Generation, Growth a>ifl Propagation of Waves 93 



to the average values and to the observations. More recent observations 

 suggest that a rapid increase of the wave steepness in the first stages of 

 development up to the maximum value 6 = -1. This value seems to remain 

 constant to about /S = \. Since this first stage is passed very fast, it will 

 probably have only little influence on the following development of the 

 wave pattern. 



With these relations, the equations (IV. 38) and (IV. 40) can be solved. 

 These solutions can be given as relations between non-dimensional parameters 

 and can be compared with observations. It is not possible to give here the 

 rather long derivations of these solutions and we must refer to the original 

 paper. 



The results are presented in the form of diagrams giving the most important 

 wave quantities and their dependence on fetch and duration. Bretschneider 

 (1952) has summarized the results into one diagram (Fig. 43), which presents 

 all the basic information on the generation of waves. It contains, first of 

 all, the dependence of c/U and gH/U 2 on the quantity gF/U 2 that is, on F 

 the fetch and U the wave velocity, for an unlimited wind duration. The 

 variation of these quantities is so large that, to represent it graphically, one 

 needs a 4 x 5 cycle logarithmic paper. The curves represent the theory, the 

 points the observations. The latter were greatly increased by Bretschneider, 

 who included data which have been obtained from wave-channel experiments, 

 from measurements on small lakes, and from observations and wave-recorder 

 measurements of ocean waves. Thus, the entire curve is covered by ob- 

 servations points. 



The duration curve tU/F and the curve H/L versus gF/U 2 also have been 

 presented in Fig. 43. These curves give in a similar manner the wave velocity 

 and the wave height as a function of wind duration t for unlimited fetch. 

 Also here there is a close agreement with the observations. The curve tU/F 

 was determined assuming that for a given fetch length, waves generated by 

 a particular wind velocity reach their limiting values in a particular time. 

 This time increases as the fetch length increases, resulting in increased limit- 

 ing values of wave height and period. For any particular fetch length and 

 wind velocity, there is a duration of wind, after which no future increase 

 in height and period occurs. This limited time is the time required for the 

 energy front associated with the significant waves to advance at the variable 

 group velocity from the beginning of the fetch to the end of the fetch. 



5. Observations on the Growth of Waves 



We have seen previously that there is a fair agreement between the theory 

 and observations on wave velocity, wave length and the wave period. It was 

 always more difficult to reconcile observations and theory on the wave height, 

 especially now that it is recognized that one has to consider the fetch and 

 the duration of the wind. The theory developed by Sverdrup and Munk has 



