Second-harmonic generation in focused shear wave beams in tissue-like media with arbitrary amplitude shading at the source

Second-harmonic generation in a focused shear wave beam with Gaussian amplitude shading and radial polarization can be described analytically using a perturbation solution [Kaufinger et al., JASA 155, A350 (2024)]. Recently, focused radially polarized shear wave beams were generated in tissue-mimicking gelatin phantoms with an oscillating concave circular piston [Cormack et al., IEEE TBME 71, 621 (2024)]. To model second-harmonic generation in a focused shear wave beam with source conditions more relevant to the experiments, i.e., without Gaussian amplitude shading, the nonlinear evolution equation is solved numerically by successive approximations in k-space using a fourth-order Runge–Kutta stepping scheme. The linear solution is obtained first, which is then substituted into the quadratic terms to solve for the nonlinearly generated second harmonic. Emphasis is placed on how more realistic source conditions, including super-Gaussian amplitude shading or source conditions estimated from measurements, affect the generation of the second harmonic. Attention is also devoted to the longitudinal displacement field that is calculated locally from the transverse displacement field and the assumption of material incompressibility. [PGK is supported by the ARL:UT Chester M. McKinney Graduate Fellowship in Acoustics.]