Point spectra and normal modes of the Rayleigh loaded string with damping

We describe the point spectra of some dissipative version of the celebrated “Rayleigh loaded string”, an elastic string of finite length carrying a number $n\ge1$ of equally spaced, equal point masses, which is a basic model that exhibits a band structure and appears in many applied areas. We consider the case in which the dissipation is due to a viscous damping due to the interaction string-environment, a standard model for internal visco-elastic dissipation (the Kelvin–Voigt model), and their combined presence. We show that the point spectrum of each of these damped versions of the Rayleigh loaded string is a continuous deformation of the point spectrum of the unloaded elastic strings with that damping and that presents a band structure similar to that of the undamped case. We also provide explicit analytical expressions for the eigenfunctions, for any value of $n$.