electromagnetic scattering by an inhomogeneous conducting or dielectric layer on a perfectly conducting plate we consider a two dimensional problem of scattering of a time harmonic electromagnetic plane wave by an inhomogeneous conducting or dielectric layer on a perfectly conducting plate the magnetic permeability is assumed to be a fixed positive constant in the media the material properties of the media are characterized completely by an index of refraction which is a bounded measurable function in the layer and a positive constant above the layer corresponding to a homogeneous dielectric medium in this paper we only examine the tm transverse magnetic polarization case a radiation condition is introduced and equivalence with a second kind lippmann schwinger type integral equation is shown with additional assumptions on the index of refraction in the layer uniqueness of solution is proved existence of solution is then established by employing a form of fredholm alternative using a general result on the solvability of integral equations on unbounded domains published earlier by chandler wilde and zhang an approximate analytic solution for the case of a thin inhomogeneous layer is obtained from the integral equation formulation and is used to show that if the index of refraction is appropriately chosen the scattered field can grow with distance from the plate

variable delay feedback control of unstable steady states in retarded time delayed systems we study the stability of unstable steady states in scalar retarded time delayed systems subjected to a variable delay feedback control the important aspect of such a control problem is that time delayed systems are already infinite dimensional before the delayed feedback control is turned on when the frequency of the modulation is large compared to the system s dynamics the analytic approach consists of relating the stability properties of the resulting variable delay system with those of an analogous distributed delay system otherwise the stability domains are obtained by a numerical integration of the linearized variable delay system the analysis shows that the control domains are significantly larger than those in the usual time delayed feedback control and that the complexity of the domain structure depends on the form and the frequency of the delay modulation

weighted sobolev spaces and exterior problems for the helmholtz equation weighted sobolev spaces are used to settle questions of existence and uniqueness of solutions to exterior problems for the helmholtz equation furthermore it is shown that this approach can cater for inhomogeneous terms in the problem that are only required to vanish asymptotically at infinity in contrast to the rellich sommerfeld radiation condition which in a hilbert space setting requires that all radiating solutions of the helmholtz equation should satisfy a condition of the form r i k u l 2 r x r n it is shown here that radiating solutions satisfy a condition of the form 1 r 1 2 ln e r 1 2 u l 2 0 1 2 and moreover such solutions satisfy the classical sommerfeld condition u o r 1 2 n 1 r furthermore the approach avoids many of the difficulties usually associated with applications of the poincare inequality and the sobolev embedding theorems

acoustic scattering by mildly rough unbounded surfaces in three dimensions for a nonlocally perturbed half space we consider the scattering of time harmonic acoustic waves a second kind boundary integral equation formulation is proposed for the sound soft case based on a standard ansatz as a combined single and double layer potential but replacing the usual fundamental solution of the helmholtz equation with an appropriate half space green s function due to the unboundedness of the surface the integral operators are noncompact in contrast to the two dimensional case the integral operators are also strongly singular due to the slow decay at infinity of the fundamental solution of the three dimensional helmholtz equation in the case when the surface is sufficiently smooth lyapunov we show that the integral operators are nevertheless bounded as operators on l 2 gamma and on l 2 gamma cap bc gamma and that the operators depend continuously in norm on the wave number and on gamma we further show that for emph mild roughness i e a surface gamma which d

diffraction by a defect in an open waveguide a mathematical analysis based on a modal radiation condition we consider the scattering of a time harmonic acoustic wave by a defect in a twodimensional open waveguide the scattered wave satisfies the helmholtz equation in a perturbed layered half plane we introduce a modal radiation condition based on a generalized fourier transform which diagonalizes the transverse contribution of the helmholtz operator the uniqueness of the solution is proved by an original technique which combines a property of the energy flux with an argument of analyticity with respect to the generalized fourier variable the existence is then deduced classically from fredholm s alternative by reformulating the scattering problem as a lippmann schwinger equation by means of the green s function for the layered half plane 2009 society for industrial and applied mathematics

a well posed integral equation formulation for three dimensional rough surface scattering we consider the problem of scattering of time harmonic acoustic waves by an unbounded sound soft rough surface recently a brakhage werner type integral equation formulation of this problem has been proposed based on an ansatz as a combined single and double layer potential but replacing the usual fundamental solution of the helmholtz equation with an appropriate half space green s function moreover it has been shown in the three dimensional case that thi si ntegral equation is uniquely solvable in the space l 2 g when the scattering surface g does not differ too much from a plane in this paper we show that this integral equation is uniquely solvable with no restriction on the surface elevation or slope moreover we construct explicit bounds on the inverse of the associated boundary integral operator as a function of the wave number the parameter coupling the single and double layer potentials and the maximum surface slope these bounds show that the norm of the inverse operator is bounded uniformly in the wave number k f orko0 if the coupling parameter h is chosen proportional to the wave number in the case when g is a plane we show that the choice hzk 2 is nearly optimal in terms of minimizing the condition number

