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).

The boundary layer equations for plane, incompressible, and steady flow are 

$$\matrix{ {u{{\partial u} \over {\partial x}} + v{{\partial u} \over {\partial y}} = - {1 \over \varrho }{{\partial p} \over {\partial x}} + v{{{\partial ^2}u} \over {\partial {y^2}}},} \cr {0 = {{\partial p} \over {\partial y}},} \cr {{{\partial u} \over {\partial x}} + {{\partial v} \over {\partial y}} = 0.} \cr }$$

Boundary Layer Theory

Among the current problems that hydrogeologists face, perhaps there is none as challenging as the characterization of fractured rock (Faybishenko and Benson 2000). This paper discusses issues associated with the quantification of flow and transport through fractured rocks on scales not exceeding those typically associated with single- and multi-well pressure (or flow) and tracer tests. As much of the corresponding literature has focused on fractured crystalline rocks and hard sedimentary rocks such as sandstones, limestones (karst is excluded) and chalk, so by default does this paper. Direct quantification of flow and transport in such rocks is commonly done on the basis of fracture geometric data coupled with pressure (or flow) and tracer tests, which therefore form the main focus. Geological, geophysical and geochemical (including isotope) data are critical for the qualitative conceptualization of flow and transport in fractured rocks, and are being gradually incorporated in quantitative flow and transport models, in ways that this paper unfortunately cannot describe but in passing. The hydrogeology of fractured aquifers and other earth science aspects of fractured rock hydrology merit separate treatments. All evidence suggests that rarely can one model flow and transport in a fractured rock consistently by treating it as a uniform or mildly nonuniform isotropic continuum. Instead, one must generally account for the highly erratic heterogeneity, directional dependence, dual or multicomponent nature and multiscale behavior of fractured rocks. One way is to depict the rock as a network of discrete fractures (with permeable or impermeable matrix blocks) and another as a nonuniform (single, dual or multiple) continuum. A third way is to combine these into a hybrid model of a nonuniform continuum containing a relatively small number of discrete dominant features. In either case the description can be deterministic or stochastic. The paper contains a brief assessment of these trends in light of recent experimental and theoretical findings, ending with a short list of prospects and challenges for the future.

Trends, prospects and challenges in quantifying flow and transport through fractured rocks

A Modern Introduction to the Mathematical Theory of Water Waves. By R. S. Johnson. Cambridge University Press, 1997. xiv+445 pp. Hardback ISBN 0 521 59172 4 £55.00; paperback 0 521 59832 X £19.95.

Hydrodynamic and hydromagnetic stability

The radiation stress due to the presence of waves is considered in the equation of motion for the velocity distribution and boundary thickness after the interaction of turbulent plane jet with small amplitude waves. On the basis of the characteristics of turbulent plane jet and small amplitude waves, the velocity distribution about the average of wave period may be assumed to be a Gaussian distribution. Behavior of the fluctuation due to wave motion is also investigated. And from experimental results, the coefficient of velocity distribution c 1 is found to be 0.105.

Theoretical study of turbulent plane jet interacting with small amplitude wave

Evolution of Stokes wave side-band instability along a super tank was studied experimentally and theoretically. The initial exponential growth of the resonant sidebands is followed by asymmetrical growth rates for the sidebands: lower sideband growth is much faster, and finally the main energy is concentrated in it and the primary wave. An active breaking process increases the frequency downshift during the latter stages of the wave propagation. Some type of wave stabilization takes place during the final post-breaking stages of the process and can be characterized by a sharp decreasing of wave breaking activity and stabilization of wave modulations. Our calculations show that Tulin (1996) dissipative modification of NLS model can satisfactory describe the first several stages of the wave train evolution: wave instability, the side band asymmetry and wave breaking effects. On the other hand, continuous wave breaking dissipation presumed in the model gives significantly overestimated values of wave attenuation on the latter stages of wave propagation and can not describe the wave modulation and restabilization at sufficiently long distances of propagation. The adjusted dissipative model based on the Nonlinear SchrOdinger Equation is suggested for adequate description the obtained experimental data. Sink/Source terms due to wave breaking processes in its right side correspond to well-known Tulin (1996) model. The wave dissipation function includes the wave steepness threshold function and applied only in the regions with active wave breaking. Permanent frequency downshift as a result of wave breaking process and post-breaking wave modulations described by the model have the satisfactory quantitative correspondence to results of experiments conducted along a super tank.

Evolution of the Stokes Wave Side-Band Instability along a Super Tank: Experiments and Threshold Modification of Tulin NLS Model

When surface wave propagating over the two layer system usually induces internal wave in three different modes: they are external, internal and combination. In the present study, the nonlinear response of an initially flat sea bed, with two muddy sections, to a monochromatic surface progressive wave was investigated. From this theoretical result, it shows that a surface water wave progressing over two different muddy sections, the surface wave will excite two opposite-traveling short interfacial waves, forming a nearly standing wave at the interface of the fresh water and the muddy layer. Meanwhile, two opposite-outgoing “mud” waves each with very long wavelength will be simultaneously induced at the interface of two muddy sections. As a result, the amplitudes of the two short internal waves are found to grow exponentially in time. Furthermore, it will be much difficult to excite the internal waves when surface water wave progressing over two muddy sections with the large density gap.

/pdf/combination-mode-of-internal-waves-generated-by-surface-wave-ncw5cpoof9.pdf

Combination Mode of Internal Waves Generated by Surface Wave Propagating over Two Muddy Sea Beds

A horizontal plate laid on water surface to reduce the wave motion is proved to be effective theoretically and verified by model tests done at Tainan Hydraulics Laboratory. This principle has been put into practice on the northern coast of Taiwan for protecting a nuclear power plant cooling water intake against intruding waves. The design and construction of wave prevention works of such type are described succinctly in the paper. Also the effect of wave diminishing has been affirmed by measuring the respective waves heights outside and inside of the wave pressing plate.

/pdf/the-wave-pressing-plate-for-protecting-cooling-waterways-of-4eshbrojph.pdf

The Wave Pressing Plate for Protecting Cooling Waterways of Coastal Power Plants

A theoretical examination for the deflection of aquatic plants by water waves is presented. An integration method developed for analyzing the deflection of cantilever beams is now applied to calculate the deformation of aquatic plants. For uniform currents, it is easy to do the calculation since the direction of the fluid loading is perpendicular to the original position of plants. For water wave cases, an extra dependence, however, appears because the wave loading has two spatial dependences. This extra spatial dependence results in difficulties of the derivation of the exact representation of the relation between the bending moment and the external loading. Therefore the integration method is be modified to calculate the theoretical as well as numerical analyses. Results for different characteristics of plants and waves presented in this paper are a preliminary step for more complex flow conditions.

Hwung-Hweng Hwung

Papers

Theoretical study of turbulent plane jet interacting with small amplitude wave

Evolution of the Stokes Wave Side-Band Instability along a Super Tank: Experiments and Threshold Modification of Tulin NLS Model

Combination Mode of Internal Waves Generated by Surface Wave Propagating over Two Muddy Sea Beds

The Wave Pressing Plate for Protecting Cooling Waterways of Coastal Power Plants

Deflection of Aquatic Plant by Water Waves