P. G. Chamberlain

The modified mild-slope equation

A modified version of the mild-slope equation is derived and its predictions of wave scattering by two-dimensional topography compared with those of other equations and with experimental data. In particular, the modified mild-slope equation is shown to be capable of describing known scattering properties of singly and doubly periodic ripple beds, for which the mild-slope equation fails. The new equation compares favourably with other models of scattering which improve on the mild-slope equation, in that it is widely applicable and computationally cheap.

Wave scattering in a two-layer fluid of varying depth

The scattering of waves in a two-layer fluid of varying mean depth is examined in a three-dimensional context using linear theory. A variational technique is used to construct a particular type of approximation which has the effect of removing the vertical coordinate and reducing the problem to two coupled partial differential equations in two independent variables. A transformation of this differential equation system leads to a particularly simple approximate representation of the scattering process. The theory is applied to two-dimensional scattering, for which a set of symmetry relations is derived. A selection of numerical results is presented to illustrate the principle interest in the problem, namely the energy transfer between surface and interfacial waves induced by bed undulations.

Atlas of coronavirus replicase structure.

Abstract The international response to SARS-CoV has produced an outstanding number of protein structures in a very short time. This review summarizes the findings of functional and structural studies including those derived from cryoelectron microscopy, small angle X-ray scattering, NMR spectroscopy, and X-ray crystallography, and incorporates bioinformatics predictions where no structural data is available. Structures that shed light on the function and biological roles of the proteins in viral replication and pathogenesis are highlighted. The high percentage of novel protein folds identified among SARS-CoV proteins is discussed.

DECOMPOSITION METHODS FOR WAVE SCATTERING BY TOPOGRAPHY WITH APPLICATION TO RIPPLE BEDS

Abstract A method is described for determining those approximations to wave scattering by bed topography which are based on second-order ordinary differential equations. The development of a decomposition method allows the scattering matrix for an extended section of varying topography to be assembled in a piecemeal fashion. In particular, the scattering matrix for a ripple bed, consisting of an arbitrary number of periodic undulations, is expressed in terms of the scattering properties of a single ripple. The structure obtained reveals the main features of ripple bed scattering, including resonant reflection at certain frequencies. The analysis is allied to numerical calculations to compare five different models of ripple bed scattering.

Multi-mode approximations to wave scattering by an uneven bed

Approximations to the scattering of linear surface gravity waves on water of varying quiescent depth are investigated by means of a variational approach. Previous authors have used wave modes associated with the constant depth case to approximate the velocity potential, leading to a system of coupled differential equations. Here it is shown that a transformation of the dependent variables results in a much simplified differential equation system which in turn leads to a new multi-mode ‘mild-slope’ approximation. Further, the effect of adding a bed mode is examined and clarified. A systematic analytic method is presented for evaluating inner products that arise and numerical experiments for two-dimensional scattering are used to examine the performance of the new approximations.

Wave scattering over uneven depth using the mild-slope equation

Abstract This paper examines the scattering of a train of small-amplitude harmonic surface waves on water by one-dimensional topography, using the mild-slope equation. The associated boundary value problem is converted into a pair of integral equations whose solutions are approximated by variational techniques, which also supply error bounds. Excellent approximations to the reflection and transmission coefficients and to the free surface shape are produced with only 2- or 3-dimensional trial spaces, by choosing these spaces to be problem dependent. The bed profiles considered include localised humps and taluds which join two horizontal planes at different depths.

An Integral Equation Method for a Boundary Value Problem arising in Unsteady Water Wave Problems

In this paper we consider the 2D Dirichlet boundary value problem for Laplaces equation in a non-locally perturbed half-plane, with data in the space of bounded and continuous functions. We show uniqueness of solution, using standard Phragmen-Lindelof arguments. The main result is to propose a boundary integral equation formulation, to prove equiv- alence with the boundary value problem, and to show that the integral equation is well posed by applying a recent partial generalisation of the Fredholm alternative in Arens et al (J Int. Equ. Appl. 15 (2003) pp1-35). This then leads to an existence proof for the boundary value problem. Keywords. Boundary integral equation method, Water waves, Laplaces equation This paper is concerned with the boundary integral equation method for a problem in potential theory, namely the Dirichlet boundary value problem in a non-locally perturbed half-plane. The main aim of the paper is to discuss the well-posedness of this problem and of a novel second kind boundary integral equation formulation. Our motivation for studying this problem is that it arises in the theory of classical free surface water wave problems, for which boundary integral equation methods are well-established as a computational and theoretical tool (B1, B2, HZ, FD). In particular, accurate numerical schemes, based on boundary inte- gral equation formulations, for the time dependent water wave problem have been proposed and fully analysed in (B1, B2, HZ), these papers providing a full nonlinear stability analysis for the spatial discretisations they propose. A signif- icant component in this analysis is the well-posedness of the boundary integral equation formulation. A key restriction in the analysis in the above papers is the requirement that the free surface be periodic (in 2D) or doubly-periodic (in 3D). This restriction is helpful theoretically and computationally. It enables the boundary integral equation on the free surface to be reduced to one on a (bounded) single peri- odic cell, which can then be discretised with a finite size mesh. Moreover, the

Approximations to wave trapping by topography

The trapping of linear water waves over two-dimensional topography is investigated by using the mild-slope approximation. Two types of bed profile are considered: a local irregularity in a horizontal bed and a shelf joining two horizontal bed sections at different depths. A number of results are derived concerning the existence of trapped modes and their multiplicity. It is found, for example, that the maximum number of modes which can exist depends only on the gross properties of the topography and not on its precise shape. A range of problems is solved numerically, to inform and illustrate the analysis, using both the mild-slope equation and the recently derived modified mild-slope equation.

