Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 105 / 3 / b / Solution

The Strong Huygens principle in three spatial dimensions says that the solution at $(t,x)$ depends only on the initial data on the sphererather than on the full ball bounded by that sphere. This follows immediately from the Kirchhoff formula and its time derivative. Consequently a disturbance has no tail inside the light cone: if the initial data are supported in a compact set $K$, then $u(t,x)=0$ whenever that sphere misses $K$. This is sharper than finite propagation speed, which only excludes influence from outside the ball.