of a ripple that the effect of surface tension is probably small even 

 for waves of usual laboratory dimensions. 



27rc? 

 As d becomes very large in equation 2 the term tanh -jr approaches 



unity and the velocity expression becomes 



C=J^ (2a) 



\ 27r 



since L=CT (1) 



then: C=^=5.12T, (26^ 



and i=5.12T2 (2c) 



Equations 2b and 2c are the limiting curves in figures 1, 2, and 2a. 

 As an example of the depth at which the bottom effect becomes negli- 

 gible, let us assume d=0.5 L. Then tanh (2 ttX 0.5) = 0.9963 and 

 the wave velocity is 0.9963 times the velocity in an unlimited depth. 



. d . 

 If an error of 5 percent is permissible, the corresponding y is 0.25. 



Evidently, the limits of applicability of the deep-water wave equa- 



d 

 tions depend on the accuracy desired and y = 0.5 is a purely arbitrary 



point of division, though frequently mentioned in papers on wave 



phenomena as the basis of the definition that waves in water having 



a depth greater than half the wave length are deep-water waves, and 



waves in lesser depths are shallow-water waves. 



If the wave length is very large as compared with the depth, 



, 2ird , 2'Kd , . , , ^ 



tanh -Y~ approaches -^ and equation 3 takes the lorm: 



C=^|^2^=V»'3- (2d) 



This is the well-known equation for the velocity of long waves of small 

 amplitude in still water. 



The horizontal and vertical displacements (3) of the water particles 

 from their mean position at a distance z below the still water level 

 are functions of x and t as shown by: 



^ cosh-£(c^+2) . ^. 



^'=2 . ^2.d ^^^ '^\L~f) (^^) 



smh— p 



2t 



and F,= t: 



7 sinh-T-(d+2) 



2 . .2Trd 

 smh^-- 



sin 27r(2~7') (4^) 



