Asymptotic fourier analysis of a “sawtooth like” wave for dual frequency source excitation

Abstract

An exact weak shock zone solution was found previously for Burgers' equation for plane waves. An approximate solution for spherical waves also was obtained. Both of these solutions held for dual frequency excitation at the source. In the work reported in the present paper, an asymptotic formula was derived which yields harmonic amplitudes at a point beyond the “sawtooth like” zone. The formula's predicted harmonic amplitudes have been found to agree well with numerical results for a specific dual frequency case. Excellent agreement also was obtained with the well-known solution for a monochromatic source. The new formula predicts that for certain frequency ratios of the primary signals the amplitude of the parametrically generated difference signal will exceed that of a directly projected signal at the same frequency and with the same total input power. This result holds for infinite planar and spherical source geometries. The formula should be useful in estimating beam axis values of the harmonic amplitudes for practical source geometries by applying the plane wave version in the near field and the spherical version in the far field. Such information can be of value to those who are constructing mathematical models of parametric array behavior.