the retarded fundamental solution

Flash a light bulb on for an instant at one place. Far away, you do not see the flash until the light has had time to travel to you — the signal arrives later, delayed by distance over speed. The retarded fundamental solution is the wave operator's response to a single flash of source at one point in space and time, with that physical delay built in: nothing happens before the source fires (causality), and the disturbance spreads outward at the wave speed c.

For the wave operator (1/c^2) u_tt - Laplacian u, the retarded fundamental solution E_ret satisfies (1/c^2) E_tt - Laplacian E = delta(x) delta(t), with the causality requirement E_ret = 0 for t < 0. In three space dimensions it is a sharp shell: E_ret = delta(t - |x|/c) / (4 pi |x|) — the entire response lives exactly on the sphere |x| = c t, arriving at the time light would take to get there and gone an instant later. This sharpness is Huygens' principle. Because it carries the delay, the solution of the driven wave equation is the retarded integral u(x, t) = integral of f(y, t - |x - y|/c) / (4 pi |x - y|) dy: each source point contributes what it was doing at the earlier retarded time t - |x - y|/c. There is also an advanced fundamental solution supported for t < 0 (signals from the future); physics selects the retarded one.

This is the propagator of classical field theory: it is exactly how the electromagnetic potentials of moving charges are written (the retarded potentials of Lienard and Wiechert) and it expresses finite propagation speed in the cleanest possible way. One subtlety: the clean shell (response only on the light sphere) holds in odd space dimensions three and up; in two dimensions the response fills the whole disk inside the cone and lingers — Huygens' principle fails — so a flash in flatland is heard as a fading rumble rather than a single crack.