choice of a theory for a particular problem. It should be kept in mind
that linear (or first-order) theory applies to a wave which is symmetrical
about stillwater level and has water particles that move in closed orbits.
On the other hand, Stokes’ second-order theory predicts a wave form that
is unsymmetrical about the stillwater level but still symmetrical about a
vertical line through the crest, and has water particle orbits which are
open.
************** EXAMPLE PROBLEM **************
GIVEN: A wave traveling in water depth of d = 20 feet, with a wavelength
of L = 200 feet and a height of H = 4 feet.
FIND:
(a) Compare the wave profiles given by the first- and second-order
théories.
(b) What is the différence between the first- and second-order
horizontal velocities at the surface under both the crest and
trough?
(c) How far in the direction of wave propagation will a water particle
move from its initial position during one wave period when z = 0?
(d) What is pressure at the bottom under the wave crest as predicted
by both the first- and second-order théories?
(e) What is the wave energy per unit width of crest predicted by the
first-order theory?
SOLUTION:
(a)
The first-order profile Equation 2-10 is:
H
T? = ~ COS0 ,
2 ’
where
.
/2nx
2nt\
y = I---- —---- I,
\ L
T /
and the second-order profile Equation 2-49 is:
for
nH2
+ ■---8L
cosh (27rd/L)
sinh3 (27rd/L)
2 + cosh
cos 26
d
20
L ~ 2ÔÔ = 0,1 ’
2-40