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d'Alembert's formula

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

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

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

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

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

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

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

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

Using u_t(x,0)=h(x)\, we get cF'(x)-cG'(x)=h(x)\,.

Integrate the last equation to get

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

Now solve this system of equations to get

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

Now, using

u(x,t) = F(x+ct)+G(x-ct)\,

d´Alembert's formula becomes:

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

[edit] External links

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