New solutions of the wave equation for a bowed string

Abstract

In 1860, H. v. Helmholtz concluded, by observing the movement of each particular point of a bowed string with his vibration microscope, that the highest point of the bowed string travels along upper and lower parabolic arcs. Since then, his conclusion has been accepted and little attention has been paid to the shape of the envelope of the bowed string. The experimental work of Kondo et al., which focused on this subject [6th I.C.A., N‐2‐4 (1968)], stimulated the present work. This work re‐examines the general solution of the wave equation by giving some initial conditions and then, in addition to the normal Helmholtz solution, obtaining two new solutions, whose shapes are of an elliptic arc and of hyperbolic arc, respectively. These three solutions, parabolic, elliptic, and hyperbolic, coincide with each other when a specific parameter approaches infinity. One experimental result found in the literature seems to represent an elliptic solution.