We were hoping you'd tell us.
Reading between the lines, it's a back tire on some vehicle. If it is an independent rear suspension, then take it to an alignment shop. Rear wheel alignment is almost as important as the front. Even a solid axle needs to be square and level.
Tires should be balanced (and optionally, shaved concentric) when they are first mounted. Tighten the lugnuts once with a torque wrench, and then feel how tight they are when you loosen them by hand. You can always reproduce that afterward.
I'm going to go with 'excessive whirl'.