point-source response

Drop a single pebble in a still pond and watch the ring of ripples spread out. You have just measured the pond's response to a point disturbance. The point-source response is exactly this idea made into mathematics: the output a linear system produces when its input is an idealized spike concentrated at one location (and, for time problems, at one instant).

The point-source response is the meaning of the Green's function. Where you supply a unit point source delta(x - s), the system reacts with G(x, s); G as a function of the observation point x, for a fixed source point s, is the response field generated by that single point. In a time-dependent problem the same object is called the impulse response: feed in a delta in time and read off the resulting motion. Because the governing operator is linear, this one response is enough — knowing how the system answers a point tells you, by superposition, how it answers everything.

This is why engineers obsess over impulse responses and physicists over propagators: a single measurement (or calculation) of the point-source response characterizes the whole linear system. In signal processing the impulse response convolved with the input gives the output; in optics the response to a point of light is the point-spread function; in seismology it is the response of the earth to a sharp impact. The same word 'response' threads through all of them.