High–order resonances in a homogeneous waveguide

Abstract

Recently, we have shown (cf. Brevdo [7], [8]) that every homogeneous elastic waveguide is neutrally stable and possesses a countable set of temporally resonant frequencies \(\{ \omega_n , n\in {\Bbb N} \}.\) For each \(\omega_n\) in this set, the response of the waveguide to a spatially localised oscillatory forcing, with the time dependence \(e^{\displaystyle{-i \omega_n t}},\) grows in time at least as \(\sqrt{t},\) for \(t\to\infty.\) The growth \(\sqrt{t}\) occurs in the case of a low order resonance. In the present paper we show that, for a particular combination of the physical parameters, a high–order resonance occurs in a homogeneous waveguide for a certain frequency of oscillations. It produces a response that grows at least as \(t^{3/4}.\) Moreover, the set of physically relevant waveguides possessing high–order resonances is shown to be rather wide. The treatment is based on the asymptotic evaluation of the solution of the initial–value stability problem expressed as an inverse Laplace—Fourier integral. The results support the hypothesis in [8] that certain earthquakes can be caused by a sequence of events triggered by localised low amplitude oscillatory forcings at resonant frequencies.