###This is most certainly not easily done in blender.
It would be much simpler, easier, and less time consuming to use something designed for creating art like this: http://www.escapemotions.com/products/flamepainter/
However, similar things have been done in blender Abstract Wave with Particle System
It is much much easier not to use blender for this task. In fact, I even have a demo of something similar to this that runs in realtime in JS http://codepen.io/Thisisntme/full/KNpPEz/
If you wanted to know, my codepen works by creating a sequence of lines, and then dragging them down the screen (from left to right although it doesn't really matter) all while changing the y position with a few trig functions.