On integral equation and least squares methods for scattering by diffraction gratings

In this paper we consider the scattering of a plane acoustic or electromagnetic wave by a one-dimensional, periodic rough surface. We restrict the discussion to the case when the boundary is sound soft in the acoustic case, perfectly reflecting with TE polarization in the EM case, so that the total field vanishes on the boundary. We propose a uniquely solvable first kind integral equation formulation of the problem, which amounts to a requirement that the normal derivative of the Green's representation formula for the total field vanish on a horizontal line below the scattering surface. We then discuss the numerical solution by Galerkin's method of this (ill-posed) integral equation. We point out that, with two particular choices of the trial and test spaces, we recover the so-called SC (spectral-coordinate) and SS (spectral-spectral) numerical schemes of DeSanto et al., Waves Random Media, 8, 315-414 1998. We next propose a new Galerkin scheme, a modification of the SS method that we term the SS* method, which is an instance of the well-known dual least squares Galerkin method. We show that the SS* method is always well-defined and is optimally convergent as the size of the approximation space increases. Moreover, we make a connection with the classical least squares method, in which the coefficients in the Rayleigh expansion of the solution are determined by enforcing the boundary condition in a least squares sense, pointing out that the linear system to be solved in the SS* method is identical to that in the least squares method. Using this connection we show that (reflecting the ill-posed nature of the integral equation solved) the condition number of the linear system in the SS* and least squares methods approaches infinity as the approximation space increases in size. We also provide theoretical error bounds on the condition number and on the errors induced in the numerical solution computed as a result of ill-conditioning. Numerical results confirm the convergence of the SS* method and illustrate the ill-conditioning that arises.

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.

Perturbed Rankine vortices in surface quasi-geostrophic dynamics

An analytical dispersion relation is derived for linear perturbations to a Rankine vortex governed by surface quasi-geostrophic dynamics. Such a Rankine vortex is a circular region of uniform anomalous surface temperature evolving under quasi-geostrophic dynamics with uniform interior potential vorticity. The dispersion relation is analysed in detail and compared to the more familiar dispersion relation for a perturbed Rankine vortex governed by the Euler equations. The results are successfully verified against numerical simulations of the full equations. The dispersion relation is relevant to problems including wave propagation on surface temperature fronts and the stability of vortices in quasi-geostrophic turbulence.

Wave-driven wind jets in the marine atmospheric boundary layer

The interaction between ocean surface waves and the overlying wind leads to a transfer of momentum across the air–sea interface. Atmospheric and oceanic models typically allow for momentum transfer to be directed only downward, from the atmosphere to the ocean. Recent observations have suggested that momentum can also be transferred upward when long wavelength waves, characteristic of remotely generated swell, propagate faster than the wind speed. The effect of upward momentum transfer on the marine atmospheric boundary layer is investigated here using idealized models that solve the momentum budget above the ocean surface. A variant of the classical Ekman model that accounts for the wave-induced stress demonstrates that, although the momentum flux due to the waves penetrates only a small fraction of the depth of the boundary layer, the wind profile is profoundly changed through its whole depth. When the upward momentum transfer from surface waves sufficiently exceeds the downward turbulent momentum flux, then the near-surface wind accelerates, resulting in a low-level wave-driven wind jet. This increases the Coriolis force in the boundary layer, and so the wind turns in the opposite direction to the classical Ekman layer. Calculations of the wave-induced stress due to a wave spectrum representative of fast-moving swell demonstrate upward momentum transfer that is dominated by contributions from waves in the vicinity of the peak in the swell spectrum. This is in contrast to wind-driven waves whose wave-induced stress is dominated by very short wavelength waves. Hence the role of swell can be characterized by the inverse wave age based on the wave phase speed corresponding to the peak in the spectrum. For a spectrum of waves, the total momentum flux is found to reverse sign and become upward, from waves to wind, when the inverse wave age drops below the range 0.15–0.2, which agrees reasonably well with previously published oceanic observations.