EditorialWaves 2007 conference

This special volume contains a selection of papers, based on talks presented at the Eighth International Conference on Mathematical and Numerical Aspects of Waves (Waves 2007), which took place at the University of Reading, UK, organised jointly with INRIA, from 23rd to 27th July 2007. Wave phenomena are hugely important in many areas of science and engineering and at a whole variety of length scales, from the atomic to terrestrial and beyond. Applications include acoustic and electromagnetic scattering, seismicwave motions, the coupling of wind and ocean waves, refraction and imaging with photonic crystals, probing atomic structure with attosecond pulses, the design of low-loss optical wave guides, and travelling waves in the occurrence of epidemics. This conference is one of the main venues where significant advances in the analysis and computational modeling of wave phenomena and exciting new applications are presented, and this meeting was the eighth in a sequence which started in Strasbourg in 1991. Conference themes included forward and inverse scattering, nonlinearwave phenomena, fast computational techniques, numerical analysis, absorbing layers and approximate boundary conditions, analytic and semi-analytic techniques for wave problems, domain decomposition, guided waves and random media. The invited speakers were Mark Ablowitz (Colorado, USA), Annalisa Buffa (Pavia, Italy), Weng Cho Chew (Illinois, USA), Tom Hagstrom (NewMexico, USA), Andreas Kirsch (Karlsruhe, Germany), Ross McPhedran (Sydney, Australia) and John Toland (Bath, UK). In addition, there were approximately 180 varied and interesting contributed talks, with speakers travelling from all over the world to attend the meeting. The conference featured an embeddedWorkshop on High Frequency Propagation and Scattering, supported by the Isaac Newton Institute, Cambridge, UK, as a satellite meeting of the INI Programme on Highly Oscillatory Problems: Computation, Theory and Application. Waves 2007 was also a satellite conference of ICIAM 07, the 6th International Congress on Industrial andAppliedMathematics. In addition to theworkshop, therewere sevenminisymposia: brainwaves and cognitive neurodynamics; inverse problems; nonlinear waves; periodic and random media; photonic crystals and metamaterials; resonances and trappedmodes; time domainmethods. These areas, and others, are represented amongst the 51 papers that appear in this volume (from62 originally submitted), each ofwhichwas subjected to an extensive reviewprocess.Wewould like to thank the many anonymous referees who gave up their time to contribute to this effort. It is a pleasure also to thank the following for their direct or indirect financial support for theWaves 2007meeting and its embedded workshop: the Isaac Newton Institute, Cambridge; the Institute of Mathematics and its Applications; the Society for Industrial and Applied Mathematics (SIAM), UK section; INRIA, and the University of Reading Computational Sciences Theme. We sincerely hope you find this an interesting and illuminating volume of papers related to wave phenomena, and we look forward to seeing you at the Ninth International Conference on Mathematical and Numerical Aspects of Waves, which will take place in Pau, France, in June 2009.

Editorial: Waves 2007 conference

This special volume contains a selection of papers, based on talks presented at the Eighth International Conference on Mathematical and Numerical Aspects of Waves (Waves 2007), which took place at the University of Reading, UK, organised jointly with INRIA, from 23rd to 27th July 2007. Wave phenomena are hugely important in many areas of science and engineering and at a whole variety of length scales, from the atomic to terrestrial and beyond. Applications include acoustic and electromagnetic scattering, seismicwave motions, the coupling of wind and ocean waves, refraction and imaging with photonic crystals, probing atomic structure with attosecond pulses, the design of low-loss optical wave guides, and travelling waves in the occurrence of epidemics. This conference is one of the main venues where significant advances in the analysis and computational modeling of wave phenomena and exciting new applications are presented, and this meeting was the eighth in a sequence which started in Strasbourg in 1991. Conference themes included forward and inverse scattering, nonlinearwave phenomena, fast computational techniques, numerical analysis, absorbing layers and approximate boundary conditions, analytic and semi-analytic techniques for wave problems, domain decomposition, guided waves and random media. The invited speakers were Mark Ablowitz (Colorado, USA), Annalisa Buffa (Pavia, Italy), Weng Cho Chew (Illinois, USA), Tom Hagstrom (NewMexico, USA), Andreas Kirsch (Karlsruhe, Germany), Ross McPhedran (Sydney, Australia) and John Toland (Bath, UK). In addition, there were approximately 180 varied and interesting contributed talks, with speakers travelling from all over the world to attend the meeting. The conference featured an embeddedWorkshop on High Frequency Propagation and Scattering, supported by the Isaac Newton Institute, Cambridge, UK, as a satellite meeting of the INI Programme on Highly Oscillatory Problems: Computation, Theory and Application. Waves 2007 was also a satellite conference of ICIAM 07, the 6th International Congress on Industrial andAppliedMathematics. In addition to theworkshop, therewere sevenminisymposia: brainwaves and cognitive neurodynamics; inverse problems; nonlinear waves; periodic and random media; photonic crystals and metamaterials; resonances and trappedmodes; time domainmethods. These areas, and others, are represented amongst the 51 papers that appear in this volume (from62 originally submitted), each ofwhichwas subjected to an extensive reviewprocess.Wewould like to thank the many anonymous referees who gave up their time to contribute to this effort. It is a pleasure also to thank the following for their direct or indirect financial support for theWaves 2007meeting and its embedded workshop: the Isaac Newton Institute, Cambridge; the Institute of Mathematics and its Applications; the Society for Industrial and Applied Mathematics (SIAM), UK section; INRIA, and the University of Reading Computational Sciences Theme. We sincerely hope you find this an interesting and illuminating volume of papers related to wave phenomena, and we look forward to seeing you at the Ninth International Conference on Mathematical and Numerical Aspects of Waves, which will take place in Pau, France, in June 2009.