22 



THEORY OF SEAKEEPING 



Fig. 16 Beam factor of f (9o) relating beam width 

 60 to form drag. Upper curve: high-frequency 

 spectrum; lower curve: Neumann spectrum (Sec- 

 tion 6.2). Beam width of a tradewind sea is 130 

 (from Munk, 1955) 



Fig. 17 Contributions to mean-square elevation, mean- 

 square slope, and form drag by waves of different wave 

 age |8 (from Munk, 195 5) 



X I cos e (cos e - 13)13-' 5(/3, 9) d/3 dQ (47) 



Jo J -w 



= Sb /(Go) (const) W 

 where 



/(eo) = o 1 + 



K' 



sin f)„ 



~e7 





Quoting from Munk (1955a) : "In Section (above) we 

 have dealt with the case of a 'fully-arisen sea' (Neumann 

 1953) where it is assumed that all spectral components 

 have attained their equililjrium value. If the wind 

 fetch or duration is small, the effect will be that only tiie 

 high frequencies (.small 0) have attained their equilib- 

 rium \'alues. For prediction purposes Neumann intro- 

 duces what is essentially an abrujjt high-pass filter. 

 This is equivalent to specifying some critical wave age, 

 say /3„,, which depends on wind speed, fetch, and dura- 

 tion. The spectrum is presumed to be fully developed 

 for 13 < P„ and zero for /J > (3„. 



"Suppose now that /3„ = 0.5. It will be seen from 

 Fig. 17 that the mean-square elevation is a very small 

 fraction of the equilibrium value. The mean-square 

 slope has more than half its eciuilibrium value. The 

 form drag, however, has nearly its equilibrium value. 

 The form drag is therefore much less affected by limita- 

 tions in fetch and duration than the wave amplitude .... 



"The opposite example, that of a low-pass filter, is 

 provided by the action of oil or detergent spread on the 

 water surface. Suppose for the moment that the 

 spectrum, for 0„ less than 0.5, is dissipated by the sur- 

 face-active agent. This would leave the mean-square 

 elevation virtually unchanged, reduce the mean-square 

 slope by a large factor and essentially eliminate the form 

 drag. In general this corresponds to what is ob- 

 served ..." 



Figs. 16, 17 and the foregoing ciuotation probably 



represent the most important results of Munk's (1955) 

 work. Together with the work of Cox and Munk (1954) 

 and Van Dorn (1953) it demonstrates the predominating 

 importance of the small high-frequency waves or ripples 

 in the transmission of energy from wind to water. 



An equally important conclusion is that wave slopes 

 are much more significant for the wind stress than wave 

 elevations. 



The conclusion, from equations (45), (46) and (47), 

 that form drag is proportional to U^ is in direct contra- 

 diction to Neumann (1948, 1949a, b) who shows the 

 force to be proportional to C/'-^ In either case the basic 

 relationship is that the force is proportional to U^ times a 

 certain drag coefficient. Neumann finds this coefficient 

 to be proportional to U~'^-^, and Munk to a" or U. Con- 

 ceivably this discrepancy may have been caused by en- 

 vironmental conditions. Neumann's observations were 

 made in open seas. Cox and Munk's observations, on 

 the other hand, apparently were made in an area sur- 

 rounded by islands. The fetch may have been too short 

 for a fully arisen sea to develop at higher wind 

 velocities, and the waves therefore may have been 

 steeper than they would be in an open ocean. This situa- 

 tion appears to be indicated by the c/U and X/H ratios 

 listed in Table 5. It is possible, therefore, that the 

 proportionality of the drag t to U^ is valid for a "young 

 sea," but will not neccs.sarily apply to the more de- 

 veloped sea in an open ocean. 



Theoretically, the proportionality of the drag coefficient, 

 or Ca, to the wind velocity U was indicated in equations 

 (45) , (46) and (47) on the assumption of a constant value 

 for the pressure coefficient s. However Jeffreys' expres- 

 sion (37) setting the air pressure per unit area proportional 

 to the wave surface slope d-q/dx does not correspond to 

 physical facts. The coefficient s is in reality a variable de- 

 pendent on the wave form, which in turn depends on the 

 wind velocity. The equations may therefore be mislead- 

 ing in showing U^ conspicuously; the product sU' may 

 be a different function of the wind velocity. 



