Environmental Engineering Reference
In-Depth Information
Fig. 2.39 Differential pressures in the region of the surface of the water [30]
The starting point is given by the wave profile and the velocity potential of the
undisturbed primary wave in Cartesian coordinates (z
¼
0 at seabed, Figure 2.41) and
in complex notation:
H
2
e
i
k
x
v
t
ð
Þ
z
0
x
ðÞ¼
;
H
2
g
v
cosh k z
ð
Þ
Þ
e
i kxvt
ð
Þ
F
0
x
ð
;
z
;
t
Þ¼ i
cosh k d
ð
It is expedient to formulate the boundary value problem in terms of cylindrical coordinates.
Following transformation, the result is (Figure 2.41)
"
#
2
1
H
m
¼
0
e
m
i
m
e
i
v
t
z
0
r
ð
; u;
t
Þ¼
J
m
k
r
ð
Þ
cos m
u
ð
Þ
"
#
Þ
1
g
H
2 v
cosh k
ð Þ
cosh k d
m¼0
e
m
i
mþ1
e
ivt
F
0
r
ð
; u;
z
;
t
Þ¼
J
m
k r
ð
Þ cos m u
ð
Þ
ð
Fig. 2.40 Pressure distribution around circumference of cylinder (second-order terms) [30]
