Effect of moving boundaries on the vibrating elastic string

Abstract

A new derivation of a wave equation for small vibrations of elastic strings fastened at ends varying with time is presented. The model takes into account the change of length during the vibration and the nonlinear behavior of elastic strings in general. This model is a generalization of the Kirchhoff equation which contains a nonlinear term involving the displacement gradient. Numerical simulations of the model are based on finite difference approximations. Differences between linear and nonlinear aspects and the assumptions of numerical and theoretical analysis are briefly discussed and comparisons are made for linear and nonlinear elastic strings as well as the Kirchhoff model and the linear model without the term containing the displacement gradient.