# Are all waves periodic?

I'm studying waves, and I am confused about two different definitions of a wave. One place defines a wave as "a propagating dynamic disturbance of one or more quantities". Another says that "waves are the disturbances that moves with a fixed shape and with a constant velocity".

Do all waves contain a repeating pattern and have an associated period, and is it necessary that waves must have constant velocity?

• Since any (strong or weak) solution to D'Alembert equation admits a Fourier intergral decomposition, by a stretch of argument, any wave is either periodic, or made up of an infinity of periodic waves. – DanielC May 15 at 10:32
• Do the quantum tags really have anything to do with this question as asked? – DrSheldon May 16 at 7:47

No. Waves can be of any form. Just look at a super tsunami wave. Or a people-standing-up-sitting-down-wave in a football stadium (The Wave). "Wave" doesn't necessarily mean that repeating patterns are involved. It's true though that all waves can be broken up into periodic waves (extending in both space and time to infinity).

Waves do not, in general, have a constant velocity. For example, if the tsunami wave suddenly finds itself in liquid mercury the speed of the wave will change.

• "It's true though that all waves can be broken up into periodic waves" — not for all waves such expansion is meaningful though, it's only useful for linear waves—the ones that obey the superposition principle. – Ruslan May 15 at 21:07
• @Ruslan But can a non-linear wave be broken up into periodic waves? Such waves don't obey the principle of superposition, that is, you can't superimpose two of these waves. Does this mean that such an expansion doesn't exist (because you can superimpose linear periodic waves)? – Deschele Schilder May 15 at 21:59
• I've never heard of such an expansion for nonlinear waves, but who knows, maybe it does exist... At the very least it isn't the Fourier expansion. – Ruslan May 15 at 22:02
• "For example, if the tsunami wave suddenly finds itself in liquid mercury the speed of the wave will change." Or, more commonly, shallower water. A typical tsunami on the open sea is kilometers long and imperceptibly low amplitude. Then when it comes closer to the coast, the front end slows down while the back end is still in deeper waters, compacting the wave, ultimately giving it an amplitude of about 10 meters by the time it hits land. – Arthur May 15 at 22:07
• Yes, the superposition will evolve differently from the case when the two peakons started far away from each other and then "collided". – Ruslan May 15 at 23:02

Introductory texts often restrict themselves (for simplicity) to scenarios where waves have a constant uniform velocity and are periodic, so it is easy to get the impression that these conditions apply to all waves. This is not correct.

White noise (or, indeed, noise of any other colour) is not periodic.

Wave velocity depends on the local properties of the medium, and can also depend on the frequency of the wave component, so it is not necessarily uniform in space or constant in time.

• So , in reality there do exist waves with varying velocities, but in general or in the classical wave equation we take the velocity of the wave constant, by neglecting these exceptions, right? – Anubhav Rajyan May 15 at 12:44
• @AnubhavRajyan Yes, that is correct. – gandalf61 May 15 at 14:58
• good, thnx@gandalf61 – Anubhav Rajyan May 15 at 19:26

The best way to define a wave of any kind is using the Wave equation which looks like in one dimension

$$\frac{\partial^2\psi(x,t)}{\partial x^2}=\frac{1}{v^2}\frac{\partial^2 \psi(x,t)}{\partial t^2}$$

The wave equation can be used to describe any kind of waves, such as mechanical waves (e.g. water waves, sound waves, and seismic waves) or light waves. It arises in fields like acoustics, electromagnetics, and fluid dynamics.

It's not necessary that the wave should have a fixed shape, the wave consists of different modes So that you can write it as some of these modes. Look for an instant the wave packet. It's due to the fact that different might be traveling with different velocities and so the wave shows dispersion.

When talking of velocity, There are two sorts of velocities. Group velocity and Wave velocity. Limiting ourselves to wave velocity means a pure wave, It might have a varying velocity due to the fact, like tension on a string might position-dependent.

Though in wave equation, We have assume wave velcoty $$v$$ to be constant.

• Thanks,@Young Kindaichi, your answer efficiently clear my confusion to a large extent – Anubhav Rajyan May 15 at 12:39
• The wave equation can't describe breaking waves or any such waves with more than one point at a given location. – user2121 May 15 at 19:18

A lot of physics studies regular periodic waves, or wave-packets containing many near-constant periods of a wave. This makes the maths tidy, but there are some exceptions that are also tractable. Have a look at solitary waves or 'solitons'. There is a nice, historical introduction in the section on the 'Wave of Translation' in
https://en.wikipedia.org/wiki/John_Scott_Russell

The River Severn bore is an example of such a wave.