Ultrasonic measurement of the nonlinear parameter using third harmonic waves – 1. Theory

Theories and experiments related to the measurement of quadratic nonlinear parameters capable of evaluating the degree of damage or nonlinear material properties through the generation and measurement of second harmonics are relatively well established. In contrast, measurement of cubic nonlinear parameters using third harmonics is limited to relative measurement, and theories and experiments for absolute measurement have not been well developed. This paper deals with the theoretical aspects for accurately measuring cubic nonlinear parameters using third harmonics in isotropic materials. First, the amplitude of the third harmonic is obtained by applying the perturbation solution to the displacement-based wave equation, and the nonlinear parameter formula is defined. Since the amplitude of the ultrasonic wave actually measured deviates from the plane wave due to diffraction and attenuation, corrections of these effects are required. The attenuation and diffraction corrections are obtained from a multi-Gaussian beam model based on the integral solutions for the three-dimensional Westervelt equation. We present a formula for the cubic nonlinear parameter corrected for diffraction and attenuation. The results shown in this paper will be applied to the measurement of cubic nonlinear parameter in a separate paper