Klein-Gordon equation

The behavior of the Klein-Gordon equation D[u[t, x], t, t] == D[u[t, x], x, x] - u[t, x] is visually very similar to that shown for the sine-Gordon equation. For the Klein-Gordon equation, however, there is an exact solution:

u[t, x] = If[x^^{2} > t^^{2}, 0, BesselJ[0, Sqrt[t^^{2}-x^^{2}]]]