TM No. 377 



It is not clear on theoretical grounds just where the lower limit of the 

 equilibrium range should be found; however, Pierson (1959) offers evidence that 

 in sharply peaked spectra the equilibrium range can only begin at frequencies 

 greater than twice the frequency of the maximum spectral peak. 



Workers have provided evidence that the slopes of the auto-spectra of the 

 free surface function "ty (t) do, indeed, approximate the f"5 relationship indi- 

 cated in equation (V-25). This has been shown by Phillips (1958), Burling (1959): 

 Kinsman (i960), and Kinsman (196U). 



If there really exists an equilibrium range where the free surface spectra 

 are functionally related to the frequency, then this phenomenon should also be 

 evident in the particle velocity spectra. The classical free surface function 



"W (t) at a fixed horizontal position is associated as follows with the deep 

 water wave velocity functions, given by equations (II-5) and (II-6): 



t-y(+) r ^ccstri . (v-28) 



It has been demonstrated that the observed wave motions are roughly similar to 

 classical wave motions, at least with respect to approximate phase relationships 

 of the u and w wave components (see figures V-2 and V-3). It is therefore 

 reasonable to assume that the same forces affecting the spectral behavior of 

 if (t) would likewise influence the motions u(t) and w(t). Moreover, since the 

 value of "»'*■ and y»J and yjf * are directly associated with the potential and 

 kinetic wave energies, their spectra should be related tc similar dynamic 

 effects in the waves. 



According to Kinsman (i960), there should be a relatively constant decrease 

 in spectral energy with increase in frequency in the region somewhat above the 

 maximum peak energy. Also, it was inferred that a saturated spectral region 

 occurs irrespective of the absolute peak heights or wind speeds, assuming only 

 that a minimum of wave energy is available to form the equilibrium range. 



Figure V-35 shows five auto- spectra of the vertical velocity component w 

 measured at 0.5 meter depth. Wind conditions varied from 5.2 m sec _ l (serial 

 052) to lU.O m sec -1 (serial 057B). The variances are shown in the upper right 

 corner. Although both the peak values and the frequencies of peak location 

 vary considerably, the values and slopes of ^<jj are very similar in the fre- 

 quency range from 500 mcps (2 sec) through 800 mcps (1.25 sec). These curves 

 are suggestive of an equilibrium range in the wave velocity spectra. 



To explore any functional relationship of <£ w to f, a cursory examination 

 was made of some specific spectra of the vertical motions from BBELS-11. The 

 first set of observations (030, 031, 032, and 033) was made over the period 

 1622 to 1720 hours on 29 March 1965, at a depth of 0.5 meter. During this 

 period (see table IV-3) the winds were 10-11 m sec~l from the ENE, and the seas 

 were probably fully developed because of the ENE fetch limit (see figure IV-3). 



132 



