SEAWAY 



17 



Eckart. The storm is modeled mathematically by the 

 succession of traveling pressure areas, and no considera- 

 tion is given to the effect of the wind velocilij. The wind 

 velocity U enters into Eckart's theory only in that the 

 pressure areas are assumed to travel with the wind 

 velocity. Neglect of the wind velocity itself is consist- 

 ent here with the initial assumjilion of "infinitesimal" 

 waves. Quoting again from Eckart: "The term in- 

 finitesimal in this connection, refers to the neglect of non- 

 linear terms in the hydrodynamic eciuations; it seems 

 certain that this must be ultimately remedied if all 

 phenomena connected with the interaction of wind and 

 water are to be treated theoretically . . ." Particularly 

 important in this definition are the infinitely small 

 slopes of the water surface. Since the wind action on 

 water appears to depend on the square or higher power of 

 these slopes, the mechanism of the kinetic-energy trans- 

 fer from wind to water is es.sentially absent in Eckart's 

 ca.se; only changes of static pressure of the air are con- 

 sidered. It .should be clear from the preceding sections 

 that the transfer of energy from wind depends specifically 

 on the finite height of wa\-es, and particularly on the 

 existence of sharp slopes which are not considered in this 

 ca.se. 



While the foregoing paragraph represents a plea for 

 consideration of wave-correlated pressiu-e areas, it should 

 be noted that Eckart's conclusions (except for wave 

 height) probably would not he greatly affected by such 

 considerations. Since the waves themseh-es are ran- 

 domly distributed, and in fact have the appearance of 

 randomly distributed groups, it can be assiuned that 

 consideration of correlated pressures would have taken 

 a form similar to the one used by Eckart. An apparent 

 major difference would be the use of wa\'e-group \'elocity 

 in.stead of wind velocity for the propagation of pre.s.sure 

 areas. Most of Eckart's conclusions, except the mean 

 value of wave heights, can therefore be assumed to be 

 valid for the actual sea surface, at least for the time 

 being. Unfortunately, observational data on the angu- 

 lar dispersion of wave propagation and on the lengths of 

 wave crests are very meager. In particular, data on the 

 waves within the storm area appear to be almost com- 

 pletely absent. Such data as for instance Weinblum's 

 (3-193(3) stereophotographs on the San Francisco in a 

 sevei'e storm have not yet been analyzed in a form suit- 

 able for comparison with the theory discussed here. 

 Furthermore they usually cover too small an area to be 

 valid statistically. 



The work of Eckart (1953a, b, c) can be considered as 

 extremely important not only for its results, but for the 

 method of attack as well. Further work based on this 

 method, but considering the action of the wmd on the ir- 

 regular sea surface with large wa\'e slopes, should be en- 

 couraged and sponsored. 



4.2 Phillips' Theory. Apart from the mathematical 

 methods, Phillips' theory differs from Eckart's by the 

 adoption of a more general randomness. While Eckart 

 postulated random distribution of gusts in space and 

 time, he, by independent reasoning, specified the diam- 



eters and durations of gusts. He also assumed that 

 gusts move with the wind velocity U. Phillips made the 

 statement of the prolilem more general by assuming 

 that the dimensions and lifetime of gusts are also random. 

 This included the smaller gusts moving near the s^a sur- 

 face in the air stream of reduced velocity uiz) < U. 

 In stating the problem Phillips, therefore, postulated an 

 (as yet) vuiknown \'elocit3' f '<.. 



A random distribution of fluctuating pressures is de- 

 scribed by a spectrum;'" i.e., it is thought of as com- 

 posed of a superposition of sinusoidally varying pressure 

 fluctuations of different amplitudes, frequencies and 

 phase relationships. The term "spectrum" or, more 

 exactly, "spectral density" is applied to the mean am- 

 plitudes of fluctuations within a narrow frequency band. 

 Waves excited by such pressure fluctuations also are de- 

 .scribed by a spectrum; i.e., liy the summation of simisoi- 

 dal wave components of various amplitudes and fre- 

 quencies. The end result of Phillips' solution is an ex- 

 pression defining the wave amplitudes in terms of the 

 amplitudes of pressure fluctuations at all frequencies. 

 This relationship is time dependent and the wave ampli- 

 tude is .shown to increase in proportion to the elapsed 

 time. In the process of .solution, the effective gust travel 

 velocity Uc was defined. 



Quoting Phillips, "It is found that waves de\-elop 

 most rapidly by means of a resonance mechanism which 

 occurs when a component of the surface pre.s.sure distri- 

 bution moves at the same speed as the free surface wave 

 with the same wave number. 



"The development of the waves is conveniently con- 

 sidered in two stages, in which the time elapsed [from the 

 onset of a turbulent wind ] is respectively less or greater 

 than the time of development of the pressure fluctuations. 

 An expression is given for the wave spectrum in the 

 initial stage of development, and it is .shown that the 

 most prominent wa\'es are ripples ftf wa\'elength Xcr = 

 1.7 cm, corresponding to the minunum phase ^•elocity 

 c = {i g T/p)^'* '' and moving in directions cos~'(c/t'',) 

 to that of the mean wind, where U, is the 'convection 

 velocity' of the .surface pressure fluctuations of length 

 scale X.r or approximately the mean wind speed at a 

 height Xcr above the surface. Obser\'ations by Roll 

 (1951) have shown the existence, under appropriate 

 conditions, of waves qualitati\'ely .similar to those pre- 

 dicted by the theory. 



"Most of the growth of gra\'ity waves occurs in the 

 second, or principal stage of development, which con- 

 tinues until the waves grow so high that nonlinear ef- 

 fects become important. An expression for wave 

 spectrum is derived, from which the following result is 

 obtained : 



'° An outline of the notation and mathematics used in connection 

 with random processes (particuhirly sea waves) will be found in 

 .Section 8. The reader is askeil to accept the brief and incomplete 

 statements on the subject in Sections 4 and 5, and thus a tem- 

 porarily incomplete understandinK of these sections, with the 

 hope that he will return to them after perusing Section 8, "Mathe- 

 matical Representation of the Sea Surface." 



" T here is the surface tension. 



