Internat. J. Math. & Math. Sci.
Vol. 9 No. 4 (1986) 625-652
625
NONLINEAR DIFFRACTION OF WATER
WAVES BY OFFSHORE STRUCTURES
MATIUR RAHMAN
Department of Applied Mathematics,
Technical University of Nova Scotia, Halifax
Nova Scotia B3J 2X4, Canada
and
LOKENATH DEBNATH
Department of Mathematics
University of Central Florida
Orlando, Florida 32816-6990, U.S.A.
(Received January 8, 1986 and in revised form July 30, 1986)
BSTRAT.
This
paper
is
concerned
axisymmetric water wave problem.
in
with
a
cylindrical polar coordinates is derived.
current
knowledge
on
analytical
of
formulation
variational
a
non-
The full set of equations of motion for the problem
This
and
theories
is
followed by a review of the
numerical
treatments
of
nonlinear
A brief discussion is
diffraction of water waves by offshore cylindrical structures.
made on water waves incident on a circular harbor with a narrow gap.
Special emphasis
A new theoretical
is given to the resonance phenomenon associated with this problem.
analysis is also presented to estimate the wave forces on large conical structures.
Second-order (nonlinear) effects are included in the calculation of the wave forces on
the conical structures.
A list of important references is also given.
Variational principle for non-axisymmetrlc water waves, nonlinear diffraction of water waves, incident and reflected waves, wave forces, waves
incident on harbors, and Helmholtz resonance.
KEY WORDS AND PHRASES.
1980 AMS SUBJECT CLASSIFICATION CODE.
I.
I’DCTIOH.
76B15.
Diffraction of water waves by offshore structures or by natural
Due to the tremendous
boundaries is of considerable interest in ocean engineering.
need and growth of ocean exploration and extraction of wave energy from oceans, it is
becoming increasingly important to study the wave forces on the offshore structures or
natural
boundaries.
Current
methods
of
caiculating
wave
forces
on
the
offshore
structures and/or harbors are very useful for building such structures that are used
for exploration of oil and gas from the ocean floor.
In the theory of diffraction, it is important to distinguish between small and
large structures (of typical dimension b) in comparison with the characteristic
Physically, when a/b is small and kb
wavelength (2/k) and the wave amplitude a.
-I
k
is large (the characteristic dimension b of the body is large compared with k
is the wavenumber) the body becomes efficient as a generator of dipole wave radiation,
and the wave force on it becomes more resistive in nature.
separation becomes insignificant while diffraction is dominant.
This means
that flow
In other words, the
626
M. RAHMAN AND L. DEBNATH
body radiates a very large amount of scattered (or reflected) wave energy.
for small
other hand,
-l)
compared with
(the characteristic dimension
kb
and a/b is large
k
amount of scattered wave energy.
(a/b
>
0(I)),
b
On the
of the body is small
the body radiates a very small
This corresponds to a case of a rigid lld on the
ocean inhibiting wave scattering, that is, diffraction is insignificant.
Historically,
[I]
Havelock
gave
the
l+/-nearlzed
diffraction
theory
for
small
Based upon this work, MacCamy and Fuchs [2]
amplitude water waves in a deep ocean.
extended the theory for a fluid of finite depth.
These authors successfully used the
llnearlzed theory for calculation of wave loading on a vertical circular cylinder
from
extending
a
horizontal
floor
ocean
above
to
the
free
surface
of
water.
Subsequently, several authors including Mogrldge and Jamleson [3], Mei [4], Hogben et.
al.
[5],
[6-7] obtained analytical solutions of the llnearlzed diffraction
However, the llnearized theory has
Garrison
problems for simple geometrical configurations.
limited applications since it is only applicable to water waves of small steepness.
In reality, ocean waves are inherently nonlinear and often irregular in nature.
Hence, water waves of large amplitude are of special interest in estimating wave
forces on offshore structure or harbors.
,In
recent years, there has been considerable interest in the study of the hydro-
dynamic forces that ocean waves often exert on offshore structures, natural boundaries
and harbors of various geometrical shapes.
Historically, the wave loading estimation
for offshore structures was based upon the classical work of Morlson et. al. [8] or on
the linear diffraction theory of water waves due to Havelock [I] and MacCamy and Fuchs
[2].
Morlson’s formula was generally used to calculate wave forces on solid structures
According to Debnath and Rahman [9], Morlson’s equation expresses the
in oceans.
total drag
D
V U
M
force,
as a sum of the inertia force,
C
associated with the irrota-
p C A U2
tlonal flow component, and the viscous drag
related to the
D
vortex-flow component of the fluid flows under the assumption that the incident wave
field
not
is
affected
significantly
presence
the
by
of
the
structures.
Mathematically, the Morison equation is
D
P CM V U +
D C
D
A U2
(l.l)
M
where
coefficient,
U
is
the
is
D
M
fluid
C
density,
is the added mass,
M
V
(I +
a
)
is
the
(or
Morison
inertial)
is the volumetric displacement of the body,
a
the fluctuating fluid velocity along the horizontal direction,
C
projected frontal area of the wake vortex, and
D
is
A
is
the drag coefficient.
the
For
cases of the flow past a cylinder or a sphere, these coefficients can be determined
It is also assumed that inertia and
relatively simply from the potential flow theory.
viscous drag forces acting on the solid structure in an unsteady flow are independent
in the sense that there is no interaction between them.
There are several characteristic features of the Morison equation.
One deals
U.
The other
with the nature of
the inertia force which is
includes the nonlinear factor
believed
U2
linear in velocity
in the viscous
that all nonlinear effects
dra
in experimental
force term.
It is generally
data are associated with drag
NONLINEAR DIFFRACTION OF WATER WAVES BY OFFSHORE STRUCTURES
627
However, for real ocean waves with solid structures, there is a significant
forces.
nonlinear force associated with the irrotational component of the fluid flow because
These waves are of special interest in the
of the large amplitude of ocean waves.
It is important to include all significant nonlinear effects
wave loading estimation.
associated with the nonlinear free surface boundaryconditions in the irrotational flow
of
component
the
loading
wave
However,
structures.
the
on
the
of
effect
large
amplitude waves on offshore structures of small mean diameter may be insignificant,
but it is no longer true as the diameter increases in relation to the wavelength of
Consequently the Morison equation is no longer applicable,
the indident wave field.
It is necessary to distinguish between
and diffraction theory must be reformulated.
the
of
structures
small
the
large diameters,
and
diameter
being
compared
to
the
characteristic wavelength and amplitude of the wave.
Several
have
studies
that
shown
formula
the Morison
is
satisfactory.
fairly
However, several difficulties in using it in the design and construction of offshore
These are concerned with the drag
structures have been reported in the literature.
The shortage of reliable full-scale
force which has relativly large scale effects.
drag data in ocean waves is another problem.
There is another significant question
whether a linear theory of the Irrotational flow response is appropriate at all to
Despite these difficulties and short-comings,
water wave motions with a free surface.
the use of the Morison equation had extensively been documented in the past literature
throughplentiful data for determining the coefficients
Several
of
estimate
nonliner
that
indicate
studies
recent
of
diffraction
water
waves
by
the
offshore
C
and C
D.
M
second-order theories
structures
for
an
provide
the
accurate
the linearized analysis together with corrections approximating to the
effect of finite wave amplitude.
paper
This
formulation of
concerned with a variational
is
problem.
wave
water
full
The
of
set
equations
of
motion
a non-axisymmetric
for
the
problem
in
This is followed by a review of the current
cylindrical polar coordinates is derived.
knowledge on analytical theories and numerical treatments of nonlinear diffraction of
A brief discussion is made on water
water waves by offshore cylindrical structures.
waves incident on a circular harbor with a narrow gap.
