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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]

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]