Exploration of third-order nonlinear acoustics for projection of narrow-beam lower-frequency underwater beams

Abstract

Projection systems are considered where two or three frequencies (e.g., f1, f2, and/or f3) are simultaneously projected into water in a parallel fashion. High near-field amplitudes produce beams of frequencies equal to any linear combination of f1, f2, and f3, with integer coefficients n1, n2, and n3 (possibly zero or negative). Interest here is in the case where the magnitudes of the coefficients sum to three, associated with a third-order nonlinearity. The question addressed is that of how large the amplitude of the far-field signal will be. The considered causes of the nonlinearities are (1) the convective derivative term in the total time derivative of the fluid velocity, and (2) the higher coefficients in the expansion of the fluid density in terms of the deviation of the pressure from its ambient value. These coefficients are derived from data reported by Holton et al. [J. Acoust. Soc. Am. (1968)] on the sound speed in water. A perturbation technique is explored starting with the basic nonlinear equations of compressible time-dependent fluid dynamics, where at each step one has a simultaneous set of coupled linear and homogeneous equations with the source terms dependent on the solutions of the analogous equations corresponding to the previously considered orders.