Special emphasis is given to
A new theoretical analysis is
the resonance phenomenon associated with the problem.
Second-order
also presented to estimate the wave forces on large conical structures.
(nonlinear) effects are included in the calculation of the wave forces on the conical
structures.
2.
VARITIONAL PRINCIPLE FOR NON-AXISYMMETRIC NONLINEAR WATER WAVES.
We
consider
density
p
an
irrotational
Inviscid
subjected to a gravitational field
confined
(r’
in
a
region
(r,,z)
0’ ),
z
0
such
<
r
< =,
0
<
the
that
g
fluid
z
<
fluid
n,
of
flow
constant
,
acting in the negative z-axis which
The fluid with a free surface
is directed vertically downward.
potential
non-axlsymmetrlc
<
<
7.
n (r,
z
There
0 and
t)
is
velocity
V
velocity is given by
and the potential is lying in between z
a
exists
z
n (r,, t).
Then the variational principle is
61
6
f f
D
L dx dt
0
(2.1)
M. RAHMAN AND L. DEBNATH
628
D
where
-
L
and
space, and the Lagrangian
L
is
n(r,O,t)
[*t
0
0(r,0,z,t), (r,0,t)
0
(, t)
is an arbitrary region in the
(2.2)
to vary subject to the restrictions
allowed
are
D
on the boundary
(vo ) + gz] dz,
+
60
0,
of D.
According to the standard procedure of the calculus of variations, result (2.1)
yields
=
o
{[ +(v
D
gZ]z.
+
+- 00 600
+ Oz
60z
Integrating the z-integral by parts along with
r
(0 t + O r 60 r
+
0
dz} r drd0 dt
and
0-integrals,
it turns
out that
61
0
ff
I0 t
D
+
fD f {[B
O
(rlt O)z__rll
dz-
0
Or O
+
[-r 0
+
[ 0 1_
006O
r
+
+ (vO )2 + gZ]z.
0 (Orr
0 1--d o00 60
Or O
+
dz
dz
(1- 2-r 00060)z’rl]
dz
r
Ozz6O
[(Oz6O)zfrl- (Oz6O)z= 0
(nrOr60)z=r
dz
dz]} dx dt
-
In view of the fact that the first z-integral in each of the square brackets
vanishes on the boundary
0
61
ff
D
{[0 t
D,
+ (vO )z + gZ]z=n 6n + [(-n
f [Orr
0
We first choose
6n
.
we obtain
+I
0, [6O] z=0
err +
Or
+
Or
+
r
O0
[60] z=n
O00
+
Ozz
+
0;
t
-nrO r
Ozz]
r
6#
dz
6O
since
0, 0
<
z
<
n0O 0
+
z
[0z6]z= O}
dx dt
is arbitrary, we derive
n (r,0,t),
(2.3)
NONLINEAR DIFFRACTION OF WATER WAVES BY OFFSHORE STRUCTURES
Then, since
6q, [6#] z--O
6
and
z=rl
629
can be given arbitrary independent values, we
deduce
#t
Evidently,
and
the
qr #r
+
qt
(V# )2 + gz
+
-
+ r
q0#O
0,
z
n (r,O,t)
(2.4)
#z
0,
z
(r,O,t)
(2.5)
bz
O,
z
(2.3), two free-surface conditions (2.4)-(2.5)
conditon (2.6) constitute the non-axisymmetric water wave
the Laplace equation
bottom boundary
equations in cylindrical polar coordinates.
This set of equations has also been used
by several authors including Debnath [I0], Mohanti [II]
initial
(2.6)
0
and Mondal
[12], for the
value
investigation of linearized axisymmetric water wave problems.
The
elegance of the variational formulation is that within its framework, the treatment of
both linear and nonlinear problems become identical
3.
DIFFRACrlOM OF NONLINEAR WATER WAVES IN AN OCEAN BY CYLINDERS.
Several authors including Charkrabarti [13], Lighthill [14], Debnath and Rahman
[9], Rahman and his collaborators [15-18], Hunt and Baddour [19], Hunt and Williams
[20], Sabuncu and Goren [21]. Demirbilek and Gaston [22] have made an investigation of
the theory of nonlinear diffraction of water waves in a liquid of finite and infinite
depth by a circular cylinder.
numerical results.
These authors obtained some interesting theoretical and
We first discuss the basic formulation of the problem and indicate
how the problem can be solved by a perturbation method.
We formulate a nonlinear diffraction problem in an irrotational incompressible
fluid of finite depth
which
is
amplitude
acted
a
on
h.
by
a
We consider a large rigid vertical cylinder of radius
train
of
two-dimensional,
propagating in the positive
absence of the wave, the water depth is
surface elevation is
STILL WATER
LEVEL
x
h
periodic
progressive
waves
direction as shown in Figure I.
b
of
In the
and in the presence of the wave the free
above the mean surface level.
DIRECTION OF
WAVE
PROPAGATION
y
OCEAN BOT OM
FIG. 1. Schematic diagram of a cylindrical structure in waves
630
M. RAHMAN AND L. DEBNATH
(r,0,z)
In cylindrical polar coordinates
from the origin at the mean free surface,
the z-axls vertically upwards
with
the governing equation,
free surface and
boundary conditions at the rigid bottom and the body surface are given by
V2(r,B,z,t)
nt + #rnr
$
$
r
+ (r2 +
t + gn
where
1
rr + r +
=-
z
r
+
+
2OB
r
/ #8n0
0z
zz
(3.1)
>
(3.2)
z 2)
+
0, zffin,
#z
0
zfn
r>
b,
r
b,
(3.3)
0
on
z
-h
(3.4)
0
on
r
b, -h<z<rt,
(3.5)
is the velocity potential and
#
0, b<r <’, -<0<, -h<z<n,
g
is the acceleration due to gravity.
Finally, the radiation condlton is
(kr)I/2 [(r
llm
kr+
where
k
2/
Ik)
+/-
(3.6)
0
#R
(or scattered) wave,
is the wavenumber of the reflected
is the total potential,
and
#I
#R
$I
SR
+
represent the incident and reflected potentials
respectively.
We apply the Stokes expansion of the unknown functions
E
#
n=l
where
en #n’
E
n
and
#
in the form
en n
(3.Tab)
n=l
is a small parameter of the order of the wave steepness
For any given
n
the s of only the first
n
terms of the series
(3.Tab) may
be considered as the nth-order approximation to the solutions of the problem governed
by (3.1)-(3.6).
terms
m
when
The nth-order approximation is the solution subject to the neglect of
m
>
n.
.
We next carry out a Taylor-serles expansion of the nonlinear boundary conditions
(3.1)-(3.2) about
z
0,
substitute (3.7ab) into
Equating the first powers of
boundary conditions on
V251
E
(r,O,z,t)
and equate powers of
r,8 and t:
0, valid for all
z
(3.(3.2)
leads to the following equations with the llnearlzed
(3.8)
0
+ gnl
0,
for
z
0, r
>
b
(3.9)
rtlt- @Iz
0,
for
z
0, r
_>
b
(3.10)
#Iz
O,
for
z
-h
(3.11)
$1r
0,
for
r
b
(3.12)
#It
NONLINEAR DIFFRACTION OF WATER WAVES BY OFFSHORE STRUCTURES
[(r +/-
llm (kr
kr+
where
#II + #IR
#I
(3.13)
0
#IR
the total potential as the sum of the first order
representing
and the flrst-order reflected potential.
