I am interested in the interpretation of PDE that are not strictly elliptic, parabolic,or hyperbolic: ones that change type in sections of the domain, e.g. the generalised Tricomi equation

$ A(x) u_{,xx} + u_{,yy} = 0$

will be elliptic in regions where $A(x) \geq 0$ and hyperbolic when $A(x) < 0$.

I am aware there is a great deal of literature surrounding the topic (e.g. transonic flows), but mostly regarding well-posedness and such, not so much about the actual physics of what is going on.

For a start, I usually think of hyperbolic equations as having wave-like components in their solutions -- this usually means there is some kind of 'time' involved. What does it mean for a PDE in *spatial* variables to be hyperbolic? It seems weird to me to think of 'travelling waves' in the $(x,y)$ plane. What about an elliptic PDE involving both spatial and *temporal *variables?

Can someone explain what is physically happening here? In the non-linear case [$A(x) \rightarrow A(x,u)$] does this kind of elliptic-hyperbolic transition imply something about shocks?

As a simple example, if we have $A(x) =x$ and the domain is something like $D = \{x,y: -a \leq x \leq a, -b \leq y \leq b\}$, what are the 'wave-like' portions doing in the $x < 0$ region? What physics are they conveying?