## 2 Theoretical Approach  
### Governing Equation  
Consider two linear elastic media in contact subjected to a one-dimensional elastic longitudinal ultrasonic wave (Figure 1). The direction of ultrasound propagation is through the thickness of the media which is defined as the \(x\)-axis. The origin of the \(x\)-axis is the point of first contact of the surfaces. The roughness mean lines above and below the interface are located at \(X_{-}\) and \(X_{+}\), respectively (Figure 1a). The separation (mean gap) between the two surfaces under zero load is then given by (\(X_{+}-X_{-}\)). When there is no elastic wave the surfaces are in static equilibrium under a nominal contact pressure \(p_{0}\) causing an equilibrium separation of \(h_{0}\) (Figure 1b,c). When an elastic wave passes through the interface the surface separation, \(h(t)\) is cyclically reduced and then increased about the equilibrium value \(h_{0}\). The relative surface approach \(Y(t)\) is defined as the difference between the separation variation with time \(h(t)\) and equilibrium separation \(h_{0}\), (\(Y(t)=h(t)-h_{0}\)). In the absence of an elastic wave, the relative surface approach \(Y(t)\) is zero. The sign of relative surface approach \(Y(t)\) is opposite to the direction of loading/unloading by the wave. During the loading process by the elastic wave, the relative surface approach \(Y(t)\) is negative, whereas during the unloading process it is positive. In order to clearly illustrate the relative approach of the surfaces (assuming the lower body is fixed and the upper moving), the upper body is considered to be rigid and flat whilst the lower body is elastic and rough (Figure 1c). The elastic wave pressure superimposed on the nominal contact pressure \(p_{0}\) generates a relative surface approach \(Y(t)\) which varies with time (shown in Figure 1d). Increasing nominal contact pressure \(p_{0}\) decreases separation of the surfaces. The equations described in this section follow the method presented in [38; 39]. The equation of the interface subjected to a one-dimensional longitudinal ultrasonic wave is given by:  
\[\rho\frac{\partial^{2}u}{\partial t^{2}}=\frac{\partial\sigma}{\partial x} \tag{1}\]  
The stress-displacement relation is given by:  
\[\sigma(x,t)+p_{0}=E\frac{\partial u}{\partial x} \tag{2}\]  
where \(\rho\) is the density of the media, \(\sigma(x,t)\) is the normal stress generated by the ultrasonic wave and \(p_{0}\) in the media, \(u(x,t)\) is the displacement of the ultrasonic wave and \(p_{0}\) is the nominal contact pressure. The general solution of Equation (1) results in the displacement of each side of the interface:  
\[\left\{ \begin{array}{c}u(x,t)=f(x-ct)+G(x+ct)\,\ x<X_{-}\\ u(x,t)=H(x-ct)\,\ x>X_{+}\end{array} \right. \tag{3}\]  
where \(u(x,t)\) is the displacement of the interface, \(c\) is the speed of sound in the media which is \(\sqrt{E/\rho}\), \(t\) is the time of flight of ultrasound, \(f(x-ct)\) is the incident wave, \(G(x+ct)\) and \(H(x-ct)\) are the reflected and transmitted wave, respectively. The displace  
Figure 1: Schematic diagram of the asperity junctions and ultrasonic wave propagation through the media [1; 39]: (**a**) aluminium blocks and asperity contact in the absence of normal load; (**b**) origin and direction of the positive propagation axis; (**c**) contacting surfaces in the absence of elastic wave under static equilibrium contact; (**d**) relative surface approach and surface separation due to propagating elastic wave. ment through the upper and lower media is as \(x<X_{-}\) and \(x>X_{+}\), respectively. The boundary condition at the interface is defined by:  
\[\sigma(X_{+},t)=\sigma(X_{-},t)=-p(h(t))\,\ p(h_{0})=p_{0} \tag{4}\]  
where \(h(t)\) is the rough surface mean line separation (surface separation) variation with time and \(h_{0}\) is the separation under static equilibrium \(p_{0}\) in the absence of the elastic wave (as shown in Figure 1). A negative sign pressure term indicates compression at the interface.