A Nyström method for a boundary value problem arising in unsteady water wave problems

This paper is concerned with solving numerically the Dirichlet boundary value problem for Laplace’s equation in a nonlocally perturbed half-plane. This problem arises in the simulation of classical unsteady water wave problems. The starting point for the numerical scheme is the boundary integral equation reformulation of this problem as an integral equation of the second kind on the real line in Preston et al. (2008, J. Int. Equ. Appl., 20, 121–152). We present a Nystr¨om method for numerical solution of this integral equation and show stability and convergence, and we present and analyse a numerical scheme for computing the Dirichlet-to-Neumann map, i.e., for deducing the instantaneous fluid surface velocity from the velocity potential on the surface, a key computational step in unsteady water wave simulations. In particular, we show that our numerical schemes are superalgebraically convergent if the fluid surface is infinitely smooth. The theoretical results are illustrated by numerical experiments.

Optimization of electroless copper plating on polyethylene films modified by surface grafting of vinyl ether of monoethanolamine

Metallized plastics have recently received significant interest for their useful applications in electronic devices such as for integrated circuits, packaging, printed circuits and sensor applications. In this work the metallized films were developed by electroless copper plating of polyethylene films grafted with vinyl ether of monoethanoleamine. There are several techniques for metal deposition on surface of polymers such as evaporation, sputtering, electroless plating and electrolysis. In this work the metallized films were developed by electroless copper plating of polyethylene films grafted with vinyl ether of monoethanoleamine. Polyethylene films were subjected to gamma-radiation induced surface graft copolymerization with vinyl ether of monoethanolamine. Electroless copper plating was carried out effectively on the modified films. The catalytic processes for the electroless copper plating in the presence and the absence of SnCl2 sensitization were studied and the optimum activation conditions that give the highest plating rate were determined. The effect of grafting degree on the plating rate is studied. Electroless plating conditions (bath additives, pH and temperature) were optimized. Plating rate was determined gravimetrically and spectrophotometrically at different grafting degrees. The results reveal that plating rate is a function of degree of grafting and increases with increasing grafted vinyl ether of monoethanolamine onto polyethylene. It was found that pH 13 of electroless bath and plating temperature 40°C are the optimal conditions for the plating process. The increasing of grafting degree results in faster plating rate at the same pH and temperature. The surface morphology of the metallized films was investigated using scanning electron microscopy (SEM). The adhesion strength between the metallized layer and grafted polymer was studied using tensile machine. SEM photos and adhesion measurements clarified that uniform and adhered deposits were obtained under optimum conditions.

Ocean acoustic models for low frequency propagation in 2D and 3D environments

In this paper we are mainly concerned with the development of efficient computer models capable of accurately predicting the propagation of low-to-middle frequency sound in the sea, in axially symmetric (2D) and in fully 3D environments. The major physical features of the problem, i.e. a variable bottom topography, elastic properties of the subbottom structure, volume attenuation and other range inhomogeneities are efficiently treated. The computer models presented are based on normal mode solutions of the Helmholtz equation on the one hand, and on various types of numerical schemes for parabolic approximations of the Helmholtz equation on the other. A new coupled mode code is introduced to model sound propagation in range-dependent ocean environments with variable bottom topography, where the effects of an elastic bottom, of volume attenuation, surface and bottom roughness are taken into account. New computer models based on finite difference and finite element techniques for the numerical solution of parabolic approximations are also presented. They include an efficient modeling of the bottom influence via impedance boundary conditions, they cover wide angle propagation, elastic bottom effects, variable bottom topography and reverberation effects. All the models are validated on several benchmark problems and versus experimental data. Results thus obtained were compared with analogous results from standard codes in the literature.

