On the instability of resonances in parallel‐plane waveguides

It has been observed 13 that the propagation of acoustic waves in the region Ω 0 = ℝ 2 × (0, 1), which are generated by a time‐harmonic force density with compact support, leads to logarithmic resonances at the frequencies ω = 1, 2,… As we have shown 9 in the case of Dirichlet's boundary condition U = 0 on ∂Ω, the resonance at the smallest frequency ω = 1 is unstable and can be removed by a suitable small perturbation of the region. This paper contains similar instability results for all resonance frequencies ω = 1, 2,… under more restrictive assumptions on the perturbations Ω of Ω 0 . By using integral equation methods, we prove that absence of admissible standing waves in the sense of Reference 7 implies the validity of the principle of limit amplitude for every frequency ω ≥ 0 in the region Ω =Ω 0 − B , where B is a smooth bounded domain with B̄⊂Ω 0 . In particular, it follows from Reference 7 in the case of Dirichlet's boundary condition that the principle of limit amplitude holds for every frequency ω ≥ 0 if n · x ′ ⩽ 0 on ∂ B , where x ′ = ( x 1 , x 2 , 0) and n is the normal unit vector pointing into the interior B of ∂ B . In the case of Neumann's boundary condition, the logarithmic resonance at ω = 0 is stable under the perturbations considered in this paper. The asymptotic behaviour of the solution for arbitary local perturbations of Ω 0 will be discussed in a subsequent paper.