High-order resonances in a homogeneous waveguide

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 is an element of N}. For each omega(n) in this set, the response of the waveguide to a spatially localised oscillato ry forcing, with the time dependence e(-i omega nt), grows in time at least as root t, for t --> infinity. The growth root t occurs in the case of a l ow order resonance. In the present paper we show that, for a particular com bination of the physical parameters, a high-order resonance occurs in a hom ogeneous waveguide for a certain frequency of oscillations. It produces a r esponse that grows at least as t(3/4). Moreover, the set of physically rele vant waveguides possessing high-order resonances is shown to be rather wide . The treatment is based on the asymptotic evaluation of the solution of th e initial-value stability problem expressed as an inverse Laplace-Fourier i ntegral. 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.