10 



Here, z is negative measured downward from the still-water surface 

 {z=—d at the bottom). The displacements are out of phase by T/4 

 in time and i/4 in the direction of motion. The wave form is sinus- 

 oidal and apphes to waves of small amplitude only. 



The fuU amplitude of oscillation expressed as a fraction of the 

 height from trough to crest is 



cosh-J ((i+?:) 



Horizontal: \= . 'ird ^^^ 



smh-y- 



, sinhy (c^+2) 



Vertical: ^= . , 27rd ^^^ 



smh-j- 



At the surface, 2=0, t- =coth-^ and^ = l. 



Figures 3 and 4 show the values of equations 5 and 6 with ■= as the 



parameter. As y approaches zero, the horizontal displacements ap- 



proach uniformity while the vertical displacements increase hnearly 

 from bottom to surface. These circumstances permit a simplification 

 in the theory of long periodic waves which, however, does not alter 

 the fact that they are merely a limiting case of oscillatory wave 

 motion. 



Equation 5 can be converted to another form by expressing the 

 hyperbolic functions in terms of exponentials as follows: 



iri 2irZ _2rd 2tZ 



a e ^ ■ e ^ -\-e ^ ■ e ^ 



T — 2ird 2ird 



e ^ —e ^ 



_2rd 



For very large depths, d-^ » and e ^ -^0 giving 



2irZ 



i-e-TT (7) 



The quantity z is again negative measured downward from the sur- 

 face. The same result is obtained for -j- and the paths traced out by 



fluid elements are circles. 



The ratio of the vertical to horizontal ampfitudes of the surface 

 orbits is then: 



ivd 



~tmh-j-=-^^ (8) 



