How can i procedurally change ring sizes in wave texture? I got to the point where I got each ring size different by having the Wave Texture go through a Noise Texture.

How can I get each ring to not be uniform like this drawing. Not exactly like the drawing since each ring isn't exactly circular but each ring having a constantly changing width.

I wrongly assumed by changing the ring width it would have the effect of the sides running down like this drawing. I'm not sure how to go about this. Any changes to the z scale in the Mapping node removes the side lines and the top/down center circles.
You can mix some noise into the texture coordinate space.

This has the unfortunate side effect of offsetting the center of the circle, so you would have to use the Mapping node to translate it back.