Questions about waves

Another post has made me want to develop a better understanding of waves in general. Does anyone have an approach to remembering everything there is to know about waves? I need a referesher.

1. Transverse waves

2. Longitudinal waves

3. Compression waves (longitudinal synonym)

4. Shear waves (transverse synonym)

5. Standing waves

6. Traveling waves

7. Spherical waves

8. Plane waves

9. Beats

All waves have what in common exactly?

It's one thing to understand the sine function, or cosine function, its something totally different to understand a sine wave, or a cosine wave, in which you have the medium changing in time.

There are density changes, and also, in order to define wavespeed, you have to have some frame of reference, and there are all kinds of different frames of reference to choose from. All this has to be handled in a comprehensive theory of waves. Can anyone help, in the sense that they have a quick way to remember all there is to know about waves?

Thank you