Vibrating String with a Variable Length

Abstract

The propagation of waves in a string for the case of timevarying length is a problem of one-dimensional wave propagation with moving boundary conditions. Confronted with the difficulty of finding an exact analytical method in order to solve such a problem, scientists and engineers have relied on numerical and approximate analytical methods. For simple cases, there are specific methods. However, more general mathematical methods are used for analytically solving such problems. These methods lead to the solutions of the functional or integro-differential equations. The method of separation of variables used by Tadashi Kotera leads to a complex solution. Such solutions are very cumbersome and complex. Recently, Y. M. Ram and J. Caldwell introduced a new method the method of distorted images. This method ! remains limited and does not take into account the energy of the system. However, these methods do not allow the deduction and the understanding of the physical phenomena. More recently, Stefan Kaczmarczyk and Wieslaw Ostachowicz used the method of multiple scales to study the longitudinal dynamics of hoisting cables and to predict the responses near the resonance region. Nicolas Gonzales studied the stability of the energy of the wave equation for periodic motion. The method used in this paper is an exact analytical method based on analytical functions and conformal transformations. It has been developed for solving a large class of systems with non-stationary dimensions. The main advantage of this method is that it gives the exact solution for all motions (parabolic, hyperbolic, etc.). The other advantages are the easy application, the exact formulation of the energy ratio, and the possibility of deducing the energy. By means of the method cited above, we have determined the exact solution of our problem, the modes, the energy ratio, the energy and discussed the results obtained. 2. SUMMARY OF L. GAFFOUR METHOD