332 



THEORY OF SEAKEEPING 



Ci = 12.95 ni/sec €■> = 21.9 m/sec 

 = 25 knots = 42.7 knots 



amplitudes: 



fli 



;.0 m 



3.4 m 



ca 



a-j 



16 m/sec 

 31 knots 



m 



Assuming each of the three "partial waves" to be 

 sinusoidal, the resultant profile of the total waves is 

 represented as 



y 



Oi sni 



c^t 



+ flo sill 



Cit 



as 



sin 27r [ 



C3I 



+ 



(18) 



Quoting from Neumann (l-1952a, pages 106, 107) : 

 "The time-dependent fluctuations of the seaway, re- 

 sulting from the superposition of the thi'ee designated 

 waves, are shown in Fig. 5. The upper wave train a 

 represents the sequence of waves at the place ,r = for 

 the duration from t = to t = 260 sec. The lower wave 

 train b gi^'es the corresponding fluctuations for the same 

 time interval at a place x = 550 m; i.e., at a position 

 located almost 1/3 of a nautical mile from x = in the 

 direction of wave propagation. The constructed wave 

 records show that outstanding wave groups can be ex- 

 pected onl_v from time to time at a fixed place. At the 

 location x = 0, they appear at intervals of about 45 sec at 

 the beginning, at the location x = 550 m at the end of the 

 observation period. In these 'groups' certain wave crests 

 grow to particularly great heights. In nature they 

 probably do not reach their full theoretical height, since 

 after exceeding a certain raaximiun steepness they be- 

 come unstable and break. 



"Between the groups at the beginning of the wave 

 train a, and correspondingly at the end of 6, lie mostly 

 three low waves, the heights of which change with time 

 and place. Also the interference described as 'double 

 waves' appears at certain intervals of time. After pas- 

 sage of a sequence of outstanding groups the seaway 

 picture at a fixed place changes. The conspicuous dif- 

 ference between particularly high waves and the low 

 waviness between groups evens out more and more, until 

 after a certain time the typical group character of the 

 seaway again predominates." 



Quoting from Neumann (1-19526 pages 253, 254): 

 "The complex picture of the development of a seaway 

 must be strongly taken into account in the question of 

 energ}' transfer from wind to water; the shearing forces 

 of wind at the rough wave boundary surface are deter- 

 mined to a very large extent by the short wave overlay 

 of the main wave profiles." 



"In the following, an attempt will be made to calculate 

 the growth of a complex seaway for different wind veloci- 

 ties as dependent on the duration of wind and on the length 

 of effective wind path (fetch). In this, the turbulent 

 nature of the event must be taken as much as possible 

 into account, and the fact must be considered that, with 

 increasing development of a seaway, the 'width of the 



*-> 



Fig. 6 Wave profile with rough surface and resolution of ef- 

 fective drag T into normal and tangential components (from 

 Neumann, 19526) 



spectrum of periods' of characteristic waves also grows. 

 "The whipping-up of the seaway to the fully de- 

 veloped condition of all waves appears to be a not con- 

 tinuous process. Rather one must assume discontinui- 

 ties in certain stages of wave growth; the relatively 

 short 'seas' even in a fully developed state must be ac- 

 cepted as steep waves breaking from time to time. These 

 steep breaking waves (with various 'apparent' wave 

 lengths) are probably necessary for the further develop- 

 ment of the complex seaway, so that longer waves with 

 greater energy content can build up, until the energy 

 transmitted from wind and dissipated in wave confusion 

 reaches a balance. . . . The width of the spectrum of 

 periods as well as the mean and maximum wave heights 

 depend on the wind strength, and in a not-f ully developed 

 seaway, on the state of development of indi\-idual wave 

 components " 



5 Energy Transmitted from Wind 



On the basis of the foregoing descriptions, Neumann 

 formulates a quantitative wave theory, considering the 

 shearing force of wind due to small waves and ripples, 

 and the air pressures on the surfaces of three dominating 

 waves. Fig. 6 shows Neumann's concept of the total 

 horizontal force r caused by the wind as compo.sed of the 

 normal pressure component and tangential shear force. 

 The former is taken as proportional to the slope (ir}/bx of 

 the wave under consideration and the second as caused 

 by a "roughness" resulting from the overlay of smaller 

 waves and ripples. The mean energy .4 transmitted by 

 these two components is: 



For the normal component 



A„ 



ip 



„w dx 



(19) 



For the tangential component 



A, 



X jo 



r,{u + u')dx 



where u is the horizontal component of the orbital 

 velocity of a water particle at the surface, u' is the mean 

 transport \'elocity of Stokes' waves of finite height, and 

 w is the vertical component of the orbital velocity. 

 For the wave profile expressed as 



