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Priority for discovery and explanation of the Parametric Array owes to Westervelt, although experimental work was contemporaneously underwater in the former Soviet Union. The phenomenon of the parametric array was seen first experimentally by Westervelt in the 1950's and theoretically presented first in 1960 at a meeting of the Acoustical Society of America, as an extension of Westervelt's classic work on the nonlinear Scattering of Sound by Sound.
In Westervelt's original papers on the Scattering of Sound, it was concluded that two non-collinear sound beams do not scatter to produce sum or difference frequency fields to points lying outside of the interaction region of the beams. It is somewhat remarkable that this result, although quickly confirmed by Bellin and Beyer [38 ], and by experimental work by a number of other researchers, this result has been somewhat conroversial even up to the present day, being contradicted by a number of theoretical studies. However, the more recent and highly accurate experiment by Roy and Wu [6] did conclusively demonstrate that Westervelt’s original non-scattering result is correct, however, contrary theoretical predictions have been published and it would appear, have never been corrected or retracted to bring them into line with experiment. A corollary to this result is, that collinear beams of sound will produce scattering and a significant interaction, which is especially interesting at the difference frequency, effectively generating and emiting sound that survives beyond the interaction region. This phenomenon was given the name “parametric array” by Westervelt [2]. Combining these results, a theorem can be stated thus: “there is no scattering of sound by sound, except for Westervelt’s parametric array”.
The foundation for Westervelt's theory of sound generation and scattering in nonlinear acoustic media owes to the equation of Lighthill [36,37 ]. The application of Lighthill’s theory in the nonlinear acousic realm yields the Westervelt-Lighthill (nonlinear acoustic wave) Equation, or WLE [1].
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