, -,,. TM No. 377 



Using equations (V-39) and (V-kO) to calculate the magnitude r of the vector 

 having u' and w' components, given as 



fhct+u,*]'* = ±(<r±) 



and plotting the vector magnitude r ( Q~-t ) over one wave period, one obtains 

 a quasi-ellipse. Figure V-^D is a hodograph depicting r as a function of Q~t . 

 This is a quasi-ellipse (i.e., slightly pinched across the semi -minor axis) with 

 the semi-major axis tilted back at about h^° from the vertical. Thus, a stress- 

 generating Eulerian wave model has been produced which resembles the tilted ellipse 

 and is proved by the phase-shift equations of motion (V-39) and (V-UO). 



The use of sinusoidal models to represent waves is very idealistic. The 

 simple functions do, however, point up one important aspect which is true for any 

 function employed to represent waves; namely, that small systematic phase dis- 

 crepancies, by virtue of the correlation phenomena, can give rise to gross effects 

 upon the measured momentum and energy transfer properties of the fluid. 



Let us examine some wave models which provide statistical properties some- 

 what comparable to those obtained from actual wave meter measurements. Hypo- 

 thetical wave data were generated which portray three different two-dimensional 

 heuristic wave models. 



1. Random-Biased Model (REM). This wave system has orthogonal 

 velocity components, u and w, which are quasi-random, having no preferred fre- 

 quency peak in the auto-spectra. About 5 percent of the data are slightly 

 adjusted to provide a small, negative covariance. A segment of the u and w 

 traces is shown in figure V-UoA (upper traces). This first hypothetical model 

 can be envisaged as a surface wave field where the motion is quasi -random, such 

 as might be produced by many oscillatory progressives moving in many directions. 

 The value of the covariance is about -3.7 cm 2 sec -2 . It is therefore necessary 

 to simulate a value of the stress near the water surface of 3.7 dynes cm" 2 . 

 This is considered, ordinarily, to be the order of magnitude of wind stress upon 

 the ocean surface (see ' Stommel, 1958). 



2. Sinusoidal-Unbiased Model (SUM). This model is represented, 

 approximately, by simple sine-cosine functions of u and w given by equations 

 (V-36) and (V-37). The velocities were simulated by picking u and w from hand- 

 drawn (hence, imperfect) sinusoidal curves. A segment of the simulated record 

 is shown in figure V-U6B (middle traces). 



3. Sinusoidal-Biased Model (SBM) . This model, shown in figure V-U6C 

 (bottom traces), is identical to the SUM except that the u function has been 

 slightly increased at its positive maximum point. This intentional b iasi ng was 

 done for two reasons: (l) to synthesize a desired negative value of u w ; and 

 (2) to bias the SUM model by a simple mechanism, perhaps not unlike that existing 

 in some natural ocean waves. To appreciate this, examine a simple mechanism 



lk5 



