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.