Three-dimensional logarithmic resonances in a homogeneous elastic wave guide

We treat analytically the initial boundary-value linear stability problem f or three-dimensional (3-D) small localized disturbances in a homogeneous el astic wave guide by applying the Laplace transform in time and the Fourier transform in two orthogonal spatial directions. Motivated by seismological applications, we assume that the upper surface of the wave guide is free wh ile its lower surface is rigidly attached to a half-space. The outcome of t he analysis is an extension of the results of Brevdo (1996, 1998a) concerni ng the neutral exponential stability and existence of resonances in a two-d imensional (2-D) wave guide to the 3-D case. The dispersion relation functi on in the 3-D case is shown to be equal to D(root k(2) +l(2),omega), where D(k, omega) is the dispersion relation function of the same model in the 2- D case, k and l are wave numbers in two orthogonal horizontal spatial direc tions x and y, and omega is a frequency. Hence, any 3-D wave guide is neutr ally stable. For studying asymptotic responses in time of 3-D wave guides t o nearly harmonic in time sources we apply the mathematical formalism for 3 -D spatially amplifying waves. It is shown that every 3-D wave guide releva nt for modelling in seismology possesses a countable unbounded set of reson ant frequencies that coincide with the set of resonant frequencies in the 2 -D case. Sources with resonant frequencies, producing in the 2-D case respo nses growing in time like root t, in the 3-D case produce responses that gr ow either like Int or like root t. The result provides a further support to the hypothesis by Brevdo concerning a possible resonant triggering mechani sm of certain earthquakes, namely through localized low amplitude oscillato ry forcings at resonant frequencies. (C) 2000 Editions scientifiques et med icales Elsevier SAS.