D'Alembert's formula

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

${\displaystyle u_{tt}-c^{2}u_{xx}=0,\,u(x,0)=g(x),\,u_{t}(x,0)=h(x),}$

for ${\displaystyle -\infty <x<\infty ,\,\,t>0}$

It is named after the mathematician Jean le Rond d'Alembert, who derived it in 1747 as a solution to the problem of a vibrating string.^[1]