scattering by infinite one dimensional rough surfaces we consider the dirichlet boundary value problem for the helmholtz equation in a non locally perturbed half plane this problem models time harmonic electromagnetic scattering by a one dimensional infinite rough perfectly conducting surface the same problem arises in acoustic scattering by a sound soft surface chandler wilde and zhang have suggested a radiation condition for this problem a generalization of the rayleigh expansion condition for diffraction gratings and uniqueness of solution has been established recently an integral equation formulation of the problem has also been proposed and in the special case when the whole boundary is both lyapunov and a small perturbation of a flat boundary the unique solvability of this integral equation has been shown by chandler wilde and ross by operator perturbation arguments in this paper we study the general case with no limit on surface amplitudes or slopes and show that the same integral equation has exactly one solution in the space of bounded and continuous functions for all wavenumbers as an important corollary we prove that for a variety of incident fields including the incident plane wave the dirichlet boundary value problem for the scattered field has a unique solution

a uniqueness result for scattering by infinite rough surfaces consider the dirichlet boundary value problem for the helmholtz equation in a non locally perturbed half plane with an unbounded piecewise lyapunov boundary this problem models time harmonic electromagnetic scattering in transverse magnetic polarization by one dimensional rough perfectly conducting surfaces a radiation condition is introduced for the problem which is a generalization of the usual one used in the study of diffraction by gratings when the solution is quasi periodic and allows a variety of incident fields including an incident plane wave to be included in the results obtained we show in this paper that the boundary value problem for the scattered field has at most one solution for the case when the whole boundary is lyapunov and is a small perturbation of a flat boundary we also prove existence of solution and show a limiting absorption principle

on the gap metric and coprime factor perturbations abstract new conditions are derived for when the distance between two linear systems in the gap metric is less than one by including a coprimeness assumption in the coprime factor uncertainty description it is shown that an open ball in the gap metric is equivalent to an open ball phrased in terms of coprime factor perturbations a new criterion is given for robust stabilization

acoustic scattering by an inhomogeneous layer on a rigid plate the problem of scattering of time harmonic acoustic waves by an inhomogeneous fluid layer on a rigid plate in r 2 is considered the density is assumed to be unity in the media within the layer the sound speed is assumed to be an arbitrary bounded measurable function the problem is modelled by the reduced wave equation with variable wavenumber in the layer and a neumann condition on the plate to formulate the problem and prove uniqueness of solution a radiation condition appropriate for scattering by infinite rough surfaces is introduced a generalization of the rayleigh expansion condition for diffraction gratings with the help of the radiation condition the problem is reformulated as a system of two second kind integral equations over the layer and the plate under additional assumptions on the wavenumber in the layer uniqueness of solution is proved and the nonexistence of guided wave solutions of the homogeneous problem established general results on the solvability of systems of integral equations on unbounded domains are used to establish existence and continuous dependence in a weighted norm of the solution on the given data

a fast two grid and finite section method for a class of integral equations on the real line with application to an acoustic scattering problem in the half plane we consider the numerical treatment of second kind integral equations on the real line of the form phi s psi s int infty infty k s t z t phi t dt s in mathbb r abbreviated k z in which l 1 z l and bc the space of bounded continuous functions on are assumed known and bc is to be determined we first derive sharp error estimates for the finite section approximation reducing the range of integration to a a via bounds on i k z 1 as an operator on spaces of weighted continuous functions numerical solution by a simple discrete collocation method on a uniform grid on is then analysed in the case whenz is compactly supported this leads to a coefficient matrix which allows a rapid matrix vector multiply via the fft to utilise this possibility we propose a modified two grid iteration a feature of which is that the coarse grid matrix is approximated by a banded matrix and analyse convergence and computational cost in cases wherez is not compactly supported a combined finite section and two grid algorithm can be applied and we extend the analysis to this case as an application we consider acoustic scattering in the half plane with a robin or impedance boundary condition which we formulate as a boundary integral equation of the class studied our final result is that ifz related to the boundary impedance in the application takes values in an appropriate compact subsetq of the complex plane then the difference between s and its finite section approximation computed numerically using the iterative scheme proposed is c 1 khlog 1 kh 1 1 2 ka 1 2 in the interval a a 1 forkh sufficiently small wherek is the wavenumber andh the grid spacing moreover this numerical approximation can be computed in c 2 n logn operations wheren 2a h is the number of degrees of freedom the values of the constantsc 1 andc 2 depend only on the setq and not on the wavenumberk or the support ofz

