G70 BARBER AND TUCKER [CHAP. 19 



corresponding areas on the windward and lee slopes of the waves, where V is 

 the wind velocity and C is the phase velocity of the waves. He was not able to 

 calculate the absolute value of the pressure difference, but had to introduce an 

 arbitrary constant, s, which he called the "sheltering coefficient". Equating the 

 energy fed into the waves to the energy lost by dissipation due to viscosity in 

 the water, he was able to calculate that there would be a net gain of energy by 

 the waves when 



C(F_C)2 ^ 4vg{p-p')lsp'. (6) 



Here, v is the kinematic viscosity, p and p' are the densities of water and air 

 respectively. This formula allows calculation of the least wind speed which can 

 generate waves, and comparing this with observation, Jeffreys deduced that 

 s = 0.27, a value which is reasonable physically. Further deductions do not, 

 however, agree with observations. 



This type of theory has been considerably refined and elaborated, particularly 

 by Russian workers (Shuleikin, 1956; Krilov, 1958). In some cases the dif- 

 ference in tangential stress (wind drag on the surface) between the crest and 

 trough of a wave is also taken into account. However, most theories of this kind 

 contain unknown constants equivalent to the sheltering coefficient s, which 

 have to be determined by comparison with actual wave measurements. This is 

 unsatisfactory, since it is impossible to calculate from first principles whether 

 the processes postulated can feed an adequate amount of energy into the waves. 



Miles (1957, 1959) has overcome this difficulty. By considering the air flow 

 over the waves, he is able to calculate the amplitude and phase of the air- 

 pressure fluctuations in terms of the characteristics of the wind-profile, with no 

 unknown constants. He can thus calculate the energy transfer from the wind 

 to the waves, and can test his theory by comparing this with observations of 

 the rate of growth of waves. He finds order-of-magnitude agreement. He is also 

 able to work in terms of a real sea consisting of a complete spectrum of wave- 

 lengths. 



The second type of mechanism has recently been studied by Phillips (1957, 

 1958). He resolves the moving pattern of pressure fluctuations in a turbulent 

 wind into "long-crested" sinusoidal components, and some of these components 

 have a relationship between wavelength and velocity which is the same as that 

 of a free wave on the water surface. These components will build up waves by 

 resonance. Thus, if the spectrum of the turbulent pressure fluctuations in the 

 wind is known, he can predict the wave spectrum. His results explain qualita- 

 tively many observed properties of sea waves (see also, for example, Phillips, 

 1958a), but there is at present some doubt as to whether the theory can give 

 correct quantitative results. The difficulty here is mainly the lack of adequate 

 knowledge of the turbulent pressure fluctuations in wind over the sea (see, for 

 example, comments on Phillips's 1958 paper in Chapter 21). 



An outstanding feature of Phillips's theory is that it yields a theoretical 

 spectrum for the wave pattern in both frequency and direction of travel. 



It seems likely that a complete theory of the generation of waves by wind 



