Finite-size effects on the quasistatic displacement pulse in a solid specimen with quadratic nonlinearity

There is an unresolved debate in the scientific community about the shape of the quasistatic displacement pulse produced by nonlinear acoustic wave propagation in an elastic solid with quadratic nonlinearity. Early analytical and experimental studies suggested that the quasistatic pulse exhibits a right-triangular shape with the peak displacement of the leading edge being proportional to the length of the tone burst. In contrast, more recent theoretical, analytical, numerical, and experimental studies suggested that the quasistatic displacement pulse has a flat-top shape where the peak displacement is proportional to the propagation distance. This study presents rigorous mathematical analyses and numerical simulations of the quasistatic displacement pulse. In the case of semi-infinite solids, it is confirmed that the time-domain shape of the quasistatic pulse generated by a longitudinal plane wave is not a right-angle triangle. In the case of finite-size solids, the finite axial dimension of the specimen cannot simply be modeled with a linear reflection coefficient that neglects the nonlinear interaction between the combined incident and reflected fields. More profoundly, the quasistatic pulse generated by a transducer of finite aperture suffers more severe divergence than both the fundamental and second order harmonic pulses generated by the same transducer.