580 CAHTWRIGHT [CHAP. 15 



amount of information about E{k, 6) but have the advantage of greater sim- 

 phcity and compactness than any system mentioned above. One method, 

 recently used successfully at the National Institute of Oceanography and 

 originally suggested in unpublished work of Barber (1946), is to record simul- 

 taneously ^, dt^jdx and dl,ldy by means of a shallow, freely floating buoy.^ 

 Calling the signals L{t), M{t) and N{t) respectively, it can easily be seen that 



Cii = 7TAo{k), Qim = 7TkAi{k), Cmm-Cnn = 7Tk^A2{k) 



Qln = 7TkBi{k), 2Cmn = 7Tk^B2{k) 



with the further identity Cmm+Cnn = k^Cii, which can serve as a calibration 

 check. Thus, for all values of k we obtain all the harmonics of Ek{d) of order 

 0, 1 and 2 and no more ; these can be shown to be equivalent to the moments 

 nipq of E"{u, v) of order 0, 1 and 2, from which various statistical properties of 

 the wave surface can be derived as shown in section 4 of this Chapter (page 

 573). In particular, the principal direction of the waves of number k is given 

 by 6p{k) = ^ tan~i {B2IA2) and the long-crestedness, or r.m.s. angular spread, is 

 given hyyHk) = (l-i22)/(l + i?2), where R2^ = {A2^ + B2^)IAo^. 



If the angular spread of the wave system is known to be small, then a detector 

 on a ship moving on a series of straight courses at uniform speed can determine 

 their wave-number spectrum and mean direction (in fact Aq, Ai and Bi), in 

 virtue of the Doppler shift in frequency (Cartwright, 1956). In the same 

 circumstances a mean direction of waves within ± 90° of the y-axis can be 

 determined from just two recorders fixed along the rc-axis by using the phase 

 angles obtained from their cross-spectrum. Both these last two methods can 

 separate crossing swells if their wave numbers and directions differ sufficiently. 



6. Waves Recorded by a Single Detector 



A record of surface height above a fixed point gives no indication of direction 

 at all ; in fact it gives 



aO = C(0, 0,0 = 2 an cos {aj- en) 



whose energy spectrum with respect to frequency is simply 



E{a) = r" E{k{a), 6} dd = 7TAo{k{a)}. 

 Jo 



However, since the vast majority of wave records are of this type, we must 

 devote some space to discussion of their main statistical properties. We are 

 now simply dealing with a random function of time, as described in treatises 

 on random noise (e.g. Rice, 1945), but we shall emphasize a few points of 

 special interest to wave analysts. 



1 A pressure meter and a current meter recording two horizontal components of oscil- 

 latory current would give the same information. 



