On The Derivation Of A String Equation

Abstract

The traditional derivation of the wave equation for an elastic string is revised. The focus is on a rigorous implementation and subsequent analysis of the Second Newton’s Law adapted for a small string element. Asymptotic treatment of the plane strain problem for a pre-stressed elastic strip shows that the 1D classical wave equation corresponds to the leading order long-wave low-frequency approximation. At the same time, the next order approximation is not given by a hyperbolic equation supporting a dispersive transverse motion.