The secular equation for elastic surface waves under boundary conditions of impedance type: A perspective from linear algebra

Abstract

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.