I’m wanting to make a pattern that resembles a blurry bullseye target which is really a repeated circular gradient with a sine or spherical ramp between the gradient stops. It would be for 3D engraving.
similar to: 
I’m hoping to find a way to do this in Inkscape or gimp where I can vary the frequency and/or width of the waves. I’m new at both of these and so far editing the gradients in inkscape only seems to have a linear ramp. Advice please?
I can’t even look at that picture! 
(But you’ll need to rasterize the results when you get them.)
Gimp can do this easily with a custom gradient.
Use the gradient editor to create a gradient that goes from black to white. Right click on the gradient and set the Blending Function for Segment to Sinusoidal.
Then use your new gradient with the gradient tool. Set Shape to Radial and Repeat to Triangular Wave.
In the image, click and drag the gradient tool to set the width of the waves. If you want to be precise in defining your widths you can set up a couple of guides to help.
Here is an example - sinusoidal bullseyes are on top and linear bullseyes are on the bottom.
Just had a cool idea. If you duplicate your bullseye, put it on a layer above the original, and set the layer to difference mode you can create interference patterns by moving one layer relative to the other. This creates some great abstractions.
Just got this to work in GIMP and now I’m getting the “stuck in your vision” thing and seeing them everywhere!
Awesome, perfect. Thanks for writing it in steps - that really helped me learn it out! The blending of gradients on top of gradients is getting interesting! Obviously this is what I’m going to spend the rest of my day doing. 
I wonder what’s down here…
Nothing to see here…
OR…
