The statistical distribution of the curvature of a random Gaussian surface

The distribution of the total (or ‘second’) curvature of a stationary random Gaussian surface is derived on the assumption that the squares of the surface slopes are negligible. The distribution is found to depend on only two parameters, derivable from the fourth moments of the energy spectrum of the surface. Each distribution function satisfies a linear differential equation of the third order, and the distribution is asymmetrical with positive skewness, in general. A special case of zero skewness occurs when the surface consists of two intersecting systems of long-crested waves.