18 



reduces the net flow to zero. Equation 146 describes the motion 

 relative to the fluid and may be assumed as superimposed on a uni- 

 form return current of velocity as given by equation Md. In such 

 a closed basin, the observed velocity distribution relative to a fixed 

 point then would be: 



The velocity distributions corresponding to equations 146 aild 14e 

 are illustrated in figure 6. Figure 7 shows the velocity distribution 

 for a wave one foot high and with various periods in a channel of 

 unlimited length. The velocity distribution for waves of other 

 heights is obtained by multiplying by h^. 



Provided that the depth is of such magnitude that the velocity at 

 the bottom given by equation 146 is negligible, Ua in equation 14d 

 is the actual return velocity at the bottom in a channel with closed 

 end. Figure 8 shows this velocity as a function of depth and wave 

 period. 



Mitchim (10) found experimentaUy that equations 146 and 14rf 

 represent actual conditions. 



The preceding equations apply under conditions of finite wave 

 height; however, the series solution of Stokes includes only the first 

 terms and therefore the equations are theoretically accurate only if 

 the height is small as compared with the length. For deep-water 

 waves, Stokes found as the third approximation to the wave form 



y=d cos -^4--^ cos ^+-2X2" ^°^ "X" ^ ^ 



and for the wave velocity: 



Equation 16 is represented in figure 9, Regarding equation 15, 



Stokes notes, "It is remarkable that this equation coincides with that 



of the prolate cycloid if the latter equation be expanded according to 



ascending powers of the distance of the tracing point from the center 



of the rolling circle and the terms of the fourth order are omitted." 



The coincidence of the surface described by equation 15 with the 



cycloid is of interest in connection with the trochoidal theory to be 



discussed later. 



To the fourth order of approximation, Stokes obtains as the surface 



profile 



27ra; /I o , 17 , A 47rx , 3 o 3 Qttx 



y—a cos -J — ( 2^a ^-24W^^ ) cos "T' + g^^ ^^^ ~T~ 



— o^V cos -j^-j- . . . (17) 



