SECT. 5] ANALYSIS AND STATISTICS 579 



If we iioM consider the records of ^(f) from any two detectors placed along 

 the .r-axis with separation d, viz. 



L{t) = C(0, 0, = 2 «« cos {cTnt-en), 



n 



M{t) = l,{d, 0, <) = 2 «n COS {ont-dkn COS On-^n), 

 n 



the cross-spectral components at wave number k can be expressed as 



Cilia) = Cmm{<j) = r E^id) dd = TrAoik) (22) 



00 



= 7TAo{k)Jo{kd ) + 277 2 i'Ar{k)Jr{kd ), (23) 



r = l 



where Jr{x) are Bessel functions of the first kind. Assuming the ^'s converge 

 suitably so that values for r^ N can be ignored, then, by using also detectors 

 separated by 2c?, 3d, . . . Nd, we get several estimates of ^o(^) from (22), and a 

 set of N simultaneous equations like (23) which can be solved for Ai . . . A^. 

 These are sufficient to reconstruct Ek{6) if it is symmetrical about the x-axis 

 so that the B's are zero, as is assumed in the case of waves generated by wind 

 blowing along the a:;-axis. The odd-order B's could be obtained by aligning 

 the array of detectors along the y-a,xis, but not the even orders. To obtain all 

 the B's, Barber (1959) suggests also using slope recorders transverse to the x- 

 axis. Cross-spectral analysis of pairs such as ^0, 0, t){dldy) ^(d, 0, t) gives a set 

 of equations like (23) containing only the B's. To achieve best results, d should 

 be about a quarter of a wavelength, TTJ'Ik, so that different separations are 

 needed for different ranges of k. Worthy of note is that considerably fewer than 

 N detectors are needed for N separations. For example, four recorders at x = 0, 

 d, 4:d, Qd, contain all separations up to Qd, and six recorders at x = 0, d, 4d, Id, 

 lOd, I2d, contain all separations up to 12d. 



Another method, suitable when the waves are known to be travelhng within 

 ± 90° of the ^-axis, is to measure the spectral power of the sum of the signals 

 from a large number of detectors spaced evenly along the a;-axis. This heavily 

 weights the signals from wave components near 6= ±\tt and can, in fact, be 

 taken as a measure of Ek{\Tr). If, then, the signals are altered in phase, that from 

 the detector at x — nd being advanced in time by [nkdju) cos d, components in 

 the direction d are most heavily weighted, and we have in effect Ek{6). Barber 

 (1958) discusses ways of economizing on the numbers of detectors and giving 

 them various weighting factors to optimize their angular resolving power. A 

 similar principle applied to an instantaneous stereo-photograph is studied by 

 Marks (1954). 



Finally, we consider some recording systems which yield only a limited 



