What is the general mathematical definition of wave?

Question

What is the general mathematical definition of wave?

Answer 1

Waves are phenomena of the real world. In order to provide the most general definition of waves from the mathematical point of view, one should start with a careful listing of all the physical properties shared by all the phenomena we want to call "waves". I am not sure a complete list exists, but, taking into account a large set of well known linear and non linear phenomena classified as waves, I would say that a quite wide and general mathematical definition of most of the physical waves could be: any phenomen which can be described by the solution of a partial differential equation of hyperbolic type, in most of the cases second-order in time; i.e., a partial differential equation for a function $f(\vec{r},t)$ (where $\vec{r}$ is a "space" point in an $n-$dimensional configuration space), such that it has a well-posed initial value problem of the kind: $$ f(\vec{r},t=0)=\phi_0(\vec{r})\\ \frac{\partial{f}}{\partial{t}}(\vec{r},t=0)=\phi_1(\vec{r}) $$ were $\phi_0$ and $\phi_1$ are given functions.

Notice that the usual linear d'Alembert wave equation $$ \frac{\partial^2{f}}{\partial{t}^2}=v^2\nabla^2{f} $$ is a very special case, of hyperbolic equation, leaving out too many wave-like phenomena (dispersive and/or damped waves, non linear waves including solitons, quantum mechanical probability waves etc.).

A word of caution is required for some phenomena like shock waves, whose description can be based on hyperbolic pde as well, but in same cases other descriptions not using pde are possible.