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What is the general mathematical definition of wave?

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    $\begingroup$ A wave is $f(x-v t)$ for any function $f(x)$, where $v$ is the wave speed. $\endgroup$ – flippiefanus Nov 14 '18 at 9:11
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    $\begingroup$ @flippiefanus That is far too restrictive. Under that definition water waves, gravitational waves, light waves in dispersive media, etc. are all not waves. $\endgroup$ – knzhou Nov 14 '18 at 9:22
  • $\begingroup$ Somewhat related: Do all waves of any kind satisfy the principle of superposition? (though unfortunately the accepted answer is dead wrong). Making general statements about waves, in a way which applies to all waves (but without making definitions which are so formless that they apply to things to which the term doesn't apply) is a pretty hard task. $\endgroup$ – Emilio Pisanty Nov 14 '18 at 18:19
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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.

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    $\begingroup$ This sounds good, but it would be better if you include a definition of a hyperbolic pde. $\endgroup$ – garyp Nov 14 '18 at 12:04
  • $\begingroup$ A not too formal definition is the sentence starting with "i.e. a partial...". I think that if one is asking for a mathematical definition should know enough about mathematics. But you are a more experienced user of this platform, so, I have added a link to the relevant wikipedia page. $\endgroup$ – GiorgioP Nov 14 '18 at 16:44
  • $\begingroup$ Thanks for the link. I had no idea that the definition would be so involved! $\endgroup$ – garyp Nov 14 '18 at 19:36

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