Klein–Gordon equation

The behavior of the Klein–Gordon equation ∂_{tt}u[t, x] ∂_{xx}u[t, 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}]]]