537 



-^ FROM CROSS-SPECTRUM . 



) FOR INDIVIDUAL WAVES 

 AND STANDARD DEVIATION 



<-> 0.0 



2 3 4 5 6 7 8 9 10 11 12 

 FREQUENCY ( Hz ) 



FIGURE 14. An example of the comparison of one- 

 dimensional phase speeds of wind waves, determined 

 from cross-spectra of records of two wave gauges , and 

 determined for individual waves (zero-crossing trough- 

 to-trough) together with the standard deviation, and 

 the dispersion relation for water waves. At the bottom 

 is shown the coherence of the cross-spectra. The f^ 

 represents the frequency at the energy maximum. [Cited 

 from Tokuda and Toba (1978).] 



inferred that these facts strongly indicate the 

 existence of similarity in the individual waves or 

 in the field of wind waves, presumably as a result 

 of the strong nonlinear ity. 



5. APPROACH BY SIMILARITY HYPOTHESIS AND 

 DIMENSIONAL CONSIDERATION 



In cases of strongly nonlinear processes, such as 

 turbulence, it is hard to approach problems from 

 the rigorous way of solving a closed system of 

 equations. In these cases, some assumptions based 

 on physical considerations are sometimes introduced 

 to supplement the system of equations, to arrive at 

 useful results. In the case of wind waves, it 

 seems that an approach by the traditional model of 

 component irrotational free waves with their weak 

 interactions is not necessarily realistic as has 

 been shown. There is another approach, in which 

 a kind of similarity structure in the field of wind 

 waves is assumed, and a regularity in gross structure 

 is sought by invoking dimensional considerations. 

 An example of this line of approach has been 

 attempted as partly described in a paper by me 

 [Toba (1978a)] . 



Since the local wind stress distribution along 

 the surface of individual waves is as shown in 

 Figure 7, the local wind drift is forced to be 

 stronger near the crest and weaker near the trough. 

 Water particles near the surface travel a longer 

 distance when they are near the crest than when 

 near the trough. On the other hand, water waves of 

 finite amplitude cause the wave current, resulting 

 from the difference between the foreward and the 

 backward movements of the water particles. Some 

 self-adjustment should occur for individual waves 

 in such a manner than the forward and the backward 

 movements by the waves are coincident with the 

 difference in the local wind drift as to the phase. 

 The wave current Ug of the individual waves of 

 amplitude, a, and angular frequency, a, is now 

 approximated by that of the second order Stokes 

 wave : 



UQ 



a^aVg 



10' 



H" 



10" 



10 



Fetch lOOm 



170 



2A0 



310 



380 



450 



5A5 



587 



Significant \teve 

 Standard Deviation 



10° 



10' 



FIGURE 15. Examples showing that the main part of individual waves in the wind-wave tunnel (zero-crossing 

 trough-to- trough) satisfies the 3/2 power law between the normalized wave height H* and the period T*. The 

 u was 68 cm/s. [Cited from Tokuda and Toba (1978).] 



