# Fundamentally speaking, what are all the essential factors for waves (of any nature) to exist?

Considering all systems in which waves can exist, from atomic scale (such as DeBroglie Waves for example) to cosmic scale (gravitational waves for example), and those that require media for transport (sound waves for example) and those that don't (electromagnetic for example) is there a list of essential factors that determine when a wave can exist/or not?

Many textbooks/online references describe what a wave is, but I've never seen any literature that address this question over all systems. In my thinking waves are an expression of energy flowing through time and space. So after some thought I'm thinking that waves (of any nature) can exist if:

1. there is some source of available energy
2. there is some region of space (or matter) for the energy to propagate, and
3. there is some means for the energy to change form/state as it propagates through space (or matter)

Can the list be as short and simple as this, or are there deeper factors to consider?

• A wave is nothing but a solution to the wave equations. There are in the universe fields whose propagation happens in space and time and is described by $(\partial_t^2 -\partial_x^2)f(x,t)=0$. The interpretation as energy flow that does this or that other thing is incorrect. – gented Dec 11 '15 at 0:25
• @GennaroTedesco Can you provide a more detailed account of what you are suggesting - perhaps as an answer? In my experience waves are a very physical thing that are associated with the flow of energy - in many different physical incarnations. Mathematical equations - aren't these just the language we've chosen to speak of these physical entities? – docscience Dec 11 '15 at 14:52
• What characterises a wave is how it propagates in space and time, in some very specific ways (namely following the wave equation). That then it is associated with energy, well, many other things in the universe are associated with energy flows and energy changes without being waves though. However, the answer reported below provides more insights. – gented Dec 11 '15 at 15:40

There must be three things

• An inertia tendency

In mechanical waves this is simply inertia. In electromagnetic waves it is the time derivative terms in Maxwell's equations.

• A restoring tendency

The tensional forces in mechanical waves, gravitational effects in surfaces waves on a liquid and so on.

• Coupling between neighboring values

The tension again for mechanical waves, the tendency of fluids to move from from high pressure to low, and so on.

You should recognize the first two features as being those needed for oscillation; they are what generate the time variation. It is the coupling that allows the wave to move in space (it transfers energy from one region to another).

In many cases the last two features are closely related to one another because the restoring tendency is the coupling between neighboring regions plus some pre-existing equilibrium.

• Isn't there a more general term than inertia tendency which has a strong tie to mechanical systems? The inertia/restoring tendencies I believe are in line though with the third factor I listed: the form of energy. Potential/kinetic, magnetic/electric, etc. And yes the coupling has to be in place for the energy to flow. – docscience Dec 11 '15 at 14:55