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 assumed to be small, the displacement of any fluid particle from
its mean position is small. Intégration of Equations 2-13 and 2-14 gives
the horizontal and vertical particle displacement from the mean position,
respectively. (See Figure 2-4.)
Thus,
e =
_ M_2
4ttL
cosh [2n(z + d)/L]
------- ------- v.------ sin
cosh (27rd/L)
/ 2îtx
\ L
2wt\
T /
(2-17)
r =
+ M_2
4ttL
sinh [2ît(z+ d)/L]
-------------- ;------- co s
cosh (2nd/L)
/ 2ttx
\ L
2îrt\
~T~/’
(2-18)
The above équations can be simplified by using the relationship
\2
2tt£
, 2îrd
— = —- tanh ---T /
L
L
Thus,
H cosh [2n(z + d)/L] , / 2ttx
--------------------- - — sin I----sinh (27rd/L)-------y L
2nt
v
H
f = + 2
sinh [2tt(z + d)/L]
/2irx
,
,
cos I ----sinh (27rd/L)
\ L
2îrt
T
Writing Equations 2-19 and 2-20 in the following forms:
sin2 / 2nx _ 2nt\ = [l sinh (2ffd/L) ~|2
\ L
T y
a cosh [2tt(z+d)/L_
cos*
= lï sinh (2n d/L) j2
\ L
T y
a sinh [2tf(z+d)/L]
and adding, gives:
in which
H cosh [ 2?r(z 4- d)/L]
sinh (2nd/L)
(2-19)
(2-20)
(2-21)
(2-22)
H sinh [2tt(z+ d)/L]
2
sinh (2nd/L) ’
(2-23)
2-16