Integration of equations (2-13) and (2-14) gives the horizontal and vertical 

 particle displacement from the mean position, respectively (see Fig. 2-4). 



Thus, 



HgT^ cosh[2Tr(z + d)/L] / 2itx 



5 = ° — — -^- sin 



2irti 



4ttL 



cosh(2Trd/L) 



(2-17) 



HgT^ sinh[2Tr(z + d)/L] /2Trx 2Trtl 



^ = + _2 — _ __ cos 



4ttL 



cosh(2ird/L) 



(2-18) 



The above equations can be simplified by using the relationship 



/2it\2 2irg , 2ird 

 — = tanh 



Thus, 



H cosh[2Ti(z + d)/L] Ittx. 2Tit 



r = _ sin 



2 sinh(2Trd/L) \ L T 



(2-19) 



H sinh[2TT(z + d)/L] Iwx. 2TTt 



r = H ; COS 



2 sinh(2Trd/L) \ L T , 



(2-20) 



Writing equations (2-19) and (2-20) in the forms. 



and adding give 



in which 



5in2 



cos' 



2irx 2irt 



2irx 2TTt 



^ sinh(2iTd/L) 



a cosh[2ir(z + d)/L] 



sinh(2Trd/L) 



a sinh[2Tr(z + d)/L] 



"^ 



= 1 



(2-21) 



_ H cosh[2TT(z + d)/L] 

 ~1 sinh(2Trd/L) 



(2-22) 



B = 



H sinh[2ir(z + d)/L] 

 7 sinh(2iTd/L) 



(2-23) 



Equation (2-21) is the equation of an ellipse with a major (horizontal) semi- 

 axis equal to A and a minor (vertical) semiaxis equal to B. The lengths 

 of A and B are measures of the horizontal and vertical displacements of 

 the water particles. Thus, the water particles are predicted to move in 



2-16 



