The minimum power output of free field point sources and the active control of sound

Abstract

Optimization techniques are used to enable an unambiguous quantification of the degree to which the sound power output of a distribution of point primary sources can be reduced by the addition of a number of point secondary sources. The near field technique developed by Levine is used to calculate the total power output from an arbitrary number of point primary and secondary sources. This power output is a quadratic function of the complex strengths of the secondary point sources. For a given arrangement of primary and secondary sources, this quadratic function has a unique minimum value associated with an optimal set of secondary source complex strengths. Results are presented for the minimum power radiated by the combination of a single point primary source and various arrangements and numbers of point secondary sources. In particular it is demonstrated that, for the number of secondary sources considered, substantial reductions in total power output can be achieved only if the secondary sources are separated from the primary source by a distance which is less than one half wavelength at the frequency of interest. It is possible, however, to produce net source distributions of unexpectedly low radiation efficiency with a relatively small number of secondary sources placed close to the primary source.