Neumann and Dirichlet boundary conditions for the one-dimensional wave equation

Abstract

The notion of a boundary condition is typically considered to be somewhat advanced and not suitable to be introduced in high school level physics. In this article, we give a simple visual demonstration of the difference between Dirichlet and Neumann boundary conditions for a string which oscillates according to the one-dimensional wave equation. The classical Dirichlet problem for a vibrating string in the plane is mathematically deep, but we will avoid the mathematical issues here and only focus on some key physical points which are relevant for understanding how waves propagate along strings.