on the solvability of a class of second kind integral equations on unbounded domains we consider integral equations of the form x x k x y z y y dy in operator form kz where is some subset ofrn n 1 the functionsk z and are assumed known withz l and y the space of bounded continuous functions on the function yis to be determined the class of domains and kernelskconsidered includes the case rnandk x y x y with l1 rn in which case ifzis the characteristic function of some setg the integral equation is one of wiener hopf type the main theorems proved using arguments derived from collectively compact operator theory are conditions on a setw l which ensure that ifi kzis injective for allz wtheni kzis also surjective and moreover the inverse operators i kz 1onyare bounded uniformly inz these general theorems are used to recover classical results on wiener hopf integral operators of21and19 and generalisations of these results and are applied to analyse the lippmann schwinger integral equation

solvability and spectral properties of integral equations on the real line i weighted spaces of continuous functions abstract we consider in this paper the solvability of linear integral equations on the real line in operator form k where c and k is an integral operator we impose conditions on the kernel k of k which ensure that k is bounded as an operator on x bc r let xa denote the weighted space x a x s o s a as s our first result is that if additionally k s t s t with l 1 r and s o s b as s for some b 1 then the spectrum of k is the same on xa as on x for 0 k k k w which ensure that if 0 and kk has only the trivial solution in x for all k w then for 0 a b k has exactly one solution xa for every k w and xa these conditions ensure further that k 1 x a x a is bounded uniformly in k w for 0 a b as a particular application we consider the case when the kernel takes the form k s t s t z t with l 1 r z l r and s o s b as s for some b 1 as an example where kernels of this latter form occur we discuss a boundary integral equation formulation of an impedance boundary value problem for the helmholtz equation in a half plane

coupling of finite element and boundary integral methods for electromagnetic scattering in a two layered medium consider a time harmonic electromagnetic plane wave incident on an inhomogeneity embedded in a two layered medium in this paper a method of coupling of finite element and boundary integral equation methods is presented for the solutions of electromagnetic scattering in both transverse electric and magnetic polarization cases the well posedness of the continuous and discrete problems as well as optimal error estimates for the coupled variational approximations are obtained numerical results are included to illustrate the accuracy with the optimal convergence property of the proposed method and to show the wave features in a two layered medium

the mathematics of scattering by unbounded rough inhomogeneous layers in this paper we study via variational methods a boundary value problem for the helmholtz equation modelling scattering of time harmonic waves by a layer of spatially varying refractive index above an unbounded rough surface on which the field vanishes in particular in the 2d case with te polarization the boundary value problem models the scattering of time harmonic electromagnetic waves by an inhomogeneous conducting or dielectric layer above a perfectly conducting unbounded rough surface with the magnetic permeability a fixed positive constant in the medium via analysis of an equivalent variational formulation we show that this problem is well posed in two important cases when the frequency is small enough and when the medium in the layer has some energy absorption in this latter case we also establish exponential decay of the solution with depth in the layer an attractive feature is that all constants in our estimates are bounded by explicit functions of the index of refraction and the geometry of the scatterer

maxwell s equations in a periodic structure consider a diffraction of a beam of particles in r3 when the dielectric coefficient is a constant cl above a surface s and a constant e2 below a surface s and the magnetic permeability is constant throughout rii s is assumed to be periodic in the xl direction and of the form x1 f1 s x3 f3 s x2 arbitrary we prove that there exists a unique solution to the time harmonic maxwell equations in r 3 having the form of refracted waves for x3 1 and of transmitted waves for x3 1 if and only if there exists a unique solution to a certain system of two coupled fredholm equations thus in particular for all the e s except for a discrete number there exists a unique solution to the maxwell equations introduction in this paper we consider the maxwell equations for time harmonic solutions in the entire space r3 with piecewise constant dielectric coefficient having jump across a periodic surface the magnetic permeability u is assumed to be constant whereas the dielectric coefficient e is given by e 8 above a surface s x3 f xl and e e2 below the surface s e and 82 are different constants if s is a half space x3 0 then the solution eo ho can be computed explicitly we assume in this paper that s is periodic i e f xl l f xl vx1 e r l 0 we wish to find a solution e h such that e eo and h ho are superpositions of transmitted waves 0 1 in x3 a where a maxlfl in 1 7 we assume that f e c2 and we reduce the solution of the maxwell equations to a fredholm system of four integral equations in 8 we reduce it further to a fredholm system of two integral equations thus for all but a discrete sequence of values of the physical parameters there exists a unique solution to the integral equations yielding a solution of the maxwell equations received by the editors january 25 1989 1980 mathematics subject classification 1985 revision primary 78a10 78a45 secondary 35p25 47a40

