Similarity of a Fourier transform generalization of the Earnshaw solution for planar waves to an interacting wave model for finite amplitude effects in sound beams

Abstract

The Earnshaw solution for a finite amplitude planar wave, which displays amplitude dispersion, is valid for arbitrary excitation f(t) on a boundary. The form that results for small acoustic Mach numbers when f(t) is represented as the inverse of its Fourier transform F(ω) may be considered to be a coordinate straining of the linearized signal, in which the transformation has the appearance of a Fredholm integral equation in the frequency domain. In the case of an acoustic planar wave, the phase speed of all frequency components is the same, independent of frequency. A generalization of that result is obtained if one considers the possibility that the cumulative growth effect is an arbitrary function of distance. Rather than being an abstraction, this form is shown to be analogous to an earlier solution for finite amplitude sound beams [J. H. Ginsberg, H. C. Miao, and M. A. Foda, J. Acoust. Soc. Am. Suppl. 1 81, S25 (1987)]. The signal in that analysis was represented in terms of a spectrum of transverse wavenumbers, rather than frequencies, and the signal for each wavenumber was formed from interacting quasiconical waves, rather than a single planar wave. Nevertheless, the two problems share a common mathematical structure. A discussion of numerical algorithms for such models leads to some surprising observations regarding the implicit functional form of the Earnshaw solution. [Work supported by ONR.]