SECT. 5] WIND WAVES 673 



where p is the density of the water, g is the acceleration due to gravity, and 

 A is an empirical constant which he found to be 8.27 x 10~4 sec~i. 



Roll and Fischer (1956) pointed out that it is not logical to assume that H is 

 proportional to the energy per unit period range. It is more logical to assume 

 that it is proportional to the energy per unit proportional range of periods, and 

 then the same answer is obtained if a proportional frequency range is used. 

 Applying this argument to Neumann's figures, they obtained 



E'{a) = lA'pg^TT^cr-^ ex-p {-2g^la^V^). (9) 



A' here is a dimensionless constant with a value of approximately 1.35 x 10~^ 

 (Neumann and Pierson, 1957). 



At high frequencies this gives -E"(cr) oc a~^, which agrees with a conclusion 

 reached by Phillips (1958b), who shows by a dimensional argument that on the 

 reasonable assumption that gravity is the only relevant parameter, the shape 

 of the saturated frequency spectrum must be of the form 



E'{g) = Cg^G-^. 



Since the high frequencies saturate first, this would be expected to be the shape 

 of the high-frequency end of the spectrum. Experimental evidence on the whole 

 supports this conclusion, but is not precise enough to prove that the exponent 

 could not be 4.5 or 5.5 or even 6. 



It should be noted that the relationship between H and the energy spectrum 

 used by Neumann is intuitive and has not so far been derived theoretically or 

 checked empirically, though the authors understand that work on these lines 

 is under way. The predictions of mean wave height and period have, however, 

 proved reasonably satisfactory, and Neumann's formulae are the basis of the 

 prediction system which is probably the most widely used at the present time 

 (Pierson, Neumann and James, 1955). 



Darbyshire, starting in 1952, has approached the problem in a basically 

 different manner. He takes many measured wave spectra, and tries to fit 

 purely empirical formulae to them. In his latest paper on the subject (Darby- 

 shire, 1959), he has considered spectra of waves recorded by a weather ship in 

 the North Atlantic, and has found that if, considering only large durations and 

 fetches, he plots E'{a)IE against (ct — ctq), where E is the total energy in the 

 wave pattern (i.e. the mean square wave amplitude), and o-q is the angular 

 frequency of the maximum of the spectrum, all the spectra fit reasonably well 

 on to a single curve (Fig. 4). An equation which is a near fit to this curve is 



E'{g)IE = 3.82 exp -[(a-cro)/0.0531((T-CTo + 0.27)]-'i (10) 



= when (a-cro) < 0.27. 

 He also finds that 



E = 2.5x10-6 V\ (11) 



where E is the mean square wave amplitude in ft^ and V is the "surface" 

 wind speed in knots ; and that 



To = 27r/ao = 2.37F''^ + 2.5 + 5.06F4(sec). (12) 



