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Scattering by a periodic tube in $ \newcommand{\real}{\mathbb{R}} \newcommand{\re}{{\rm Re}} \mathbf{\real^3}$ : part i. The limiting absorption principle

Kirsch, Andreas 1
1 Karlsruher Institut für Technologie (KIT)


Scattering of time-harmonic waves from periodic structures at some fixed real-valued wave number becomes analytically difficult whenever there arise surface waves: These non-zero solutions to the homogeneous scattering problem physically correspond to modes propagating along the periodic structure and clearly imply non-uniqueness of any solution to the scattering problem. In this paper, we consider a medium, described by a refractive index which is periodic along the axis of an infinite cylinder in R3 and constant outside of the cylinder. We prove that there is a so-called limiting absorption solution to the associated scattering problem. By definition, such a solution is the limit of a sequence of unique solutions for artificial complex-valued wave numbers tending to the above-mentioned real-valued wave number. By the standard one-dimensional Floquet–Bloch transform and the introduction of the exterior Dirichlet–Neumann map we first reduce the scattering problem to a class of periodic problems in a bounded cell, depending on the wave number k and the Bloch parameter α¬. We use a functional analytic singular perturbation result to study this problem in a neighborhood of a singular pair (k, α¬). ... mehr

Verlagsausgabe §
DOI: 10.5445/IR/1000098680
DOI: 10.1088/1361-6420/ab2e31
Zitationen: 7
Cover der Publikation
Zugehörige Institution(en) am KIT Institut für Angewandte und Numerische Mathematik (IANM)
Publikationstyp Zeitschriftenaufsatz
Publikationsmonat/-jahr 10.2019
Sprache Englisch
Identifikator ISSN: 0266-5611, 1361-6420
KITopen-ID: 1000098680
Erschienen in Inverse problems
Verlag Institute of Physics Publishing Ltd (IOP Publishing Ltd)
Band 35
Heft 10
Seiten Art. Nr.: 104004
Vorab online veröffentlicht am 09.09.2019
Schlagwörter Keywords: Helmholtz equation, limiting absorption principle, periodic wave guide
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