Professor (now Emeritus) Peter J. Westervelt, in his original treatment of the scattering of sound by sound, noted that sound fields propagating and interacting in real (and therefore, nonlinear) media can emit, ampify, and absorb sound. This profound observation has yielded a number of important applications, most notably the parametric acoustic array [2]. The parametric array allows the generation of narrow, nearly sidelobe free beams of low frequency sound, through the mixing of higher frequency sound wave beams, effectively overcoming the diffraction limit (a kind of spatial 'uncertainty principle') associated with linear means of sound generation and scattering. Applications include underwater sound [1], medical ultrasound, underground sesimic prospecting, and directional high-fidelity commercial audio systems.
Priority for discovery and explanation of the Parametric Array owes to Westervelt, although experimental work was contemporaneously underway 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, 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 was and 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 of Westervelt's result on the scattering of sound by sound 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]. Solutions to the WLE have been developed using Green's functions and Parabolic Equation Methods, most notably the KZK equation. A mathematical formalism using Fourier Operator methods in wavenumber space, was also developed by Westervelt, and generalized by the Woodsum, for solving the WLE in a most general manner [1,3,11]. The objective in developing this solution has been to remove approximations in the treatment of field geometry, attenuation mechanisms, and order of approximation for the medium equation of state, for each order of scattering, to produce a completely accurate prediction tool for nonlinear acoustic scattering phenonmena. The solution method is formulated in Fourier (wavenumber) space in a representation related to the beam patterns of the primary fields generated by linear sources in the medium. It is worth noting that the resulting equations of the theory resemble in many respects the computational approaches of quantum field theory , most especially quantum electrodynamics, which is considered the most accurate of all physical theories[17,21].
Because such diverse fields as General Relativity [28], Nonlinear Optics [20], and particular theoretical treatments of the Scattering of Light-by-Light [18,19] all deal with similar types of nonlinear field theory and nonlinear wave equations it is though that the theoretical techniques, particularly the Fourier methods, could be potentially applical to these other domains as well as to nonlinear acoustics. In this more general context, comparison of operator solution of Westervelt, when applied to the absorption of sound by sound [15, 16] and to the redistribution of acoustic spectral energy due to nonlinear scattering to produce agreement with the phonon Black Body radiation specturm, [41] in full agreement with calculations by quantum field theory, are worth reviewing. The ability of the present theory to obtain results normally requiring a resort to quantum postulates, from a nonlinear classical theory, is indeed interesting, and perhaps deserving of further examination.
References
[1] H.C. Woodsum and P.J. Westervelt, "A General Theory for the Scattering of Sound by Sound", Journal of Sound and Vibration (1981) 76(2), 179-186.
[2] Peter J. Westervelt, "Parametric Acoustic Array", Journal of the Acoustical Society of America, Vol. 35, No. 4 (535-537), 1963.
[3] H.C. Woodsum, Ph.D. Thesis in Physics, Brown University (1979), "A General Theory for Sound-Sound Scattering with Parametric Array Applications", published by University Microfilms, Inc.
[4] Mark B. Moffett and Robert H. Mellen, "Model for Parametric Sources",
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[5] Mark B. Moffett and Robert H. Mellen, "On Parametric Source Aperture Factors", J. Acoust. Soc. Am. Vol. 60, No. 3, Sept. 1976 (see Figure 2)
[6] Ronald A. Roy and Junru Wu, "An Experimental Investigation of the Interaction of Two Non-Collinear Beams of Sound", Proceedings of the 13th International Symposium on Nonlinear Acoustics, H. Hobaek, Editor, Elsevier Science Ltd., London (1993).
[7] "Introduction to Quantum Mechanics", Chapter IX-32, Linus Pauling and E. Bright Wilson, Dover Publications, New York.
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[91 Harvey C. Woodsum and William Hogan, "Work by Virtual Sources Against In-phase and Quadrature Components of a Pressure Field: Impact on Nonlinear Attenuation", J. Acoust. Soc. Am., Vol. 88 Suppl. 1 (4PA4), Nov. 1990
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[35] Pierre Cervenka and Pierre Alais, “Optimization of a Parametric Transmitting Array”, Proceedings of the 12th International Symposium on Nonlinear Acoustics, Austin Texas, USA, 27-31 August, 1990. Elsevier Science Ltd., London (1990), Paper F3.
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[42] H.C. Woodsum, infinity, Bull. Of Am. Phys. Soc., Fall 1980; “A Boundary Condition Operator for Nonlinear Acoustics”