2.251
Wave Celerity, Length, and Surface Profile. It can be shown that,
for second-order théories, expressions for wave celerity and wavelength
are identical to those obtained by linear theory. Therefore,
C=^tanh(—]
(2-3)
2îf
\ l / ’
and
t
gT2
, /2îrd\
<0 4^
L = 2— tanh ---- .
(2-4)
2n
\ L /
The above équations, corrected to the third order, are given by:
5 + 2 cosh (4;rd/L) + 2 cosh2 (4nd/L)
8 sinh4 (2nd/L)
,(2-47)
and
gT2
L = ---- tanh
2it
5 + 2 cosh (4;rd/L) + 2 cosh2 (4nd/L)
8 sinh4 (27rd/L)
•(2-48)
The équation of the free surface for second-order theory is
H
T? = — COS
2
(2-49)
cosh (27rd/L)
sinh3 (27rd/L)
2 + cosh (47rd/L)
For deep water, (d/L > 1/2) Equation 2-49 becomes,
nH2
.
o
+ ----- cos
4Lo
(2-50)
2.252 Water Particle Velocities and Displacements. The periodic x and z
components of the water particle velocities to the second order are given
by
HgT cosh [2n(z + d)/L]
l2nx
2nt\
u = —— ------------------------- cos |----- — ---2L
cosh (2nd/L)
\ L
T /
(2-51)
3 /7rH\2
cosh [4tt(z + d)/L]
/4nx
4nt\
4 \ L /
sinh4 (27rd/L)
\ L
T /
2-37