ORIGINAL: HighPlains
True. the static tip speed is far short of the speed of sound. But you forgot a couple of factors.
1.) The engine will unload in the air and operate at higher rpm. Much higher when the airplane is dove, as gravity provides the horsepower to accelerate the prop.
2.) Tip speed is not simply a function of Pi x D, you have to include the forward travel of the prop with the unloaded tip speed. For high rpm high pitch props, this is significant.
3.) The speed of sound varies with the airfoil moving through the air.
From my example, you need to gain about 500 fps from the static condition to be in the transonic realm.
Lets take your #2 and assume airspeed is 120 mph. Add that vector to rotational speed, which is orthogonal to it. So, say 176 fps in the axial plane and 630 fps in the rotational. I get a velocity vector resultant of 654 fps, not much increase and still way short of 1125 fps. Now add #1, rpm increase. I could see a few hundred or maybe even a thou or two increase due to prop unloading, but we're short on the order of 10K rpm. I'd have to see convincing data to believe an increase of that magnitude - my ears sure don't beleive it.
As for #3, I must admit to being totally unaware of that phenomenon. Can you expand on it?