What is D Alembert's equation?
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Answer:
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}(x,t)=c^{2}u_{xx}(x,t)}{\displaystyle u_{tt}(x,t)=c^{2}u_{xx}(x,t)} (where subscript indices indicate partial differentiation, using the d'Alembert operator, the PDE becomes: {\displaystyle \Box u=0}{\displaystyle \Box u=0}).
The solution depends on the initial conditions at {\displaystyle t=0}t=0: {\displaystyle u(x,0)}u(x,0) and {\displaystyle u_{t}(x,0)}{\displaystyle u_{t}(x,0)}. It consists of separate terms for the initial conditions {\displaystyle u(x,0)}u(x,0) and {\displaystyle u_{t}(x,0)}{\displaystyle u_{t}(x,0)}:
{\displaystyle u(x,t)={\frac {1}{2}}\left[u(x-ct,0)+u(x+ct,0)\right]+{\frac {1}{2c}}\int _{x-ct}^{x+ct}u_{t}(\xi ,0)\,d\xi .}{\displaystyle u(x,t)={\frac {1}{2}}\left[u(x-ct,0)+u(x+ct,0)\right]+{\frac {1}{2c}}\int _{x-ct}^{x+ct}u_{t}(\xi ,0)\,d\xi .}
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