D'Alembert's formula

In mathematics, and specifically partial differential equations (PDEs), d'Alembert's formula is the general solution to the one-dimensional wave equation:

for ${\displaystyle -\infty <x<\infty ,\,\,t>0}$. It is named after the mathematician Jean le Rond d'Alembert.

The characteristics of the PDE are ${\displaystyle x\pm ct=\mathrm {const} \,}$, so we can use the change of variables ${\displaystyle \mu =x+ct,\eta =x-ct\,}$ to transform the PDE to ${\displaystyle u_{\mu \eta }=0\,}$. The general solution of this PDE is ${\displaystyle u(\mu ,\eta )=F(\mu )+G(\eta )\,}$ where ${\displaystyle F\,}$ and ${\displaystyle G\,}$ are ${\displaystyle C^{1}\,}$ functions. Back in ${\displaystyle x,t\,}$ coordinates,

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