576 CAKTWBIGHT [CHAP. 15 



All these probabilities are independent of time and space over the zone for 

 which the wave system is assumed stationary, provided t, and its derivatives 

 are measured simultaneously at the same point. Quantities taken at fixed 

 spacings or time intervals have similar distributions, but will not be discussed 

 until the next section. 



Probability distributions of many quantities describing the geometry of the 

 surface can be evaluated from the normal distribution of the derivatives of ^ 

 (Longuet-Higgins, 1957), but are mostly outside the scope of this chapter. We 

 shall, however, mention a few t3rpical results involving only moments of order 

 2 or 4. The distribution of absolute slope a is 



p{a) = aZJa"'^' exp {-a2(wi2o + mo2)/4J2}/o{a2(m2or*^02)/4/l2}, (16) 



where Io{x) is the Bessel function Jo{ix). The mean length per unit area of 

 contour of height ^ is 



7r-i(mo2 + W2o/moo)'/^ exp (-^2/2moo)(l +y2)-'/2E{(l-y2)-/.), (17) 



where E is an elliptic integral and y^ is defined by (13). 



The distribution of curvature is important when considering the nature of 

 light reflection in the wave surface. In Longuet-Higgins (1958), it is shown that 

 while the "mean curvature" 



J = dmdx^+emdy^ 



has, as is easily seen, the normal distribution 



p{J) = (27ri))-'/^e-'^'/2^, D = m4o + 2m22 + mo4, 

 the "total curvature" 



dx^ dy^ \dx By} 



determining the intensity of reflections and the sizes of images, is far from 

 normal. The actual distribution of Q is sharply peaked at the origin and skew, 

 and involves an awkward integral evaluated by Catton and Millis (1958). 

 However, it depends only on moments of order 4. 



Other properties (derived in Longuet-Higgins, 1957) involve the spatial 

 densities of maxima and minima (whose sum, by an important theorem, is 

 equal to the density of "saddle points"), the velocities of zeros, of contours, and 

 of points with a given gradient, and various properties of the wave "envelope" 

 (an important concept when the spectrum is narrow). All these are expressed in 

 terms of moments nipq and mpq^^'> up to a finite order. 



5. Estimating the Directional Energy Spectrum 



We have just seen that, given the spectrum E{k, 6) or E"{u, v) of a wave 

 system, many statistical properties of the surface can be deduced in terms of 

 the moments of the spectrum. Conversely, from statistical analysis of various 



