A discrete model for the vibrating string

Abstract

The problem of the vibrating string is a very old one in mathematical physics. Its modern solution consists of solving the one‐dimensional wave equation with the appropriate boundary conditions. In this paper it is shown how the problem can be solved from fundamental principles. Instead of the continuous string, a discrete model is taken, consisting of a chain of n equal masses situated at equal distances along a flexible string, itself assumed to be massless. The eigenfrequencies of small transverse vibrations of this chain are calculated, and from this the normal mode eigenvectors are found. The limiting expressions for the continuous string are then obtained by letting n?8. Although this idea is not new, it has been based in the past mainly on guessing the right solution; here we establish the results rigorously from matrix algebra.