Mode matching methods for spectral and scattering problems

Abstract

We present several applications of mode matching methods in spectral and scattering problems. First, we consider the eigenvalue problem for the Dirichlet Laplacian in a finite cylindrical domain that is split into two subdomains by a "perforated" barrier. We prove that the first eigenfunction is localized in the larger subdomain, i.e., its L 2 norm in the smaller subdomain can be made arbitrarily small by setting the diameter of the "holes" in the barrier small enough. This result extends the well known localization of Laplacian eigenfunctions in dumbbell domains. We also discuss an extension to noncylindrical domains with radial symmetry. Second, we study a scattering problem in an infinite cylindrical domain with two identical perforated barriers. If the holes are small, there exists a low frequency at which an incident wave is fully transmitted through both barriers. This result is counter-intuitive as a single barrier with the same holes would fully reflect incident waves with low frequences.This domain can be naturally decomposed into two rectangular subdomains Ω 1 = (−a 1 , 0) × (0, h 1 ) and Ω 2 = (0, a 2 ) × (0, h 2 ). Without loss of generality, we assume h 1 ≥ h 2 . For each of these subdomains, one can explicitly write a general solution of the equation in (1.1) due to a separation of variables in perpendicular directions x

Resonance scattering in a waveguide with identical thick barriers

“…We study wave propagation through the waveguide Q 0 when the barriers are closing, i.e., the opening part of the barriers, Γ = Ω\D, is vanishing. As a similar problem for infinitely thin barriers was studied in [18], the main focus and novelty of the present paper is a finite thickness w of barriers. We consider the stationary wave equation…”

Design of a mode converter using thin resonant ligaments

“…The decomposition with the Neumann and Dirichlet problems would be the same and the asymptotic procedure would be very similar. This is an interest of the method proposed here compared for example to the technique of [9,10] based on decomposition in Fourier series or the one of [28,26] relying on integral equations with an explicit kernel, we do not need separation of variables in Π ± . But as already announced in Remarks 6.1, 6.2, for a generic domain, we can not position the ligaments to get (17).…”

Abnormal acoustic transmission in a waveguide with perforated screens

“…More precisely, for certain choices of L 0 , L , L in L ε := L 0 + εL + ε 2 L and for a certain condition (31) on the shape of the holes, we can obtain lim ε→0 R ε = 0 and lim ε→0 T ε = T 0 with |T 0 | = 1 (see Section 5). Similar problems have been considered in [1,3] but with only one hole and with Dirichlet boundary conditions. In [3], the authors work with decompositions in Fourier series which are hard to generalize.…”