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



333 



7, = a sin />■(.!■ - ct), (20) 



the vertical velocity of a water particle is 



w = dri dt = —akc cos k{x — rt) (21) 



and the horizmilal xclncity (including; llic mean mass 

 transport) 



(/ + '/' = ((/.:(• sin A-(.r - ct) + a-l:-r (22) 



The wiutl velocity U over a sinnsoidal \va\c' prolile is 

 expressed as 



[/ = r[l + ,r5sin /,■(.(■ - ct)] 



(23) 



where 5 is the wave steepness 2a/X, and C a constant 

 mean value of the velocity over the wave profile. Neg- 

 lecting the d'-, L"- is written as 



f/2 = {7-.[] _^ Owd sin /,■(.(■ - ct)] (24) 



and the tangential stress 



r, = p'f.r- = p'!,U-[l + '2iv5 sin kix - ct)] (25) 



where/, is the "effective" tangential-iorce coefficient. 



The assymmetry of the normal pressure distrihution 

 over the windward and leeward wave faces is assumed to be 



T„ = f„ + T,,' cos /,-(.r - ct) (2(5) 



where f„ denotes a mean value of the normal force oxer 

 the wave profile. 



Substitution of eijuations (23), (25) and (20) into equa- 

 tions (19) and integration over the wa\'e length leads to 



An = V2ir5r,/c 

 .1, = 2iv-pTfdJ-c 



(27) 



Since data on the coellicients t„' ami/, are not avail- 

 able, Neumann writes finally for the total energy {Ep 

 = An + Ai) tran.sferred from wind to waves 



Ep = p'UWi^) 



(28) 



where U is now the wind velocity at anemometer height, 

 |8 = c/U, and C{IS) is a coefficient which is a function of 

 (3. To the energy equation (28) corresponds the effective 

 tangential stress 



('(0)p'U^ 



(29) 



By comparison with ecjuation (1), C(lS) is seen to be 

 identical with y which was previouslj^ evaluated, equa- 

 tion (3), empirically as a function of U on the basis of 

 water-surface inclination and wind- velocity gradient 

 at sea. Neumann, however, points out repeatedly that 

 evaluation of the seaway in terms of a single mean wave 

 has relatively little value, since it does not explain many 

 features of the complex wave formation. He proceeds 

 therefore with an analysis of seaway growth due to wind 

 in three stages corresponding to the plot of wave steep- 

 ness 5 = H/'\ versus wave age (3 = c/U as shown in Fig. 

 1-19. The stages are /3 < 1/3, 1/3 < /3 < 1.37 and (3 > 

 1.37. The objective of this work is to derive functional 

 relationships for three dominant wave systems. The im- 



portant features of the complex .seaway (irregularity, 

 wa\-e-group formations) can be approximated by con- 

 sidering the joint action of the.se wave .systems. 



Quoting from Neumann (1-1952/*): "Under the a.s- 

 sumption that the seaway pa.sses consecutively through 

 three main stages of development until it is fully ari.sen, 

 the relationships among wave characteristics (wind 

 speed, duration of its action, the fetch) can be derived; 

 these represent a pragmatic approximation for the pre- 

 diction of wave conditions. The waves defined at a given 

 point, Neumann (l-1952a), (3,„, ,3(1), li,„* with the periods 

 Ti, To, Ti, must be visualized as the characteristic mean 

 values for certain intervals of periods, when we think of 

 the continuous spectrum as subdivided into these three 

 major intervals; the heights Hi, H-2, Hi are then visual- 

 ized as 'e(iui\-alent' heights for these major intervals." 

 In the foregoing, /3(1) is the "intermediate wa\'e" with 

 c = U, i3„, the ",s(>a" with c < U, and (3,„* the long wave 

 withe = 1.37 r. 



6 Energy in Terms of Wave Age and Slope 



The starting point is empirical ex'aluation of 5 = /(/5). 

 At the beginning of the development outlined here, very 

 little empirical data were available for the range fi < 1/3, 

 as can be seen from the plot in Fig. 18. At the same time, 

 on theoretical groinids, maximum \alues of 5 could be 

 associated with (3 = 1/3 at which the maximum energy 

 transfer from wind occurs according to Jeffreys (1925). 

 Neumami a.ssumed therefore that 



1.111/3= for 13 < 1/3 



(30) 



Later Fig. 19 was constructed in which many experi- 

 mental points at (3 < 1/3 were added, mostly from Neu- 

 mann and from Roll (1-1951). On this basis Neumann 



sets 



5 = eon.st = 0.124 for /3 < 1/3 

 For the other stages of wuxe development, 



6 = 0.215c-'-''"'* for 1/3 < /3 < 1.37 

 and 



const 



0.022 for (3 > 1.37 



(31) 



(32) 

 (33) 



Next it is assimied that the effective shear-force co- 

 efficient is a finiction of the maximum wave slope 

 2ira,\ = ird, in terms of a coefficient of proportionality s 

 (similar to Jeffreys'), and the relative air-wa\-e \-elocity 

 U — c. Neumann (1-1948) estimated the coefficient s to 

 have approximately a constant value of 0.095. With this 

 value for s and the previously established expressions for 

 5, the mean value of the coefficient C'(|8) is computed by 

 integrations over the relevant ranges of variation of (3. 

 The "effective friction" force is caused mostly by the work 

 of the normal forces on the sharp profiles of small waves 

 by which the mean wave is covered. The true skin fric- 

 tion of a smooth water surface is here neglected in com- 

 parison with the much higher effective force due to 



