NEUMANN'S 1948-1952 WORK ON WAVE GENERATION BY WIND 



335 



respdiiiliii^ III the fully tievelopcd seaway. At earlier 

 stages (if wave development the turbulent viscosity, 

 designated as ij.*(I3), is an unlcnown function of the wave 

 age. Neumann assumes for it an exponenlial relation- 

 ship: fori </3,„*< 1.37, 



., .,.,, l.:]7 - IS,,*' 



M*((3»,»,) 



fi exp 



and for 0.1 < l3 < 1, 



H*{I3) = ^*{\) exp [-.3.:m(l 



^)1 



(43) 



(44) 



where ti*(l) is the value of ti*(l3) at /3 = 0„* = 1. 

 Table 5 shows the values of n* versus wind speed. Table 

 6 shows the coefficient m*(/3) (cm^'g sec~') at a wind 

 speed of 10 m sec at various stages of seaway de\-elop- 

 ment. 



Author's Remark: The rapid growth of the coef- 

 ficient IX* with wind speed U and wave age 13 gives a dis- 

 torted impression as to the magnitude of the actual energy 

 dissipation iJrf,. This growth of ^u* results from the 

 presence of 5- in the denominator of ef|uation (41), and the 

 fact that p. is referred to the very low 8 of the "long wave," 

 asshownby the value for /Jm* = 1.37. The actual energy 

 dissipation, however, according to equation (40) is 

 mostly caused by "young waves" of low (3 and large 5, 

 and is proportional to the product /j*(/3)6-. It would ap- 

 pear that consistency would require the e\'aluation of 

 M*(/3) by integration over all values of /^, following the 

 same procedure as was used earlier for the evaluation of 

 y{0) in equation (35). 



8 Energy Balance 



With the nece.s.sary expressions ile\-eloi)ed for the 

 energy Ep received from the wind, and E^s dissipated by 

 turbulence, the energy balance can be set up and from 

 this the rate of wave growth can be computed. The 

 symbols Ep and E^s designate the energy per unit area 

 of sea surface per second. Neumann, following the 

 procedure previously used by Sverdrup and Munk, 

 writes the expression for the rate of change in energy for 

 an area of unit width and wave length X in length. The 

 wave energy advances at the group \'elocity c/'2. The 

 wave length X and amplitude a change with time t and 

 position X so that 



'^ (E\) = {Ep - E,.)\ (45) 



dt 



and 



^^D/^ c5/;X /5X C5X 



(Ep - E,i.)\ 



(46) 



Two cases are now considered: In Ca.se A a constant 

 wind of velocity U blows over an unlimited space. 

 Waves in all positions grow with time at an ('(jual rate. 

 The space derivatives vanish and 



dE , E dX „ „ 



—: + ~— = Ep - E,, 

 ot X ot 



(47) 



or, substituting the expressions for E (equation 58, 

 Appendix A) and X (Table 1, Appendix A) 



da pga- dc 



dt c dt 



(48) 



In Ca.se B a wind has blown for a sufficiently long time 

 along a fetch of limited length x. Thus, time derivatives 

 vanish and 



or 



c /dE , E ^ 

 2\dx \dx 



, / o £>c , c da 

 dx 2 dx 



E, 



E„ 



Eds 



(49) 



(50) 



Introducing now l3 = fid) according to eiiuations (31) 

 and (32) : 



Case A for (5 < 1/3, 



g dt 



(ol) 



and for 1/3 < l3 < (3 

 4 



d/3 



p — n-'U'li'e--'^{:i - rl3) ^ 

 g dt 



Ep - E„. (52) 



Ca.se B, for (i < 1/3, 



6'^^ U^P^ f = Ep- Ed. 

 g dx 



(53) 



and for 1/3 < /? < /3, 



P ^' n-U'ii' e--'^ (3 - r/3) ^ 

 g dx 



Ep - £„, (54) 



In all cases {Ep — Eds) is a function of /3. In writing the 

 foregoing, the symbol j) was u.sed for the number 0.0(i2, 

 /• for the number 1.667 and n for the number 0. 1075 found 

 in equations (31) and (32), and /3,„* = 1.37. 



Substituting the previously derived expressions for 

 Ep and Eds and transposing leads to differential ecjuations 

 as follows: 



Case .4, for /3 < 1/3 



dl 



p^ 127r-p- jj 



li' 



where 



Bi(/3) = 



C\{I3) - 5i(/3) 

 8^-gp- M*(/3) 



dli 



U'I3 



Case A, for 1/3 < /3 < 13 „ 



p4ir" 



-irff(o _ 



(3 - r,3) 



dt = '—- "" f^ 



p'g cm - B-Afi) 



dl3 



where 



B-m 



8T^-gn' M*(/J)e-^"^ 

 p' U'fi 



(55) 



(56) 



(5/) 



(58) 



