Difference frequency pressure within the interaction region of a parametric array

Abstract

A theoretical model that predicts difference frequency pressure within the interaction region of a parametric array is presented. Linear and nonlinear attenuation are neglected, and it is assumed that the primary signals are spherically spreading throughout the interaction region. It is shown that with these assumptions the three-dimensional scattering integral can be reduced to a single integral if the primary pressure fields are axially symmetric. Analyses of the error introduced by assuming the primary fields are spherically spreading and by neglecting linear attenuation are presented. Theoretical results compare favorably to p’revious experimental results. The results indicate that diffraction effects are more important than attenuation when the observation point is within the interaction region and when primary signal level reduction is due more to spherical spreading than attenuation. Subject Classification: 25.35; 20.30, 20.15.

Nonlinear generation of elastic waves in crystalline rock

“…Accordingly, we searched for a signal at the difference frequency that occurred when the frequencies of the driving transducers were varied. A second criterion is that the amplitude of the difference frequency beam Aa• c should be proportional to the product of the amplitudes of the driving signals [Jones and Kobett, 1963;Rolleigh, 1975]. Because these two tests did not necessarily distinguish between nonlinear signals originating in the rock and those produced in the external transducers and electronics or through surface wave interaction, we also checked for directional effects.…”

“…The signal at the difference frequency Aris of particular interest since its amplitude decays more slowly with distance because of its longer wavelength. Thus despite the fact that the energy conversion from the primary input beams to the difference frequency beam is inefficient [Taylor and Rollins, 1964], the combined effects of collimation and lower spatial attenuation produce a useful low-frequency acoustic source; exploitation of nonlinear beam generation has led to develCopyright 1987 by the American Geophysical Union.…”

The Angular Response of Parametric Arrays—general Numerical Solution

“…However, Rolleigh [5] proved that the error introduced by approximating these collimated plane waves by spherical waves inside the Rayleigh distance R P at the primary frequency is negligible if the observation point is slightly beyond R P . The calculation of the three-dimensional integral is accomplished in the MATLAB environment.…”

“…It is found that the suggested method for the three-dimensional numerical integration has good convergence. Rolleigh [21] pointed out that there were controversial results concerning the di!erence frequency beamwidth of parametric arrays. For instance, Rolleigh's results show a decrease in di!erence frequency beamwidth as the observation distance increases.…”

“…This accounts for the di!raction e!ects due to spherically spreading primary waves, and he showed that the second order perturbation solution of this equation is in the form of a three-dimensional scattering integral similar to Westervelt's scattering integral. Moreover, a theoretical model that predicts the di!erence frequency pressure within the interaction region of a parametric array is given by Rolleigh [5]. Here, both linear and non-linear absorptions are neglected and it is assumed that the primary waves are spherically spreading throughout the interaction region.…”