Resonance Phenomena in Local Perturbations of Parallel-plane Waveguides

We study initial and boundary value problems for the wave equation ∂ t 2 u - Δu = fe -iωt with Dirichlet or Neumann boundary data in smooth domains Ω, which coincide with Ω o = R 2 x (0,1) outside a sufficiently large sphere. The concept of a standing wave, introduced in [7], seems to be of special relevance. A standing wave of frequency ω is defined as a non-trivial solution U of the equation ΔU + ω 2 U = 0 in Ω, which satisfies on ∂Ω the prescribed boundary condition U = 0 or ∂U/∂n = 0, respectively, and a suitable condition at infinity. For instance, U(x) = sin πkx 3 is a standing wave of frequency πk in the unperturbed domain Ω o with Dirichlet boundary data. As shown in a series of joint papers with K. Morgenrother, u(x,t) is bounded as t → ∞ if Ω does not admit standing waves of frequency ω. The main purpose of the present paper is the proof of the converse statement: If standing waves of frequency ω exist in Ω, then u(x,t) is unbounded as t → ∞ for suitably chosen f ∈ C 0 ∞(Ω). Thus the appearance of resonances is closely related to the presence of standing waves. The leading term of the asymptotic expansion of u(x, t) as t → ∞ will be specified. In particular, it turns out that the resonance rate is either t or In t. The rate t appears if and only if ω 2 is an eigenvalue of the spatial operator, while the rate In t can only occur if ω = πk. In the case of Neumann boundary data, U = 1 is a standing wave of frequency 0 for every local perturbation Ω of Ω 0 . The corresponding resonance has already been studied in [21].