  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).
