On wave reflections and energetics for a semi-infinite traveling string with a nonclassical boundary support

The free, linear, transverse vibrations of a semi-infinite axially moving string are considered in order to study the reflection of an incident wave and the damping properties of different types of boundary conditions such as free, fixed, spring–dashpot, and mass–spring–dashpot. To obtain the response to the initial conditions, the method of d׳Alembert is applied. Furthermore, analytical expressions for the time-rate of change of the total mechanical energy of the system are derived. The obtained results give insight into the most efficient way of placing a boundary support depending on the direction of the transport velocity. Moreover, for nonclassical boundary conditions, the dynamics of the string is described by the relative values of the system parameters.