Civil Engineering Reference
In-Depth Information
Figure 2.5
shows a two-dimensional, simple progressive wave propagation
in the positive x-direction. The symbol
η
denotes the displacement of the water
surface relative to the SWL, which is a function of x and time, t, at the wave
crest,
, equal to one-half of the wave height.
Water particle displacement is presented in
Figure 2.6
for deep and shallow
water. Water particle displacement is an important factor in linear wave
mechanics, which deal with the displacement of individual water particles within
a wave. Water particles generally move in elliptical paths in shallow water or
transitional water and in circular paths in deep water (see
Figure 2.6
). If the
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. Therefore, the displacement of the fluid particle
is small if the wave height is small.
Figure 2.7
presents the horizontal and vertical velocities and acceleration for
various locations of the particles, which is very important in calculating the
wave forces on any subsea structural member, as the drag force and inertia force
are functions of the particle velocity and acceleration, respectively. The following
equations are used to calculate the wave velocity and acceleration.
F
1
=
2
η
L
πð
z
+
d
Þ=
(2.8)
F
2
= ð
2
π
=
L
Þ
d
(2.9)
F
3
= ð
2
π
x
=
L
Þ − ð
2
π
t
=
T
Þ
(2.10)
Velocity
●
U
= ½ð
H
=
2
Þð
gT
=
L
Þ
cosh F
1
=
cosh F
2
cos F
3
(2.11)
W
= ½ð
H
=
2
Þð
gT
=
L
Þ
sinh F
1
=
cosh F
2
sin F
3
(2.12)
C
Direction of propagation
L
Crest
z
a
η
SWL
SWL
0
x
H
a
Trough
2
π
x
L
2
π
L
T
Note: (a)
η
=
a
cos
−
d
(b) For given origin (
x
=
0) wave profile
is shown for
t
=
3
T
/4, 7
T
/4, 11
T
/4......
(c)
H
/2 at wave crest
η
=−
a
=−
H
/2 at wave trough
η
=
a
=
Bottom,
z
=−
d
FIGURE 2.5
Simple sinusoidal progressive wave-propagation curve.
Search WWH ::
Custom Search