#II
incident potential,
ik)
E
leads to the following system of
#2
and
Similarly, equating the second powers of
express the second-order terms
that
equations
631
as functions of
2
#I
and
i:
V2# 2
+ (#Ir
#2t + nl#Itz
gn2 +
+
n2t
(3.14)
0
+
#Irnlr
r
2
#I 02 + #Iz 2)
+
(#2z
#IOnl8
+ n # Izz
O, z- 0
0
0, z
r
r
>
>
(3.15)
b,
(3.16)
b
#2z
0
for
z
-h,
(3.17)
#2r
0
for
r
b,
(3.18)
with the radiation condition
(k2 r)I/2 (r +/- Ik2)(#2
llm
k
2
o
#21 )]
(3.19)
is the wavenumber corresponding to second-order wave theory, and
2
the second-order term of the incident potential
where
k
#lit
Similarly,
#2tt + gO2z
+
-n
for
z [# ltt + g# Iz
>
(3.20)
b
#
[#r
a
-{
[#Ir
2
2
+
(#8 )2
+
( #18 )2
#z 2]
as a power series in
P#lt
+
+
#z
2
(3.21)
]’ for z-o, r>_b
can be determined from the Bernoulli equation
p(r,8,z,t)
gz + # t +
-pgz
O, r
z
can be eliminated from equations (3.15) and (3.16) to obtain
n2
Substituting
p
O,
g#Iz
The pressure
P+
p
can be eliminated from (3.9)-(3.10) to derive
q
The function
#21
2
e p
{#2t
+
into
[#Ir
2
(3.22)
0
+
(3.22), we can express
( #18
2
+
2]}
+ 0 (e
p
3)
as
(3.23)
M. RAHMAN AND L. DEBNATH
632
The total horlzonatal force is
2 q
F
q
where
/
x
[p] r--b (-b cos) dzd0
-h
0
is given by the perturbation expansion
Substituting (3.23)
into
(3.7b).
(3.24) and expressing the z-integral as the s of
q
0
f
(3.24)
f,
+
we obtain
0
-h
F
f
bO
x
[gz +
qlg2q2
+
2
[gz +
0
+ e2
eit
-h
{2t
(Iz
+
I@
2
2
(Iz
It + {2t + 12+I
I )]rffib dz
It is noted that condition (3.12) is used to derive (3.25).
tha’t
the integral of
e
F
Fxl
x
3
cos8
term and
2
gql +
z
2q2
which would only introduce
may be written as
F
Thus
etc.
+
(3.25)
Also, the upper limit of the z-integral of the second-order
may be taken at z=0 in place of
hlgher-order terms
d
It is clear from (3.25)
the hydrostatic term, up to z=0 contain no
gz,
hence may be neglected.
terms
cos
(3 26)
Fx2
where the flrst-order contribution is
27
Fxl
0
[ (it)r_b_
-h
bo
0
cos
dz]
(3.27)
d0
and the second-order contribution is
E
+
2
2
0
{gz +
E,it}r__b
0
0
{*2t +7 (*Iz
-h
4.
[I
bP
Fx2
2
+
*I
2
1}
dz
dZ]rffib cose
de
(3 28)
FIRST-ORDER WAVE POTENTIAL AND FREE SURFACE ELEVATION FUNCTION.
MacCamy and Fuchs [2] solved the system (3.8)-(3.13) and obtained the first-order
solution which can be expressed in the complex form
mcosh k(z+h)
k
2
sinh kh
e- it
Z
m=0
i
m
A (kr)
m m
cosme,
(4.1)
NONLINEAR DIFFRACTION OF WATER WAVES BY OFFSHORE STRUCTURES
e
B1
Z
k
where
6
[I,
2,
m
i
m+l
A (kr) costa0,
when
when
m
m
0
0
*
(4.3)
J ’(kb)
A (kr)
m
and the frequency
2
m
(I)
H
m
m
J (kr)
m
m
(4.2)
mm
m=0
H
633
(I)(kb)
m
and wavenumber
H
(I)
m
k
(kr),
(4.4)
satisfy the dispersion relation
(4.5)
gk tanh kh,
is the ruth-order Hankel function of the first kind defined by
(i)
H
(kr)
m
in which J
m
(kr)
and
J
respectively,
m
J (kr) + i Y (kr)
m
(4.6)
m
Y (kr) are the Bessel functions of the first and the second kind
m
(x)
denotes the first derivative.
It is noted that result (4.2) represents the complex form of a plane wave of
amplitude
k
-I
propagating in the x-direction and represented
Jm(kr),
radial dependence is given by
together with a reflected component described
Hm(1)(kr).
by the terms whose radial dependence is given by
e
5.
by the terms whose
The first order solution
includes the complex form of an incident plane wave of amplitude e.
SECOND-ORDER WAVE POTENTIAL AND FREE-SURFACE ELEVATION FUNCTION.
With
known values
for
assumes the following form for
2tt + g2z
Iz
g -2 imt
e
2k
and
n
(4.2) and (4.3), equation (3.2)
given in
>
b:
0
and
Z
B (kr) cos mD,
m
m=0
r
(5.1)
where
E
m=0
B (kr) cos mO
m
g 5 5 im+n-I A
E
m n
mn
m=O n=O
coth kh
+ 2
k2r 2
E
Z
m=0 n=0
m n
AmA n (mn) [cos (m-n) 0
[cos (re+n)0 + cos(m-n)e]
i
m+n-I
cos (m+n) O]
(5.2)
with
Amn
(3 tanh kh
coth kh) A A
m n
+ 2 coth kh A A
m n
(5.3)
M. RAHMAN AND L. DEBNATH
634
and a prime in (5.3) denotes differentiation with respect to
kr.
(5.1) indicates that the general solution of (3.14) with (3.171-
form of
The
(3.18) can be written as
we2k m=OE [0 Din(k2) Am (rk2)
2
where
in
k
2
cosh
k2(z+h) dk2]
(5.4)
cos m0,
denotes the wavenumber of a second-order wave taking only continuous values
2
(0, (R)).
-
We next substitute (5.4) into (5.1) to obtain a relation between
the form
2
k2h] Am(rk 2) Dm(k 2)
4k tanh kh cosh
k2h-
sinh
dk
m
and
B
Bm(kr)
2
in
m
(5.5)
B (r) can be obtained by equating similar terms of the Fourier Series (5.2).
m
in the form
Equation (3.15) gives the second-order free surface elevation r
2
where
r12
where
-
[k
D
ql’
,32
I
[---
+
and
2
2
2
2
,1, + ,1, + ,;1, }1,
21 + 7 {t’r
tr--)
tz
nl
O, r
z
>_
(5.6/
b,
are determined earlier.
In order to compute the total horizontal force on the cylinder as given by (3.26)
combined with
(3.27)-(3.28),
it is necessary to solve
(5.5) for the case of
m-l.
A tedious, but straight forward, algebraic manipulation gives the value of the
Bl(r)
complex quantity
B
l(r)
Since
the
(I
+
2
)"
_where
6
m
m
2"
cosh k(z+h) E
2
2k sinh kh m
m
is defined by (4.3) and
+
(I
m
i e
-i t
m
m
it
I’
discuss
I
@I
(5.7)
Am+
is
linear,
its
real
is the complex conjugate of
is not possible to express the
physical meaningful second-order
from (4.1) in the form
Am (kr) cos
m
(5.8)
is given by
(5.9ab)
m<0-
-0,
and the function
relation
where
We next
We first write a real solution
-I
I
+
is nonlinear in
(2
solution for
k2r2
problem described by (3.8)-(3.131
is given by
However, since (3.211
I"
solution of (3.20 as
+ 2 coth kh m(m+l) Am
[Am,m+
first-order
solution
physical
(-I) m+l
m=Or
8
as
A (kr)
m
is defined by
(4.3) for positive values of
m, and by the
635
NONLINEAR DIFFRACTION OF WATER WAVES BY OFFSHORE STRUCTURES
A
-m
(kr)
A
m
(kr)
(5.L0)
The stnmation in (5.8) includes both m
m.
