On the Transverse Vibrations of Strings and Beams on Semi-Infinite Domains

In this paper, we study the transverse vibrations of a string and of a beam which are infinitely long in one direction. These vibration problems can be used as a toy model for rain-wind induced oscillations of cables. In order to suppress undesired vibrations in the string (or beam), dampers are used at the boundary. The main aim of this paper is to show how solutions for these string and beam problems on a semi-infinite domain can be computed. We derive explicit solutions for a linear string problem which is attached to a mass-spring-dashpot system at x = 0 by using the D’Alembert method, and for a transversally vibrating beam problem which has a pinned, sliding, clamped or damping boundary, respectively, at x = 0 by using the method of Laplace transforms. It will be shown how waves are reflected for different types of boundaries.