Reflection and damping properties for semi-infinite string equations with non-classical boundary conditions

In this paper, initial-boundary-value problems for a linear wave (string) equation are considered. The main objective is to study boundary reflection and damping properties of waves in semi-infinite strings. This problem is of considerable practical interest in the context of vibration suppression at boundaries of elastic structures. Solutions of wave equations will be constructed for two different classes of boundary conditions. In the first class, a massless system consisting of a spring and damper will be considered at the boundary. In the second class, an additional mass will be added to the system at the boundary. The D׳Alembert method will be used to construct explicit solutions of the boundary value problem for the one-dimensional wave equation on the semi-infinite domain. It will also be shown how waves are damped and reflected at these boundaries, and how much energy is dissipated at the boundary.