309 



The velocity profiles for air were taken primarily at 15-2 cm water 

 depth, but there should be li_ttle change in air flow at 5.1 cm. One should 

 bear in mind, however, that Cg calculated from the data in Table I represents 

 Ts in a narrow slice along the center section of the channel. Strictly 

 speaking, Goodwin's estimates of Xg include the variation of Cg' in the 

 y direction. From Figure 5> it is clear that the shear on the water surface 

 near the maocima in the cross-sectional distribution of air velocity will be 

 larger than that at the center section. This may account for the differences 

 between values of Cg at d = 15-2 cm as estimated by Goodwin's method and the 

 momentum integral technique. 



V. WAVE SPECTOA 



Autocorrelation Functions and the Frequency Spectra 



The time correlations between displacements of the water surface were 

 calculated from the digitized depth gauge data. The autocorrelation function 

 R(t) is defined as: 



R(t) = C(tj^) iit^) , T = t^ - t^ (10) 



where CCt^) and C(t2) a-^e surface displacements taken at the same point 

 for two different times, ti and t2. The averaging technique in Eq. (lO) was 

 carried out after the method given in Blackraan and Tukey (1958)' 



The function R(t) for waves in the channel was found to exhibit certain 

 interesting features. A typical example is shown in Figure 13- R(t) was 

 generally found to oscillate regularly about the R(t) = line with in- 

 creasing T . Its amplitude decreased sharply initially, but it became 

 fairly steady at higher values of t , though sometimes it varied slowly 

 as if a lower harmonic was present. The behavior of R (t) suggests that 

 there is a tendency for the mutual action of the two fluids to force a 

 nearly periodic, regular disturbance to develop on the water at a given 

 fetch in the channel. On the regular waves are superimposed small, random 

 disturbances which are related to the larger values of R(t) for small t. 

 This, of course, is precisely the physical picture which developed from 

 visual observations of the development of the significant waves. 



It is well known that the autocorrelation of a periodic function of 

 period P is another periodic function with period P and a zero mean. Hence, 

 the period of the significant waves can be estimated from the zero crossings 

 of the autocorrelation function at large values of t where the effects of 

 the random component are small. As seen in Figure 13, the components of 

 "noise" tend to damp out rapidly, so that the period of the significant 

 waves also can be calculated approximately from zero crossings of R(t) 

 over the whole range of t • Typical values of l/P found in this m an ner 

 are listed in Table II. 



