we may find, by taking the sine-transform, the function 



y 2 [<r(u,v)E(u,v) — <t{ — u, — v)E{ — u, — v)] 

 But cr(u, v) is known, even function. Thus on division we find 



%[E(u,v) -E(- u, - v)] 



(15) 

 (16) 



and hence, by addition of (13) and (16), the spectrum E(u, v) itself. 



It is planned to measure £(*, y, along an arbitrary straight line x sin 9 = 

 y cos 6 by means of a radar altimeter mounted in an aircraft and directed vertically 

 downwards to the sea surface. A second altimeter, mounted directly aft of the first, 

 and therefore passing over the position occupied by the first a short time before, would 

 provide a measure of d£/dt along the same path. 



The method has certain advantages over stereophography. In the first place a 

 large number of wavelengths may be included in each straight cross-section of the 

 surface, so enabling a more accurate determination of the autocorrelation function to 

 be made. In the second place the complete directional spectrum E(u,v) is obtained, 

 not merely its even part V2[E(u,v) -f E( — u, — v)]. 



Even without performing the correlation analysis, the data may be made to yield 

 some useful information. For example, it can be shown (see reference [1]) that the 

 average number N of zero-crossings of the surface £(x,y,t) along a plane section in 

 direction 9 is given by 



N 2 = (ra 2 o cos2 + 2wn cos sin + ra 2 sin2 ff), (17) 



7r 2 m o 



where 



+ 00+00 



m pq = J J E(u,v)u p v g dudv. 



(18) 



Thus N is a maximum and a minimum in two directions at right-angles. The ratio 



•A^O min 



iVc 



(19) 



is equal to the r.m.s. angular spread in direction of the energy about the mean direction, 

 provided this is small (see reference [1]). So by measuring N as a function of 9 the 

 r.m.s. spread in direction may be found. 



Similarly, the number N 1 of crests and troughs per unit distance in the direction 

 9 can be shown to be given by 



1 

 JVi 2 = (m 40 cos 4 -f 4ra 31 cos 3 sin -f- • • • + m 0i sin 4 0). (20) 



7rWo 2 m o 



From this the moments m 40 , m 31 . . . m 04 can be determined. If we write 



w 4 o w 3 i m 22 



A 4 = WI31 W22 W13 



(21) 



and 



W22 mn W04 



SH = m 4 oW 4 — 4m 3 irai 3 + 3w 22 2 

 57 



(22) 



