Applying an image to an extruded (and beveled) mesh

I am new to Blender and trying to figure the right terms to use to best ask my question.

What I want to do is to have an image, e.g.:

And then map it to a circle (or cylinder) mesh:

And then extrude and bevel the shape so that I could end up with this:

I have figured out how to approximate my shape, but not how to get the image to be applied as a texture map, or whatever. The tutorials on UV mapping don’t seem applicable, since what I want is for the “grain” of this image (it could be an arbitrary image, not just wood) to map consistently across the 2D space (x,y) of the object.

In other words, picture thousands of the same picture laminated together directly one over the other to create a cylinder and then put the cylinder/dowel on a lathe. The effect I am looking for would be that result; any cross section of the object prior to turning on the lathe would look the same.

Pointers to either threads on this type of problem or how to best phrase my question welcome. Thanks!

planar coordinates should work, but UV mapping could also work, by choosing the ‘project from view’ option.

Thanks. If you have time, would you help me with my terminology? Is what I’m attempting a “texture map” (my old-days term) or something else?

I’m new enough to Blender that “planar coordinates” doesn’t make me think, “aha!” as to where I would be in the UI. I’ll certainly search the doc/forums/wikis, but if you have a moment to be a little more explicit - here or in a pointer to another thread - I’d really appreciate it.

sorry, in texture buttons just set map input to ‘plane’.