Difference-frequency generation in nonlinear scattering of acoustic waves by a rigid sphere

TL;DR

This paper models the generation of difference-frequency acoustic waves when two primary waves scatter off a rigid sphere, analyzing various configurations and identifying key factors influencing the nonlinear interaction and pressure decay.

Contribution

It introduces a partial-wave expansion method to analyze difference-frequency generation in nonlinear acoustic scattering involving a rigid sphere and multiple wave configurations.

Findings

Difference-frequency generation is mainly due to nonlinear interactions between scattered waves when sphere size is comparable to wavelengths.

In collinear scattering, primary incident wave interactions also significantly contribute.

Difference-frequency scattered pressure decays as r^{-1} log r and 1/r with distance.

Abstract

In this paper, the partial-wave expansion method is applied to describe the difference-frequency pressure generated in a nonlinear scattering of two acoustic waves with an arbitrary wavefront by means of a rigid sphere. Particularly, the difference-frequency generation is analyzed in the nonlinear scattering with a spherical scatterer involving two intersecting plane waves in the following configurations: collinear, crossing at right angles, and counter-propagating. For the sake simplicity, the plane waves are assumed to be spatially located in a spherical region which diameter is smaller than the difference-frequency wavelength. Such arrangements can be experimentally accomplished in vibro-acoustography and nonlinear acoustic tomography techniques. It turns out to be that when the sphere radius is of the order of the primary wavelengths, and the downshift ratio (i.e. the ratio between the fundamental frequency and the difference-frequency) is larger than five, difference-frequency generation is mostly due to a nonlinear interaction between the primary scattered waves. The exception to this is the collinear scattering for which the nonlinear interaction of the primary incident waves is also relevant. In addition, the difference-frequency scattered pressure in all scattering configurations decays as $r^{-1} \ln r$ and $1/r$, whereas $r$ is the radial distance from the scatterer to the observation point.