Asymptotic analysis of SH-wave modes in geologic resonators (sedimentary basins) of non-separable geometry

Abstract

The problem investigated in this paper is that of estimating the normal modes of oscillation (resonant frequencies and wave functions) of scalar waves in a two-dimensional enclosure or resonator. Because the study is aimed to applications in geologic (seismic) situations, we consider SH-waves and wave resonators limited above by a flat reflector - the earth's free surface - and below by a curved one. The curved reflector is smooth, but otherwise of arbitrary shape. Commonly occuring sediment-filled valleys or sedimentary basins with such configuration act as seismic wave resonators to incoming earthquake waves. Determining the eigenmodes of basins of arbitrary geometry is a difficult problem in seismic wave propagation; even to simulate the dynamic response of basin models of simple non-separable geometry often requires the use of sophisticated numeric codes and high-performance computers. The accurate determination of the eigenfrequencies is important because in practice aby buildings or engineered structures with natural frequencies of oscillation close or equal to those of the resonant basin, may themselves resonate with the ground motion. This coincidental match of oscillation frequencies, known as “double resonance” is potentially very destructive, and in most cases leads to the total collapse of the building. It thus becomes important to clearly understand the effects of realistic basin geometries on seismic waves, and especially to accurately predict the resulting predominant frequencies of ground motion. We simplify the calculation of ground motions through asymptotic and numeric schemes that can be programmed on a personal computer. For this purpose we use analytic approximations based upon semiclassical quantization; the so-called EBK (Einstein-Brillouin-Keller) method for estimating the eigenmodes of non-linearly coupled mechanical oscillators. The EBK method solves for the eigenvalues (resonant frequencies) by imposing resonance conditions on the accumulated phase of the trapped waves (seismic ray trajectories) over topologically independent closed paths in phase space. To obtain the wave functions associated with the resonant frequencies, we propose a class of equations of motion whose asymptotic solutions are described by Hermite polynomials with an argument that is itself a function of the valley's geometry. The accuracy of the resonant frequencies and wave functions thus obtained is demonstrated by comparisons with finite element results. The approach can be easily applied to resonating alluvial valleys with an upper surface free of tractopm and a lower boundary that can be assumed rigid.