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Mathematics LibreTexts

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

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    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.