Statistical properties of an isotropic random surface

A number of statistical properties of a random, moving surface are obtained in the special case when the surface is Gaussian and isotropic. The results may be stated with special simplicity for a 'ring' spectrum when the energy in the spectrum is confined to one particular wavelength $\overline{\lambda}$. In particular, the average density of maxima per unit area equals $\pi /(2\surd 3\overline{\lambda}{}^{2})$, and the average length, per unit area, of the contour drawn at the mean level equals $\pi /(\surd 2\overline{\lambda})$.