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

This paper presents a new Boussinesq-type model equations for describing nonlinear surface wave motions in porous media. The mathematical model based on perturbation approach reported by Hsiao et al. is derived. The drag force and turbulence effect suggested by Sollitt and Cross are incorporated for observing the flow behaviors within porous media. Additionally, the approach of Chen for eliminating the depth-dependent terms in the momentum equations is also adopted. The model capability on an applicable water depth range is satisfactorily validated against the linear wave theory. The nonlinear properties of model equations are numerically confirmed by the weakly nonlinear theory of Liu and Wen. Numerical experiments of regular waves propagating in porous media over an impermeable submerged breakwater are performed and the nonlinear behaviors of wave energy transfer between different harmonics are also examined.

Extended Boussinesq Equations for Water-Wave Propagation in Porous Media

We provide experiments that had been conducted to investigate the particle trajectory beneath a solitary water wave. Experimental results show that the surface drift is larger than the bottom drift. Meanwhile, the ratio of the net horizontal displacement to the total height of the trajectory at the free surface and subsurface will decrease with the initial vertical position and increase with the relative wave height.

https://royalsocietypublishing.org/doi/pdf/10.1098/rsta.2011.0445

Experimental study of the particle paths in solitary water waves

An exploration of stakeholder conflict over the Taiwanese marine protected area

To investigate the evolution of wave modulation and wave breaking, a series of elaborate experiments were conducted in a super wave flume (300 m × 5.0 m × 5.2 m) at Tainan Hydraulics Laboratory. Two wave trains of initial regular waves and initial imposed sidebands were generated by a wave maker with active wave absorption using an electro-hydraulic server feedback control system. The wave profiles were recorded by 66 high-resolution capacitance-type wave gauges positioned along the wave flume, and connected to a specially designed data acquisition system to collect a large number of parallel inputs in the long flume. Experimental data were obtained on wave modulation and related wave breaking. The long time evolution of the nonlinear wave train and the corresponding wave spectra are demonstrated for both breaking and non-breaking cases. Specifically, the periodic modulation and demodulation of wave train is observed at post-breaking stage.

https://iopscience.iop.org/article/10.1088/0957-0233/16/10/006/pdf

Measurements of wave modulation and breaking

The boundary layer flows and wave-induced pore pressures inside porous beds are both investigated in wave tank. Two kinds of porous beds are composed of two different grades of quartz sands which the porosities are 0.437 and 0.472 respectively. According to the experimental results, it is found that the larger pore pressure in permeable bed is induced by the longer wave period, and the pore pressure has an exponential attenuation with the wave steepness increasing. From the elaborate measurements in the boundary layer, the overshooting induced by wave motion is effected by bed porosity and it becomes smooth. In the experimental cases, the maximum horizontal overshooting occurs at 2 / 5 « 1.2 ,which is lower than the impermeable case of Z 15 ~2.1 obtained by Hwung and Lin (1989). Furthermore, the vertical overshooting has also been found in our experiments.

/pdf/wave-boundary-layer-flows-and-pore-pressure-in-permeable-54dte2eh33.pdf

Hwung-Hweng Hwung

Papers

Extended Boussinesq Equations for Water-Wave Propagation in Porous Media

Experimental study of the particle paths in solitary water waves

An exploration of stakeholder conflict over the Taiwanese marine protected area

Measurements of wave modulation and breaking

Wave boundary layer flows and pore pressure in permeable beds