-it
i
e
A (kr) and e
A * (kr).
for negative values of
in order to incorporate
O
=-
(r,0,t)
6m Ym im+
E
m
0
O
The corresponding real solution for
r
0 + and
e
is given by
[
-i t
m
Am(kr)
(5.11)
cos m0
where
m>
l,
Ym
-1,
0+]
<
m
5. 2ab)
0
Equation (3.21) then takes the following form which is similar to (5.[):
322 + g 2
32
where
[e
-2it
Y.
B (kr) cos m0 + c.c]
(5.13)
m
m=O
4k
z
stands for the complex conjugate, and
c.c.
Z B (kr) cos m0 + e 2 imt l B * (kr) cos m0
m
m
m=0
m=0
-2 it
e
gm
-i( + )t
m+n+1
I l
4
m=.-o
Z
6
m
6
l
e
n
n
A
mn
[cos (re+n)0 + cos(m-n)0]
AA
m n
cos(m-n)O]
[cos(m+n)O
A
where
is given by
mn
[ m (tanh kh- coth kh) +
A
mn
(m +
+ I__
m
m
It is noted that for
definition
,
A
-m,-n
(5.14)
of
A
m,m
A
mn
m, n
is
(m
n
m
+
tanh kh] A A
m n
n
coth kh A
m
A
(5.15)"
n
this expression is identical with (5.3) and hence the
0
consistent
with
the
previous
definition.
Furthermore,
It can be verified that the double series in (5.14) contains terms
that are independent of time
t
and hence correspond to standing waves.
However, it
can readily be checked that these terms add up to zero and hence there are no standing
waves in the solution.
Finally,
the form
the solution for
2
satisfying the required boundary conditions has
M. RAHMAN AND L. DEBNATH
636
e
0
-2i t
4k
e
+
result
B (kr)
m
6.
[f Dm(k 2) Am(rk 2)
0
+2 i0 t
4k
This
E
m=O
2
2
Z
[/
m=0
0
* (k
*
2) Am
D
(rk2)
m
(z+h)
2
cosh k
2
dk2]
(z+h)
Dm(k2)
(5.4), and
similar to that of
is
cosh k
cos m@
dk2]
(5.16)
cos m0
is the solution of
(5.5) with
given in (5.14) which is analogous to (5.2).
RESULTAST HORIZONTAL FORCES 08 THE LfLIN’DER.
For
whose
wave
diffracted
a
first-order
potential
hydrodynamic pressure evaluated at the cylinder
b
r
of
is
form
(5.8),
the
4[, 2’
etc.
The
the
depends on
flrst-order horizontal force on the cylinder is obtained from (3.27) in the form
Fxl
4g
tanh kh
3
{HI( I>’ (kb{
k
(6 l)
cos (mt- a
where
J1’ (kb)
tan-|
[Y
(6.2)
(kb)
This result was obtained earlier by other researchers including MacCamy and Fuchs
Lighthill
[14] and Rahman [17].
In the limit
kh
=,
[2],
(6.[) corresponds to the
result for deep water waves which are in agreement with Lighthill [14], and Hunt and
Baddour [19].
We
summarize
next
force given by (].28).
below
the second-order contribution to
We first put values of
I
and
[
the total horizontal
from (5.8) and (5.11)
into (].28) and then evaluate the z-integral to obtain the coefficient of
(].28), apart from the
16k
2
E
E
m=0 n=0
-(I+
mn
b2k2
+ cos
2/t
m n
2kh
cos
sinh 2kh
(1 +
(m
2kk h)
[{(3
R R
m n
cos0
in
term, in the form
sinh
2kh
sinh 2kh
n+
n+
(m
am + an+
cos [-2mt +
(m-n))}
cos (-2t +
cos (m+n)0 + cos
m+n+
(m-n))}{cos (re+n)0
cos
(re+n)]
(m-n)0]
(m+n))
(r-n)0}]
where the Wronskian property of Bessel functions yields
A (kb)
m
2i [kb
HKl’(kb)"
]-I
m
R
m
e
m
(6.4)
_
NONLINEAR DIFFRACTION OF WATER WAVES BY OFFSHORE STRUCTURES
with
R
m
-I/2
[Jm2"
Y’2.kb.j(!
m
(kb) +
2g
2
k
+
2
e
(s+l).)
2
s
{(I
l
s=O
2kh
+
E
sin kh
s(s+.l) (I + 2kh
si 2kh
b2k 2
+
sinh 2kh
(1
2k
b
2kh
(-l)s [(3-
(6.5ab)
r
in the double series (6.3) and hence obtain the following
cos0
(-)
J’ (kb)
[]
t)
0-integration describe([ in (3.28), we need only the
In view of the subsequent
coefficient of
_|
tan
m
637
][Cs
cos 2t0t-S sin2t]
S
(6.6)
where
--[(Js’Y’s+l Js+l Y’)s
[Es, Cs’ Ss]
M
with
IJ s’2 + Y’2)Ij’2
s+1
s
s
+
(Y’J’
s s+l
S
Y’s+l J’l
s
J’s J’s+l )],
(Y’Y’s+[
s
y,2
s+l)
+
(6.8)
and the argument of the Bessel functions involved in (6.6)-(6.7abc) is
The
$2
of
part
proportional
(6.7abc)
given by (5.16), contributes to the
cos 0,
to
kb.
z-integral in (3.28) the term
g tanh kh e -2imt
0
kb
where
Dl(k 2)
D (k
f
is related to
Combining (6.3)
2)
sinh k h
2
k22 (1)’(k2b)
dk
H
Bl(kb)
and (6.9),
(6.9)
+ c.c.
2
through (5.5).
integrating with respect
to
0,
the result can be
expressed as the sum of steady and oscillatory componeuts:
Fx2
F
S
O
+ F
x2
x2
(6.10)
where
F
S
x2
k
2
(k
2
E
s=O
{[I-
(s+l)][
b2k 2
s
+ sinh2kh2k
Es}’
(6.11)
and
F
O
x2
[og
2bO_g
k
2
tanh kh e -2it
k
(k)
f
0
Z (-I) s
s--O
Dl(k2)
k
2
2
[(3
H
sinh
(ii)
(kb
2kh
k2h
dk
2
+ c.c.]
2
")(I
sinh 2kh
cos 2rot- S
+
2kh
s
(s+l)
sinh 2kh
sin 2mtJ,
(6.12)
M. RAHMAN AND L. DEBNATH
638
S
x2
0
correspond to results (6.1), (6.2) and
x2
(6.3) obtained by Hunt and Williams [20] for the diffraction of nonlinear progressive
These
x
the
on
together
oscillatory
an
F
and
each
consist
term
having
of
components,
two
the
twice
frequency
Hunt and Williams calculated the maximum value of
term.
wave
cylinder
vertical
with
F
The second-order contributions to the total horizontal force
in shallow water.
waves
F
for
expressions
steepness,
found
to
theory.
slender
depths and
water
significantly
higher
The
second-order
effects
but
cylinders,
These
diameter.
in
deep
predictions
that
than
are
found
they
water
supported
are
by
predicted
be
to
are
by
steady
the
component
first-order
for various values of
The maxim[ value of
cylinder diameters.
be
F
a
of
the
greatest
greatest
some
for
existing
F
is
diffraction
linear
shallow water
in
of
cylinders
experimental
for
larger
results
which, for finite wave steepness, shows an increase over the linear solution.
