Resonance phenomena in compound cylindrical waveguides
Günter Heinzelmann, Peter Christian Werner · Mathematical Methods in the Applied Sciences · 2006
Abstract We study the large time asymptotics of the solutions u(x,t) of the Dirichlet and the Neumann initial boundary value problem for the wave equation with time‐harmonic right‐hand side in domains Ω which are composed of a finite number of disjoint half‐cylinders Ω1,…,Ωr with cross‐sections Ω′1,…,Ω′r and a bounded part (‘compound cylindrical waveguides’). We show that resonances of orders t and t1/2 may occur at a finite or countable discrete set of frequencies ω, while u(x,t) is bounded as t→∞ for the remaining frequencies. A resonance of order t occurs at ω if and only if ω2 is an eigenvalue of the Laplacian −Δ in Ω with regard to the given boundary condition u=0 or ∂u/∂n=0, respectively. A resonance of order t1/2 occurs at ω if and only if (i) ω2 is an eigenvalue of at least one of the Laplacians for the cross‐sections Ω′1,…,Ωr′ with regard to the respective boundary condition and (ii) the respective homogeneous boundary value problem for the reduced wave equation ΔU+ω2U=0 in Ω has non‐trivial solutions with suitable asymptotic properties as | x | →∞ (‘standing waves’). Copyright © 2006 John Wiley & Sons, Ltd.