SEAWAY 



21 



water surface is formed by suijerposition of many waves, 

 the shorter ones richng the surface of the Itinger ones, the 

 definition of c is not immediately evident. Munk has in- 

 troduced the concept of "facet velocity," i.e., he defined 

 c for the present purjjo.se as the horiz(jntal velocity of 

 translation of a small element of water surface; this 

 velocity will be designed here as C. The vertical veloc- 

 ity of water surface and the horizontal facet velocity C 

 are connected with the wave slope by the relationship 



b-n/bt = -C'dTj/d.f 



(39) 



Equation (37) can be simplified liy assuming C to be 

 much smaller than U , and neglecting the square of it. 

 Then 



V 



sp' U{U - 2C) d-o/dx 



(40) 



iMunk omits the factor of 2 in equation (40), and bj' 

 using ecjuation (39) writes 



p = sp'UiU - C) dv/dx = sp'u(u ^_ + ^) (41) 



The form drag force per vuiit area becomes then 



r = sp'f/t/^^ + ^\ 



bx dx dx dr 



(42) 



For a complex sea formed by superposition fif waves of 

 all amplitudes, directions, and lengths, the differential 

 expressions in the foregoing are averaged by statistical 

 methods (Eckart, 19536) and the drag force is expressed 

 as 



T = sp'C 



X I I (f'' cos e - c(/,:)]/c= cose, S'(/'-,0)/.-f/A-f/e (43) 



Jo J -n 



where k = 27r/A is the wave number and 9 is the direction 

 of wave propagation. The expression S{k,0)k dk dO 

 designates the contribution to the total mean-square 

 elevation by wa\-es with wa\'e numbers in the interval 

 k - (1/2) dk to k + (1/2) 5k, w^hich travel in the direction 

 e - (1/2)59 to 9 + (1/2)59 relative to wind direction. 



The wind stress is then evaluated for different a.s.sump- 

 tions as to the form of the function S{k,Q). By consider- 

 ing waves propagating in one direction, i.e., 9 = 0, and 

 assuming c to be negligibly small, Munk arrives at the 

 expression 



T = sp U-a- 



(44) 



where a- is the mean square slope. From Cox and ]\Iunk 

 (see Section 4.3) a- is taken as proportional to the wind 

 velocity U. Neglect of the small celerity c leads finally 

 to 



T = Sp' (const) U^ 



(45) 



For a range of wave directions 9, within the sector 

 ■ eo/2 < < 9o/2, equation (42) yields 



r = .sp' (const) /(9o) f/3 



■2 



Fig. 15 Probability distributions of wave slopes 

 along crosswind axis (upper) and up/downwind 

 axis (lower curves, positive upwind). Solid curves 

 refer to observed distribution; dashed to a Gaussian 

 distribution (from Cox and Munk, 1954) 



where 



.f(eo 



sin 9c 



(46) 



The values of f{Qo) are plotted ver.sus 90 in Fig. 16. 

 The dotted vertical line indicates the value of Bo derived 

 by Cox and Munk (1954) from the sun glitter measure- 

 ments. 



A third form of ,S'(A-, 9) considered by Munk corresponds 

 to the Neumann spectrum. A special section will be 

 devoted to this spectrum later on ; suffice it to say here 

 that it includes wa\'es of a wide range of wave numbers 

 k. For the present purpose Munk (1955a) replaces the 

 spectrum in terms of k by one in terms of /3 where fi = 

 c/U. The .spectra ,S'(/v,9) or <S(^,9) define the contribu- 

 tion of all waves to the mean-square amplitude. Simi- 

 larly .spectra can be expressed for the wave slopes. To 

 the mean-square elevation spectrum S{k, 9) corresponds 

 the mean-square slope spectrum k-S{k, 9). Fig. 17 

 taken from Munk (1955a) shows these two spectra in 

 terms of /3 = c/U, as well as the integrand of the cor- 

 responding expression for the tangential force 