In order to establish the fact that (6.12) represents the oscillatory component
of the second-order solution, it is necessary to evaluate the complex form of
from the integral equation (5.5)
for
m=l
and in conjunction with (5.7).
the analyses of Hunt and Baddour [19] and Hunt and Williams
[20],
D1(k 2)
Following
we obtain from
(5.5)
and (5.7) that
Dl(k2)
k
2
kk2 Al(k2r) Bl(r)dr
sinh
k2h 4k tanh kh cosh k2h’
(6.13)
which san be written as
l(k 2)
D
sinh
A
k
k2h
b
l(k2r) B l(r)dr
tanh kh
k2(k2- 4ka
2
k22 HI(1)ik2 b)
H
(I)’
(6.14)
(k2b)
The nondimensional form of (6.14) is given by
D
G(bk2)
l(bk 2)
sinh
k
Clearly,
root
k
2
2
tanh
kb
(be2)2 Hl (1)ibk2)
It is noted that the function
is singular when
f
k2h
k
2
k2h
4 bk tanh kh.
n 2h
(bk2) [be2
G2(bk 2)
is analytic near and at
(6.15)
(bk 2)
H
k
2
However, it
0.
is a root of the equation
(6.16)
4 k tanh kh
The
2k for shallow water problem.
4k for deep water case and k
2
2
of (6.16) lles between 2k and 4k and hence may be regarded as correspond-
k
An argument similar to that of Grlffith [23]
ing to an ocean of intermediate depth.
shows
A 1(bk 2) B l(kr) d(kr)
that
the
G(k
Integrand in the integral
0
a particular deep water wave, and at
k
2
2k
2)
dk
2
is singular at
k
2
4k
for
for a particular shallow water wave.
The non-dimensional forms of the first-order and the second-order forces can be
expressed as
NONLINEAR DIFFRACTION OF WATER WAVES BY OFFSHORE STRUCTURES
FI
pgD
][
3
2(kb)
3
Fx2
pgD
cos (rot
tanh kh
tanh kh e
IHI
/
47 (kb)
0
G(k 2) de
E (-1) s [(3
s=0
4
(6.17)
(kb)
-2imt
8 (kb)
3
I
(I)
47(kb)
D
4
2
+ c.c.]
2kh
-)
+ s(s+l)
2k2"
sinh 2kh
cos 2mt- S
where
639
Z
s=O
2kh
+
sinh 2kh
2tJ
sin
s
(I
)(I
[(I- s(s+t)
2
b2k
+
2kh
)
sinh 2kh
E
(6.18)
s
is the diameter of the cylinder.
Thus the total horizontal force in nondimensional form is expressed as
F= F
+F
(6.19)
2
wher e
(i_8J(HIL
F1
CM
F2
[{/8}(H/LJ
D/LJ
2
tanh kh
e- 2 it
2
(HIL)
E (-I) s
3
4 (D/L)(kb) s=O
x
(C s
cos 2rot
(H/L) 2
4(D/e)(kb) 3
H
where
and
C
M
2a
G(k
s+/-nh 2kh
S
l
s=0
s
to be
+ c.c.
2) dk2
2kh
{(3- ’)
/
s(s+l)
b2k2
(I
/
2kh
sin 2kh
sin 2mr)
s(s+l)
[I
is the total waveheight and
is defined
(6.20)
a[)
tanh kh cos(t
D/L
b2k2
L
](I
+
2kh__)E s
(6.21)
sin 2kh
is the wavelength of the basic wave,
the Morison coefficient due to
linearlzed theory and is
given by
CM
4/[(kb)21H(ll)’(kb)l
(6.22)
In ocean engineering problems, wave forces on the structures depend essentially
H/L, D/L and h/L. However, Hunt and Williams
on three dimensionless parameters
have pointed out that many experimental studies of wave forces have been published
640
with
M. RAHMAN AND L. DEBNATH
such a variation of
parameters
that precise experimental verification is not
possible.
Recent findings of Rahman and Heaps have been compared
with experimental
data collected by Mogridge and Jamieson [3].
An agreement between theory and
experiment is quite satisfactory as shown
in Figures 2
4.
Another comparison s
made in Fig. 5 with the experimental data due
to Raman and Venkatanarasaiah [24]. The
second-order results of Rahman and Heaps [17] seem
to compare well with these
experimental data.
In Fig. 6, both the first-order and the
second-order solutions are
compared with force measurements of Chakrabarti
[13] which are generally found to be
closer to the second-order theory.
***
Experiment
0.20
Linear Theory
Second-Order Theory
D/L=O.057
h/L--O.090
0.15
0.10
0.05
/
0
H/L
I,
0.02
0.01
0
with
FIG. 2. Comparison of linear and second-order wave forces
experimental data of Mogridge and Jamieson l3].
0.40
/
D/L=O.086
/
h/L--0.136
f
0.30
x
u.
0.20
,,
/
.,<
0.10
0
0.02
0.04
0.06
FIG. 3. Comparison of linear and second-order wave forces with
experimental data of Mogrldge and Jarnieson [3].
NONLINEAR DIFFRACTION OF WATER WAVES BY OFFSHORE STRUCTURES
Experiment
**,
641
Linear Theory
Second-Order Theory
0.40
DIL=O-080
h/L--O. 153
0.30
x
=
0.20
E
0.10
.I
0
0.04
0.02
0
0.06
FIG.4 Comparison of linear and econd-order wave force with
experimental data of Mogridge and Jamleson [3].
4.5
4.0
/
"
D/L--O.0493
h/L=0.123
Experiment* **
Linear Theory
Second-Order
Theory
3.0
-
2.0
1.0
0
0.01
0.02
0.03
0.04
FIG. 5.Comparion of linear and zecond-order wave forcee
with experimental data of Ram=zn and Venk=tan=r=z=d=h [24].
A flnal comment on the singular nature of
structure, the wavenumber
the
root
of
the equation
G(k2)
Is In order. For a cylindrical
of the second-order wave theory must not coincide
2
with
(6.16) unless the corresponding integral in (6.15) van-
k
ishes.
Otherwise, the structure will experience a resonant response at the
wavenumber
k
This kind of resonance is predicted by the second-order
2.
diffraction theory but
not by the linear wave theory.
In real situations involving ocean waves, such nonlinear resonant phenomenon is frequently observed.
Hence the correct values of the
wave forces on the offshore structures cannot be predicted
by the linear wave theory.
According to Rahman and Heaps’ analysis, the cylindrical
structure will ex-
642
M. RAHMAN AND L. DEBNATH
perience a resonance when
k
4k for the case of deep water waves, and when k
2
2k
2
Obviously, there is a need for modification of
for the case of shallow water waves.
the existing theories in order to obtain a meaningful solution at the resonant wave-
A partial answer to the resonant behavior related to the shallow water case
number.
has been given by Rahman [25].
Recently,
Sabuncu and Goren [21]have
studied the problem of nonlinear diffrac-
tion of a progressive wave in finite deep water, incident on a fixed circular dock.
This study shows that the second-order contribution to the horizontal force is also
Their numerical
highly significant.
results
for
the vertical
and horizontal wave
forces on the dock are in excellent agreement with those of others.
Gaston
Demirbilek and
[22] have also reported some improvements on the existing results concerning
In spite of various ana-
the nonlinear wave loading on a vertical circular cylinder.
lytical and numerical treatments of the problems, further study is desirable in order
to resolve certain discrepancies of the predicted results.
