Scalar Wave Equation in Two Space Dimensions
In two space dimensions, the wave equation is
We can use the three-dimensional theory to solve this problem if we regard u as a function in three dimensions that is independent of the third dimension. If
then the three-dimensional solution formula becomes
where α and β are the first two coordinates on the unit sphere, and dω is the area element on the sphere. This integral may be rewritten as an integral over the disc D with center (x,y) and radius ct:
It is apparent that the solution at (t,x,y) depends not only on the data on the light cone where
but also on data that are interior to that cone.
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