Klein–Gordon equation

The behavior of the Klein–Gordon equation ∂_ttu[t, x]  ∂_xxu[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[x² > t², 0, BesselJ[0, Sqrt[t² - x²]]]