Acoustic trapped modes in a three-dimensional waveguide of slowly varying cross section

In this paper we develop an asymptotic scheme to approximate the trapped mode solutions to the time harmonic wave equation in a three-dimensional waveguide with a smooth but otherwise arbitrarily shaped cross section and a single, slowly varying `bulge', symmetric in the longitudinal direction. Extending the work in Biggs (2012), we first employ a WKBJ-type ansatz to identify the possible quasi-mode solutions which propagate only in the thicker region, and hence find a finite cut-on region of oscillatory behaviour and asymptotic decay elsewhere. The WKBJ expansions are used to identify a turning point between the cut-on and cut-on regions. We note that the expansions are nonuniform in an interior layer centred on this point, and we use the method of matched asymptotic expansions to connect the cut-on and cut-on regions within this layer. The behaviour of the expansions within the interior layer then motivates the construction of a uniformly valid asymptotic expansion. Finally, we use this expansion and the symmetry of the waveguide around the longitudinal centre, x = 0, to extract trapped mode wavenumbers, which are compared with those found using a numerical scheme and seen to be extremely accurate, even to relatively large values of the small parameter.

Variational approach in weighted Sobolev spaces to scattering by unbounded rough surfaces

We consider the problem of scattering of time harmonic acoustic waves by an unbounded sound soft surface which is assumed to lie within a finite distance of some plane. The paper is concerned with the study of an equivalent variational formulation of this problem set in a scale of weighted Sobolev spaces. We prove well-posedness of this variational formulation in an energy space with weights which extends previous results in the unweighted setting [S. Chandler-Wilde and P. Monk, SIAM J. Math. Anal., 37 (2005), pp. 598–618] to more general inhomogeneous terms in the Helmholtz equation. In particular, in the two-dimensional case, our approach covers the problem of plane wave incidence, whereas in the three-dimensional case, incident spherical and cylindrical waves can be treated. As a further application of our results, we analyze a finite section type approximation, whereby the variational problem posed on an infinite layer is approximated by a variational problem on a bounded region.

Fabrication and characterization of micromachined rectangular waveguide components for use at millimeter-wave and terahertz frequencies

The fabrication and characterization of micromachined reduced-height air-filled rectangular waveguide components suitable for integration is reported in this paper. The lithographic technique used permits structures with heights of up to 100 μm to be successfully constructed in a repeatable manner. Waveguide S-parameter measurements at frequencies between 75-110 GHz using a vector network analyzer demonstrate low loss propagation in the TE10 mode reaching 0.2 dB per wavelength. Scanning electron microscope photographs of conventional and micromachined waveguides show that the fabrication technique can provide a superior surface finish than possible with commercially available components. In order to circumvent problems in efficiently coupling free-space propagating beams to the reduced-height G-band waveguides, as well as to characterize them using quasi-optical techniques, a novel integrated micromachined slotted horn antenna has been designed and fabricated, E-, H-, and D-plane far-field antenna pattern measurements at different frequencies using a quasi-optical setup show that the fabricated structures are optimized for 180-GHz operation with an E-plane half-power beamwidth of 32° elevated 35° above the substrate, a symmetrical H-plane pattern with a half-power beamwidth of 23° and a maximum D-plane cross-polar level of -33 dB. Far-field pattern simulations using HFSS show good agreement with experimental results.

The dependence of clear-sky outgoing longwave radiation on surface temperature and relative humidity

A simulation of the earth's clear-sky long-wave radiation budget is used to examine the dependence of clear-sky outgoing long-wave radiation (OLR) on surface temperature and relative humidity. the simulation uses the European Centre for Medium-Range Weather Forecasts global reanalysed fields to calculate clear-sky OLR over the period from January 1979 to December 1993, thus allowing the seasonal and interannual time-scales to be resolved. the clear-sky OLR is shown to be primarily dependent on temperature changes at high latitudes and on changes in relative humidity at lower latitudes. Regions exhibiting a ‘super-greenhouse’ effect are identified and are explained by considering the changes in the convective regime associated with the Hadley circulation over the seasonal cycle, and with the Walker circulation over the interannual time-scale. the sensitivity of clear-sky OLR to changes in relative humidity diminishes with increasing relative humidity. This is explained by the increasing saturation of the water-vapour absorption bands with increased moisture. By allowing the relative humidity to vary in specified vertical slabs of the troposphere over an interannual time-scale it is shown that changes in humidity in the mid troposphere (400 to 700 hPa) are of most importance in explaining clear-sky OLR variations. Relative humidity variations do not appear to affect the positive thermodynamic water-vapour feedback significantly in response to surface temperature changes.

