Preamble: I have come to believe that alot of difficulties in explaining physics to people of all levels comes from the relatively mundane idea of a wave equation with a mass gap $$\left(-\partial^2_t +\nabla^2 -m^2\right)\phi = 0$$ or more generally a field that does not have propagating modes in some frequency band. Being able to demonstrate this behavior would be useful in explaining the differences between conductors and insulators, the difference between the Higgs field and the Higgs boson and other things - it would even be useful explaining gapless fields like EM to be able to refer to a gapped field.
However despite being an extremely common and tame phenomenon I can't think of any approachable examples of gapped wave equations. I can't think of anything where I could say, even to a junior undergraduate "its a gapped field, just like X". Nor can I think of a system I could show a video of, or a demonstration that would give intuitive understanding of gapped fields.
So my question is: does anyone know of a "gapped system" that would be useful for pedagogical purposes? That is, a system that has a continuum of propogating frequencies and a continuum of non-propogating frequencies.
The best I can think of is the following experiment: by scattering debris on the bottom of a pan filled with shallow water, you can localize the surface waves, but I assume the high frequency waves should still propagate. I don't like this for a variety of reasons, least of which is you are using Anderson localization to explain a mass gap.
To reiterate I am looking for something which can give understanding - so something intuitive or something that can be played with until it becomes intuitive. I know there are lots of common things that are gapped (metals and EM radiation for example), but I can't think of anything that is pedagogically useful.