Interaction of elastic waves in incompressible soft materials containing weak quadratic and cubic nonlinearity

Scattering of sound from sound describes the process of mixing two or more waves to produce additional waves at the sum and difference frequencies of the original waves. Such wave mixing is well known within the topic of nonlinear acoustics. For compressible solids, Korneev and Demčenko [J. Acoust. Soc. Am. 135, 591-598 (2014)] developed a full accounting of all possible two wave interactions in weakly nonlinear and compressible elastic materials with inclusion of first-order nonlinear effects. The theory developed for compressible materials prevents those results from direct application to soft or nearly incompressible materials, which are important for biomedical ultrasound diagnostics. This article develops the full accounting of wave mixing configurations for two shear waves in weakly nonlinear incompressible elastic materials, including effects from first- and second-order nonlinearity. In turn, the topic of nonlinear wave mixing is fully extended into a previously unexplored class of materials. Quantitative results based on parameters from pig brain highlight the potential of new wave mixing configurations for diagnostic ultrasound.