Determination of the absolute nonlinear parameters using a pulse-echo method in a terminal impedance-mismatched system

The pulse-echo method is desirable in practice because it enables single-sided measurement of the absolute nonlinear parameter (β), allowing quantitative characterisation of material properties, such as early damage and elastic constants. Transducer calibration is an essential step to determine the absolute β. However, the current transducer calibration method requires dedicated systems or specific apparatuses for the pulse-echo setup. Impedance-mismatch configurations involving different types of apparatuses have been employed in various studies, but there has not been work investigating and solving the effects of terminal impedance on the determination of β. In this study, the theory of the terminal impedance effect on the transfer function, as well as the β measurements is derived. A practical calibration method is proposed to eliminate the influence of the impedance-mismatched problem and the restrictions of specific apparatuses or experimental configurations. Three simple and impedance-mismatched experimental configurations were designed with general apparatuses. Nonlinear experiments were performed to determine the β of water using planar and focused transducers in the pulse-echo systems with these configurations. The measured β agrees with the reference value, which verifies the effectiveness of the proposed correction method. The measured results using the three configurations are also discussed to give practical advice for building a nonlinear pulse-echo setup.