LECTURE 4 - THE EQUATION FOR A VIBRATING STRING AND ITS SOLUTION BY D'ALEMBERT'S METHOD

Abstract

This chapter discusses the equation for a vibrating string and its solution by D’Alembert's method. It examines the vibrations of an infinite string in detail by describing two simple cases. In case 1, the function ϕ1(X) is identically zero, and the function ϕ0(X is zero except in a finite interval — k □ × □ k. In case 2, the function ϕ0(X) is identically zero, and the function ϕ1(X) is zero except in a finite interval — k □ × ▪ k. In such a case, one may say that the string has an initial impulse but no initial disturbance. In this case, two waves travel along the string, one forward and one reverse. They differ only in sign. Where both the forward and reverse waves have already passed, the string will have reached a state of rest, but it will not, in general, have returned to its original position. A so-called residual displacement will remain in the string. It is possible to produce a wave travelling in one direction only by giving the string a suitable initial disturbance and impulse. It is only necessary to ensure that the reverse waves evoked by the initial displacement and by the initial impulse differ only in sign. The equation for a vibrating string and equations of hyperbolic type in two independent variables similar to it are often encountered in various problems of mathematical physics.