# 6: D’Alembert’s Solution to the Wave Equation

- Page ID
- 8315

\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)

\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)

It is usually not useful to study the general solution of a partial differential equation. As any such sweeping statement it needs to be qualified, since there are some exceptions. One of these is the one-dimensional wave equation which has a general solution, due to the French mathematician d’Alembert.

- 6.1: Background to D’Alembert’s Solution
- The wave equation describes waves that propagate with the speed c (the speed of sound, or light, or whatever). Thus any perturbation to the one dimensional medium will propagate either right- or leftwards with such a speed.

- 6.2: New Variables
- To understand the solution in all mathematical details involved in D’Alembert’s solution to the wave equation we make a change of variables.

- 6.3: Examples
- Now let me look at two examples.