303 



The Phase Speed and th' Drift Current 



The phase speed of significant waves c as measured with respect to a 

 fixed point was compared with values calculated from the theory for small 

 amplitude gravity waves. The theoretical phase speed is: 



c^ = [^gk tanh(kd)J (2) 



where k = 2ir/X . In all cases, the values of c^ were larger than values cal- 

 culated by Eq. (2) . This effect has also been observed by Francis (1951) and 

 Cox (1958)' Francis qualitatively accounted for the deviation by considering 

 an increase in wave velocity associated with the surface drift, and the fact 

 that the waves are finite in amplitude. Cox, on the other hand, attributed 

 the difference to the combined effects of finite amplitude, orbital velocity 

 of low-frequency wavelets, drift currents, and dynamic effects of the wind. 

 Cox only analyzed in detail the finite amplitude effect as calculated by 

 Sekerzh-Zenkovich (1956). Cox found that the finite amplitude effect could 

 only explain his observed increase in phase velocity for waves larger than 

 X = 7 cm. The observed differences in phase speed for wavelets of length 

 smaller than 'V' 7 cm could not be accounted for by the influence of finite 

 amplitudes alone . 



For strict comparison to Cg, Cq should be corrected for the mean motion 

 of the water and not the surface drift since Cq should be measured relative 

 to an average transport in the water. Because the orbital movement of water 

 particles associated with the waves extends downward to some depth, the sur- 

 face drift uo is not the proper correction factor. The correction should be 

 proportional to a weighted average water velocity over some depth below the 

 surface . 



Lilly (196^4-) has proposed a drift correction for waves traveling on 

 water at finite depth. Assuming that the vertical profile of the drift cur- 

 rent is parabolic (laminar flow), and that the waves have infinitesimal ampli- 

 tudes, Lilly found that 



1 + -3 _ 1^ 1 + 2 cosh(2kd)\ ■ ^^^ 



2(kd.)2 I (kd)sinh(2kd) 



For deep water, kd -»• » and Equation (2) implies that the waves travel with 

 the surface flow only (i.e., ct = Cq + Uq ). However, for shallow 

 water, kd ■*• 0, and Eq. (3) predicts, as expected, that c_ -*■ c . 



The values of cp as calculated by Eq. (3) were compared to the corres- 

 ponding experimental data, and the results are shown in Figure 11. Experiment 

 and theory agree within + 15 percent. This error is approximately that_ 

 expected on the basis of experimental errors in estimation of c^, and c 

 using \ and Uq. 



