SEAWAY 



25 



chosen intuitively liy the individual researcher. In par- 

 ticular, the effects of several separate parameters are 

 usually combined in groups; it has been found that 

 functional relationships between judiciously selected 

 groups of parameters can be e\-aluated cnipirically with 

 much greater reliability than relationships between in- 

 dividual parametei's. In this connection, the short- 

 comings of the partially developed theoretical I'easoning 

 are impliritly compensated for in establishing empirical 

 relationships. The Inilk of this kind of information has 

 been accumulated since 1943, and had its incenti\'e in 

 the need for predicting sui'f conditions at Euro]3ean inva- 

 sion beaches in World War II. Hie prolilems of wave 

 formation in a storm area and of wave decay or dispersion 

 outside of the storm area had therefore to be treated. 

 The objective was to arrive at the condilions existing at 

 a beach at a gi\'en time due to a storm or .several storms 

 which had occurred s(jme time earlier, often at a distance 

 many hundreds of miles away. In the present text only 

 the wave conditions in or near the storm area will be dis- 

 cussed. Only a liricf outline of the most important 

 methods used to obtain information on the.se conditions 

 will be given. Details will be found in the readily avail- 

 able references. 



5.1 Method of Sverdrup and Munic (1946, 1947). 

 Quoting from Sverdrup and Alunk (1947): "Within the 

 generating area there always exist a large number of such 

 trains of waves of different length, traveling with the 

 wind or at small angles with the wind direction. From 

 interference and criss-crossing there results an extremely 

 irregular appearance of the sea surface, but the larger 

 waves can be recognized and the theoretical relationships 

 between period, length, and velocity apply to the.se 

 (Kriimmcl 191 1, S\'erdrup, et al, 1942). 



"Because of the simultaneous presence of many 

 trains, the wave characteristics ha\'e to be described by 

 some statistical terms. For that purpose it has been 

 found convenient to introduce the average height and 

 period of the one-third highest waves. The waves de- 

 fined in this manner are called 'the significant waves,' but 

 the definition recjuires further refinement because the 

 composition of the 'one-third highest waves' depends on 

 the extent to which the lower wa\'es have been considered. 

 Experience so far indicates that a careful obser\'er who 

 attempts to establish the character of the higher wa\'es 

 will record \'alues which approximatelv fit the defini- 

 tion '"9 



The "significant wa\-es" arc assumed to have all the 

 properties of simple waves of finite height (Stokes' 

 waves) except that their length, height and energy con- 

 tent do not remain constant as is the case with simple 

 waves. These cjuantities increase with the time and the 

 distance over which the energy of the wind is transmitted 

 to waves. The details of this process of wave growth are 

 not known, but certain energy relationships can be 

 established. In this connecticjn, Sverdrup and Munk 



make the following observation: ". . . When a wind of 

 constant velocity has blown for a long time oxer a limited 

 stretch of water, such as a lake, a steady state is estab- 

 lished. At any fixed locality the significant waves do not 

 change with time, l)Ut on the downwind sidi' of the lake 

 they arc higher and longer than on the upwind side. If 

 on the other hand a uniform wind blows o\'er a wide 

 ocean, waves grow just as fast in one region as in any 

 other region and the significant waves change with lime 

 but do not vary in a horizontal direction." 



The energ.v balance is then written for the.se conditions 

 by equating enei'gy growth to energy received from wind. 

 The dissipaticni of enei'gy by molecular or turbulent \'is- 

 cosity is neglected. The energy growth is represented liy 

 the growth of wave height, length, and celerity all of 

 which are taken as functions of time t and distance x 

 along the fetch. The energy is transmitted from wind 

 to wa\'es by the work of the normal pressure times the 

 vertical water .surface \-elocity 



E, = l r P>" fl-^- (52) 



where X is the wave length and w is the vertical xclocity 

 of the water surface. Following Jeff'reys, the out-of- 

 phase component of pressure is written as 



P 



±sp' (U 



dr)/d.r 



(515) 



This (Miuation indicates wave growth for [' > c, and 

 decay for U < c. To reconcile this with the fact that 

 waves are often observed to tra\'el faster than wind, the 

 energy transfer by the frictional drag force is introduced 



Et = I f rUdx . (54) 



where u denotes the horizontal component of water- 

 particle velocity (orbital and drift \('locity) at the sea 

 surface, t is the stress which wind exerts on the sea .sur- 

 face, and is evaluated as 



= yp'U^ 



(55) 



where for U measured at a height of 8 to 10 m, the co- 

 efficient 7 is taken as 0.002(5. Owing to the symmetry 

 of an harnKjnic wave the integral in (54) vanishes, but 

 for Stokes' waves of finite height, a characteristic of 

 which is mass transport (see Appendix A, Section 3.3), 

 the integral has a finite value. Since the water-particle 

 velocity is low in comparison with wind velocity, energy 

 is shown to be transmitted to water, and wa\'es can 

 grow even if [/ < c.-" 



The energy balance equations are now written : 



For the transient state, 



(IE Edc 

 dt cdt 



Et ± Ep 



(56) 



and for steady state, 



" For a comparison of instrumontal and vLsual observations see 

 Roll (195.5). 



'" It has been found that small wave com|ionents or ripples are 

 instrumental in absorbing the energy from the wind, making the 

 assumption of skin friction unnecessary as has been discussed in 

 previous sections. 



