Domain of influence for wave equation in 2 dimensions

Hello, I have some trouble seeing why the solution of the wave equation in 2 dimensions exist at all later times once it passes an initial disturbance....

For example, take a simple case where the initial position is zero, and the initial velocity equals some function inside some circle domain. The solution would be:

[tex]\frac{1}{2\pi }\int \int \frac{\psi (x,y)\partial x\partial y}{\sqrt{t_{o}^{2}-(x-x_{0})^{2}-(y-y_{o})^{2}}}[/tex]

1) Where in that equation tells you that the solutions continues to exist at all later times?
2) If the initial velocity was zero outside the circle domain, why would the solution continue to exist? If we plug in Ψ = 0, wouldn't the solution be zero instead?
3) Can a solution be negative?
 
Maybe I am not being clear. What I just want to know is how this solution shows that it extends outwards at t>0 and why it continues to exist at all later times. In other words, can somebody prove to me why Hyugen's principle fails at dimension 2?
 
Last edited:
bump...two days and no answer at all?
 
I've read that many times and still do not understand. Can someone explain it from the formula I posted above?
 

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