198] WAVES IN DEEP WATER. 531 



waves long enough, in comparison with their height to neglect , 2 -> 

 we have 



145) . =. 



If s is the height from which a body must fall to acquire a velocity 

 equal to the wave - velocity, since a 2 = 2gs, the equation becomes 



146) I = xs, 



accordingly the velocity of propagation of long waves in deep water 

 is equal to the velocity acquired by a body falling freely from a 

 height equal to one -half the radius of a circle whose circumference 

 is the wave-length. 



In order to study the motions of individual particles of water 

 let us now impress upon the motion given by 137) a uniform velocity a 

 in the X- direction. Equations 137) now give the motion with respect 

 to moving axes travelling with the waves, so that in order to obtain 

 the motion with respect to fixed axes we have to add a to the u 

 of 137) and replace x by x at, obtaining 



u = A~ke* y sin k (x at], 



147) 



ij = Ake k y cosk(x at), 



for the equations of the unsteady motion of the actual wave -propa- 

 gation. For the velocity of a particle we have 



148) q = -\/u 2 + v 2 = AJce*y 



showing that the velocity decreases rapidly as we go below the 

 surface, so that for every increase of depth of one wave-length it is 

 reduced in the ratio e~ 27t = .001867. If the displacement of a particle 

 which when at rest was at x, y is |, 17 we have 



-FT = A~ke k v sin~k(x at), 

 149) 



if we neglect the small change of velocity from x,y to x -\-%,y -\- r}, 

 so that we obtain by integration 



| = 



1 F\A\ 



il = 

 Thus each particle performs a uniform revolution in a circle of 



2 it, >L 



radius Be ky in the periodic time y = We thus see how the 



ka a 



motion is confined to the surface layers. The direction of the motion 

 in the orbit is such that particles at the crest of the wave move in 

 the direction of the wave -propagation, those at a trough in the 

 opposite direction. 



34* 



