# What is the general mathematical definition of wave?

What is the general mathematical definition of wave?

• A wave is $f(x-v t)$ for any function $f(x)$, where $v$ is the wave speed. – flippiefanus Nov 14 '18 at 9:11
• @flippiefanus That is far too restrictive. Under that definition water waves, gravitational waves, light waves in dispersive media, etc. are all not waves. – knzhou Nov 14 '18 at 9:22
• 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. – Emilio Pisanty Nov 14 '18 at 18:19

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.).