An asymptotic method for solving the problems of wave propagation in a thread

An asymptotic method for solving the problems of wave propagation in a thread is proposed. The method uses the magnitude of characteristic deformation as a small parameter. The use of the method is demonstrated by the example of point action upon a fixed particle of the thread. This action, usually called transverse impact, is simulated by the particle's nonstationary motion. In the case being considered, the transverse waves propagating in the thread at a velocity a0 produce the appropriate tension before the head transverse wave whose velocity is much less than a0. The momentum equation in the initial direction of the thread, in the zero approximation, provides for the tension to be independent from the coordinate in the region of transverse wave propagation. Once the transverse displacements in this region are determined, the field of longitudinal velocities and longitudinal deformations is being found. The longitudinal and transverse components of deformation, in the zero approximation, exceed considerably the general deformation (their order is lower than the latter's). For the case of an exponential time dependence of the velocity of the action (transverse impact) applied, in the absence of initial tension in the thread, the field of transverse motions is of self-simulating nature.