Active control of the sound power scattered by a locally-reacting sphere

Active control of the sound power scattered by a sphere is theoretically investigated using spherical harmonic expansions of the primary and secondary fields. The sphere has a surface impedance that is uniform, real, and locally reacting while being subjected to an incident monochromatic plane wave. The scattered power is controlled with a number of monopole sources, initially on the surface of the sphere, and is expressed as the sum of squared amplitudes of the spherical harmonics due to both the scattered and control fields. This quadratic function is minimized to identify the optimal strengths for different numbers of control sources. At low frequencies, the scattered field is dominated by the first few spherical harmonic terms, and its power can be significantly reduced with a single controlling monopole for a soft or absorbent sphere and with two monopoles for a hard sphere. The number of secondary sources required to significantly attenuate the scattered field at higher frequencies is found to be proportional to the square of the frequency, and the attenuation also falls off rapidly if the secondary sources are moved away from the surface of the sphere, no matter what its surface impedance.