized to a dependence on a normalized bed slope, controlling whether a so-called mild-slope regime or a steep-slope regime prevails, in which enhanced incident forced waves dominate over breakpoint-generated waves or vice versa.

/pdf/shoaling-of-subharmonic-gravity-waves-3wlwr79dd0.pdf

Shoaling of subharmonic gravity waves

Cross-shore hydrodynamics within an unsaturated surf zone

Dissipation in random wave groups incident on a beach

Changes in the mean horizontal-velocity profile in combined wave-current motion, outside the bottom boundary-layer, are essential for the correct prediction of cross-shore sediment transport outside the surf zone and the transport of dissolved matter, e.g. Klopman (1992). To be able to verify mathematical and numerical models (see Klopman, 1992) experimental data is needed. This is especially the case with the vertical structure of the wave and turbulence Reynolds stresses where data is lacking. For this reason, a laboratory experiment was conducted to study the flow kinematics under combined wave-current motion, for waves propagating in the current direction and for waves opposing the current. The wave-current facility in which the tests have been carried out is equipped with two computer-controlled wave boards, one generating waves and the other absorbing waves. Both wave boards have active wave-absorption systems which eliminate spurious waves. A constant discharge was provided by a flow-circulation circuit. Special care was taken in the design of the inflow and outflow structures, in order to introduce the current smoothly into the channel and to minimize unwanted reflections of the waves at the inflow and outflow. Flow velocities were measured in one vertical cross-section of the channel, with two laserDoppler velocimetry (LDV) systems, mounted at a fixed distance above each other. Water surface elevations were measured with six resistance-type wave-height meters, and the discharge was measured with an electro-magnetic flow (EMF) meter. Tests were performed with mono-chromatic, bi-chromatic and random waves without current, following the current and opposing the current. Also a test series was performed for a steady current without waves.

Vertical structure of the flow due to waves and currents, part 2: Laser-Doppler flow measurements for waves following or opposing a current

In this paper we present a new method to derive Boussinesq-type equations from a variational principle. These equations are valid for nonlinear surface-water waves propagating over bathymetry. The vertical structure of the flow, required in the Hamiltonian, is approximated by a (series of) vertical shape functions associated with unknown parameter(s). It is not necessary to make approximations with respect to the nonlinearity of the waves. The resulting approximate Hamiltonian is positive definite, contributing to the good dynamical behaviour of the resulting equations. The resulting flow equations consist of temporal equations for the surface elevation and potential, as well as a (set of) elliptic equations for some auxiliary parameter(s). All equations only contain low-order spatial derivatives and no mixed time–space derivatives. Since one of the parameters, the surface potential, can be associated with a uniform shape function, the resulting equations are very well suited for wave–current interacting flows.
The variational method is applied to two simple models, one with a parabolic vertical shape function and the other with a hyperbolic-cosine vertical structure. For both, as well as the general series model, the flow equations are derived. Linear dispersion and shoaling are studied using the average Lagrangian. The model with a parabolic vertical shape function has improved frequency dispersion, as compared to classical Boussinesq models. The model with a hyperbolic-cosine vertical structure can be made to have exact phase and group velocity, as well as shoaling, for a certain frequency.
For the model with a parabolic vertical structure, numerical computations are done with a one-dimensional pseudo-spectral code. These show the nonlinear capabilities for periodic waves over a horizontal bed and an underwater bar. Further some long-distance computations for soliton wave groups over bathymetry are presented.

A variational approach to Boussinesq modelling of fully nonlinear water waves

Using a variational principle and a parabolic approximation to the vertical structure of the velocity potential, the equations of motion for surface gravity waves over mildly sloping bathymetry are derived. No approximations are made concerning the non-linearity of the waves. The resulting model equations conserve mass, momentum and positive-definite energy. They are shown to have improved frequency-dispersion characteristics, as compared to classical Boussinesq-type of wave equations.

/pdf/a-variational-model-for-fully-non-linear-water-waves-of-2xzv4yrplk.pdf

G. Klopman

Papers

Vertical structure of the flow due to waves and currents, part 2: Laser-Doppler flow measurements for waves following or opposing a current

A variational approach to Boussinesq modelling of fully nonlinear water waves

A variational model for fully non-linear water waves of Boussinesq type

Vertical structure of the flow due to waves and currents - Laser-doppler flow measurements for waves following or opposing a current

Secondary circulation of the flow due to waves and current: Laser-Doppler flow measurements for waves following or opposing a current