SECT. 5] WIND WAVES 693 



Periodogram analysis on a digital computer follows the standard method for 

 extracting Fourier harmonics, which can be found in most elementary text- 

 books of physics. The computing time is large, however, because of the necessity 

 for computing the sine and cosine multipliers and because of the large number 

 of multiplications involved. This time can be greatly reduced by clever pro- 

 gramming, and a programme, due to J. Watt, is available for "Deuce" com- 

 puters which is remarkable in this respect. The alternative wide-band method 

 due to Tukey (1949) using correlation is usually faster, and gives directly the 

 smoothed spectrum which is usually required, so that it is more commonly 

 used. This forms in the first place a "lag correlation", 



</-(t) = mat + r), (22) 



where t is the time lag, and i^{t) is the instantaneous amplitude of the wave 

 record. 



If ^(^) is written down as a Fourier series, 



C{t) = 2 (^n cos {ant + 8n); (23) 



then 



(/((t) = 2 «ra^ cos anT. (24) 



It will be seen that the phase S„, which is usually unimportant, has been lost. 

 A periodogram analysis of a short length of 0(t), analysing only for the cosine 

 component, gives an estimate of the spectral density, 



E'{am) = I r^{(7n)an^ (25) 



where rmicn) is the filter function of the process. 



The shorter the length of (/((r) analysed, the wider becomes the pass-band of 

 f?n{<yn)- The shape of the filter function can be greatly improved either by 

 multiplying 0(t) by a simple function before analysis, or by taking a weighted 

 mean such as 



0.25E'{Gm-i) + 0.5E'{am) + 0.25E'{a,n+i). (26) 



The reader is referred to Munk et al. (1959) for details of this method. 



10. Methods of Observation and Analysis — The Directional Power Spectrum 



Photographs of the sea taken from an aircraft can show the directions of 

 wave travel. The directional power spectrum of such a photographic trans- 

 parency can readily be displayed by diffraction methods (Barber, 1949), but it 

 is very doubtful whether the transmission of the photograph itself correctly 

 represents the pattern of elevation or slope in the waves. Stereophotography 

 from an aircraft (Chase et al., 1957) is a more certain means of discovering the 

 pattern of water elevation. The example of a directional spectrum given in 

 Fig, 14 was produced by this means. At present this method demands much 

 time in stereoanalysis and in computation. Neither method can conveniently 

 distinguish between similar waves travelling in opposite directions. 



