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D’Alembert’s formula; w </math>

In mathematics, and specifically partial differential equations, d´Alembert’s formula is the general solution to the one-dimensional wave equation: <math>u_{tt}-c^2u_{xx}=0, u(x,0)=g(x), u_t(x,0)=h(x)\,</math>. It is named after the mathematician Jean le Rond d’Alembert.

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

<math>u(x,t)=F(x+ct)+G(x-ct)\,</math>
<math>u\,</math> is <math>C^2\,</math> if <math>F\,</math> and <math>G\,</math> are <math>C^2\,</math>.

This solution <math>u\,</math> can be interpreted as two waves with constant velocity <math>c\,</math> moving in opposite directions along the x-axis.

Now consider this solution with the Cauchy data <math>u(x,0)=g(x), u_t(x,0)=h(x)\,</math>.

Using <math>u(x,0)=g(x)\,</math> we get <math>F(x)+G(x)=g(x)\,</math>.

Using <math>u_t(x,0)=h(x)\,</math> we get <math>cF’(x)-cG’(x)=h(x)\,</math>.

Integrate the last equation to get

<math>cF(x)-cG(x)=\int_{-\infty}^x h(\xi) d\xi + c_1\,</math>

Now solve this system of equations to get

<math>F(x) = \frac{-1}{2c}\left(-cg(x)-\left(\int_{-\infty}^x h(\xi) d\xi +c_1 \right)\right)\,</math>
<math>G(x) = \frac{-1}{2c}\left(-cg(x)+\left(\int_{-\infty}^x h(\xi) d\xi +c_1 \right)\right)\,</math>

Now, using

<math>u(x,t) = F(x+ct)+G(x-ct)\,</math>

d´Alembert’s formula becomes:

<math>u(x,t) = \frac{1}{2}\left[g(x-ct) + g(x+ct)\right] + \frac{1}{2c} \int_{x-ct}^{x+ct} h(\xi) d\xi</math>


External links

  • An example of solving a nonhomogeneous wave equation from www.exampleproblems.com

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