15 



The analysis by Stokes involved a series solution which was shown 

 to be convergent for "deep-water waves" by Levi Civita (7) and for 

 "shallow-water waves" by Struik (8). To a second order of approxima- 

 tion, Stokes found the wave velocity, for any depth of water, to be 

 given by: 



2vd _ 2vd 



e ^ +e ^ 

 The equation of the surface of the wave is: 



y=a cos-^+-^cos-^^ / g^d _2j^y (13) 



or 



y=a cos mx-^Ka^ cos 2mx 



where: 



/ 2vd _ 2jrd\ / 47rd _ 1e^ \ 



2\e^-e ^ ) 



The velocity of forward transport (also known as "mass transport") of 

 a particle at a depth d below the still water surface is: 



U=^-C' ' ....'■,.,.. (14) 





The height between trough and crest is 2a. The midheight between 

 trough and crest is Ka^ above y=0. Measured along the level 2/=0 

 (still water surface), the portion of the wave length above the still 



water surface is 7: — -^ — while the portion below is -^-\ — ^ — For 

 2 27r ^ 2 2r 



any value of L, K increases as the depth decreases, producing sharp- 

 ened crests and flattened troughs. The surface curves (equation 13) 



for different values of y and y appear in figure 5. 



When the water depth is great in comparison to the length of the 

 wave, these equations simplify to 



Wave velocity: C^J^ (12a) 



Wave surface: y=a cos "^ — | — y- cos -y- (136) 



4ira^ i^ 



Mass transport velocity: U=^r-2-eL-C (14a) 



At x=0, and x=L, y=a-\ — y-; while at x=-^) y=—ci-\—jr 



