## 2 Theory
### Single Element Transducer  
When a finite-size transducer is used to generate and receive nonlinear waves, the wave diffraction effect should be included in the wave equation. Therefore, we introduce a nonlinear wave equation in the three-dimensional coordinate system to model the propagating and reflecting waves. The attenuation effects will not be considered in this study. In terms of the particle velocity, \(v\), the Westervelt-like equation for longitudinal waves can be expressed as follows [15]:  
\[\left(\
abla^{2}-\frac{1}{c^{2}}\frac{\partial^{2}}{\partial t^{2}}\right)v=- \frac{\beta}{c^{3}}\frac{\partial^{2}v^{2}}{\partial t^{2}}\ \, \tag{7}\]  
where the operator \(\
abla^{2}\) is the Laplacian in the \((x,y,z)\) space and \(z\) is taken as the direction of propagation.  
The quasilinear solution is assumed to have the following form:  
\[v=v_{1}\exp(-i\omega t)+v_{2}\exp(-2i\omega t) \tag{8}\]  
Substitution of Equation (8) into Equation (7) yields the following two equations:  
\[\
abla^{2}v_{1}+k^{2}v_{1}=0\ \ \, \tag{9}\]  
\[\
abla^{2}v_{2}+4k^{2}v_{2}=\frac{2\beta k^{2}}{c}v_{1}^{2}. \tag{10}\]  
The solutions of the above equations can be obtained by using the [PERSON] integral method and [PERSON]'s function approach:  
\[v_{1i}(x,y,z)=-2 ik\int_{S}v_{1}(x^{\prime},y^{\prime},0)G_{1}(x,y,z|x^{\prime},y^{\prime},0)\,dS \tag{11}\]  
\[v_{2i}(x,y,z)=\frac{2\beta k^{2}}{c}\int_{0}^{z}\int_{-\infty}^{+\infty}\int_ {-\infty}^{+\infty}v_{1i}^{2}(x^{\prime},y^{\prime},z^{\prime})\ G_{2}(x,y,z|x^ {\prime},y^{\prime},z^{\prime})\ dx^{\prime}dy^{\prime}dz^{\prime} \tag{12}\]  
The Green's function for the \(n\)th harmonic is given as follows:  
\[G_{n}(x,y,z|x^{\prime},y^{\prime},z^{\prime})=\frac{1}{4\pi r}\exp(nikr),\ \ \ \ n=1,2 \tag{13}\]where \(r=\sqrt{(x-x^{\prime})^{2}+(y-y^{\prime})^{2}+(z-z^{\prime})^{2}}\) is the distance from the source point \((x^{\prime},y^{\prime},z^{\prime})\) to the target point \((x,y,z)\). The sound source on the transducer surface is defined as the following:  
\[\left\{\begin{array}{c}v_{1}(x^{\prime},y^{\prime},z^{\prime}=0)=v_{0}(x^{ \prime},y^{\prime}),\;0\leq{x^{\prime}}^{2}+{y^{\prime}}^{2}\leq a^{2}\\ v_{2}(x^{\prime},y^{\prime},z^{\prime}=0)=0\end{array}\right. \tag{14}\]  
where \(a\) is the radius of the circular transmitting element.  
The reflected wave fields can be derived from the stationary phase method when a wave is reflected from a large planar interface [17]. Then, the solution in the backward direction can be expressed as follows:  
\[v_{1r}(x,y,z)=R\times(-2 ik)\int_{S}v_{0}(x^{\prime},y^{\prime},z=0)G_{1}(x,y,z \big{|}x^{\prime},y^{\prime},0)\,dS \tag{15}\]  
\[\begin{array}{rl}v_{2r}(x,y,z)&=v_{2r1}(x,y,z)+v_{2r2}(x,y,z)\\ &=R\times\frac{2\beta k^{2}}{c}\int_{0}^{z_{0}}\int_{-\infty}^{+\infty}\int_{ -\infty}^{+\infty}v_{1i}^{2}(x^{\prime},y^{\prime},z^{\prime})\;G_{2}(x,y,z \big{|}x^{\prime},y^{\prime},z^{\prime})\;dx^{\prime}dy^{\prime}dz^{\prime}+\\ &R^{2}\times\frac{2\beta k^{2}}{c}\int_{2_{0}}^{z}\int_{-\infty}^{+\infty}\int _{-\infty}^{+\infty}v_{1i}^{2}(x^{\prime},y^{\prime},z^{\prime})\;G_{2}(x,y,z \big{|}x^{\prime},y^{\prime},z^{\prime})\;dx^{\prime}dy^{\prime}dz^{\prime}\end{array} \tag{16}\]  
The average velocity received by the same transducer at \(z=2z_{0}\) can be obtained by the following integral:  
\[\widetilde{v}_{n}(2z_{0})=\frac{1}{S}\int_{S}v_{nr}(x,y,2z_{0})dS,n=1,2 \tag{17}\]  
where \(dS\) is the receiver area.  
Now that the received velocities of the fundamental and second harmonics are available, the nonlinear parameter \(\beta^{\prime}\) in the pulse-echo setup can be defined in terms of displacement. Using the relationship between velocity and displacement, \(v_{n}=-n\omega u_{n}\), \(n=1,2\), \(\beta^{\prime}\) in the pulse-echo mode at \(z=2z_{0}\) can be expressed as follows:  
\[\beta^{\prime}=\frac{8\widetilde{u}_{2}}{k^{2}z_{0}\widetilde{u}_{1}^{2}} \tag{18}\]  
where \(\beta^{\prime}\) denotes the \"uncorrected\" or \"relative\" nonlinear parameter.