A collocation method for high frequency scattering by convex polygons

We consider the problem of scattering of a time-harmonic acoustic incident plane wave by a sound soft convex polygon. For standard boundary or finite element methods, with a piecewise polynomial approximation space, the computational cost required to achieve a prescribed level of accuracy grows linearly with respect to the frequency of the incident wave. Recently Chandler–Wilde and Langdon proposed a novel Galerkin boundary element method for this problem for which, by incorporating the products of plane wave basis functions with piecewise polynomials supported on a graded mesh into the approximation space, they were able to demonstrate that the number of degrees of freedom required to achieve a prescribed level of accuracy grows only logarithmically with respect to the frequency. Here we propose a related collocation method, using the same approximation space, for which we demonstrate via numerical experiments a convergence rate identical to that achieved with the Galerkin scheme, but with a substantially reduced computational cost.

A review of dissolved oxygen modelling techniques for lowland rivers

This review introduces the methods used to simulate the processes affecting dissolved oxygen (DO) in lowland rivers. The important processes are described and this provides a modelling framework to describe those processes in the context of a mass-balance model. The process equations that are introduced all require (reaction) rate parameters and a variety of common procedures for identifying those parameters are reviewed. This is important because there is a wide range of estimation techniques for many of the parameters. These different techniques elicit different estimates of the parameter value and so there is the potential for a significant uncertainty in the model's inputs and therefore in the output too. Finally, the data requirements for modelling DO in lowland rivers are summarised on the basis of modelling the processes described in this review using a mass-balance model. This is reviewed with regard to what data are available and from where they might be obtained. (C) 2003 Elsevier Science B.V. All rights reserved.

Aqueous geochemistry and oxygen isotope compositions of acid mine drainage from the Río Tinto, SW Spain, highlight inconsistencies in current models

The Rio Tinto river in SW Spain is a classic example of acid mine drainage and the focus of an increasing amount of research including environmental geochemistry, extremophile microbiology and Mars-analogue studies. Its 5000-year mining legacy has resulted in a wide range of point inputs including spoil heaps and tunnels draining underground workings. The variety of inputs and importance of the river as a research site make it an ideal location for investigating sulphide oxidation mechanisms at the field scale. Mass balance calculations showed that pyrite oxidation accounts for over 93% of the dissolved sulphate derived from sulphide oxidation in the Rio Tinto point inputs. Oxygen isotopes in water and sulphate were analysed from a variety of drainage sources and displayed delta O-18((SO4-H2O)) values from 3.9 to 13.6 parts per thousand, indicating that different oxidation pathways occurred at different sites within the catchment. The most commonly used approach to interpreting field oxygen isotope data applies water and oxygen fractionation factors derived from laboratory experiments. We demonstrate that this approach cannot explain high delta O-18((SO4-H2O)) values in a manner that is consistent with recent models of pyrite and sulphoxyanion oxidation. In the Rio Tinto, high delta O-18((SO4-H2O)) values (11.2-13.6 parts per thousand) occur in concentrated (Fe = 172-829 mM), low pH (0.88-1.4), ferrous iron (68-91% of total Fe) waters and are most simply explained by a mechanism involving a dissolved sulphite intermediate, sulphite-water oxygen equilibrium exchange and finally sulphite oxidation to sulphate with O-2. In contrast, drainage from large waste blocks of acid volcanic tuff with pyritiferous veins also had low pH (1.7). but had a low delta O-18((SO4-H2O)) value of 4.0 parts per thousand and high concentrations of ferric iron (Fe(III) = 185 mM, total Fe = 186 mM), suggesting a pathway where ferric iron is the primary oxidant, water is the primary source of oxygen in the sulphate and where sulphate is released directly from the pyrite surface. However, problems remain with the sulphite-water oxygen exchange model and recommendations are therefore made for future experiments to refine our understanding of oxygen isotopes in pyrite oxidation. (C) 2009 Elsevier B.V. All rights reserved.

C---H local modes in cyclobutene, part I: FTIR studies, 600 to 10000 cm–1

The vibrational structure of C---H stretching states in gas-phase cyclobutene was studied using FTIR spectroscopy in the range 700–9000 cm−1. The structure was modelled using two effective vibrational Hamiltonians, one for each type of C---H bond present, consisting of local mode basis functions subject to coupling with symmetrically equivalent bonds and to Fermi resonances with suitable low frequency vibrations. Best-fit model parameters were determined using least-squares routines and the model predictions are compared to the observed band positions and intensities. Some discussion is given of the relevance of the observed couplings to intramolecular vibrational redistribution (IVR) which results in the observation of statistical behaviour in cyclobutene isomerization induced by excitation of C---H stretching overtones in the visible region.

The Klein-Gordon equation on the half line: a Riemann-Hilbert approach

We solve an initial-boundary problem for the Klein-Gordon equation on the half line using the Riemann-Hilbert approach to solving linear boundary value problems advocated by Fokas. The approach we present can be also used to solve more complicated boundary value problems for this equation, such as problems posed on time-dependent domains. Furthermore, it can be extended to treat integrable nonlinearisations of the Klein-Gordon equation. In this respect, we briefly discuss how our results could motivate a novel treatment of the sine-Gordon equation.

Vibrational perturbation theory

Expressions are derived for the Jacobian of the coriolis ζ interaction constants and the centrifugal stretching constants (DJ, DJK, etc.) with respect to the force constants in a vibrating-rotating molecule.