mean particle position is considered to be at the center of the ellipse or 

 circle, then vertical particle displacement with respect to the mean 

 position cannot exceed one-half the wave height. Thus, since the wave 

 height is assigned to be small, the displacement of any fluid particle from 

 its mean position is small. Integration of Equations 2-13 and 2-14 gives 

 the horizontal and vertical particle displacement from the mean position, 

 respectively. (See Figure 2-4.) 



Thus, 



I = - 



HgT^ cosh[27r(z + d)/L] 



47rL 



cosh (27rd/L) 



sin 



(2-17) 



HgT^ smh[27:(z + d)/L] /27rx l-nt^ 



f = + —2 ; ; cos 



^ 477 L cosh (277 d/L) \ L T ; 



(2-18) 



The above equations can be simplified by using the relationship 



Thus, 



2ire , 277d 

 — ^ tanh . 



H cosh [277(z + d)/L] . /27rx 277t 



— ; sin I — — — 



2 sinh(277d/L) \ L T 



(2-19) 



H sinh [277(z + d)/L] /277X 277t \ 



— ; cos I — - I. 



2 sinh (277d/L) \ L T / 



(2-20) 



Writing Equations 2-19 and 2-20 in the following forms: 



sin 



cos 



277X 277t 



27rt 

 T 



i, sinh (277d/L) 



a cosh [277(z+d)/L. 



j sinh (27rd/L) 



a sinh[277(z+d)/L]_ 



and adding, gives: 



— + ^r = 1, 



(2-21) 



in which 



A = 



H cosh[277(z+d)/L] 

 2 sinh (277d/L) 



(2-22) 



H sinh[277(z+d)/L] 

 ^ " 2 sinh(277d/L) ' 



(2-23) 



2-16 