Finally,
close this section by citing a somewhat related problem of waves
we
incident on harbors.
A recent study of Burrows [26] on linear waves incident on a
circular harbor with a narrow gap demonstrates that the wave amplitude inside the har-
boT
is
At
significantly affected by the frequency of the incident waves.
4.0
.--
Linear Theory
""
3.5
certain
Second-Order
Theory
3.0
2.0
2
1.5
E
1.0
0.5
0
0
1.0
0.5
1.5
2.0
2.5
FIG.6. Comparison of linear and second-orderwave
force with experimental date of Chakrabarti
frequencies
the
harbor
acts
as
a
resonator
and
the
wave
amplitude
3.
becomes
very
If the harbor is closed and the damping neglected, the free-wave motion is the
superposltlon of normal modes of standing waves with a discrete spectrum of char-
large.
acteristic frequencies.
With a circular harbor wlth a narrow opening,
a resonance
occurs whenever the frequency of the Incident waves approaches to a characteristic
A resonance of a different klnd is given by the so
frequency of the closed harbor.
called Helmholtz mode when the oscillatory motion inside the harbor is much slower
NONLINEAR DIFFRACTION OF WATER WAVES BY OFFSHORE
643
STRUCTURES
Burrows determined the steady-state response of the
than each of the normal modes.
harbor with a narrow gap of angular width
to an incident wave of a single
2
frequency under the assumption of small width compared to the wavelength.
The re-
sponse is a function of frequency and has a large value (a resonance) at the frequency of
.
the
Helmholtz
closed harbor.
2
mode and
also near
the
characteristic
frequencies of
The actual nature of the response near these frequencies depends on
It is shown that the peak value at each resonance increases as
is the harbor paradox for a single incident wave frequency.
slow.
the
,
The peak width also depends on
decreases, that
However, the increase is
decreases, but the
and decreases as
decrease for the Helmholtz mode is less than for the higher modes.
Some
authors
flow
through
frequency.
the
This
gap
is
[27]
Lee
including
problem inside the harbor.
gave a numerical
treatment
of
the
resonance
In approximate calculations it is assumed that the total
will
effectively
flow
the
determine
resonance,
near the Helmholtz
correct
higher resonances where the through-flow is small.
Helmholtz resonance was based on the linear theory.
near
the
but incorrect
resonant
the
near
Most of the work on the subject of
The question remains whether or
not the circular harbor is a Helmholtz resonator for nonlinear water waves.
7.
NONLI%RWAVE DIFFRACTION CAUSED BY LARGE CONICAL STECTRES.
We consider a rigid conical structure in waves as depicted in Figure 7.
With
(b-z) tan u
reference
to
where
is the distance between the vertex of the cone and undisturbed water sur-
b
face,
this figure, the equation of the cone may be given by
is the semi-vertical angle of the cone and
-
r
r
is the radial distance of the
(r,0,z). The fluid occupies the space (b-z) tan < r < =,
0
h
z
< <
< < (r,0,t), where h is the height of the undisturbed free
surface from the ocean-bed and (r,0,t) is the vertical elevation of the free surface.
cylindrical
,
The
coordinates
partial
governing
(r,0,z,t)
is
V2 82q2 +
r
r
+
differential
<
(b-z) tan u
within the region
+-2
02
r
r
z
<
,
equation
for
the
velocity
potential
(7.1)
0
<
0
<
n, -h< z<.
The free surface conditions are
))
-f+
for
z
))
2
+ [(-r
gn
and
(b-z) tan a
an
a
an
))
+
(’’)
<
r;
an
8)
2
2
+
0
a
(7.2)
(7.3)
r
for
z
-
(b-z) tan a
n and
<
r.
The boundary condition at the ocean-bed is
z
0
at
z
-h
(7.4)
M. RAHMAN AND L. DEBNATH
644
DIRECTION OF
WAVE PROPAGATION
SWL
O
FIG.7. Schematic Diagram of a Conical Structure in wave.
is
The boundary condition on the body surface
n
at
r
(b-z)
r
tan
+z sin a
cos a
a, -h
z
q
(7.5)
0
where
n
suris the distance normal to the body
is needed for the unique solution of
There is another boundary condition which
radiation
This condition is known as the Sommefeld
this boundary value problem.
This is briefly deduced as follows:
condition which is discussed by Stoker [28].
face.
The velocity potential
$(r,O,z,t)
where
Re
may be expressed as
Re[(r,,z)
e
imt]
stands for the real part and
(7.6)
is the frequency.
We assume
NONLINEAR DIFFRACTION
(r,0,z)
I Re[l
such that
Therefore,
where
Then
I
the
e
OFFSHORE STRUCTURES
645
S
-imt
]’
S-- Re[s
e
-ira t
I + #S
#
S
and
(7.7)
are the incident wave and
scattered wave potentials respectively.
radiation condition is written as
lira
This
+
I
OF WATER WAVES BY
(_Sv,
#V
condition
may
i k
be
1/_
#S
(7.8)
generally
(r) -v2exp(-ikr).
proportional to
0
satisfied
when
)
S
takes
an
asymptotic
form
Here
k
is a wave number.
The linear incident wave potential
I
may be obtained from the solution of the
Laplace’ s equation,
V2 I
2I + 2I +
0
x
2
(7.9)
y2
subject to the linear boundary conditions (Sarpkaya and Isaacson [29])
l(x,y,z,t)
Re[l(x,y,z
e
-imt]
where
I
C
cosh k(z+h)
exp(i(kx cos y + ky sin
cosh kh
2m
)’ Y
Y)),
(7 I0)
is the direction of propogation of the incident wave in the
x-y plane.
The famous dispersion relation
for water waves is
m
2
gk t anh kh
(7.11)
Using this relation, we find
exp (i(kx cos y + ky sin 7))
--exp (i(kr cos(0-y))
E
mffiO
where
80
-
Bm J (kr) cos m(8-y)
m
I,
and
m
m
exp (i
m
/2),
50
I, m0; 6
m
The incident wave expression (7.10)
can be written as
2, m
>
646
M. RAHMAN AND L. DEBNATH
C
$I
cosh k(z+h)
cosh kh
l
13
m=0
m
J
(kr) cos m(0-y)
m
(7 12)
We are now in a position to construct the scattered potential
S
where
-cShcoshk(Z+h)kh m=0r. 13m Bm Hm(1)
C
S
which is given by
(kr) cos m(0-y)
(7.13)
B
is a constant.
m
It can be easily verified that
(7.13) satisfies the radiation condition (7.8).
The surface boundary condition (7.5) gives that
n
I
3n
(b-z) tan a
r
at
3n
I
(7.14)
I
r cos = + z sin a
kC
cosh kh
7.
13
m
[J’m(kr)
cosh k(z+h) cos a
+ J (kr) sinh k(z+h) sin a] cos m(0-y)
m
where
y
is the angle made by unit normal with the radial distance
(7.15)
r.
Similarly, we
get
3
3
S
S
Dr
3n
3
cos a +
coshkCkh
r.
m--O
S
s in a
+ Hm (I) (kr)
J
m
(kr) + J
B
at
m
[Hm
m
(I)
(kr) + H
m
It is to be noted from (7.17) that
-
instead of a constant.
B
m
cosh k(z+h) cos a
k(z+h) sin ] cos m(0-y)
using the conditions
(7.16)
(7.14), we obtain
(kr) tanh k(z+h) tan
(b-z) tan a,
r
sinh
cos re(O-y),
13
Comparing the coefficients of
[am(1)’(kr)
13m Bm
(1)(kr)
h
<
z
<
tanh k(z+h) tan a]
(7.17)
n.
