16 



THEORY OF SEAKFEPING 



mann, and James (H). On the other hand, the broad- 

 ness of the subject is overlooked in the recent "ad- 

 vanced" work discussed in Section 4, and attention is 

 concentrated entirely on the transfer of the energy from 

 the wind. 



4 Generation of Waves by Wind— Advanced Rational Approach 



• Under this heading several recent papers can be re- 

 viewed briefly. The.'^e are elaborate mathematical de- 

 velopments and the reader is referred to the readily ob- 

 tainable original papers for the complete exposition of 

 the subject. Only a general outline of the principles 

 used will be gi\'en here, primarily in order to indicate the 

 apparent shortcomings and desirable directions of further 

 development." 



The ad\'anced appr((ach to the subject of wave forma- 

 tion b.y wind has taken two broad directions. Eckart 

 (1953a, b, c) and Phillips (1957) took air-pressure fluc- 

 tuations in a gusty wind as the primary cause of wave 

 formation. The pressure fluctuations in this case are 

 uncorrelated with the wave form. In another ajiproach, 

 Munk (1955) and ;\Iiles (1957), following in principle the 

 elementary method of Jeffreys (Section 2), considered the 

 air pressiu'es as caused bj^ the air flow about the wave 

 profile. In this case the air-pressure variations are com- 

 pletely correlated with the waves. Two such difTerent 

 concepts can be entertained only in the early stages of 

 development of a subject. It is probable that at a later 

 stage of the theoretical development a concept of partial, 

 correlation will be introduced. In these early stages of 

 development, also, the energy dissipation in wa\'es has 

 not been considered. 



4.1 Eckari's Theory. The wind pressure is assumed 

 to be every whei'e normal to the undisturbed surface of 

 water and caused by an en.scmble of "gusts." A gust is 

 defined as an area of high or low pressure, which moves 

 with the mean wind speed, and has a radius D/2 and a 

 duration or life T. At the end of the time period T a 

 gust "blows itself out" leaving its wake to di.ssipate in 

 the form of free gravity waves. First, the theory of this 

 phenomenon is developed for a single gust. Ne.xt the ef- 

 fect of the ensemble of gusts is treated on the basis of the 

 time average, as commonly used in the theory of turbu- 

 lence. A very large number of gusts of uniform diameter 

 and intensity is assumed to be distributed at random 

 through space and t'me. This ensemble of gusts makes a 

 storm. 



Quoting from Eckart (1953c): "In the generating 

 area the wa\'es may be man}' meters high, and thus repre- 

 sent a large surface density of energy. This energy can- 

 not be supposed to have been obtained from the air in- 

 stantaneously and locally. Much of it will have been ob- 

 tained from the air earlier and at a considerable distance 

 from the point of obser\'ation (though still in the storm 

 area). It will have been transported by the water es- 

 sentially according to the laws of free wave motion. 



' Only in the realm of basic ideas. Mathematical techniques 

 will not be discussed. 



This has long been recognized by the use of the concept 

 of fetch ..." The effect of the few gusts near to a point 

 of interest on the sea .surface is, therefore, negligible com- 

 pared to the many distant gusts the effect of which has 

 accumulated with fetch. 



The waves caused by the ensemble of randomly distrib- 

 uted and fluctuating pressure areas represent the sum- 

 mation of many comjDonent waves of different wave 

 lengths, heights, phases and directions of propagation. 

 Such waves are described bj' a spectrum. As will be dis- 

 cussed in greater detail later, the resultant appearance of 

 the .sea is that of groups of waves separated by calmer 

 regions, each group consisting of a few wa\-csof varying 

 heights. Quoting further from Eckart (195oc): "Since 

 the surface disturbance has a random character, no 

 unique value of wave number and frequency can be as- 

 signed to it; the.se dominant values correspond rctughly 

 to a maxinuun in the .spectrum." From a consideration 

 of the empirically observed number of 5 to 10 waves in 

 such groups, Eckart concluded that in a wind of 20 mps 

 (about 39 knots) for in.stance, the life T of the gust is 15 

 to 30 sec, and the typical gust radius D/2 is 40 m 

 (130 ft). 



Eckart's solution covers regions in.side and outside of 

 the storm area; the latter case is simpler and the solution 

 is more precise. Outside the storm area the spectrum of 

 wave directions is symmetrically dispo.sed with re.spect 

 to the radial direction from the storm center, i.e., the 

 velocity of propagation of the dominant wave is in the 

 radial direction. The existence of a .spectrum of wave 

 directions causes wave short-crestedness, and at the 

 radial distance r of 10 storm diameters D, for instance, 

 the average length of wave cre.sts is 2.2 of wave length X 

 (between succeeding crests). 



Inside the storm, the wave components have not yet 

 .separated, and there is no similarly dominant direction. 

 Each point is tra^'ersed bj' waves tra\-elling in many di- 

 rections. The conditions are particularly confusing in 

 the center of the storm area. The predominating direc- 

 tion of wave propagation becomes more clearly defined 

 as a point imder consideration mo\'es from the center to 

 the periphery of the storm area. Generally, the sector of 

 wa\'e directions inside the storm is not symmetric with 

 respect to the wind direction, the asymmetry decreasing 

 toward the edges of the storm. 



Outside of the storm area the wave remains constant, 

 since it is no longer influenced by atmospheric disturb- 

 ances. Inside the .storm area the formulas derived by 

 Eckart imply that the spectrum is a function of position 

 in the area, and in particular, that the efl'ect of a given 

 fetch depends on its position in the storm area. 



While the theory developed by Eckart explains many 

 of the obser\'ed characteristics of storm-generated waves, 

 it fails to predict correctly the wave height. The air 

 pressure needed to generate the observed waves accord- 

 ing to the theory is shown to be possibly ten times 

 greater than its probable value. 



One of the possible reasons for this discrepancy is con- 

 tained in the initial formulation of the problem by 



