On transversal vibrations of an axially moving string with a time-varying velocity

In this paper an initial-boundary value problem for a linear equation describing an axially moving string will be considered for which the bending stiffness will be neglected. The velocity of the string is assumed to be time-varying and to be of the same order of magnitude as the wave speed. A two time-scales perturbation method and the Laplace transform method will be used to construct formal asymptotic approximations of the solutions. It will be shown that the linear axially moving string model already has complicated dynamical behavior and that the truncation method can not be applied to this problem in order to obtain approximations which are valid on long time-scales.