A measure of the area of a homogeneous random surface in space
Introduction. We are given an indefinitely large space containing random surface or surfaces homogeneously located in the mean. The problem is to relate the average area of surface per unit volume of space to a simpler statistical quantity, in particular the average number of cuts per unit length made by a straight randomly directed sampling line with the surface. The plane case will be studied first. After the three dimensional case, illustrative application will be made to the problem of extending to two and three dimensional variables a theorem of S. 0. Rice on the average rate of occurrence of any particular value of a one dimensional random variable. Possible use in describing fluid mixing is also indicated. Two dimensions. Given a plane "homogeneously" inscribed with contour or contours of arbitrary shape. The homogeneity is statistical, i.e. any statistical function associated with the contours is invariant to a translation of coordinate system in the plane. Let £ be the average contour length enclosed in unit area and let n be the average number of cuts per unit length made by an arbitrary straight traverse line crossing the plane. For a non-isotropic field n is averaged over all traverse directions with uniform weighting; for an isotropic field, any single line will do. Draw a "very large" square in the plane, L on a side, and subdivide it into "very narrow" traverse strips parallel to one pair of sides. "Very large" here denotes L so large that averages over L or L2 are satisfactorily close to their asymptotic values. For example it requires that each traverse strip cross the contours a very large number of times and that the length of contour in L2 divided by L2 be as close as we like to <£. "Very narrow" denotes S so small that virtually all of the intercepted contour segments in a strip can be approximated by secants. This gives restriction on the permissible number of corners and contour intersections. We imagine the square and strip structure rotated through 180° for averaging purposes in case the field is not isotropic. Then the average number of crossings in one