Skip to main content
\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)
[ "article:topic-guide", "wave equation", "D\u2019Alembert solution", "authorname:nwalet", "license:ccbyncsa", "showtoc:no" ]
Mathematics LibreTexts

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.