Second-harmonic generation in focused beam fields of phased-array transducers in a nonlinear solid with a stress-free boundary

In nonlinear acoustic harmonic generation in solids with stress-free boundaries,such a boundary is known to destructively change the second harmonic generation,and the pulse-echo method is not practically applicable.Focused beams have often been used for fluid nonlinearity and biomechanical imaging in pulse-echo test setups.This paper considers the focused beam fields of linear phased-array transducers to ensure that pulse-echo harmonic generation can be applied to solids with stress-free boundaries.The fundamental and second-harmonic beam fields that are focused and reflected at the stress-free and rigid boundaries are calculated and their properties are investigated in terms of the received average fields.The phase difference between the two second-harmonic components after reflection from the boundary—that is,the reflected and the newly generated second harmonic—is emphasized.The phase difference is used to explain the improved and accumulated second harmonic observed in the simulation results.