Reciprocity Measurements in Acoustical and Mechano-Acoustical Systems. Review of Theory and Applications

In 1873 Lord Rayleigh defined the General Reciprocity Theorem for stable, finite, lumped, passive, linear dynamical systems which only contain reversible (bilateral) elements. In spite of Rayleigh's authority, there have been a number of papers expressing doubts about the validity of the theorem for certain acoustical and mechano-acoustical systems. It can be shown however, that these papers are either based on too primitive experiments or on too simple theoretical models. On the other hand there are many publications showing the validity of the theorem. It can be concluded that Rayleigh's theorem is correct.

Flow disturbs reciprocity because it introduces non-reversible elements. Nevertheless, in many systems with flow, the disturbing effect is limited and reciprocity can still be applied.

The most straightforward application of reciprocity is the measurement of transfer functions. Second in order, because it requires two measurements instead of one, is the measurement of acoustical or mechanical source strength by reciprocal substitution. Third in order, because it needs three measurements, is the reciprocity calibration of microphones and hydrophones. Until 1970, only the latter application was known. In the period between 1970 and 1988 the author of this paper and his colleagues developed a number of methods for the reciprocal measurement of transfer functions and source strength in mechano-acoustical and acoustical systems. It was shown that the reciprocal measurement of transfer functions and of source strength has often great advantages over the direct alternatives. Since 1985, others joined the efforts. Particularly J. W. Verheij, F. J. Fahy and I. L. Vér have done very much for the further development and dissemination of the methods. Nowadays they are widely applied in the automotive industry and in the shipbuilding industry, where they are used for transfer path analysis, data gathering for prediction methods, source identification, source characterisation and for radiation measurements. The methods are also applied to aircraft, trains and buildings, but the dissemination of the methods for those systems is less than for cars, trucks and ships. Further applications concern the development of quiet gearboxes, the development of quiet fans, the prediction of the effects of sonic booms and the design of musical instruments.