Verifying Kirchhoff formula

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  • Thread starter jostpuur
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  • #1
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Main Question or Discussion Point

I've read the derivation of Kirchhoff's formula for three dimensional wave equation using the Euler-Poisson-Darboux equation. The derivation seems to be fine, but I thought that the Kirchhoff's formula as a final result is a kind of result that you should be able to verify it by substituting it back to the wave equation. Well I substituted it into the wave equation, and tried to manipulate the expressions in hope of all the terms cancelling, but I couldn't make it work. My question is that has anyone ever succeeded in verifying the Kirchhoff's formula by substituting it into the wave equation?

The wave equation:

[tex]
\partial_t^2u(t,x) - \nabla_x^2u(t,x) = 0
[/tex]

The Kirchhoff's formula:

[tex]
u(t,x) = \frac{1}{4\pi t^2}\int\limits_{\partial B(x,t)} \big(t\partial_tu(0,y) + u(0,y) + (y-x)\cdot\nabla_xu(0,y)\big)d^2y
[/tex]
 

Answers and Replies

  • #2
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Thanks for the post! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post?
 

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