21 



This equation does not agree with the trochoid to the same order of 



2 

 approximation, the difference in ordinates being ~"o^* cos 2x the 



sm-face curve lying "a little above the trochoid at the trough and a 

 little below it at the shoulders" (2). 



Levi-Civita (7) showed that Stokes' series was convergent and 

 confirmed the surface curve of equation 17 for waves in deep water. 

 For the velocity of the wave, to the fifth approximation, he obtained 



where 



a=j{h) 



or expressing Cn in terms of h and L the foregoing equation takes the 

 form 



a 



.-i(^+f +'^+ • ■ •) a«) 



Figure 9 shows the correction for finite wave height. The fact that 

 the series is convergent demonstrates, at least theoretically, the 

 possibility of periodic waves of permanent form. 

 Regarding the form of the waves, Stokes remarked: 



It appears also that, whatever be the order of approximation the waves will be 

 symmetrical with respect to vertical planes passing through their ridges, as also 

 with respect to vertical planes through their lowest points. 



This conclusion was also confirmed by Levi-Civita. 



When the depth of the water is about the same magnitude as the 

 length of the wave, or less, the terms involving the water depth in 

 the equations must be retained. The analysis employed in this case 

 is similar to that followed for the case of deep water but the solution 

 is somewhat more complicated. Stokes (2) and Struik (8) both 

 obtained solutions to a third approximation. The equations will be 

 given here without derivation. 



The expression for the velocity of propagation of the wave is: 



2ird 2irdr~ 2ird 2-ird / 47rd ivdX 



(._QT e^-e 



2ir — 27rd 



/ 2vd 2wd\ 



The equation of the surface of the wave is: 



/ 2vd 2vd\ / ivd _Ud \ 



y=a cos ^ j-^ 2J _2_^d\4 cos-^ 



/ 12nd 127rd \ / Sjrd 8ird\ / Ud _4jr^\ 



xW 5Ve ^ +e ^ )-\-U \e^ -^e ^ ) + l9\e^ -}-e ^/ + 32 _„ Qirx 



/ 2ird _2]rd\t 



~l J2 / 2nd 27rd\6 COS j^ 



