My dad has a brand new Gary Fisher Pirhana with a rigid and suspension corrected fork from Surley. When he's going from 16-24 miles an hour and takes his hands off the handlebar, the wheel and handle bar vibrate left and right (as if it's steering left and right 1-2 times a second) of their own accord.

New tires effected the magnitude, smaller tires resonating to a degree of about 10-15, and thicker tires resonating at avout 20.

We placed weights inside of the handlebars, but that only dampened it.

Sometime today or tomorrow we are going to switch my wheel and his wheel and see if it still happens.

I'm beginning to suspect that it's just this frame/wheel/fork/handlebar combination that happens to resonate badly, but if anyone can shed light on how to fix this it would be greatly appreciated.

My hardtail mtb tourer/commuter with rigid fork did the same thing when I ran Conti Town & Country tires. The vibration stopped after swapping out those tires for another brand and type.

It's quite amazing you should say that, because that's the tire that my dad put on the bike before he even rode it. The tire we switched to was a smaller (and pretty worn) Conit Avenue.

I think he's experiencing "shimmy"
I don't know much about it, exept what I read here, and here.

Shimmy can be dangerous so be careful and good luck.

Sounds to me like he has too much trail and would do better with a straighter or taller fork.

Shimmy usually occurs at higher bike speed and is usually at a much higher frequency than once or twice a second.

Along with the wheel change, check to be sure the headset is properly adjusted a too tight headset may be a contributor.