One-Dimensional Wave-Transfer Functions of the Linear Visco-Elastic Multi-Layered Half-Space

Abstract

The most important problem in the earthquake response analysis of a structure is to suppose reasonable earthquake excitations depending on the seismicity and the dynamic characteristics of ground at the site of the structure. In this paper, as one of the basic studies related to the supposition of random earthquake excitations for the dynamic aseismic design of structures, the authors will deal with the analytical expressions of the one-dimensional wave-transfer functions of a general class of linear visco-elastic, horizontally multi-layered half-space to vertically incident plane waves at the bottom boundary of the layered media through the half-space, and also discuss the properties of such wave-transfer functions in the complex plane. Both the one-dimensional wave-transfer functions and the associate characteristic equation are expressed in the successive product forms involving some kind of symbolic operator, which are suitable for finding out the properties of those functions in the complex plane as well as for discussing the eigen-value problems of such layered media and also for carrying out the numerical calculation of the wave-transfer functions. For the usually encountered linear visco-elastic layered half-space including the purely elastic case, it is found that the singular points of the wave-transfer functions consist of a finite number of branch points and a denumerably infinite number of poles having positive immaginary parts, which are zeros of the characteristic equation. And also, it is found that the wave-transfer functions are finite in the neighbourhood of the branch points and vanish in exponential order at infinity as far as the inner points of the layered media are concerned. These properties of the one-dimensional wave-transfer functions may guarantee the validity of the residue theorem in estimating the impulsive responses as well as the variances and co-variances of the random responses of the linear visco-elastic multi-layered half-space.