Regarding wave shapes, it depends on the domain. Basically you have a wave equation

D^2 u = d^2u/dt^2

inside the domain and some boundary condition on the outside. (For example, if you were looking at a vibrating drum head, the domain would be a disk and the boundary condition would be u=0, since the drum head is fixed to some base.)

You might expect a second-order PDE to have solutions built from exp, which as we know are sines and cosines (or hyperbolic ones, but signs of stuff don't allow those). You can do a Fourier treatment and look in frequency space as well.

In one-dimension, we have u_{xx} = u_{tt}, and separating as u(x,t)=X(x)T(t) gives

X''/X = T''/T

As the left-hand depends only on x and the right-hand side depends only on t, we conclude both are a constant k, so

X'' = kX,

and we see why we have exponentials. Boundary constraints fix the sign of k (for real-word scenarios it is negative), and so we have sines and cosines. Boundary conditions also give us the frequency, and we basically have a sequence of eigenvalues giving rise to the fundamental frequencies (what you see in a Fourier series expansion).

So you don't just have one sine function, you have a whole series of them, but their amplitude decreases (and pretty quickly, else the series would diverge). So you have the primary one, and then you have overtones/harmonics. This works basically the same way for a one-dimensional wave on a string or a musical instrument.

Detail can be found here:

https://en.wikipedia.org/wiki/Wave_equationPPE: I think wave equation (maybe you need an extra damping term) is approximate enough for surface waves. For full blown fluid dynamics in a viscous liquid, you'd need Navier-Stokes. I believe Navier-Stokes reduces to wave equations under various simplifying assumptions.