I'm going to give a presentation on partial differential equations next week. The audience has no affiliation to mathematics, which is why I thought it would be nice to have some visually appealing material (e.g. a video of a solution of a hyperbolic PDE) possibly related to some interesting physical situations. I tried some searches on YouTube and Google, but didnt find much material I liked, so I'm hoping that maybe some of you working in that area would be willing to share some of their results.
On a related note, how would you explain what a PDE is to somebody that does not have a clear understanding of the notion of a derivative?
You might have some luck with the (unfinished, but still quite nice) PDE Coffee Table Book. At the very least it has lots of nice pictures and some good heuristics. If you're okay with talking about PDE in fluid mechanics, you can also try The Gallery of Fluid Motion.
For your second question I usually say something like "PDE is the language of physics."