
# 6.1: Background to D’Alembert’s Solution

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I have argued before that 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

$\dfrac{\partial^2}{\partial x^2} u(x,t) - \frac{1}{c^2} \dfrac{\partial^2}{\partial t^2} u(x,t) = 0,$ which has a general solution, due to the French mathematician d’Alembert.

The reason for this solution becomes obvious when we consider the physics of the problem: 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. This means that we would expect the solutions to propagate along the characteristics $$x\pm ct=\text{constant}$$, as seen in Fig. $$\PageIndex{1}$$.

Figure $$\PageIndex{1}$$:The change of variables from $$x$$ and $$t$$ to $$w=x+ct$$ and $$z=x-ct$$.