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

TL;DR

This paper introduces a linear algebra-based method to analyze the secular equation for elastic surface waves under complex impedance boundary conditions, ensuring well-posedness and broadening understanding of wave propagation in elastic media.

Contribution

It presents a novel linear algebra approach to handle complex impedance boundary conditions for elastic surface waves, extending analysis to non-standard cases including Godoy's boundary conditions.

Findings

Secular equation does not vanish in the upper complex half-plane.

Full impedance boundary conditions are a particular limit case.

The secular equation remains non-zero outside the real axis, ensuring well-posedness.

Abstract

The study of elastic surface waves under impedance boundary conditions has become an intensive field of research due to their potential to model a wide range of problems. However, even when the secular equation, which provides the speed of the surface wave, can be explicitly derived, the analysis is limited to specific cases due to its cumbersome final expression. In this work, we present 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 non-standard boundary conditions of impedance type. They are defined by proportional relationships between both the stress and velocity components at the surface, with complex proportional ratios. Our analysis shows that the associated secular equation does not vanish in the upper complex half-plane including the real axis. Interestingly, the full impedance boundary conditions proposed by Godoy et al. [Wave Motion 49 (2012), 585-594] arise as a particular limit case. An approximation technique is introduced, in order to extend the analysis from the original problem to Godoy's impedance boundary conditions. As a result, it is shows that the secular equation with full Godoy's impedance boundary condition does not vanish outside the real axis. 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. Due to the cumbersome secular equation, this property has been verified only for particular cases of the impedance boundary condition, namely the stress-free boundary condition (zero impedance) and when either one of the impedance parameter is set to zero (normal and tangential impedance cases).