# A Feasibility Study for a Nonlinear Guided Wave Mixing Technique
## 2 Theory  
### Guided Wave Mixing Theory  
A bulk wave consists of a longitudinal wave and transverse wave, and has the same velocity, regardless of frequency. The characteristics of bulk waves make studying and applying them easy. Guided wave study is more difficult to use for experiments. The most important feature of a guided wave is dispersion. The wave velocity has a variety of value ranges that depend on the mode and frequency, and the specimen depth makes the magnitude of displacement different. Guided wave displacement consists of in-plane displacement \((u_{1})\), and out-of-plane displacement \((u_{3})\). In-plane displacement takes place horizontal to the propagation direction, while out-of-plane displacement takes place vertical to that direction. Figure 1 shows the in-plane and out-of-plane placement coordinate diagram. In the figure, \(x\) and \(t\) denote the propagation direction and thickness, respectively. These components are expressed in Equations (1) and (2) [30; 31]. These indicate scalar potential (\(\varphi\)) and vector potential (\(\psi\)), as in Equation (3). This is called Helmholtz decomposition. Equations (1) and (2) can be simplified into Equations (4) and (5):  
\[u_{1}=\left(\frac{\partial\phi}{\partial x_{1}}+\frac{\partial\varphi_{2}}{ \partial x_{3}}\right)=\left(ik\Phi(x_{3})+\frac{\partial}{\partial x_{3}}( \Psi(x_{3}))\right)\exp(i(kx_{1}-\omega t)) \tag{1}\]  
\[u_{3}=\left(\frac{\partial\phi}{\partial x_{3}}-\frac{\partial\varphi_{2}}{ \partial x_{1}}\right)=\left(\frac{\partial}{\partial x_{3}}(\Phi(x_{3}))-ik \Psi(x_{3})\right)\exp(i(kx_{1}-\omega t)) \tag{2}\]  
\[u=\
abla\phi+\
abla\times\varphi \tag{3}\]  
\[u_{1}^{(1)}=A_{1}\exp i(k_{1}x-\omega_{1}t) \tag{4}\]  
\[u_{3}^{(1)}=A_{2}\exp i(k_{3}x-\omega_{1}t) \tag{5}\]  
where, \(k\) and \(\omega\) are the wave vector and frequency, respectively, and \(A\) shows the positive wave amplitude. In the bulk wave, the longitudinal component is horizontal to the propagation direction, while the transverse component is vertical to the propagation direction. This concept becomes equivalent to the in-plane and out-of-plane displacement of the guided wave. These displacement components cannot be separated out in one guided wave signal, but the ratio of the two components changes with the mode and frequency. Analyzing wave structure helps to determine the component ratio. \[u_{tt}-c_{l}^{2}u_{aa}=\left(3c_{l}^{2}+C_{111}/\rho\right)u_{a}u_{aa}+\left( c_{l}^{2}+C_{166}/\rho\right)(v_{a}v_{aa}+w_{a}w_{aa}) \tag{6}\]  
\[v_{tt}-c_{s}^{2}v_{aa}=\left(c_{l}^{2}+C_{166}/\rho\right)(u_{a}v_{aa}+v_{a}u_ {aa}) \tag{7}\]  
\[w_{tt}-c_{s}^{2}w_{aa}=\left(c_{l}^{2}+C_{166}/\rho\right)(u_{a}w_{aa}+w_{a}u_ {aa}) \tag{8}\]  
\[u^{(1)}=A\sin(k_{l}x-\omega_{l}t) \tag{9}\]  
\[v^{(1)}=B\sin(k_{s}x+\omega_{s}t) \tag{10}\]  
\[u_{3}=\left(\frac{\partial\phi}{\partial x_{3}}-\frac{\partial\varphi_{2}}{ \partial x_{1}}\right)=\left(\frac{\partial}{\partial x_{3}}(\Phi(x_{3}))-ik \Psi(x_{3})\right)\exp(i(kx_{1}-\omega t)) \tag{2}\]0 In-plane (\(u_{1}\)) and out-of-plane (\(u_{3}\)) displacement coordinate diagram. where, \(C_{111}\) and \(C_{116}\) indicate the third order elastic moduli, and \(c_{l}\) and \(c_{s}\) indicate the longitudinal and transverse wave's velocities respectively. \(\rho\) denotes the constant density. Subscripts \(a\) and \(t\) mean the spatial coordinate and time, and '(1)' indicates primary waves. Some conditions are assumed to use some parts of bulk wave mixing theory in guided wave mixing theory. If the in-plane displacement is greater than the out-of-plane displacement, the guided wave signal is treated as a longitudinal wave. If the out-of-plane displacement is greater, the signal is treated as a transverse wave in this paper. Figure 2 illustrates the guided wave mixing concept. The case in which one component has dominant in-plane displacement, and another component has dominant out-of-plane displacement, is shown in the plate. Each signal is defined by Equations (9) and (10).