Revisiting the problem of existence of surface Rayleigh waves with impedance boundary conditions

Abstract This paper considers the problem of surface waves in an isotropic elastic half-space endowed with impedance boundary conditions as first proposed by Godoy et al. (Wave Motion 49:585–594, 2012. https://doi.org/10.1016/j.wavemoti.2012.03.005 ). These conditions are controlled by two impedance parameters, where the standard stress-free boundary condition is retrieved for zero impedance. While the existence of a unique surface wave (called Rayleigh wave) is well-established for the standard stress-free boundary condition, the introduction of more general boundary conditions may lead to the absence of surface waves or even cause the PDE boundary value problem to become ill-posed. For the case of Godoy’s impedance boundary conditions, the problem of existence and uniqueness of a surface wave of Rayleigh type was investigated by means of the complex function method based on Cauchy-type integrals. However, this method is quite cumbersome and hard to apply. In this work, we present an alternative method based on elementary tools from calculus to deal with the problem. We consider a particular case, where both impedance parameters are non-zero and demonstrate the existence and uniqueness of the surface wave for all material and boundary parameter values. Numerical examples are presented to illustrate the effect of the impedance parameter on the speed of the surface wave.