Propagator matrices in elastic wave and vibration problems
Propagator matrices in elastic wave and vibration problems
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DOI：
10.1190/1.1439771
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发表时间：
1966-04
影响因子：
0.9
通讯作者：
F. Gilbert;G. Backus
中科院分区：
文献类型：
--
作者：
F. Gilbert;G. Backus
分层介质中弹性波传播研究中最常见的边值问题可以用有限个变系数线性一阶常微分方程来表达。沃尔泰拉（1887）指出，这类方程组的解可以方便地用系数矩阵的乘积积分或传播子表示，本文综述了传播子的一些较好的性质及其数值计算方法。当色散关系是积分矩阵的某个m阶子式时，可以处理m阶子传播子，使色散关系是m阶子式积分矩阵的单个元素。本文给出了各向同性和横向各向同性介质中SH波和P-SV波的频散方程。此外，还给出了P-SV波的第二次传播子方程。利用平均系数法，利用Cayley-Hamilton定理和Lagrange-Sylvester插值公式，得到了传播子的矩阵多项式逼近。
The boundary value problems most frequently encountered in studies of elastic wave propagation in stratified media can be formulated in terms of a finite number of linear, first order and ordinary differential equations with variable coefficients. Volterra (1887) has shown that solutions to such a system of equations are conveniently represented by the product integral, or propagator, of the matrix of coefficients.In this paper we summarize some of the better known properties of propagators plus numerica methods for their computation. When the dispersion relation is somemth order minor of the integral matrix it is possible to deal withmth minor propagators so that the dispersion relation is a single element of themth minor integral matrix. In this way one of the major sources of loss of numerical accuracy in computing the dispersion relation is avoided.Propagator equations forSH and forP-SV waves are given for both isotropic and transversely isotropic media. In addition, the second minor propagator equations forP-SV waves are given. Matrix polynomial approximations to the propagators, obtained from the method of mean coefficients by the Cayley-Hamilton theorem and the Lagrange-Sylvester, interpolation formula, are derived.