B
the constant
m
turns out to be a function of
z
In order to overcome this difficulty, we estimate the constant
by taking the depth average value, which is obtained by integrating both sides of
(7.17) with respect to
z
from
z
-h
to
O, such that
z
0
Bm
-h
[Jm(k(b-z)tan
fv [Hm(1)
-h
where
m
0,I,2,
a) +
Jm(k(b-z)tan
a)tanh k(z+h)tan a] dz
(7 18)
(k(b-z)tan =) +
H(1)(k(b-z)tanm
=)tanh k(z+h)tan =] dz
NONLINEAR DIFFRACTION OF WATER WAVES BY OFFSHORE STRUCTURES
Once the scattered potential
structures.
S
is determined, we can formulate the wave forces on the
may be written as
Therefore, the total complex potential
%(x,y,z)
cosh k(z+h.#
cosh kh
C
647
y.
+ B m Hm
[Jm(kr)
m
0
(1)(kr)]
(7.19)
cos m(0-y)
The formulation of the wave forces is given in the next section.
8.
WAVE FORCES FORIILATION.
Lighthill
[14] demonstrated that second order wave forces on arbitrary shaped
structures may be determined from the knowledge of linear velocity potential alone.
The exact
calculation of
second order
by Debnath and
obtained
[9]
Rahman
forces on right circular cylinders has been
Lighthill’s
the
using
The
technique.
total
potential has been obtained in the following form:
where
F
+ F d + Fw + F
q
F
F
F
force,
is the linear
.
force,
is the second order dynamic
d
F
second order waterline force, and
(8.1)
is the quadratic force.
q
are all functions of the linear diffraction potential
the following formulas:
Fw
is the
These force components
They may be obtained using
The linear force is
F fS
(-P
--)
n
(8.2)
dS
x
which can be subsequently written as
where
F
Re [-Ipe
#
I + #S"
-it
f
n
S
(8.3)
dS]
x
The second order dynamic force is
Making use of the identity
where,
s
Re [z
e
-I t
s
nx dS
(V)
p
Fd
s
], s 2
[
Re
2
Re [z
2
z
e
(8.4)
z
2
-I t
e
-2it
+
z
z
*
2]
and the asterick denotes the complex
conjugate, we can write
0 Re
Fd
[e
-21t
fS (V)
n
x
as]
p [
IV#1 2 nx dS.
(8.5)
The waterline force is
F
w
f
z--0
a
(Pl2g)(--)
which can be subsequently written as
2
dy
(8.6)
648
M. RAHMAN AND L. DEBNATH
F
w---
(p
m2 /g)
Re
-2 it
[
e
()
2
(p2/g)
dy] +
z=O
z=0
19
12 dy.
(8.7)
dy]
(8.8)
Making reference to Rahman [15] the quadratic force may be written as
F
q
Re [(-2pro
2/go
e
-2imt
z=O
a
(z) z=0
42/g
o
where
4k tanh kh
and
-
(_2
2-
{(V
+ g
az 2
dx
is the complex time independent potential gener-
ated by the structure surging at a frequency of
The vertical particle velocity
(z)
2m.
0
z
on
may be written in a series form
for finite water depth
(-z)z=0
where
{-
j=l
+
(4m2/g)
(mir)
Kl(mja) (m.h +
2K
z
2H2) (vr)
H 2)’(va)(vh +
sin
2
(re.h)
sin m.h cos
3
sinh
m.h)
2vh
(8.9)
cos 0
sinh h cosh vh)
.,
for j
2
and o
v tanh uh
mj tan m.h
4k tanh kh.
J
Expression (8.9) is valid only for
right circular cylindrical structures.
The wave
drift forces on the structures may be
obtained from the equations (8.5) and (8.7)
collecting the steady state components of the forces
F
and F
Thus
the
drift
forces
d
w
on the structure is
Fdrift
4
p
n dS +
x
(p2/g) f
z=0
9.
CALCULATION
OF
ifAV’g
i@
12 dy.
(8. O)
FORCES.
The total wave forces may be obtained
from the formulas (8.3), (8.5), (8.7) and
(8.8). The linear resultant force can be
obtained from (8.3) and is given by
F
Re [-Ipm e -imt
@
n
S
Re [-Ip( e-iuat
2
0
f
f
x dS]
O=0 z=-h
Re
where
tan
[2__pC
cosh kh
A (kr)
m
f
H(1)(kr)
m
m
m
tan a dz}(-cos
O)dO]
0
cos y e-it
J (kr) + B
m
{(b-z)
(b-z)eosh k(z+h)A (k(b-z)tan e)dz ]
2 3
NONLINEAR DIFFRACTION OF WATER WAVES BY OFFSHORE STRUCTURES
Therefore, the horizontal
649
and vertical forces can be obtained
respectively as
FX F
a,
cos
FZ
F
sin a
(9.2ab)
The resultant dynamic force can be calculated
from (8.5) and is given by
F
P
d
0
4
e-2i t f (V#) 2 n
S
Re
Re
[e- 2 imt
2
0
f
f
2 0
2
S [V[
f
IV#12((b-z)tan =)dz
-h
n dS
(V)2((b-z)tan )dz
0=0 z=-h
P-P-4 f0 f
P
dS]
(-cos 0)d0]
(-cos 0) dO
(9.3)
After extensive algebraic calculations, the dynamic
force can be written as
(P
Fd
cos
4
_)
e-2 it
Re
_
0
+ k2 sec 2 a sinh2 k(z+h))(e
+
(cosh
2
(b-z) 2
(P
k(z+h)
tan2
)
2
E / 2(b-z)tan
2
=0-h
cosh kh
(+I) + k 2 sec 2
(+I)
-(2+I)---
0
cos
4
4C2 (-csh2k(z+h
cosh2kh (b_z) 2 tan2
E / (b-z)tan
=0 -h
AA%+ I)
dz}]
ICI 2
sinh2k(z+h)
(e 2
*
AA+I+
e
-
AA+l)dZ*
(9.4)
Therefore, the horizontal and vertical dynamic forces can be obtained respectively as
F
Fdx
d
cos
a,
Fdz
F
d
sin a
(9.5ab)
The resultant waterline force can be obtained from (8.7) and is given by
2
F
W
2
(--)
+
Re
Re [e
[e-2it(-4C 2
-2it
(,)2dy]
A
E
cos T)(b tan )
E (e
=0
2
AAA+
-
gOm zfO I 2dy
-(2+1)
cos )(b tan a)
=0
(4---)(-2, ,C12
2
+
A%A+
e
+ e
2
*
A A+ I)
(9.6)
Therefore, the horizontal and vertical forces can be obtained respectively as
FwX Fw cos a, FwZ Fw sin
(9.Tab)
650
M. RAHMAN AND L. DEBNATH
Thus the drift force as defined by (8.10) can be obtained as follows:
2
0
((b-z)tan a)dz (-cos 0)d0
Fdrif t
0=0 z=-h
2
2
po
+g Oj- 0 i,I) 12z=0
(0
( (%+i)
(e
0
)l /
=0 -h
csh2k(z+h)
AA+
cosh
kh
inh2k (z+h))
*
1)dz
AA+
2
+ e
21CI 2 (b-z)tan
2
+ k 2 sec 2 s s
(b_z)2t an2
*
2
cos Y
4
(-cos 0)d0
b tan
2
2
*
--2
+ e
E (e
I) (9.8)
=0
Th6refore, the horizontal and vertical drift forces may be respectively obtained as
(0o
4-)(2IcI 2
(F
d
(F
rift)X
AA+
I*
cos y)(b tan a)
cos a
drif t
Fdrlft
Z
Fdrift
A A+
(9.9ab)
sin a
The resultant quadratic force can be obtained from the formula (8.8) and is given by
F
2pro2
Re[-
q
go
After extensive algebraic calculations, the quadratic force can be written as
F
Re[-
q
2pro2
e
-2it
f
(rdr)
r=b tan
2
I0=0 (@_)
8z z=0
[4C
2
k
2
+
A A+ I}
e
-(2+I)
x
7.
