and from Table C-2
I
cosh I---\ L
1.2040 ,
cosh
sinh
= 0.6705
4nd
1.8991 ,
7tH2
8L~
cosh (27rd/L)
sinh3 (27rd/L)
2 + cosh
= 0.48
Therefore
r] = 2 cos 6 + 0.48 cos 2 0 ,
ric>2 = 2.48 ft.,
2 = — 1*52 ft.
where nc 2 an^
2 are t^ie values of n at the crest
(i.e. cos 0=1, cos 20 = 1) and trough (i.e. cos 0 = - 1,
cos 20 = 1) according to second-order theory.
Figure 2-8 shows the surface profile n as a function of 0. The
second-order profile is more peaked at the crest and' flatter at the
trough than the first-order profile. The height of the crest above
SWL is greater than one-half the wave height; conséquently the
distance below the SWL of the trough is less than one-half the
height. Moreover, for linear theory, the élévation of the water
surface above the SWL is equal to the élévation below the SWL;
however, for second-order theory there is more height above SWL
than below.
(b) For convenience, let
uc j = value of u
u£ 2 = value of u
uc 2 = value of u
u^ 2 ~ value of u
at crest according to first-order theory,
at a trough according to first-order theory,
at a crest according to second-order theory,
at a trough according to second-order theory.
2-41