D'Alembert's formula

D'Alembert's formula

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 < x0. It is named after the mathematician Jean le Rond d'Alembert.

The characteristics of the PDE are xpm ct=mathrm{const},, so use the change of variables mu=x+ct, eta=x-ct, to transform the PDE to u_{mueta}=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) dxi + c_1,

Now solve this system of equations to get

:F(x) = frac{-1}{2c}left(-cg(x)-left(int_{-infty}^x h(xi) dxi +c_1 ight) ight),

:G(x) = frac{-1}{2c}left(-cg(x)+left(int_{-infty}^x h(xi) dxi +c_1 ight) ight),

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) ight] + frac{1}{2c} int_{x-ct}^{x+ct} h(xi) dxi

External links

* [http://www.exampleproblems.com/wiki/index.php/PDE27 An example] of solving a nonhomogeneous wave equation from www.exampleproblems.com


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