## 2 Non-Linear Parameters of Ultrasonic Waves  
Higher harmonic is a phenomenon related to the non-linear elastic behavior of the material, and the material properties of the medium determine the magnitude of the harmonics. As a result, the microscopic changes (creak, pore, etc.) of materials can be assessed by monitoring the generation of the higher harmonics in ultrasound [7; 12; 13]. For most media, the stress \(\sigma\) and strain \(\varepsilon\) of the material are non-linearly related. In a very small interval, the non-linear relationship between stress \(\sigma\) and strain \(\varepsilon\) can be described by [PERSON]'s law. Equation (1) shows the non-linear expression of [PERSON]'s law [7; 14; 15]:  
\[\sigma=E\cdot\varepsilon\cdot(1+\beta\varepsilon+\cdots) \tag{1}\]  
where \(E\) is [PERSON]'s modulus of the material and \(\beta\) is the second-order elasticity coefficient of the material, also known as the non-linear parameter.  
When ultrasound waves propagate in a non-linear medium, the incident wave at one end of the medium is a single-frequency ultrasound longitudinal wave and the signal is received by an ultrasound transducer at the other end. The second-order equation of motion simplified to one dimension, is as follows:  
\[\rho\cdot\frac{\partial^{2}l}{\partial l^{2}}=\frac{\partial\sigma_{xx}}{ \partial x} \tag{2}\]where \(\rho\) is the medium density, \(x\) is the medium coordinates, \(\sigma_{xx}\) is the normal stress along \(x\), \(t\) is time, and \(l\) is the displacement of the mass point located at \(x\) within the medium. Then the non-linear fluctuation equation for the displacement \(l(x,t)\) of the mass is [16]:  
\[\frac{\partial^{2}l}{\partial t^{2}}=\
u^{2}\cdot(1+2\beta\cdot\frac{\partial l }{\partial x})\cdot\frac{\partial^{2}l}{\partial x^{2}} \tag{3}\]  
where \(\
u=\sqrt{E/\rho}\) is the longitudinal wave velocity in the medium, \(\beta\) is a non-linear parameter and depends on the structural properties of the material [17]. Applying perturbation theory in Equation (3), the displacement \(l\) is as shown in Equation (4):  
\[l=l_{0}+l_{n1} \tag{4}\]  
where \(l_{0}\) represents the initial excitation wave and \(l_{n1}\) is the first-order perturbation term, then Equation (3) can be written as:  
\[\frac{\partial^{2}l_{0}}{\partial t^{2}}+\frac{\partial^{2}l_{n1}}{\partial t ^{2}}=\
u^{2}\cdot(\frac{\partial^{2}l_{0}}{\partial x^{2}}+\frac{\partial^{2 }l_{n1}}{\partial x^{2}})\cdot[1+2\beta\cdot(\frac{\partial l_{0}}{\partial x }+\frac{\partial l_{n1}}{\partial x})] \tag{5}\]  
Assume that the waveform function expression for the initial excitation wave \(l_{0}\) is:  
\[l_{0}=A_{1}\cos(kx-\omega t) \tag{6}\]  
where \(\omega=2\pi f\) is the angular frequency of the excitation wave and \(f\) is the angular frequency of the excitation wave. Since \(\lambda=\
u/f\) is the wavelength, then \(k=\omega/\
u=2\pi/\lambda\) is the number of waves. By solving Equations (3)-(6) and ignoring high-order small quantities in the process of solving, the displacement expression 1 with the first-order disturbance solution can be obtained [7; 16]:  
\[l=l_{0}+l_{n1}=A_{1}\cos(kx-\omega t)-A_{2}\sin(2 kx-2\omega t) \tag{7}\]  
with  
\[A_{2}=\frac{\beta}{8}A_{1}^{2}k^{2}x \tag{8}\]  
It can be seen that Equation (8) is the second-harmonic component, the magnitude of which depends on \(\beta\). This means that the parameter \(\beta\) can be evaluated in terms of the magnitude of the second-harmonic component, as shown in Equation (9):  
\[\beta=\frac{8}{k^{2}x}\cdot\frac{A_{2}}{A_{1}^{2}} \tag{9}\]  
The positions for the ultrasonic transmission and reception in this study were fixed. Therefore, the relative non-linear parameter \(\beta_{r}\) in Equation (10) was used to denote the non-linear parameter required in this study.  
\[\beta_{r}=\frac{A_{2}}{A_{1}^{2}} \tag{10}\]  
In this study, the units of the fundamental amplitude \(A_{1}\) and the second-harmonic amplitude \(A_{2}\) are V. The unit of the relative non-linear parameter is V\({}^{-1}\).