阻抗型边界条件下弹性表面波的长期方程:线性代数的视角

IF 2.1 3区 物理与天体物理 Q2 ACOUSTICS
Fabio Vallejo
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引用次数: 0

摘要

阻抗边界条件下的弹性表面波在许多问题中都引起了人们的极大兴趣。然而,对提供表面波速度的相关长期方程的分析由于其复杂性而局限于具体情况。本工作提出了一种基于线性代数工具的替代方法,用于处理受阻抗型边界条件约束的各向同性弹性半空间中表面波的长期方程。我们的分析表明,相关的长期方程在包含实轴的上复半平面上不会消失。这暗示了问题的适当性。有趣的是,Godoy等人(2012)提出的全阻抗边界条件是一种极限情况。引入一种近似技术,将所考虑问题的分析扩展到Godoy阻抗边界条件。结果表明,对于任意非零阻抗参数值,具有完全Godoy阻抗边界条件的长期方程在实轴外不会消失。这是偏微分方程边值问题的适定性的一个关键性质,因此对于解释表面波传播的模型至关重要。然而,它只对后一类边界条件的特殊情况进行了验证,包括无应力情况。在一个特殊情况下,证明了具有复速度值的表面波的存在性。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
The secular equation for elastic surface waves under boundary conditions of impedance type: A perspective from linear algebra
Elastic surface waves under impedance boundary conditions are of great interest in a wide range of problems. However, the analysis of the associated secular equation, which provides the speed of the surface wave, is limited to specific cases due to its complicated nature. This work presents an alternative method, based on linear algebra tools, to deal with the secular equation for surface waves in an isotropic elastic half-space subjected to boundary conditions of impedance type. Our analysis shows that the associated secular equation does not vanish in the upper complex half-plane including the real axis. This implies the well-posedness of the problem. Interestingly, the full impedance boundary conditions proposed by Godoy et al. (2012) arise as a limit case. An approximation technique is introduced to extend the analysis from the considered problem to Godoy’s impedance boundary conditions. As a result, it is showed that the secular equation with full Godoy’s impedance boundary conditions does not vanish outside the real axis for arbitrary non-zero impedance parameter values. This is a crucial property for the well-posedness of the boundary value problem of partial differential equations, and thus crucial for the model to explain surface wave propagation. However, it has been verified only for particular cases of the latter class of boundary conditions including the stress-free case. The existence of a surface wave with a complex valued velocity is proved for a particular case.
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来源期刊
Wave Motion
Wave Motion 物理-力学
CiteScore
4.10
自引率
8.30%
发文量
118
审稿时长
3 months
期刊介绍: Wave Motion is devoted to the cross fertilization of ideas, and to stimulating interaction between workers in various research areas in which wave propagation phenomena play a dominant role. The description and analysis of wave propagation phenomena provides a unifying thread connecting diverse areas of engineering and the physical sciences such as acoustics, optics, geophysics, seismology, electromagnetic theory, solid and fluid mechanics. The journal publishes papers on analytical, numerical and experimental methods. Papers that address fundamentally new topics in wave phenomena or develop wave propagation methods for solving direct and inverse problems are of interest to the journal.
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