Environmental Engineering Reference
In-Depth Information
where
4
1
J
0
1
k
a
q
c
M
1
¼
2
2
2
p
ð
k
a
Þ
þ
Y
0
1
k
a
ð
Þ
ð
Þ
J
0
1
k ð Þ
Y
0
1
k a
e
¼ arctan
ð
Þ
0
¼ v
@
u
H
2
cosh k z
ð
Þ
2
ðÞ
@
t
sinh k d
ð
Þ
and
ð Þ
0
is the amplitude of the acceleration of the undisturbed primary wave.
The force per unit height can be transformed into the following form:
@
u
=@
x
¼
0
þ
b
r p
a
2
ðÞ
dz
;
@
dF
x
z
u
¼
c
M
r p
a
2
ðÞ
z
;
v
uz
ð j
x
¼
0
;
@
t
where
Y
0
1
k
ð Þ
4
c
M
¼
c
M
1
cos
e
¼
2
q
J
0
1
k
a
2
2
p
ð
k
a
Þ
þ
Y
0
1
k
a
ð
Þ
ð
Þ
J
0
1
k ð Þ
4
q
J
0
1
k a
b ¼ c
M
1
sin
e
¼
2
2
2
p
ð
k a
Þ
þ Y
0
1
k a
ð
Þ
ð
Þ
H
2
cosh k z
ð
Þ
2
uz
ð j
x¼0
¼ v
;
Þ
cos
ð
v t
Þ
sinh k d
ð
x¼0
¼v
@
u
@
H
2
cosh k z
ð
Þ
2
ðÞ
z
;
Þ
sin
ð
v t
Þ
t
sinh k d
ð
Here, u
j
x
¼
0
ð Þj
x
¼
0
are the amplitudes of the velocity and acceleration of the
undisturbed primary wave on the axis of the cylinder (x
¼
0).
and
@
u
=@
This format expresses the wave force resulting from diffraction theory in a similar way
to the Morison formula (modified Morison formula).
The dependency of the variables c
M1
,c
M
, b and
e
on the dimensionless parameter (k
a)
is plotted in Figure 2.41. This parameter expresses the diameter/wavelength ratio in
physical terms:
¼
p
D
l
Using the above format, the wave force can be considered as the sum of two components.
The first of these is proportional to the acceleration and the second is proportional to the
velocity of the undisturbed primary wave (inertia and potential damping).
2
p
a
l
k
a
¼
