On resonances and the applicability of Galerkin׳s truncation method for an axially moving string with time-varying velocity

In this paper, an initial-boundary value problem for a linear-homogeneous equation describing an axially moving string is investigated. The time-dependent velocity is assumed to vary harmonically about a constant mean velocity with a small fluctuation amplitude. The two timescales perturbation method and the Laplace transform method have been employed to the equation of motion in search of infinite mode approximate solutions. All resonance cases are investigated in detail. It will be shown that Galerkin׳s truncation method cannot be applied to this problem in all cases to obtain approximations valid on long timescales.