328 



THEORY OF SEAKEEPING 



T = O.ODp'f/^/'- (4) 



The final plot of the coefficient 7 ver.sus U is shown in 

 Fig. 2. 



The drag coefficient 7 diminishes with stronger winds; 

 i.e., from the point of view of the energy transmission 

 from wind, the sea becomes less rough in a stronger wind. 

 While startling at first, this result is consistent with the 

 fact shown earlier by Sverdrup and Munk (1-1946, 1- 

 1947) that wave steepness decreases with increase of wind 

 strength (see Fig. 1-18). Neumann (1-1948) inde- 

 pendently obtains a rapid reduction of wave steepness 

 from the limiting value of \/H = 7.33 at U = 1 m per 

 sec to the maximum \/H = 25 at 12 m per sec. Darby- 

 shire (1-1952), attempting to use .Jeffreys' (1-1925) theory 

 to explain some of his findings, came to the conclusion 

 that the sheltering coefficient s is not constant, but can 

 be assumed to be proportional to the wave slope dr;/dj. 

 This slope decreases with H/\ and with increa.se of wave 

 length at increasing c/U ratio. 



Additional confirmation of the decrease of drag-force 

 coefficient with increase of wind is found in the measure- 

 ments of wind-velocity gradient made by Wiist (1-1937) 

 and Jeffreys (1-1920). The theory of such measurements 

 is outlined in Section 1-2.7, but will now be repeated 

 following Neumann (1-1948). ^ It is assumed that the flow 

 of air along the sea surface is of the same nature as along 

 a rovigh plate. L. Prandtl expressed the tangential force 

 per unit of plate area as 



T = Adu/dz = p'Hdu/dzy- (5) 



where u is the air velocity at the distance 2 from the 

 plate, and I is the "mixing length," expressed in the 

 vicinity of the plate as 



I = koiz + 20) (6) 



Here ko is a constant evaluated theoretically by \'on 

 Karman as 0.38, and experimentally measured as 0.40; 

 2o is a length dimension which defines the "roughness" 

 of the plate. I'rom (5 and 6), it follows that 



T = p'AV-(2 + zor-(du/dzy- (7) 



or 



du 



1 



1/2 



(8) 



dz ko{z -f- Zo) \p 



Assuming that shear stress in the vicinity of the plate 

 can be taken as independent of 2, it follows after inte- 

 gration, that 



1 



and 



_ V/= 2 + 20 



ko \p J Zo 



P /lO" 



In 



z -\- Zi^ 



2o 



(9) 



(10) 



Comparing (10) with Taylor's expression (1), it follows 

 that 



('»^") 



(11) 



From measurements of velocity u at several heights 2 the 

 tangential force r and the roughness parameter Za can 

 l)e computed. The results of such measurements and 

 computations made by Wiist (1-1937) and Jeffrey.? 

 (1-1920) are given in Table 3. The roughness parameter 

 2o is seen to diminish very rapidly with increase in wind 

 speed at low wind speeds, and more slowly at higher wind 

 speeds, when it generally has a very low value. 



Table 3 Roughness Parameter Z[ Versus Wind Speed 



Source 

 Wiist 

 (1-1937 



Method 

 I'ind velocity gradient in 

 lowest air layers 



Jeffreys 

 (l-i920) 



JefTrevs 

 (1-1920) 



.\ngle Ijetwcen surface wind 

 and gradient wind 



Relationship between wind 

 velocity on the surface 

 and strength of gradient 

 wind 



Author's Remark: The reduction of the tangential- 

 force coefhcient 7 with wind speed can logically be ex- 

 plained on the basis of Section 1-2, taking into considera- 

 tion the complex nature of sea wa\'es. In moderate to 

 strong winds, it can be expected that the surface of 

 larger waves is entirely covered by small waves and rip- 

 ples, for which l3 = c/U < 0.4 and which attain the limit- 

 ing steepness of 1/7 and de\'elop sharp crests. The drag 

 coefficient Ca, on the basis of Motzfeld's (1-1937) wind 

 tunnel tests as well as the wind flume tests of Francis 

 (1-1951) and of Johnson and Rice (1-1952), can be ex- 

 pected to have attained its maximum value and to re- 

 main approximately constant. The coefficient Ca is de- 

 fined, however, with respect to the relative velocity 

 U-C, where C is the facet velocity of Munk (1-1955) and 

 probably can be identified with the formula velocity of 

 Keuligan (1-1951) and Van Dorn (1-1953). The co- 

 efficient 7 on the other hand is referred to U. It follows 

 then that 



7 « c,v'/{u - cy- 



(12) 



For a recent review of this subject see Neumann (1-1955). 



and decrease of 7 with increase of U occurs if C increases 

 less rapidly than U, even in the case of constant maxi- 

 minii value of Ca- 



Moreover, the description of a complex sea given by 

 Neumann in the references here considered indicates that 

 the small ripples arc not uniformly distributed over the 

 surface of the sea. They are more developed on the wind- 



