## 2 Theory  
Consider nonlinear wave fields reflected from a planar stress-free boundary, as shown in Figure 1, where a pulse-echo testing setup is schematically displayed. In the forward propagation direction of Figure 1, \(v_{1i}\) is the particle velocity of the fundamental wave and \(v_{2i}\) is the generated second harmonic due to the forcing of \(v_{2i}\). In the backward propagation direction, \(v_{1r}\) represents the boundary-reflected wave of \(v_{1i}\). As discussed in our previous study [21], the nonlinear wave after reflection consists of two components, \(v_{2r1}\) and \(v_{2r2}\), where \(v_{2r1}\) is the second harmonic generated by the reflected \(v_{1r}\) and \(v_{2r2}\) is the reflected second harmonic when \(v_{2i}\) hits the boundary. The total second harmonic after reflection, \(v_{2r}\), is then obtained by adding \(v_{2r1}\) and \(v_{2r2}\). The stress-free boundary is located at \(z=z_{0}\). The acoustic impedance of the solid sample (medium 1) is \(z_{1}=\rho_{1}c_{1}\) and the acoustic impedance of medium 2 (air) is \(z_{2}=\rho_{2}c_{2}\approx 0\). Therefore, the reflection coefficient of the fundamental and the second harmonic is assumed to be \(R_{1}=R_{2}\approx-1\).  
Nonlinear wave equations are needed to study wave propagation, reflection and SHG in nonlinear solids. For this purpose, the Westervelt equation or the KZK (Khokhlov-Zabolotskaya-Kuznetsov) equation is a suitable model equation because they can account for the effects of nonlinearity, diffraction and attenuation [6]. Compared to the KZK equation, the Westervelt equation is described by three-dimensional coordinates and the solution of the quasilinear equation is valid for all axial ranges. These equations were used to model the nonlinear wave fields in fluids, but they are also suitable for use in solid materials when longitudinal waves are employed exclusively. In this work, we introduce the Westervelt-like equation for calculating the longitudinal wave fields in nonlinear solids.  
Several types of planar transducers are considered in this study, as shown in Figure 2. Figure 2 is a single element transducer that will be used for comparison purposes. Since this transducer has only one element, it works both as a transmitter and as a receiver. The annular array transducer of Figure 2 has four concentric elements of equal width [29]. This transducer will be used to apply the transmission phase shift to each element to improve SHG at the stress-free boundary. The emitted waves from the transducer elements will have different phases. The phase shift of each element can be controlled independently. Compared to a two-element transducer, a four-element transducer allows finer phase shifting and is, therefore, much more effective for SHG.  
A dual element transducer with phase-shifted radiation (Figure 2c) was shown to be useful in generating improved SHG in pulse-echo testing of thick samples [28] and will be used here for comparison purposes. We will also consider a focused beam using the four-element array transducer of Figure 2 and compare the received wave fields with the phase-shifted radiation. The array beam will be focused on the reflecting boundary by applying appropriate time delays to each element [30].  
Figure 1: Schematic of a pulse-echo testing configuration with a stress-free boundary.