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acoustic Aerodynamic angle of attack approximation assumed asymptotic expansions Bessel functions blade row blade slap boundary condition calculate the sound chord component of velocity composite solution compressible dipoles consider dimensional dipole of strength distance distribution of sources effect error example extended source factor far-field pressure Fluid Mechanics force distribution free stream frequency Garrick and Watkins gives Graph Helicopter rotor incompressible flow incompressible singularities infinite inner flow inner solution integral Kutta condition applied Laplace's equation lattice of airfoils length scales lift force Mach number MAE method method of MAE Muller and Obermeier neglected nondimensional nonlinear obtain oscillating outer solution plate Poisson summation formula pressible pressure field due procedure Pulsating airfoil quadrupole rotor blade shed vortices singularity distribution sinusoidal small perturbation sound field sound produced Sound wave incident source of strength sources and dipoles steepening theory three dimensions three-dimensional trailing edge velocity potential wake wave equation wavelength zero