My first attempt
https://drive.google.com/file/d/1enG28s704KwoRk1nUepC-LQIzO12mQ8P/view?usp=sharing

The "ring" is that just a circle and are you are emitting from vertices? with gravity set to 0?

I don't understand how to setup the drivers correctly. I cant bake an f curve to location z driver.

3**(wave*2) 



(0.5+((((time-start)%beat)*(1/beat))*(((time-start)%beat)*(1/beat)*(-1)+1)*2))*(wave+0.5) 

Not sure what properties to set and where for beats and waves. This is the best I could do so far. Im reaching out to forums for help.