## 2 Theoretical Approach
### Determination of Contact Acoustic Nonlinearity (CAN)  
An interface consists of asperity contacts and air gaps. In the regions of the interface with the air gap, there is no contact between the surfaces (Figure 2a). The approach of the two surfaces depends on the magnitude of the applied pressure [40]. The applied pressure is the sum of the externally applied pressure and the pressure applied by the incident wave. Figure 2: Schematic diagram of higher harmonic generation at an incomplete interface [41]: (**a**) rough surface interface modelled as array of gaps; (**b**) incidents ultrasonic wave interacts with the interface; (**c**) approaching surfaces; (**d**) being apart surfaces. A longitudinal incident cosinusoidal wave with centre angular frequency \(\omega\) and amplitude \(A\) strikes the interface at time \(t_{0}\). \[f(x-ct)=Acos\Big{\{}\frac{\omega}{c}(x-X_{-}-ct)\Big{\}} \tag{18}\]  
Substituting Equation (18) into Equation (15) at the interface gives:  
\[\dot{h}=2A\omega\ \sin\omega t+\frac{2}{\rho c}\{\ p(h)-p_{0}\} \tag{19}\]  
It is assumed at time \(t_{0}=0\) that the gap is closed and remains closed until just before time \(t_{1}\). For the sake of derivation, the gap is assumed to be fully closed; however, in reality even under high pressure there will remain air gaps and the contact does not fully close. Therefore, the gap separation \(h\) and relative surface separation \(Y\) are zero, subsequently, the initial surface separation \(h_{0}\) is zero. When the gap is open at time \(t_{1}\), the pressure at the interface is zero, so the boundary condition in Equation (4) is:  
\[\sigma(X_{+},t)=\sigma(X_{-},t)=-p(h(t))=0 \tag{20}\]  
Substituting Equation (20) into Equation (19) and integrating with respect to time gives time \(t_{1}\):  
\[cos\omega t_{1}+\frac{p_{0}}{\rho cA}\ t_{1}-1=0 \tag{21}\]  
Equation (21) is different from those presented in [38]. Equation (21) is a nonlinear equation and in the present work it is solved by the Newton-Raphson method. The gap remains open until time \(t_{2}\). Integration of Equation (19) with respect to time and substituting Equation (20) in the interval \([t_{1},\ t_{2}]\) gives opening interval:  
\[h(t)-h(t_{1})=-2A\{cos\omega t-cos\omega t_{1}\}-\frac{2}{\rho c}p_{0}(t-t_{1} )\ ;\ t_{1}\leq t\leq t_{2} \tag{22}\]  
Hence at time \(t_{2}\) the gap is just about to be closed; the right-hand side of Equation (21) is zero. Time \(t_{2}\) can be found using the Newton-Raphson method:  
\[cos\omega t_{2}+\frac{p_{0}}{\rho cA}t_{2}-cos\omega t_{1}-\frac{p_{0}}{\rho cA }t_{1}=0 \tag{23}\]  
The gap remains closed until time \(t_{3}\). The opening-closing gap transition is repeated periodically:  
\[h(t)=\left\{ \begin{array}{c}h(t_{2n})\ ;\ t_{2n}\leq t<t_{1}+nT\\ h(t_{2n})-2A\{\cos\omega t-\cos\omega(t_{1}+nT)\}-\frac{2}{\rho c}p_{0}(t-t_{ 1}-nT);\ t_{1}+nT\leq t<t_{2}+nT\end{array} \right. \tag{24}\]  
where \(n=0,1,2,\ldots\) and \(T\) is the period of the incident wave. A single longitudinal incident ultrasonic wave cycle consists of two parts causing tension and compression (Figure 2b). Figure 3 shows surface separation at the interface of aluminium blocks subjected to a longitudinal wave with centre frequency 2 MHz and amplitude 20 nm. The gap is initially assumed to be closed at time \(t_{0}=0\). It is seen from Figure 3a that, when the compressional part of wave reaches the interface (corresponding to zero to point **a** on the incident wave), the applied pressure is sufficiently large to keep the gap closed in the interval \([t_{0},t_{1}]\) (also see Figure 2c). As the wave passes (corresponding to part of the wave between point **a** and **b** on the incident wave), the applied pressure reduces so the surfaces of the gap are open at time \(t_{1}\) and be pulled apart away from the mean position (also see Figure 2d). After that, the applied pressure increases (corresponding to part of the wave between point **b** and **c** on the incident wave) and presses the surfaces of the gap closer together.