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



337 



(^uiiliiifi from Xeumaiin (l-l'.)52/>, page '27')}, in (■oii- 

 nectiou with the foregoing; figures: "Tlie curves . . . gi\-e 

 the conditions of growth, and the horizontal liranchiug 

 lines the condition of tlie fully arisen tfu-ee main waves 

 in a complex sea. In the region of lower ^'alues g.v/U'-, 

 the observed fi and gH/U- values arrange themselves 

 very well about the cur\-es. At larger values of the fetch 

 parameter gx/U- the observations deviate to the right 

 from the computed cur-\'es ; this can be expected when the 

 first mean wave (the /3„, wave) has reached its fully de- 

 veloped state, and remains as a dominant wave inde- 

 pendent of the buikkip of longer waves. In the diagrams 

 it is indicated by straight lines branching to the right 

 from the curve. As the development of the seaway pro- 

 ceeds further, at sufficiently long fetches and duration of 

 wind action, the /3(1) waves and (3,,* waves (with (3,„* 

 = 1.37) also appear in addition to the /S™ waves. For 

 these, the corresponding values of x and t for a given wind 

 strength can be taken from the upward branching curves. 

 In a fully developed .seaway, l3 numbers and correspond- 

 ing periods can thus be expected, which lie sulistantialiy 

 between the upper horizontal line and the lower lines for 

 the corresponding wind strength. 



"In order to give an example of application of the dia- 

 grams, let us consider the de\'eloi)ment of a seaway in 

 wind strength of 16 m.sec. [.3I knots]. ^ 



"The first 'main wave' with li = 0.81, c = IH //(/sec will 

 be reached at the fetch parameter gx/U- = 11,500 and 

 time parameter gt/U = ■■->7,200; this wave is fully built 

 up after a fetch of 300 km [102 sea miles] or after Hi.S hr 

 with a .sufficientljr long fetch. It has the foll(.)wing 

 'dimensions' : 



Xi = 107 m [350 ft], Hi = 5.9 m [19.3 ft], the cor- 

 responding period of 8.3 sec indicates the most frequently 

 occurring 'period,' which is observed between wave crests 

 following one-after-another at a certain particular point 

 on the sea surface. 



"The fid) wave appears at gX/U- = 13,210 and gt/U 

 = 40,5(10 or after 345 km (186 sm) and 18.4 hr. Its 

 dimensions are X3 = 163 m [535 ft], H3 = 6.6 m [21.6 ft], 

 period = 10.2 sec. Development of the waves proceeds 

 further under action of the wind, and at gx/U- = 18,060 

 and gt/U = 48,610 the longest wave in a complex sea- 

 way — age;8* = 1.35 becomes fully developed; with it the 

 seaway is practically fully developed. This occurs at a 

 fetch X = 472 km [254 sm] and with a minimum wind 

 duration of t = 22.1 hr. The dimensions of this longest 

 'characteristc' wave in a seaway is then about 



X2 = 298 m [980 ft], H-, = 6.75 m [22.2 ft], 



period 13.8 sec 



Were we to take, for the longest wave, the I3„* wave with 

 c = 1.37t/, then X-, would be 307 m [1002 ft] and //-. 

 = 6.8 m [22.3 ft] at .r > 500 km [271 sm]. 



"These three 'main weaves' characterize the seaway 

 at 16 m/.sec [31 knots] and embrace the region of 'sig- 

 nificant waves. ' The characteristic periods lie between 8 



' Convfr.sion to knots, nautical miles and fci't which are in.sc-rti'd 

 in square brackets not in original te.\t. 



and 15 sec. The obser\'e(l fre(iuency distriljution oi the 

 time inter\'als between wave crests, following one-after- 

 another at a fixed point of the sea sin-face, gave a scatter 

 of periods between 6 and 16 sec, with the maximmn oi the 

 frequency distribution at 8.3 sec at the mean wa\-e height 

 of 6 m [{9.7 ft]." 



10 Summary 



The observational evidence on which Xeumaini's de- 

 velopment is based consists of 



(a) The plot of 5 = H/\ \'ersus /3 = c/U originated by 

 Sverdrup and Munk (1-1946) and supplemented by ex- 

 tensive observations by Neumann himself, and (for low 

 13) by Roll (1-1951) and Francis (1-1951). Empirical re- 

 lationships were fitted to these plots, as shown l)V erjua- 

 tions (30) to (33). 



(6) Evaluation of the hoi'izontal drag of the .sea sur- 

 face, Neumann (1-1948) on the basis of the inclination 

 of the mean water le\'el, supplemented qualitatively by 

 observations of the wind-velocity gradient. The in- 

 clination of the water surface was measured on large sea 

 areas, The resultant empirical relationships are given 

 by ecjuations (1), (3) and (4). 



With reference to (a), many observational points were 

 added to the original jilot of Sverdrup and Alunk (1-1946) 

 by Bretschneider (1-1952) and the Neumann (l-19.52a). 

 For the range fi > 1 '3 this mtiterial represents proliably 

 the most reliable empirical information available on sea 

 waves. The values of 5 corresponding to the range 13 < 

 l/'3 are rather uncertain, but this does not represent an 

 important factor in the over-all pattern of Neumann's 

 work. The evaluation of 6 for l3 < 1/3 will be important 

 in the future for the estimates of the distribution of very 

 short and steep waves which will be needed for a more 

 rational evaluation of the wind-drag force. 



With reference to (6), the number of ob.servations made 

 and the care exercised by Neumann in choosing the data 

 can scarcely leave any doubt as to the validity of the 

 evaluation for wave lengths smaller than double the 

 water depth, i.e., X < 140 m [4(50 ft]. On the basis of 

 Neumann's (1-1948) data this corresponds to a wind 

 speed of about 35 knots. For stronger winds and longer 

 waves the results can be considered as extrapolation. 



The most important first step, in working out the 

 aforementioned empirical data, is establishment of the 

 drag exerted by wind as a function of wa\'e steepness 6. 

 Jeffreys' expression (14) was not based on ob.served 

 physical facts, and Neumann finds that, in order to 

 agree with available empirical data, it has to be modified 

 as shown by e(iuation (17); the drag is expre.s.sed as 

 directly proportional to the maximum wave slope, rather 

 than to its square. In this eciuation the coefficient s was 

 found to have a constant value of 0.095. It should be 

 noted, however, that this conclusion resulted from the 

 assumption of c = C'/3; i.e., a constant value of (3 = 

 1/3. Expression (17) was used thereafter in disregard 

 of its limited applicability. This is mentioned here in 

 order to warn future investigators against indiscriminate 

 u.se of these conclusions. 



