## 2 Theory
### Pure Plane Wave  
As can be seen in Figure 2a, if the size of the transducer is large enough, the propagation wave can be treated as a plane wave. When the nonlinear parameter \(\beta\) is introduced for a solid medium, the one-dimensional longitudinal wave equation can be expressed as follows [9]:  
\[\frac{\partial^{2}u}{\partial t^{2}}=c^{2}\frac{\partial^{2}u}{\partial z^{2}} \left(1+\beta\frac{\partial u}{\partial z}\right) \tag{1}\]  
where \(u\) is the wave particle displacement, \(t\) is the time, \(c\) is the wave velocity, and \(z\) is the one-dimensional coordinate. In Equation (1), \(\beta\) is the nonlinear parameter of solids and defined as \(\beta=-(3+C_{111}/C_{11})\) where \(C_{11}\) and \(C_{111}\) are the second- and third-order elastic constants. Using a standard perturbation theory yields the quasilinear system of equations for the fundamental and second harmonic displacements \(u_{1}\) and \(u_{2}\), respectively. The solutions are then obtained as follows:  
\[u=U\sin(kz-\omega t)+\frac{\beta U^{2}k^{2}z}{8}\cos 2(kz-\omega t) \tag{2}\]  
where \(U\) is the uniform source displacement at the transmitter surface, \(k=\omega/c\) is the wave number and \(\omega\) is the angular frequency. If the amplitudes of the fundamental and second harmonic are represented as \(A_{1}=U\) and \(A_{2}=\beta U^{2}k^{2}z/8\), the nonlinear parameter \(\beta\) can be obtained as follows:  
\[\beta=\frac{8A_{2}}{k^{2}zA_{1}^{2}} \tag{3}\]  
When the fundamental wave encounters the boundary in the propagation path, it will reflect with its amplitude determined by the reflection coefficient. In the case of the second harmonic wave, as stated previously in Figure 1, two second harmonic components will reflect from the boundary and propagate back to the initial source position [19]. The stationary phase method can be used [20] to write the reflected wave fields in terms of the initial sound source and the coordinate system. Thus, the reflected fundamental and second harmonic waves propagating in the backward direction can be expressed as follows:  
\[u_{1r}=R_{1}\times U\sin(kz-\omega t) \tag{4}\]  
\[ \begin{array}{ll}u_{2r}&=u_{2r1}+u_{2r2}\\ &=R_{2}\times\left(\frac{\beta U^{2}k^{2}z_{0}}{8}\right)\cos 2(kz-\omega t)+R_{1 }^{2}\times\left(\frac{\beta U^{2}k^{2}(z-z_{0})}{8}\right)\cos 2(kz-\omega t)\\ &=\frac{\beta U^{2}k^{2}}{8}\big{(}R_{2}z_{0}+R_{1}^{2}(z-z_{0})\big{)}\cos 2 (kz-\omega t)\end{array}  \tag{5}\]  
where \(z_{0}\) is the distance between the transmitting plane and the boundary, and \(R_{1}\) and \(R_{2}\) are the reflection coefficients for the fundamental and second harmonics, respectively. In Equation (5), the first term represents the reflected second harmonic and the second term represents the newly generated one. The reflection coefficients of the fundamental and second harmonic waves can be assumed to be \(R_{1}=R_{2}=-1\) for the stress-free  
Figure 2: Examples of transducers used for simulation of second harmonic generation in the pulse-echo setup: (**a**) single element, (**b**) four—element annular array, and (**c**) dual-element annular array. boundary, and \(R_{1}=R_{2}=1\) for the rigid boundary when the longitudinal wave is incident perpendicular to the boundary. The nonlinear parameter in the pulse-echo mode can be defined from Equations (4) and (5) as the following:  
\[\beta=\frac{8A_{2}}{k^{2}A_{1}^{2}}\left[\frac{R_{1}^{2}}{R_{2}z_{0}+R_{1}^{2}(z -z_{0})}\right] \tag{6}\]  
where \(A_{1}\) and \(A_{2}\) are the amplitudes of the fundamental and second harmonic waves after being reflected from the boundary. In Equation (6), \(\beta\) is not defined at \(z=2z_{0}\) for the stress-free boundary since the total second harmonic amplitude is 0 in this case. In the case of the rigid boundary, the \(\beta\) corresponds to that of the through-transmission mode with the sample thickness \(2z_{0}\).