## 2 Theoretical Approach
### Governing Equation
Substituting Equation (4) into Equation (2) gives:  
\[\left\{ \begin{array}{c}\frac{\partial u(X_{-},t)}{\partial x}=\frac{-1}{E}p (h(t))+\frac{1}{E}p_{0}\\ \frac{\partial u(X_{+},t)}{\partial x}=\frac{-1}{E}p(h(t))+\frac{1}{E}p_{0} \end{array} \right. \tag{5}\]  
The relative surface approach \(Y(t)\) of the mating surfaces is defined as the difference in the wave displacement \(u(x,t)\) of both sides of the interface (Figure 1d):  
\[Y(t)=\{u(X_{+},t)-u(X_{-},t)\}=h(t)-h_{0} \tag{6}\]  
Equation (6) shows that the relative surface approach \(Y(t)\) can be used instead of the surface separation \(h(t)\). The translational motion of the interface is defined by:  
\[X(t)=\{u(X_{+},t)+u(X_{-},t)\}/2 \tag{7}\]  
In order to simplify the calculation, the displacement of the interface is represented only in terms of incident wave [38]:  
\[X(t)=f(x-ct) \tag{8}\]  
Differentiation of Equations (6) and (8) with respect to time gives:  
\[\dot{X}=\frac{-c\ \partial f(X_{-}-ct\ )}{\partial(X_{-}-ct)} \tag{9}\]  
where dots denote the differentiation with respect to time. \[\dot{Y}=c\bigg{\{}-\frac{\partial H(X_{+}-ct)}{\partial(X_{+}-ct)}+\frac{ \partial f(X_{-}-ct)}{\partial(X_{-}-ct)}-\frac{\partial G(X_{-}+ct)}{\partial (X_{-}+ct)}\bigg{\}} \tag{10}\]  
Equation (10) is expressed in terms of incident, reflected, and transmitted ultrasonic waves; it is necessary to reduce this in terms of only the incident wave and nominal contact pressure. To do this Equation (3) is differentiated with respect to \(x\):  
\[\left\{ \begin{array}{c}\frac{\partial u(X_{-},t)}{\partial x}=\frac{ \partial f(X_{-},-ct)}{\partial(X_{-},-ct)}+\frac{\partial G(X_{-}+ct)}{ \partial(X_{-}+ct)}\\ \frac{\partial u(X_{+},t)}{\partial x}=\frac{\partial H(X_{+}-ct)}{\partial(X _{+}-ct)}\end{array} \right. \tag{11}\]  
Rearranging Equation (11) to make the reflected and transmitted waves the subject gives:  
\[\left\{ \begin{array}{c}\frac{\partial G(X_{-}+ct)}{\partial(X_{-}+ct)}= \frac{\partial u(X_{-},t)}{\partial x}-\frac{\partial f(X_{-},-ct)}{\partial( X_{-}-ct)}\\ \frac{\partial H(X_{+}-ct)}{\partial(X_{+}-ct)}=\frac{\partial u(X_{+},t)}{ \partial x}\end{array} \right. \tag{12}\]  
Substituting Equation (5) into Equation  
Substituting Equation (13) into Equation (10) gives:  
\[\dot{Y}=2c\frac{\partial f(X_{-}-ct)}{\partial(X_{-}-ct)}+\frac{2}{\rho c}\{\ p(h_{0}+Y)-p_{0}\} \tag{14}\]  
Equation (14) is a first order differential equation describing the relative surface approach in terms of the incident wave and the nominal contact pressure \(p(h_{0}+Y)\). Solving this differential equation results in an equation for the relative separation of the interface. Equation (14) can be written in terms of surface separation \(h(t)\):  
\[\dot{h}=2c\frac{\partial f(X_{-}-ct)}{\partial(X_{-}-ct)}+\frac{2}{\rho c}\{\ p(h)-p_{0}\} \tag{15}\]  
The reflected ultrasonic wave is now defined in terms of relative surface approach. Equations (5) and (6) give the relationship between relative surface approach \(Y(t)\) and translational motion of the contact interface \(X\):  
\[u(X_{-},t)=X(t)-\frac{1}{2}Y(t) \tag{16}\]  
Substituting Equation (3) (for \(x<X_{-}\)) and Equation (7) into Equation (15) gives:  
\[G(x+ct)=-\frac{1}{2}Y\bigg{(}t+\frac{x-X_{-}}{c}\bigg{)} \tag{17}\]  
It can be seen from Equation (17) that the reflected ultrasonic wave \(G(x+ct)\) is expressed only in terms of the relative surface approach. The negative sign of the reflected wave indicates propagation in the opposite direction to the incident wave through the upper body.