Cylindrically converging nonlinear shear waves

Abstract

The low shear moduli of soft elastic media permit the generation of shear waves with large acoustic Mach numbers that can exhibit waveform distortion and even shock formation over short distances. Waves that converge onto a cylindrical focus experience significant dispersion, causing waveforms at the focus and in the post-focal region to differ significantly from the source waveform even in the absence of nonlinear distortion. A full-wave model for nonlinear shear waves in cylindrical coordinates that accounts for both quadratic and cubic nonlinearity is developed from first principles. For the special case of an infinite cylindrical source with particle motion parallel to the axis, for which nonlinearity is purely cubic, the nonlinear wave equation is solved numerically with a finite-difference scheme. The full-wave model is compared with a piecewise model based on a generalized Burgers equation for cylindrically converging waves outside of the focal region and linear diffraction theory in the focal region. For waveforms with wavelength much smaller than the source radius, conditions are explored for which the approximate piecewise model shows good agreement with the full-wave model. [Work supported by the ARL:UT McKinney Fellowship in Acoustics.]