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



329 



ward side of large wave crests and appear much smoother 

 oil the lee side. The turbulence caused by breaking 

 wave crests also damps out small wa\'es temporarily, but 

 they are again regenerated by wind. Tiierefore, with the 

 de\'elopment of larger waves the mean value of d can 

 be expected to assume values smaller than the maximum 

 and a further reduction of 7 can be expected. 



3 Drag as Function of Wave Slope 



For later use, Xeumann de\-elops an expression for the 

 tangential stress as a function of the maximum wave 

 surface slope (dij/d.r)„,„x = 27ra/X = al;, where k is the 

 wave number 2tt/\. In terms of the drag coefficients, 



c. '- a- 



0- 



(13) 



and in terms of the wave slope and a coefficient .s (similar 

 to Jeffreys' except for the factor of 1/2), 



It follows that 



(rk~{U 



Ch = sa-k' 



(14) 



(15) 



Noting that Motzfeld (1-1<):]7) obtained different 

 powers for a and k from his wind-tunnel experiments, 

 Neumann (1-1949) writes Ca generally as 



Cd = sa"k" 



(16) 



Using equation (4) for the total stress r, Ca is com- 

 puted by equation (13) on the assumption of c = U /'i. 

 By using the empirical relationship for o/X derived in 

 Neumann (1-1948), simultaneous solutions for s and n 

 are obtained for a number of wind speeds. These show 

 that mean constant values of n = 1 and s = 0.095 can 

 be assumed. The final relationship 



s ^ ak{U - cy- 



(17) 



is then adopted. The restriction to c = U/3 in the 

 derivation of C'a appears to be neglected in the subse- 

 quent work, and a constant value of s = 0.095 is used 

 independently of 13 = c/U. 



4 Description of the Sea 



As mentioned under (</) in the introduction, Neu- 

 mann's work is valuable not only for its formal deriva- 

 tions, but also for the descriptions of the observed sea 

 conditions. These descriptions are inserted thi'oughout 

 the cited references, and it would be impractical to repro- 

 duce them with any degree of completeness. However, 

 a few quotations (in an informal translation) will be use- 

 ful here. Thus from Neumann (1-1950, page 42), speak- 

 ing about initial waves: ". . . at a wind of about 1 m/sec 

 they are about 7 cm long, and the height of these 'ripples' 



amounts to about 0.5 cm. As the; wind becomes strongei', 

 the sea surface takes on a pronounced 'rough' appear- 

 ance. The early regularity of initial wave is destroyed; 

 with relatively rapidly growing waves, all possible wave 

 lengths appear beginning with small ripples up to the 

 maximal wave. In this wa\'e confusion certain waves 

 show signs of breaking, or the beginning of breaking, of 

 sharp crests. This condition establishes itself relatively 

 fast with freshening wind. The steepness of waves {H/X 

 = 8) is relatively large. According to the results of various 

 observers, the maximum ratio 8 appears to lie at 1/8, 

 which comes close to the theoretical maximmn 5m„x. 

 = 1/7, which Michell calculated for the steepest waves 

 according to Stokes' theory." 



"When the wind strength further increases, the wave 

 lengths and wave heights grow rapidly. A fact known to 

 all sea travellers is that with a suddenly developing wind, 

 and quickly whipped-up sea, the wave crests are steeper 

 than with an older sea at corresponding wind strength. 

 With a steady wind \'elocity and with waves reaching 

 full development, a certain maximal wave corresponding 

 to the wind strength always develops. It is what a sea- 

 man designates as 'sea.' These seas can l)e individually 

 recognized, sometimes more, sometimes less clearly, in 

 the wave confusion in a seaway. But the mijre fully de- 

 veloped the sea is, the more conspicuously these waves 

 appear; further indication of a fully developed seaway in 

 a storm wind is the increasing length of wave crests. 

 The 'sea' is entirely overlaid by shorter waves down to 

 ripples. The stereophotographs in the work on "^leteor" 

 (Schumacher, 1-1925) show how the wave-agitated sea 

 surface consists of overlaying of the large forms by the 

 smaller and the smallest. Particularly in 'storm seas' 

 this is a well known occurrence. The long 'rolling' sea is 

 almost always clearly seen in a seaway, and its surface 

 apijears to be strongly roughened." 



"The wave confusion of overlaying small waves stands 

 close to a 'continuous spectrum,' which can be observed 

 in a young seawaj^ thrown up by a sudden onset of wind. 

 Again and again these small waves are rebuilt by absorp- 

 tion of the energy from wind, grow to maximum steep- 

 ness and break; they represent an important factor in 

 the cjuestion of energy transfer fi'om wind to water. In 

 the development of the complex sea they act in a certain 

 sense as 'roughness protuberances' for the wind sweeping 

 over the main wave profile of the sea. The resulting 

 shearing forces form a substantial part of the total thrust 

 exerted by wind on the sea surface. They also can be 

 effective in the sense of energy transfer when those 

 waves occur whose celerity exceeds the wind velocity." 



On the basis of the plot of 5 = H/\ versus c/U, shown 

 in Fig. 1-19, Neumann (1-1950, page 45) writes: "Among 

 the wa\'es in the appearance of the complex seaway three 

 types will play particular roles: — 



"1 The short but steep breaking c = U/3 wave.' It 

 is of small practical significance, but theoretically it 



^ Symbols are given here in the notation of the present mono- 

 graph. 



