The Dirichlet Problem for the Vibrating String Equation

This note considers the Dirichlet and Neumann type boundary value problem for the simple vibrating string equation. The detailed study for a special boundary is timely in view of certain categorical statements in the recent literature.* The results obtained below indicate how such statements are to be modified.f Of independent interest is the novel procedure, stemming from Lemma 1, for proving uniqueness in Theorems 1 and 2. The method is of wide utility and leads to interesting generalizations. For convenience we use t for vr, where v and r refer to the velocity of wave propagation and the time, respectively. The string equation is then