# A Feasibility Study for a Nonlinear Guided Wave Mixing Technique
## 2 Theory
### Guided Wave Mixing Theory
These are input primary waves in this mixing theory study. Because the directions show opposite, the marks inside of the sine functions in Equations (9) and (10) are different. Moreover, Equations (6)-(8) express nonlinear wave equations ([PERSON], 1961) [32]. These nonlinear wave equations represent wave behavior in nonlinear conditions. The nonlinearity \((\beta_{s})\) is defined as \(\beta_{s}=-\big{(}c_{l}^{2}+C_{166}/\rho\big{)}\). Based on these derived equations and two input primary waves (Equations (9) and (10)), mixing wave components can be derived. Equation (11) can be derived by substituting two primary waves based on Equations (9) and (10) in Equation (7). After a few rearrangement steps, Equation (12) is obtained. Here, there are two out-of-plane dominant mixing waves. The frequency of the first one is the difference of the two primary wave frequencies \((\omega_{1}-\omega_{2})\), while that of the second one is the sum \((\omega_{1}+\omega_{2})\). The point of interest is the difference-frequency mixing signal for dominant out-of-plane displacement, so the sum-frequency component is ignored. \[u_{3}=\left(\frac{\partial\phi}{\partial x_{3}}-\frac{\partial\varphi_{2}}{ \partial x_{1}}\right)=\left(\frac{\partial}{\partial x_{3}}(\Phi(x_{3}))-ik \Psi(x_{3})\right)\exp(i(kx_{1}-\omega t)) \tag{2}\]1  
\[u_{3}=\left(\frac{\partial\phi}{\partial x_{3}}-\frac{\partial\varphi_{2}}{ \partial x_{1}}\right)=\left(\frac{\partial}{\partial x_{3}}(\Phi(x_{3}))-ik \Psi(x_{3})\right)\exp(i(kx_{1}-\omega t)) \tag{2}\]2  
After perturbation method processes and several steps, the final form of the mixing component is expressed as Equation (13) [33; 34]. It contains the difference frequency of primary waves. One in-plane dominant wave is mixed with the other one out-of-plane dominant wave to generate an out-of-plane dominant mixing signal with the difference of the primary waves' frequencies: \(u(\omega_{1})+v(\omega_{2})\to v(\omega_{1}-\omega_{2})\). Whether the propagation direction is the same as that of a longitudinal component defined can be determined by checking the mark in the sine function of Equation (13). Figure 3 explains this. \[u_{3}=\left(\frac{\partial\phi}{\partial x_{3}}-\frac{\partial\varphi_{2}}{ \partial x_{1}}\right)=\left(\frac{\partial}{\partial x_{3}}(\Phi(x_{3}))-ik \Psi(x_{3})\right)\exp(i(kx_{1}-\omega t)) \tag{2}\]3 Diagram of the guided wave mixing concept on a plate. Equation (14) shows the amplitude term of a mixing signal (Equation (13)). Even though the concept of the mixing technique is similar to second harmonic generation, experimental signal detection is much more difficult. A few considerations are therefore needed to detect a mixing signal. \[u_{3}=\left(\frac{\partial\phi}{\partial x_{3}}-\frac{\partial\varphi_{2}}{ \partial x_{1}}\right)=\left(\frac{\partial}{\partial x_{3}}(\Phi(x_{3}))-ik \Psi(x_{3})\right)\exp(i(kx_{1}-\omega t)) \tag{2}\]4  
\[u_{3}=\left(\frac{\partial\phi}{\partial x_{3}}-\frac{\partial\varphi_{2}}{ \partial x_{1}}\right)=\left(\frac{\partial}{\partial x_{3}}(\Phi(x_{3}))-ik \Psi(x_{3})\right)\exp(i(kx_{1}-\omega t)) \tag{2}\]5  
\[u_{3}=\left(\frac{\partial\phi}{\partial x_{3}}-\frac{\partial\varphi_{2}}{ \partial x_{1}}\right)=\left(\frac{\partial}{\partial x_{3}}(\Phi(x_{3}))-ik \Psi(x_{3})\right)\exp(i(kx_{1}-\omega t)) \tag{2}\]6  
The mixing signal amplitude (Equation (14)) is a function of the primary wave frequencies. Additionally, when the denominator becomes zero, the value becomes infinite. Based on these concepts, the primary wave frequencies are decided. Equation (15) denotes the frequency ratio for the theoretical maximum amplitude. If a specimen is set, the longitudinal and transverse wave velocities are constants, because longitudinal and transverse wave velocities are determined by the material property. Therefore, their ratio is also constant. For an aluminum specimen (Al1050), the longitudinal and transverse wave velocities are (6.36 and 3.01) mm/\(\upmu\)s, respectively.