Three-dimensional logarithmic resonances in a homogeneous elastic wave guide

We treat analytically the initial boundary-value linear stability problem for three-dimensional (3-D) small localized disturbances in a homogeneous elastic 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 while its lower surface is rigidly attached to a half-space. The outcome of the analysis is an extension of the results of Brevdo (1996, 1998a) concerning the neutral exponential stability and existence of resonances in a two-dimensional (2-D) wave guide to the 3-D case. The dispersion relation function in the 3-D case is shown to be equal to D(k2+l2,ω), where D(k,ω) 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 directions x and y, and ω is a frequency. Hence, any 3-D wave guide is neutrally stable. For studying asymptotic responses in time of 3-D wave guides to 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 relevant for modelling in seismology possesses a countable unbounded set of resonant frequencies that coincide with the set of resonant frequencies in the 2-D case. Sources with resonant frequencies, producing in the 2-D case responses growing in time like t, in the 3-D case produce responses that grow either like lnt or like t. The result provides a further support to the hypothesis by Brevdo concerning a possible resonant triggering mechanism of certain earthquakes, namely through localized low amplitude oscillatory forcings at resonant frequencies.