1. v(e) = A eime (4-1) m then a perfect mrt' order spinning mode would be generated, and no energy would be lost to spurious spinning modes (there may still not he a radial mode match). An electromechanical loudspeaker, however, must couple to the duct through a finite-sizcd port, over which the velocity perturbation will bc nc;irly constant, but probably not sinusoidal in shape. In the same manner that the Fourier expansion of :I sqiiarc wave generates many harmonics of the fundaiiicntal period, the non-sinusoidal wall velocity will gcnerate spurious modes.
2. We assume that each ANC input port is being drivenbyanelectromechanical loudspeaker-typcdriver that provides a velocity source at the surface of the duct, over the port opening. The source is assuined to have a sufficiently high impedance that thc souicc is not affectedby the back-reactionofduct pressurc, cvcn up to 150-160 dB SPL, so that an arbitrary velocity boundary condition can he maintained.
3. v(e) = I/eiMo (4-4)