The transverse correlation length for randomly rough surfaces

It is shown by numerical simulations for a random, one-dimensional surface defined by the equationx₃=(x₁), where the surface profile function (x₁) is a stationary, stochastic, Gaussian process, that the transverse correlation lengtha of the surface roughness is a good measure of the mean distance d between consecutive peaks and valleys on the surface. In the case that the surface height correlation function (x₁)(x₁)/²(x₁)=W (|x₁–x₁|) has the Lorentzian formW(|x₁|)=a²/(x₁²+a²) we find that d=0.9080a; when it has the Gaussian formW(|x₁|)=exp(–x₁²/a²), we find that d=1.2837a; and when it has the nonmonotonic formW(|x₁|)=sin(x₁/a)/(x₁/a), we find that d=1.2883a. These results suggest that d is larger, the faster the surface structure factorg(|Q|) [the Fourier transform ofW(|x₁|)] decays to zero with increasing |Q|. We also obtain the functionP(itx₁), which is defined in such a way that, ifx₁=0 is a zero of (x₁),P(x₁)dx₁ is the probability that the nearest zero of (x₁) for positivex₁ lies betweenx₁ andx₁+dx₁.