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?
 A: 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.
A: 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.
A: 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.
A: 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.
