A radiation condition arising from the limiting absorption principle for a closed full‐ or half‐waveguide problem

Abstract

new approach, which is, in our opinion, simpler and opens the possibility to treat also more general cases, such as, the scattering by waveguides in R 2 (ie, open waveguide problems, see Kirsch and Lechleiter 4 ) or the scattering by infinite cylinders in R 3 with periodic refractive indices (see Kirsch 10 ). The basic difference to all of the traditional approaches (known to the authors) is that we do not use at all the spectrum of the Floquet-Bloch transformed differential operator and its dependence on the Bloch parameter. Instead, we consider the transformed boundary value problem as a family of problems that depends singularly on 2 parameters, namely, the imaginary part ε ≥ 0 of the wave number k ¼k þ iε and the Bloch parameter α in neighbourhoods of ðε; αÞ ¼ ð0;αÞ whereα is one of the "exceptional values" or "propagative wave numbers" (in the sense of Fliss and Joly 3 ). We prove a general functional analytic theorem about the structure of the solutions of certain families of linear problems depending on 2 parameters ðε; αÞ≠ð0;αÞ and can explicitly compute the limits in the inverse Floquet-Bloch transform when ε→0. This proves the limiting absorption principle and provides an explicit form of the solution as a sum of propagating modes and a decaying part. This motivates the definition of a radiation condition, which is equivalent to the well-known formulations as, eg, in Fliss and Joly 3 (see also Joly et al 7 ) but takes a slightly different form. In our approach, we distinguish between the right-going and left-going propagation modes by the sign of the eigenvalues of an eigenvalue problem in the finite dimensional space spanned by all propagating modes corresponding to a fixed propagative wave number.Having developed the radiation condition, we can "forget" the limiting absorption principle and take the radiation condition as given. Then we are able to prove uniqueness and also existence of a solution by a direct proof, which is independent of the limiting absorption principle. The proof of existence is based again on the Floquet-Bloch transform and an abstract singular perturbation result of Colton and Kress. 14 The methods we apply are all well known and, in principle, simple enough to extend our analysis to more involved scattering problems in linear elasticity or electromagnetics (which, however, has to be done). To reduce technical difficulties, in this paper, we are however merely considering the simple Helmholtz equation in R 2 .To briefly comment on this paper's structure, the following Section 2 discusses the Floquet-Bloch transform and uses it to transform the given boundary value problem to a family of operator equations of the form ũ ε; α −K ε; α ũ ε; α ¼f ε;α in a Sobolev space depending on the parameters ε = Im k and the Bloch parameter α. In Section 3, we prove the limiting absorption principle by applying an abstract functional analytic result, while in Section 4, we formulate our radiation condition and prove uniqueness and existence in a direct way, that is, without using the limiting ab...

A High Order Numerical Method for Scattering from Locally Perturbed Periodic Surfaces

“…, m j . In [19], a positive wave number k such that this condition is satisfied is also called regular. With above notations, we recall the radiation condition which is equivalent, but more explicit, compared to that defined in Definition 1.…”

“…Then the limit of solution when ε → 0 + is set to be the physical solution, which is called an LAP solution in this paper. From [10,19,40], the radiation condition for LAP solutions in periodic waveguides have been described in different forms, and the forms are actually equivalent. Definition 1 (Radiation Condition).…”

“…Remark 3. For perfectly periodic waveguide, an important result is that Assumption 2 always holds (see [10,19]). However, this case may happen for a locally perturbed periodic waveguide.…”

Numerical methods for scattering problems in periodic waveguides

“…In this paper, the limit is called the LAP solution. It has been proved that LAP solutions exist for planar waveguides filled up with periodic material, and we refer to [1][2][3][4] for different proofs.…”

“…Recently, the Floquet-Bloch transform has been applied to both theoretical analysis and numerical simulation for scattering problems in periodic structures. We refer to [9,10,19] for scattering problems with (perturbed) periodic media, to [11][12][13][14] for problems with periodic surfaces, and to [2][3][4] for periodic waveguides. From the Floquet-Bloch theory, the unique solution of the periodic waveguide problem with absorption is written as a contour integral on the unit circle, where the integrand is a family of quasi-periodic solutions depending analytically on the quasi-periodicities.…”