## 2 Theoretical Approach
### Theoretical Analysis
#### 2.3.2 Incident and Reflected Elastic Waves  
Differentiation of Equation (18) with respect to argument of the function at the interface \(x=X_{-}\) results in:  
\[\frac{\partial f(X_{-}-ct)}{\partial(X_{-}-ct)}=A\ \omega\sin\omega t \tag{29}\]  
Substituting Equations (26) and (29) into (14) gives:  
\[\dot{Y}+\frac{2K_{1}}{\rho c}Y-\frac{2K_{2}}{\rho c}\ Y^{2}=2A\omega\ sin\omega t \tag{30}\]  
which is a first order nonlinear differential equation. An approximate solution, such as the homotopy perturbation method (HPM) [43], is required to derive the particular solution of the relative approach of the contacting surfaces. To do this, the solution of the differential equation is defined as the sum of the linearized equations of the differential equation which is known as a perturbation series [43]:  
\[Y=Y_{0}+\alpha\ Y_{1}+\alpha^{2}\ Y_{2}+\alpha^{3}\ Y_{3} \tag{31}\]  
where \(Y_{0}\), \(Y_{1}\), \(Y_{2}\) and \(Y_{3}\) are the first, second, third and fourth terms of the perturbation series and \(\alpha\) is an imbedding parameter which is assumed to be 1 to derive the particular solution of the differential equation. It should be noted that \(\alpha\) is a small parameter in the range of [0, 1] [43]. As \(\alpha\) approaches 1, Equation (31) gives an approximate solution to the nonlinear differential Equation (30) [43]. In order to solve the nonlinear differential equation (30), the differentiation of Equation (31) with respect to time is substituted into Equation (30). A series of differential equations in terms of only \(Y_{0}\), \(Y_{1}\), \(Y_{2}\) and \(Y_{3}\) are derived. The differential equations containing \(Y_{0}\) and \(Y_{2}\) are in transient state (homogeneous differential equations). The initial surface approach is zero, subsequently, \(Y_{0}=Y_{2}=0\). Therefore, the transient behaviour of the differential equation was ignored and only the steady state behaviour was considered. \[Y_{1}=\frac{2A}{\sqrt{1+\left(2K_{1}/\rho c\omega\right)^{2}}}\sin(\omega t- \delta_{1}) \tag{32}\]  
\[Y_{3}=\frac{2K_{2}A^{2}}{K_{1}\Big{\{}1+\left(2K_{1}/\rho c\omega\right)^{2} \Big{\}}}\ \Bigg{\{}1-\frac{\sin\left(2\omega t-2\delta_{1}+\delta_{2}\right)}{\sqrt{1+ \left(K_{1}/\rho c\omega\right)^{2}}}\Bigg{\}} \tag{33}\]Substituting Equations (32) and (33) into Equation (31) gives the relative surface approach \(Y(t)\):  
\[Y(t)=\frac{2A}{\sqrt{1+(2K_{1}/\rho c\omega)^{2}}}\sin(\omega t-\delta_{1})+\frac {2K_{2}A^{2}}{K_{1}\Big{\{}1+(2K_{1}/\rho c\omega)^{2}\Big{\}}}\left\{1-\frac{ \sin\left(2\omega t-2\delta_{1}+\delta_{2}\right)}{\sqrt{1+(K_{1}/\rho c\omega )^{2}}}\right\} \tag{34}\]  
where \(\delta_{1}=\tan^{-1}(\rho c\omega/2K_{1})\) and \(\delta_{2}=\tan^{-1}\left(K_{1}/\rho c\omega\right)\). Substituting Equation (34) into Equation (17) results in an analytical equation for the reflected wave:  
\[ \begin{array}{rl}G(x+ct)=&-\frac{K_{3}A^{2}}{\kappa_{1}\Big{\{}1+\Big{(} \frac{2K_{1}}{\rho c\omega}\Big{)}^{2}\Big{\}}-\frac{A}{\sqrt{1+\Big{(}\frac{2 K_{1}}{\rho c\omega}\Big{)}^{2}}}}sin\big{\{}\omega t+\frac{\omega}{c}(x-X_{-})- \delta_{1}\big{\}}\\ &+\frac{K_{2}A^{2}}{\rho c\omega\sqrt{1+\Big{(}\frac{K_{1}}{\rho c\omega} \Big{)}^{2}}\left\{1+\Big{(}\frac{2K_{1}}{\rho c\omega}\Big{)}^{2}\right\}} sin\big{\{}2\omega t+\frac{2\omega}{c}(x-X_{-})-2\delta_{1}+\delta_{2} \big{\}}\end{array}  \tag{35}\]  
The terms in this equation with a frequency \(\omega\) and \(2\omega\) represent the fundamental and second order harmonics, respectively. So, the coefficient of terms containing \(\omega\) and \(2\omega\) are the amplitude of the fundamental frequency \(A_{1}\) and second harmonic \(A_{2}\), respectively. The reflection coefficient \(R\) is defined as the ratio of the amplitude of the reflected pulse from the interface to the amplitude of the incident wave [2]. The reflection coefficient \(R\) at imperfect contact is dependent on the interfacial stiffness per unit area of the interface (where the wavelength of elastic wave is large compared to the air gap).