One-Way Waves

Abstract

One representation of the solution of the one-dimensional nonhomogeneous wave equation $u_{tt} - c^2 u_{xx} = g(x,t)$, with $u = u_t = 0$ at $t = 0$, is $u = ({1/2}c)\iint_\Delta gdxdt$, where the integral is taken over a certain triangle in the $x - t$ plane. This form of solution, known as the d’Alembert solution, is illustrated by an example of a $g(x, t)$ which produces a wave of finite length traveling only in one direction. For more general $g(x, t)$ it is shown how to decompose $g(x, t)$ into four components each of which produces a component wave with a simple character.