stable and efficient evaluation of periodized green s functions for the helmholtz equation at high frequencies a difficulty that arises in the context of infinite d periodic rough surface scattering relates to the effective numerical evaluation of the corresponding quasi periodic green function g q p due to its relevance in a variety of applications this problem has generated significant interest over the last 40 years and a variety of numerical methods have been devised for this purpose none of these methods to evaluate g q p however were designed for high frequency calculations as a result in this regime these methods become prohibitively expensive and or unstable here we present a novel scheme that can be shown to outperform every alternative numerical evaluation procedure and is especially effective for high frequency calculations our new algorithm is based on the use of some exact integrals that arise on judicious manipulation of the integral representation of g q p and which reduce the overall problem to that of evaluation of a sequence of simpler integrals that can be effectively handled by standard quadrature formulas we include a variety of numerical results that confirm that indeed our algorithm compares favorably with alternative methods

the numerical solution of scattering by infinite rough interfaces based on the integral equation method in this paper we describe a nystrom integration method for the integral operator t which is the normal derivative of the double layer potential arising in problems of two dimensional acoustic scattering by infinite rough interfaces the hypersingular kernel and unbounded integral interval of t are the key difficulties by using a mollifier we separately deal with these two difficulties and propose its nystrom integration method furthermore we establish convergence of the method finally we apply the method to the scattering problem by infinite rough interfaces and carry out some numerical experiments to show the validity

mathematical analysis of the junction of two acoustic open waveguides the present paper concerns the scattering of a time harmonic acoustic wave by the junction of two open uniform waveguides where the junction is limited to a bounded region we consider a two dimensional problem for which wave propagation is described by the scalar helmholtz equation the main difficulty in the modeling of the scattering problem lies in the choice of conditions which characterize the outgoing behavior of a scattered wave we use here modal radiation conditions which extend the classical conditions used for closed waveguides they are based on the generalized fourier transforms which diagonalize the transverse contributions of the helmholtz operator on both sides of the junction we prove the existence and uniqueness of the solution which seems to be the first result in this context the originality lies in the proof of uniqueness which combines a natural property related to energy fluxes with an argument of analyticity with respect to the generalized fourier variable

the time harmonic maxwell equations in a doubly periodic structure abstract consider the diffraction of a beam of particles in r 3 when the dielectric coefficient is a constant e1 above a surface s and a different constant e2 below s and the magnetic permeability is constant throughout r 3 s is assumed to be a doubly periodic surface say z f x y with f x ml1 y nl2 f x y for integers m n the existence and uniqueness of a solution satisfying a radiation condition at infinity is reduced to a system of fredholm equations thus for all but a discrete set of e s there exists a unique solution

the domain derivative in rough surface scattering and rigorous estimates for first order perturbation theory we investigate frechet differentiability of the scattered field with respect to variation in the boundary in the case of time harmonic acoustic scattering by an unbounded sound soft one dimensional rough surface we rigorously prove the differentiability of the scattered field and derive a characterization of the frechet derivative as the solution to a dirichlet boundary value problem as an application of these results we give rigorous error estimates for first order perturbation theory justifying small perturbation methods that have a long history in the engineering literature as an application of our rigorous estimates we show that a plane acoustic wave incident on a sound soft rough surface can produce an unbounded scattered field

boundary integral equation methods for the scattering problem by an unbounded sound soft rough surface with tapered wave incidence in this paper we consider the scattering problem of tapered acoustic wave by an unbounded sound soft surface the scattering problem is modeled as a boundary value problem governed by the helmholtz equation with dirichlet boundary condition although the tapered wave is often introduced to realize asymptotic truncation for unbounded rough surface the standard helmholtz integral equations which derive for the scattering of plane waves by an arbitrary bounded obstacle are often used to generate benchmark numerical solutions different from the scattering of plane waves by an arbitrary bounded obstacle we use the angular spectrum representation radiation condition to replace the sommerfeld radiation condition and derive a boundary integral equation for studying the scattering problem then we study the integral equation by the truncation method whereby the integral equation posed on an unbounded region is approximated by an integral equation on a bounded region some properties of the integral equation in an energy space with weights are proved then the collocation method is used to solve the integral equation on a bounded region and its convergence is also obtained a rigorous derivation of bie is presented for studying the scattering problem some properties of the integral equation in an energy space with weights are proved the convergence of the numerical method is also obtained

finite element approximation of time harmonic waves in periodic structures consider the diffraction of a time harmonic wave incident on a periodic surface of some inhomogeneous material it is shown that the scattering diffraction problem may be modeled by a helmholtz equation with transparent boundary conditions the diffraction problem may be solved by a finite element method in this paper optimal error estimates for the finite element method are established the error estimates are also established when the truncation of the nonlocal transparent boundary operators takes place