Eigenvector Expansions of Green's Dyads with Applications to Geophysical Theory
Eigenvector Expansions of Green's Dyads with Applications to Geophysical Theory
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DOI：
10.1111/j.1365-246x.1968.tb00234.x
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发表时间：
1968-11
影响因子：
2.8
通讯作者：
A. Ben-menahem;Sarva Jit Singh
中科院分区：
文献类型：
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作者：
A. Ben-menahem;Sarva Jit Singh
小结 通过使用相应格林张量的特征向量展开，可以统一并简化各种向量可分离体系中边值问题的处理。特别是，该方法对于球面和柱面极坐标中的矢量拉普拉斯、泊松、亥姆霍兹和纳维方程非常有用。给出了几个例子；其中，需要特别提及弹性球体对剪切位错的动态和静态响应的评估。研究发现，与一次勒让德多项式（l=1）相关的球体静态位移场存在一些问题。在这种情况下，必须考虑附加条件，即球体绕其中心的角动量为零并且球体的质心不发生位移。建议在地球物理理论中采用特征向量展开式。其固有的优雅和紧凑性使其成为构建理论地球模型的绝佳工具。
Summary The treatment of boundary value problems in various vector-separable regimes is unified and facilitated with the use of the eigenvector expansion of their corresponding Green's tensors. In particular, the method is useful for the vector Laplace, Poisson, Helmholtz and Navier equations in spherical and cylindrical polars. Several examples are given; among them, an evaluation of the dynamic and static response of an elastic sphere to shear dislocations requires particular mention. It is found that the static displacement field in a sphere which is associated with the Legendre polynomial of the first degree (l= 1) poses some problems. In such a case, one must incorporate additional conditions, namely, that the angular momentum of the sphere about its centre is zero and that the centre of mass of the sphere is not displaced. It is recommended that eigenvector expansions be adopted in geophysical theory. Its inherent elegance and compactness make it an excellent tool for the construction of theoretical Earth-models.