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.
Read more about this topic: Wave Equation
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