It might also help to know that the scattering matrix has classical counterparts in engineering. For example the “transfer function” in circuits and signal processing, etc, and the “response amplitude operator” (RAO) in ship design. I can speak from experience only about the RAO, but once you’ve got one built and for your system, you know quite a lot about the response of the system to input (in this case typically water waves). A higher order transfer function in ship design might be constructed to give you say, drift response (or force) as a function of wave frequency, and give you wave drift response spectra when “hit with” a wave spectrum.

(First order response might regular 6DOF motions (motion spectra) in response to wave spectra)

Here the analogies are pretty fun, because you are scattering and radiating waves in the water. Very physical!