A new frequency-uniform coercive boundary integral equation for acoustic scattering

A new boundary integral operator is introduced for the solution of the soundsoft acoustic scattering problem, i.e., for the exterior problem for the Helmholtz equation with Dirichlet boundary conditions. We prove that this integral operator is coercive in L2(Γ) (where Γ is the surface of the scatterer) for all Lipschitz star-shaped domains. Moreover, the coercivity is uniform in the wavenumber k = ω/c, where ω is the frequency and c is the speed of sound. The new boundary integral operator, which we call the “star-combined” potential operator, is a slight modification of the standard combined potential operator, and is shown to be as easy to implement as the standard one. Additionally, to the authors' knowledge, it is the only second-kind integral operator for which convergence of the Galerkin method in L2(Γ) is proved without smoothness assumptions on Γ except that it is Lipschitz. The coercivity of the star-combined operator implies frequency-explicit error bounds for the Galerkin method for any approximation space. In particular, these error estimates apply to several hybrid asymptoticnumerical methods developed recently that provide robust approximations in the high-frequency case. The proof of coercivity of the star-combined operator critically relies on an identity first introduced by Morawetz and Ludwig in 1968, supplemented further by more recent harmonic analysis techniques for Lipschitz domains.

Examination of long-wave radiative bias in general circulation models over North Africa during May–July

Satellite data are used to quantify and examine the bias in the outgoing long-wave (LW) radiation over North Africa during May–July simulated by a range of climate models and the Met Office global numerical weather prediction (NWP) model. Simulations from an ensemble-mean of multiple climate models overestimate outgoing clear-sky long-wave radiation (LWc) by more than 20 W m−2 relative to observations from Clouds and the Earth's Radiant Energy System (CERES) for May–July 2000 over parts of the west Sahara, and by 9 W m−2 for the North Africa region (20°W–30°E, 10–40°N). Experiments with the atmosphere-only version of the High-resolution Hadley Centre Global Environment Model (HiGEM), suggest that including mineral dust radiative effects removes this bias. Furthermore, only by reducing surface temperature and emissivity by unrealistic amounts is it possible to explain the magnitude of the bias. Comparing simulations from the Met Office NWP model with satellite observations from Geostationary Earth Radiation Budget (GERB) instruments suggests that the model overestimates the LW by 20–40 W m−2 during North African summer. The bias declines over the period 2003–2008, although this is likely to relate to improvements in the model and inhomogeneity in the satellite time series. The bias in LWc coincides with high aerosol dust loading estimated from the Ozone Monitoring Instrument (OMI), including during the GERBILS field campaign (18–28 June 2007) where model overestimates in LWc greater than 20 W m−2 and OMI-estimated aerosol optical depth (AOD) greater than 0.8 are concurrent around 20°N, 0–20°W. A model-minus-GERB LW bias of around 30 W m−2 coincides with high AOD during the period 18–21 June 2007, although differences in cloud cover also impact the model–GERB differences. Copyright © Royal Meteorological Society and Crown Copyright, 2010

Numerical-asymptotic boundary integral methods in high-frequency acoustic scattering

In this article we describe recent progress on the design, analysis and implementation of hybrid numerical-asymptotic boundary integral methods for boundary value problems for the Helmholtz equation that model time harmonic acoustic wave scattering in domains exterior to impenetrable obstacles. These hybrid methods combine conventional piecewise polynomial approximations with high-frequency asymptotics to build basis functions suitable for representing the oscillatory solutions. They have the potential to solve scattering problems accurately in a computation time that is (almost) independent of frequency and this has been realized for many model problems. The design and analysis of this class of methods requires new results on the analysis and numerical analysis of highly oscillatory boundary integral operators and on the high-frequency asymptotics of scattering problems. The implementation requires the development of appropriate quadrature rules for highly oscillatory integrals. This article contains a historical account of the development of this currently very active field, a detailed account of recent progress and, in addition, a number of original research results on the design, analysis and implementation of these methods.