Klein–Gordon equation

The behavior of the Klein–Gordon equation δ[u[t, x], t, t] == δ[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}]]]