# In general, what is the nature of a wave? [closed]

Is the concept of "a wave" limited for only light, electricity, and voice?
Are they just means to serve our senses of sight and hearing or can their concept be extended to a wider notion? As a hypothetical question, can one prove that everything around and even us are constructed of various types of waves carrying multiple sorts of information?

## closed as too broad by sammy gerbil, peterh, Jon Custer, Gert, AccidentalFourierTransformDec 4 '16 at 10:30

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• This question is much too broad, and possibly also too philosophical. – sammy gerbil Dec 3 '16 at 22:36
• A wave is a very useful mathematical model, like the related simple harmonic motion concept, in that it can be extended to a much wider area than it's orginal applications. – user108787 Dec 3 '16 at 22:41
• Wiki says: "In physics, a wave is an oscillation accompanied by a transfer of energy that travels through a medium (space or mass).". If we consider this correct (note: in general, don't trust Wikipedia!), then the answer to your question is obvious. – peterh Dec 3 '16 at 23:44

## 1 Answer

Firstly, don't get too hung up on terminology - don't try to make terminology do too much. Like most English and natural language words, "wave" is a vague term (indeed the French term whence English got "vague" from "une vague" means a wave :), so it's the epitome of vagueness! ). We therefore need to qualify it carefully before we use it in science; a good way to do this is in mathematical terms.

One kind of object that I would definitely always call a wave is any scalar field that fulfills D'Alembert's wave equation:

$$\left(\nabla^2 - \frac{1}{c^2}\,\frac{\partial^2}{\partial\,t^2}\right)\psi(\vec{x},\,t)=0\tag{1}$$

But this is not the only kind of thing a physicist would call a wave. There are many wave different equations - a very important example aside from (1) would be nonlinear equations that have self re-inforcing solitonic solutions i.e. the disturbance keeps itself together notwithstanding dispersion.

So, aside from fields fulfilling (1), how shall we generally define a "wave"? I would suggest the word doesn't have a totally consistent usage throughout science so one has some freedom (within reason) how one chooses to define it.

But a good generalization I think can be found by looking at the properties of solutions of Hyperbolic Differential Equations, which are linear, and then calling any field with these properties a wave, whether its underlying propagation equation be linear or nonlinear; from Wikipedia:

" [The field] has a well-posed initial value problem for the first n−1 derivatives. More precisely, the Cauchy problem can be locally solved for arbitrary initial data along any non-characteristic hypersurface."

AND

"If a disturbance is made in the initial data of a hyperbolic differential equation, then not every point of space feels the disturbance at once. Relative to a fixed time coordinate, disturbances have a finite propagation speed. They travel along the characteristics of the equation. This feature qualitatively distinguishes hyperbolic equations from elliptic partial differential equations and parabolic partial differential equations. A perturbation of the initial (or boundary) data of an elliptic or parabolic equation is felt at once by essentially all points in the domain."

I hope you'll agree that the second criterion sounds what we'd intuitively understand a "wave" to be. Importantly, note that the second criterion includes, or allows for, a notion of locality: i.e. a maximum signal speed which is crucial for mathematical models consistent with special relativity.