334 



THEORY OF SEAKEEPING 



Table 4 



pres.'^ure di.'^tributions; this follows from an examination 

 of the figures for Motsfeld's Model No. 4 in Table 1-1. 

 The stress acting on the simple wave profile is 



r = const ip72)iU - c)- 



(34) 



and the mean stress resulting from contributions by 

 waves of all lengths; i.e., all celerities c', using relation- 

 ships (:30) and (32), is 



r = .i-n- I — I I c'- f 1 — ^_ 1 dc' 



-(0^'r-(-r;) 



\2 J {c - U/Z) J un V U) 



+ .^ 



' dc' (35) 



where c' is the celerity of the overlaying waves and c the 

 celerity of the main wave. Taking the lower limit of the 

 first integral as instead of the few centimeters per second 

 of an initial wave represents a negligible error, .s' is 

 taken as s'2 = 0.0475 in order to allow for the nonuni- 

 formity of distribution of the steepest small waves along 

 the profile of the main wave. After integration , int roduc- 

 ing ^ = c'U and defining the effective tangential force as 



r' = p'yim' 



the coefficient* y(fi) is evaluated as 



1037(^) = 1-75 + 16.2 



L^ |g-I.667(3 [o.48(/3 + 0.()) 



^ — 1/3 

 - 0.0(1 + /3'-)] + 0.12(>| (36) 



The resulting values of 7((3) versus /3 are given in Table 4. 



Adding the work of the normal forces on the mean wave 

 profile to the work of "effective friction" due to smaller 

 wave components (evaluated in the foregoing) the total 

 coefficient of the efl'ective wind force is given as follows: 



For the range 1/3 < /3 < 1, 



C-m = 7((3) + 0.1075s'7re-'-6«'^(l - /S^) (37) 



Designating by 0„ the values of ji in the range 1/3 

 < i8 < 1, and by (3„,* the values of ,8 > 1, the expression 

 for the range 1 < /3„,* < 1.37 is 



« The notation o(/3J is used in this monograph for y-(0] written 

 b\' Neumann. 



(\ili) = yil3,„) + 0.10757rs'e-'-'''^"*'"(] - fim)- 



- 0.10757rs*e-'-«"'''"*(l - /i^*)- (38) 



where Xeiunann takes s* = 2s. In a fully developed 



.seaway /3„,* = 1.37. 



7 Energy Dissipation 



With the coefficient C{J3) now evaluated for various 

 stages of wave development (wave age) /3, the energy 

 input from wind can be computed by equation (28). 

 The next step in evaluating the energy balance is to 

 establish the rate of energy dissipation by waves. The 

 dissipation by viscous forces was given by equation (66) 

 in Appendix A as 



£,,, = 2M(2ir/X)'V=a- 



(39) 



Substituting c- = gX 27r and 2a X = d, and replacing ju 

 by n*, the energy dissipated per second per unit area is 



E,, = 2n*w'gd- (40) 



where fi*, which may be called turbulent viscosity, is no 

 longer a constant but a function of (3, or, more generally, 

 depends on the wind and conditions of a seaway. 



In a fully developed seaway, i.e., with fully developed 

 "sea" and fully developed "long wave," the energy is 

 in balance; i.e., Ep = Ea,. n* no longer depends on /3, 

 but only on wind strength, and is designated /Z. By 

 using expression (40) for Eis and (28) for the energy Ep, 

 letting C{fi) = yiU) from equation (1), and using equa- 

 tion (32) for 6, at /3 = 1.37 the expression for ji is obtained 

 as 



2ir-g&- 



l.S7p'y(U)U> ^ 

 8x0.1075'-gir2' 



,3.3.14X 1.37 



(41) 



y{U) is taken as evaluated empirically by equation (3), 

 except that in the present case Neumann changes the co- 

 efficient 0.09 to 0.10. With p' = 0.00125 and g = 980 



cm/sec", 



il = 0.00001825^^'- (in cm^i g sec^') (42) 



The foregoing value of jl is the maximum value, cor- 



