This problem comes from equation (4) of 1. As shown in the following figures, there are two stochastic variables $y_1(x)$, representing the upper profile, and $y_2(x)$, representing the bottom profile, whose standard deviations are $\sigma_1$ and $\sigma_2$, respectively. Their slopes, $m_1$ and $m_2$, are defined in the figure. The absolute mean asperity slope is defined as $m=\frac{1}{L}\int_0^L|\frac{\mathrm{d}y(x)}{\mathrm{d}x}|\mathrm{d}x$. The author of 1 then claims that $m=\sqrt{m_1^2+m_2^2}$. What are the steps to derive this relationship?
I also referred to the appendix of paper 2 cited in 1, but the description there seems quite obscure to me, as shown in the following figure, especially where those standard deviations of the slopes and the coefficient $\sqrt{\frac{2}{\pi}}$ come from. Here $y'(x)$ denotes $\frac{\mathrm{d}y(x)}{\mathrm{d}x}$, $p'(y')$ denotes the probability distribution of the slope and $|\tan(\theta)|$ is just $m$ mentioned above. Could you please help me understand it? Thanks!
1 Yovanovich, M. Michael. "Four decades of research on thermal contact, gap, and joint resistance in microelectronics." IEEE transactions on components and packaging technologies 28, no. 2 (2005): 182-206.
2 Cooper, M. G., B. B. Mikic, and M. M. Yovanovich. "Thermal contact conductance." International Journal of heat and mass transfer 12, no. 3 (1969): 279-300.