=0
A A+
z
{((+I)
k2r2.
=0
-(2+I)
AA+
-
cos(0-Y)
]d0]
e
+
+ tanh 2 kh)
2C2k2(tanh2kh-l)
cos(0-y)x
(9.11)
After simplifying, (9.11) reduces to
F
x
q
Re[-P
2C2k2
z
=0
e
2
-2 im t
(rdr)
r=b tan
2
[..(z+I) + 3 tanh kh-l}
k2r2
8?
f (zz)
z=0
O=0
cos(0-Y)
-(2+1)
AA+
+ 2
AA+ I]
e
-
dO}
651
NONLINEAR DIFFRACTION OF WATER WAVES BY OFFSHORE STRUCTURES
Therefore, the horizontal and vertical quadratic forces may be respectively obtained
as
FqX Fq
lO.
cos
,
FqZ Fq
(9.12ab)
sin
CONCLUDING
Second order nonlinear effects are included in the derivation of the wave forces
The second order theory is consistent because it
on the large conical structures.
satisfies all the necessary boundary conditions
Theoretical
could be
for
expressions
improved by
adding
to
waterline and quadratic forces.
it
including the radiation condition.
forces have been obtained;
the wave
the
linear
the second order contributions namely,
forces
dynamic,
It would be of considerable value if the theoretical
results presented in this paper could be checked experimentally under laboratory conditions.
are made
Plans
in
future
research to check the accuracy of the predicted
results with the experimental measurements.
Science
and Engineering
Research Council of Canada for its financial support for the project.
The work of the
ACKNOWLEDGEMENT;
The
first
author
is
thankful
to
Natural
second author is partially supported by the University of Central Florida.
REFERENCES
1.
2.
HAVELOCK, T. H. The Pressure of Water Waves on a Fixed Obstacle, Proc. Roy. Soc.
A175 (1940), 409-421.
NacCANY, R. C. and FUCHS, R. A. Wave Forces on Piles, A Diffraction Theory;
Beach Erosion Board Tech. Memo NO 69 (1954), 17 pages, U.S. Army Corps of
Engineers, Washington, D.C.
3.
4.
Wave Loads on Large Circular Cylinders, A
Design Method. National Research Council of Canada, Division of Mechanical
Engineering, Rep. MH-III (1976) 34 pages.
NI, C.C. Numerical Methods in Water-wave Diffraction and Radiation, Ann. Rev.
Fluid Mech. I0 (1978), 393-416.
IRIDE, G. R., amd JANIESON, W. W.
6.
[OEN, N. el. al., Estimation of Fluid Loading on Offshore Structures, Proc.
Instn. Civil Engrs. 63 (1977), 515-562.
GARRISON, C. J. Hydrodynamic Loading of Large Offshore Structures- Three-dimensional Source Distribution Methods, In Numerical Methods in Offshore Engineering (Ed. Zienkiewicz, et. ai.)(1978), 97-140.
7.
GARRISON, C. J.
8.
NORISON, J.R., O’BRIEN, N.P., JOHNSON, J.W., and SCHAAF, S.A. The Forces Exerted
by Surface Waves on Piles, Petroleum Transactions, AIME, Vol. 189 (1950),
149-154.
I)EHATH, L. and RAHNAH, N. A Theory of Nonlinear Wave Loading on Offshore Structures, Internat. J. Math. and Math. Sci. 4 (1981), 589-613.
5.
9.
I0.
II.
12.
13.
Forces on Semi-submerged Structures, Proc. Ocean Structural
amics Symposium, Oregon State University (1982).
Dyn-
I)EBN&TH, L. On Initial Development of Axisymmetric Waves in Fluids of Finite
Depth, Proc. Natn. Inst. Sci. India 35A (1969), 665-674.
C. Small-amplitude Internal Wave Due to an Oscillatory Pressure,
NAHAHTI,
Quart. Appl. Math. 37 (1979), 92-97.
.
Uniform Asymptotic Analysis of Shallow-water Waves Due to a Periodic
Surface Pressure, Quart. Appl. Math. XLIV (1986), 133-140.
CRABARTI, S. K. Second Order Wave Force on Large Vertical Cylinder, Proc. ASCE
I01 WW3 (1975), 311-317.
HDHDAL, C.R.
M. RAHMAN AND L. DEBNATH
652
14.
15.
16.
17.
18.
19.
20.
LIGHT,fILL, l. J. Waves and Hydrodynamic Loading, Proc. Second International Conference on the Behaviour of Offshore Structures,
(1979), 1-40.
P,, I. and CIRAI/ARTY, I. C. Hydrodynamic Loading Calculations for Offshore
Structures, SIAM J. Appl. Math. 41 (1981), 445-458.
RAI, M. and CHEIL, D. $. Second-order Wave Force on a Large Cylinder, Acta.
Mech. 44 (1982), 127-136.
RAI, M. and EAP$, H. $. Wave Forces on Offshore Structures: Nonlinear Diffraction by Large Cylinders, J. Phys. Ocenogra. 13 (1983), 2225-2235.
I, H. Wave Diffraction by Large Offshore Structures; An Exact Second-order
Theory, Appl. Ocean. Res. 6 (1986), 90-100.
IJIT, J. N. and BADDOIJR, R. E. The Diffraction of Nonlinear Progressive Waves by
a Vertical Cylinder, Quart. J. Mech. and Appl. Math. XXXIV (1981), 69-87.
IINT, J. N. and WILLIARS, A. I. Nonlinear Diffraction of Stokes Water Waves by a
(1982),
Circular Cylinder for Arbitrary Uniform Depth, J. Mech. Theor Appl.
429-449.
21.
22.
23.
24.
25.
26.
27.
29.
SbIIIII, T. and GOREI, O. Second-Order Vertical and Horizontal Wave Forces On a
Circular Dock, Ocean. Engng. 12 (1985), 341-361.
DEHIRBILEK, Z. and GASTON, J. D. Nonlinear Wave Loads On a Vertical Cylinder,
Ocean. Engng. 12 (1985), 375-385.
.
GRIFFIT, J. L. On Webber Transforms, J. Proc. R. Soc. N.S.W. 89 (1956), 232.
and -EIT/ARASAIA, P. Forces Due to Nonlinear Waves on Vertical
RA,
Cylinders, Proc. ASCE 102 WW 3 (1976), 301-316.
I, H. Nonlinear Wave Loads on Large Circular Cylinders: A Perturbation
Technique, Advances In Water Resources 4 (1981), 9-19, Academic Press.
BI]RROWS, &. Waves Incident On a Circular Harbour, Proc. Roy. Soc. London A401
(1985), 349-371.
LEE, J. J. Wave Induced Oscillations in Harbours of Arbitrary Geometry, J. Fluid
Mech. 45 (1971), 375-394.
STOKER, J. J. Water Waves, Interscience Publishers, New York (1957).
SARPKAYA, T. and ISAAC$ON, I.
Mechanics of
Van Nostrand Reinhold, New York (1957).
Wave Forces on Offshore
Structures.