Axisymmetric resonant disturbances in a homogeneous wave guide

The initial boundary-value linear stability problem for small localised axi symmetric disturbances in a homogeneous elastic wave guide, with the free u pper surface and the lower surface being rigidly attached to a half-space, is formally solved by applying the Laplace transform in time and the Hankel transforms of zero and first orders in space. An asymptotic evaluation of the solution, expressed as a sum of inverse Laplace-Hankel integrals, is ca rried out by using the approach of the mathematical formalism of absolute a nd convective instabilities. It is shown that the dispersion-relation funct ion of the problem D-0(kappa, omega), where the Hankel parameter kappa is s ubstituted by a wave number (and the Fourier parameter) k, coincides with t he dispersion-relation function D-0(kappa, omega) for two-dimensional (2-D) disturbances in a homogeneous wave guide, where omega is the frequency (an d the Laplace parameter) in both cases. An analysis for localised 2-D distu rbances in a homogeneous wave guide is then applied. We obtain asymptotic e xpressions for wave packets, triggered by axisymmetric perturbations locali sed in space and finite in time, as well as for responses to axisymmetric s ources localised in space, with the time dependence satisfying e(-i omega 0 t) + O(e(-epsilon t)) for t --> infinity, where Im omega(0) = 0, epsilon > 0, and t denotes time, i.e. for signalling with frequency omega(0). We demo nstrate that, for certain combinations of physical parameters, axisymmetric wave packets with an algebraic temporal decay and axisymmetric signalling with an algebraic temporal growth, as root t, i.e., axisymmetric temporal r esonances, are present in a neutrally stable homogeneous wave guide. The se t of physically relevant wave guides having axisymmetric resonances is show n to be fairly wide. Furthermore, since an axisymmetric part of any source is L-2-orthogonal to its non-axisymmetric part, a 3-D signalling with a non -vanishing axisymmetric component at an axisymmetric resonant frequency wil l generally grow algebraically in time. These results support our hypothesi s concerning a possible resonant triggering mechanism of certain earthquake s, see Brevdo, 1998, J. Elasticity, 49, 201-237. (C) Elsevier, Paris.