How does the V22 Osprey manage such high rotor speeds without shock waves?
The Osprey's engines drive the rotors at 412 RPM in heli mode or 333 RPM in forward-flight mode, according to this article. With a rotor radius of 5.8m, that puts the tip velocity at 250m/s, or about 75% the speed of sound, at the higher RPM. So they aren't near supersonic yet because the blades are short enough.

But there is a possibility of reducing blade length for higher forward speed and RPM, and there is research into how to make that possible on-the-fly:


  Helicopter performance depends on the length of the rotor blades. For heavy lifting, a large rotor works best, but short blades reduce drag and ultimately allow for higher maximum speeds. Farhan Gandhi, a Penn State University professor of aerospace engineering, has devised an elegant, simple way to achieve both configurations in the same aircraft, using the same rotor.