Compressible potential flows over nonlifting, hovering helicopter blades are described by suitable linear and nonlinear equations of motion for subsonic and transonic cases, respectively. Analytical and numerical results are presented for the linearized subsonic three-dimensional flow in the tip region. When the tip Mach number is transonic, the flowfield is calculated using a computational method that is a formal extension to three dimensions of recently developed nonlinear two-dimensional relaxation schemes. Calculations are presented for rectangular blades with 6% thick biconvex sections. Calculations show the relative importance of tip Mach number and aspect ratio on the growth and extent of shock waves in the tip region, and indicate a significant reduction in shock strength with decreasing aspect ratio. ELICOPTER rotor blades that can operate at high tip Mach numbers without the penalties usually associated with super-critical flow would permit high-forward speeds at relatively low-advance ratios. As the advance ratio decreases, the portion of the retreating blade that is stalled would be diminished, the portion of the retreating blade that is unstalled could operate at smaller angles of attack, and the advancing blade could generate lift without also generating an excessive rolling moment. Vibratory loads would be diminished, and severe unsteady pitching moments on the lifting portion of the retreating blade would decrease if that portion could usefully operate at higher tip speeds and smaller angles of attack. The design of rotor blade geometries that can support a transonic flowfield that is nearly shock-free requires a method for computing the detailed pressure distribution on the blade for a specified rotor geometry and advance ratio. However, the unsteady aerodynamic environment of a lifting rotor that is operating at or near the critical Mach number is extremely complicated. It is easy to appreciate the difficulty of describing such flowfields when one recalls that we do not yet have an adequate model for describing the details of the flow about a lifting, hovering rotor operating in an essentially incompressible flow. Our purpose here is therefore to focus on a portion of the high-speed rotor problem that is sufficiently limited to allow for some analytical and numerical treatment, but which retains important linear and nonlinear features of the flow. We consider a two-bladed nonlifting and hovering rotor operating in the range of tip Mach numbers where compressibility effects are important. The pressure distribution near the rotor tip is studied. In this region, the flowfield is fully three-dimensional, and blade element (strip) theory is in no sense a reasonable approximation to the rotor aerodynamics. It is assumed that the flow is inviscid. The flow is steady relative to a coordinate system attached to the blades. Analytical and numerical results presented here are based on suitable approximations to the full three-dimensional potential equation for compressible flow. When the tip Mach number is subcritical, the potential equation is the usual acoustic