A theory is given for the phase speed in the presence of an adverse 

 wind. Using a stream-function approach in the inviscid momentum 

 equations leads to a well-known form of the Orr-Sommerfeld equation. 

 Solving these for the air and water flow separately, and combining them 

 by the dynamic (pressure) and kinematical boundary condition at the 

 interface, a perturbation technique yields the following expression for 

 the phase speed C: 



C/C„ = 1 + c„/c„ + cjc. 



in which C„ is the phase speed in the absence of the wind, C„ the 



O '^ '^ ' w 



correction for the surface current, and C the correction for the wave- 

 induced pressure fluctuations in the wind at the water surface. In the 

 solution for waterflow, a parabolic profile going down to zero was 

 employed (Kato, 1972), and for airflow, a logarithmic profile was used 

 up to a uniform free stream. For the wave-induced aerodynamic pressure 

 at the water surface. Miles' (1957) expression was introduced. 



For a fixed windspeed, -C /C^ grows rapidly with wave frequency, 

 while for ~C„/Cq the increase is much more gradual. In the example 

 given, the two curves follow each other up to about 1 hertz. 



The first correction term in the phase speed expression is 

 proportional to the surface current and thus to windspeed; it is, 

 therefore, positive or negative according to a favorable or adverse 

 wind. The aerodynamic term C^/Cq, however, is always negative. This 

 means that for waves of the same period, the wave speed will decrease 

 more significantly for an adverse wind than it will increase in a 

 favorable wind. Measurements by Shemdin (1972) with a favorable wind, 

 compared with the authors' results, confirm this theoretical prediction. 



The results of the theoretical analysis were compared with the 

 experimental results. There was fair agreement for periods 0.8 and 1.0 

 second, but some discrepancy for 1.2 seconds. According to the authors, 

 this discrepancy may be attributed to the chosen parabolic current 

 profile. They refer to measurements by Dobroklonskiy and Lesnikov which 

 show a logarithmic profile. Since the parabolic profile implies laminar 

 flow, and the Reynolds number for the drift current is quite large, it 

 seems that the flow was in fact turbulent in the water also, and thus a 

 logarithmic profile is expected. (The wave speed solution for a 

 logarithmic drift current was later obtained by Kato, 1974.) 



Coastal Engineering Significance . Apart from the weakness in the chosen 

 drift current profile, the investigation shows that for the wave charac- 

 teristics in question, it is important to include the aerodynamic term 

 Cg/Cg in the expression for the phase velocity of a wave over which a 

 wind is blowing. Lilly's (1966) solution considers only the drift 

 current correction. 



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