Normal modes of a laterally heterogeneous body: A one-dimensional example
Normal modes of a laterally heterogeneous body: A one-dimensional example
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横向异质体的简正模态：一维示例
DOI：
10.1785/bssa0680010103
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发表时间：
1978
期刊：
影响因子：
--
通讯作者：
S. Stein
中科院分区：
文献类型：
--
作者：
R. Geller;S. Stein
各种方法，包括一阶和二阶摄动理论和变分方法已被提出来计算横向非均匀地球的正常模式。在本文中，我们测试所有这三种方法的一个简单的一维的例子，其精确的解决方案是可用的：一个最初均匀的“字符串”，其中的密度和刚度增加一半，并在其他减少等量。结果发现，一阶微扰理论（通常应用于地震学）只产生具有平均弹性性质的弦的本征值和本征函数;二阶微扰理论更糟，因为本征函数被假设为原始本征函数加上小的校正项，但实际上可能几乎完全不同。
变分法（Rayleigh-Ritz），使用未扰动模式作为试函数，成功地给出正确的本征值和本征函数，即使是高阶数的模式。对于示例问题，只有变分解正确地产生点力激励的瞬态解，包括由不连续性反射和透射的波的正确振幅。我们的研究结果表明，但并不证明，变分法可能是最合适的方法来寻找正常模式的横向非均匀地球模型，特别是如果瞬态解决方案是必要的。
Various methods, including first- and second-order perturbation theory and variational methods have been proposed for calculating the normal modes of a laterally heterogeneous earth. In this paper, we test all three of these methods for a simple one-dimensional example for which the exact solution is available: an initially homogeneous “string” in which the density and stiffness are increased in one half and decreased in the other by equal amounts. It is found that first-order perturbation theory (as commonly applied in seismology) yields only the eigenvalues and eigenfunctions for a string with the average elastic properties; second-order perturbation theory is worse, because the eigenfunction is assumed to be the original eigenfunction plus small correction terms, but actually may be almost completely different.
The variational method (Rayleigh-Ritz), using the unperturbed modes as trial functions, succeeds in giving correct eigenvalues and eigenfunctions even for modes of high-order number. For the example problem only the variational solution correctly yields the transient solution for excitation by a point force, including correct amplitudes for waves reflected by and transmitted through the discontinuity. Our result suggests but does not demonstrate, that the variational method may be the most appropriate method for finding the normal modes of a laterally heterogeneous earth model, particularly if the transient solution is desired.