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: u t t − c 2 u x x = 0 , u ( x , 0 ) = g ( x ) , u t ( x , 0 ) = h ( x ) , {\displaystyle u_{tt}-c^{2}u_{xx}=0,\,u(x,0)=g(x),\,u_{t}(x,0)=h(x),} for − ∞ < x < ∞ , t > 0 {\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. == Details == The characteristics of the PDE are x ± c t = c o n s t {\displaystyle x\pm ct=\mathrm {const} } (where ± {\displaystyle \pm } sign states the two solutions to quadratic equation), so we can use the change of variables μ = x + c t {\displaystyle \mu =x+ct} (for the positive solution) and η = x − c t {\displaystyle \eta =x-ct} (for the negative solution) to transform the PDE to u μ η = 0 {\displaystyle u_{\mu \eta }=0} .