EM 1110-2-1100 (Part II)
30 Apr 02
where the origin of x is at the wave crest. The volume of water within the wave above the still-water level
per unit crest width is
1
16 3
2
d H
V'
(II-1-90)
3
(i) An equal amount of water per unit crest length is transported forward past a vertical plane that is
perpendicular to the direction of wave advance. Several relations have been presented to determine the
celerity of a solitary wave; these equations differ depending on the degree of approximation. Laboratory
measurements suggest that the simple expression
C ' g(H % d)
(II-1-91)
gives a reasonably accurate approximation to the celerity of solitary wave.
(j)
The water particle velocities for a solitary wave (Munk 1949), are
1 % cos(My/d) cosh(Mx/d)
u ' CN
(II-1-92)
[cos(My/d) % cosh(Mx/D)]2
sin (My/d) sinh (Mx/d)
w ' CN
(II-1-93)
[cos (My/d) % cosh (Mx/D)]2
where M and N are the functions of H/d shown in Figure II-1-17, and y is measured from the bottom. The
expression for horizontal velocity u is often used to predict wave forces on marine structures situated in
shallow water. The maximum velocity umax occurs when x and t are both equal to zero; hence,
CN
umax '
(II-1-94)
1 % cos(My/d)
(h) Total energy in a solitary wave is about evenly divided between kinetic and potential energy. Total
wave energy per unit crest width is
3
3
8
2
2
d
ρgH
E'
(II-1-95)
33
and the pressure beneath a solitary wave depends on the local fluid velocity, as does the pressure under a
cnoidal wave; however, it may be approximated by
p ' ρg (ys & y)
(II-1-96)
(l)
Equation II-1-96 is identical to that used to approximate the pressure beneath a cnoidal wave.
(m) As a solitary wave moves into shoaling water it eventually becomes unstable and breaks. A solitary
wave breaks when the water particle velocity at the wave crest becomes equal to the wave celerity. This
occurs when (Miles 1980, 1981)
II-1-48
Water Wave Mechanics
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