## 1 Introduction
It transfers time-domain signals into frequency-domain to analyze the information of frequency which would be related to the discontinuities of material. This phenomenon is due to the incident wave distorted by the nonlinear elastic response of material so that the higher harmonic waves are generated in the transmitted wave [18]. This nonlinear wave behavior appears in the form of dislocation substructures, such as slip bands. The slip bands would accumulate in the grain or at the grain boundaries which may cause strain localization. A number of investigators have studied the nonlinear ultrasonic technique to measure the effect of dislocation density. In the case of precipitate, the microstrain induced plasticity would change between the precipitate and the matrix. In addition, the precipitate would interact with dislocations which lead to changes in the nonlinear ultrasonic parameter [19]. It is expected to be much more sensitive to microstructure than the linear ultrasonic wave. From the studies above, the nonlinear ultrasonic technique shows the potential for characterizing the microstructural changes. The generation of higher order harmonic waves in a solid material is related to the nonlinear relationship between the stress and strain of material and can be expressed by the nonlinear [PERSON]'s law as shown in Equation (1) [20]  
\[\sigma=E\varepsilon+E\beta\varepsilon^{2}, \tag{1}\]  
where \(\sigma\) is stress, \(\varepsilon\) is strain, \(E\) is Young's modulus, and \(\beta\) is the nonlinear acoustic coefficient. Considering the case of a single frequency ultrasonic longitudinal wave propagating in a material, and assuming that the attenuation can be neglected, the equation of motion for longitudinal waves in the material can be represented by  
\[\rho\frac{\partial^{2}u}{\partial t^{2}}=\frac{\partial\sigma}{\partial x}, \tag{2}\]  
where \(\rho\) is the material density, \(u\) is the displacement, and \(x\) is the propagation distance. The relationship between strain and displacement for a one-dimensional longitudinal wave is given as  
\[\varepsilon=\frac{\partial u}{\partial x}, \tag{3}\]Substitution of Equations (1) and (3) into Equation (2) yields  
\[\rho\frac{\partial^{2}u}{\partial t^{2}}=E\frac{\partial^{2}u}{\partial x^{2}}+2 E\beta\frac{\partial u}{\partial x}\frac{\partial^{2}u}{\partial x^{2}}, \tag{4}\]  
In order to obtain a solution, the displacement is assumed as  
\[u=u_{0}+u^{\prime}, \tag{5}\]  
where \(u_{0}\) is a solution with \(\beta=0\) and \(u^{\prime}\) is the first order perturbation solution. Then, the solution of Equation (4) can be obtained as  
\[u=A_{1}\cos(kx-wt)-\frac{\beta A_{1}^{2}k^{2}x}{8}\sin 2(kx-wt), \tag{6}\]  
where \(A_{1}\), \(k\), and \(w\) are the amplitude of the fundamental frequency component, the wave number, the angular frequency, respectively. The amplitude of the second harmonic wave can be described as  
\[A_{2}=\frac{\beta A_{1}^{2}k^{2}x}{8}, \tag{7}\]  
Therefore, the parameter \(\beta\), which is also called the nonlinear parameter, can be evaluated from \(A_{2}\), as shown in Equation (8)  
\[\beta=\frac{8A_{2}}{k^{2}xA_{1}^{2}}, \tag{8}\]  
In this paper, the frequency of longitudinal wave and the transmitting distance are fixed. The parameter \(\beta\) is proportional to \(\frac{A_{2}}{A_{1}^{2}}\). For convenience, a relative nonlinear ultrasonic parameter is adopted and determined as  
\[\beta^{\prime}=\frac{A_{2}}{A_{1}^{2}}, \tag{9}\]  
Therefore, the variation of microstructural change could be characterized by measuring \(A_{1}\) and \(A_{2}\). This is a feasibility study intended to characterize microstructural change of the cast stainless steel using the linear and nonlinear ultrasonic techniques. Measurements of the ultrasonic parameters were carried out on the cast stainless steel specimens with different aging times and cold work. The relationship between ultrasonic parameters and the material property were correlated based on the ultrasonic responses to the variations of aging times.