579
Internat. J. Math. & Math. Sci.
VOL. 13 NO. 3 (1990) 579-590
THREE DIMENSIONAL GREEN’S FUNCTION FOR SHIP MOTION AT FORWARD SPEED
MATIUR RAHMAN
Department of Applied Mmthematics
Technical University of Nova Scotia
P.O. BOX I000
Halifax, Nova Scotia
Canada B3J 2X4
(Received December 12, 1988)
ABSTRACT.
The Green’s function formulation for ship motion at forward speed contains
double integrals with singularities in the path of integrations with respect to the
wave
number.
In this
the
study,
double
have
integrals
analysis
provides
an
efficient
way
of
computing
been
replaced by single
It has been found that this
integrals with the use of complex exponential integrals.
the
resistance
wave
for
three
dimensional potential problem of ship motion with forward speed.
KEY WORDS AND PHRASES.
Ship motion, Green’s function, Hydrodynamics, Wave Resistance
and Wave Responses.
I.
INTRODUCTION.
In ship hydrodynamics, Green’s functions play a very important role in predicting
the wave resistance, wave induced responses at zero forward speed, and the motions of
a
vessel advancing in waves.
The Green’s function formulation for ship motions at
forward speed is the most difficult part of the problem partly because it contains
double integrals and partly because of the presence of the singularities in the path
of integrations with respect to the wave numbers.
been paid
to
evaluate
Nowadays, considerable interest has
the three dimensional Green’s
function
for ship motions at
forward speed.
Many authors including Haskind [I] and Havelock [2] have expressed the Green’s
function having a constant forward speed as a double integral.
This form of
Green’s
function is not suitable for numerical analysis because the detailed computation of
the double integral is very expensive.
Therefore,
in the
present study,
we have
replaced the double integral by a single integral (see Wu and Taylor [3]) involving a
complex exponential integral, and it is found that it is more efficient to calculate
the Green’s function numerically.
2.
A FORM OF THE GREEN’S FUNCTION.
U
which is moving at constant forward speed
Consider the coordinate system oxyz
(see
surface
free
upwards from the mean
along the x axis and z measured positive
Figure I).
580
M. RAHMAN
Z
Figure 1" The Coodinate System
It is assumed that a ship is travelling at a constant forward speed U along the
Ox direction and oscillating at a frequency
Ialtone
[4]
-
Wehausen and
as in the form of e
1960 have shown that the Green’s function which satisfies the exact
in
free surface condition can be written as
=I
G(x y,z;a,b c)
+_K
7
7/2
d6
L
2
/(x-a)2 + (y-b)2
’
dO
o
o
F(0,k)dk + 2_
7
7/2
F(8,k)dk
d8
L
+ (z-c)
2
Rankine source located at
(a,b,c)
2
2
/(x-a) 2 + (y-b) + (z+c) Image about the mean free surface
R
g
7
F(6,k)dk
where
R
1+2g
R
k e
k[(z+c) +
i(x-a)cSe]cos[k(y-b)sine|
gk-( w+kUcos e)
2
(x,y,z) is the field point and (a,b,c) is the source distribution point.
parameters in Equation (2.1) are defined by
o
cos
T
(a,b,-c)
acceleration due to gravity
F(,k)
where
at
(-)
if
J
is called the Strouhal number
g
The contours L
and L
2
are defined as follows:
The other
<
(2.3)
t
(2.4)
if T
-I
(2.2)
581
SPEED
THREE DIMENSIONAL GREEN’S FUNCTION FOR SHIP MOTION AT FORWARD
kl
k2
k3
k4
and two singular points in the L
There are two singular points in L
Equation (2.1).
2
integral of
These singular points can be obtained as follows:
cose
(2.5)
+ /i-4 COS8
(2.6)
/1-4
2
/gkl /gk3
/g--72K
cos-----m
/gk--4
The alternative forms of these singularities k I, k 2, k and k can be written as
4
3
k
for /2
e
2
k
2 cosS) : /1-4 cos8 v
2
e
2
2
(I
w; and where v
g
It should be noted here that
the ranges indicated.
0
(
e
(2.8)
T2cos
kl=k 3
and
k2=k 4.
These singularities are real in
It is, however, worth mentioning here that in the range
y, the singularities k
and k
2
become complex quantities and are either given
by
+/-
/gk2’ /gkl
i/4T cos
2T cos0
(2.9)
or
k2 ’kl
2z cos0
2
+/-
i/4 cos-I
z2cos2 0
(2.10)
v
Thus the integrand in the integral
F(8,k)dk]de
O
O
contains no real singular points in the path of integration from 0 to
3.
(R).
EVALUATION OF INTEGRALS.
The
Green’s
function
given
in
equation
(2. I)
is
difficult
to
numerically because as we have seen in the previous section, the contours L
have singularities at k l,k2,k
3
and k
4.
2
This difficulty can be overcome by introducing
the Cauchy Principal Value (PV) integrals.
k.i
o-
F
integrate
and L
k2
’k >
M. RAHMAN
582
The first contour integral along the path L
G
2-K
L
can be rewritten as
712
F(0,k)dk
d0
L
F( 0,k)dk
0, equation (3.1) can be written as
When z
2_g
GL1
where
f dO
y
the
evaluate
f
o
(V.V.)
f F( 0,k)dk
o
(P.V.)
To
7/2
F(0,k)dk
k
k
2
f f
fy dO {+
F( O,k)dk
F(O,k)dk
(3.2)
0
(3.3)
deformationsandf
the
along
integral F(0,k) in terms of its singularities.
F(O,k)
7/2
kl- k2-e
f+ f + f
0
k1+e k2+e
Lim
integral
2g
/
we
decompose
We write
exp{k[(z+c) + i(x-a)cosO]} cos(k(y-b)sin0)
g II-4 zcos O [Z_k Z_f
the
(3.4)
which can be put in the following compact form
kl
F(8,k)
2g,/1-4"rcosO
k2
[[2kl k-k2]
[exp(kXl)
+
exp(kx2)]
(3.5)
where
xl=
w+
(z+c) +
iw+,
x
2
(z+c) + i w.
(x-a)cosO + (y-b)sin0,
w_
(x-a)cosO- (y-b)sln O
We know that (z+c)0 so we can redefine x
x
{(z+c)
iw+},
x
and x
{(z+c)
2
2
(3.6)
as follows
(3.7)
iw_}.
Thus, equation (3.5) can be rewritten as
F(0,k)
2g,/I-4 z
cos
{exp(-kXl) + exp(-kx2)
[kl
o
k2 {exp(-kXl) + exp(-kx2)
k-k2
k-kl
(3.8)
The integration along the deformatlons.and fIn equation (3.2) can be obtained
according to the residue theorem.
f+ f
2g,/I-4 zcos O
Thus
F(8,k)dk
{kl(exp(-klX I)
Thus, equation (3.9) reduces to
+
exp(-klX2))
+
k2(exp(-k2x I)
+
exp(-k2x2))}
(3.9)
THREE DIMENSIONAL GREEN’S FUNCTION FOR SHIP MOTION AT FORWARD SPEED
=2__K 12
/ d0
Gel
/
(P.V.)
/2
i
,/I-4 os
F(e,k)dk
0
y
+
583
e
{kl(e-klXl
+ e -k IX2) + k 2
(e-k 2Xl
+
e-k2x2)}
de
(3.10)
In a similar manner the second contour integral along path L 2 equation (2.1) can
be obtained
GL 2 2g /2f de
F(0,k)dk
0
{k3(k3Xl
,/1-4 cos e /2
f
(P.V.)
k4(e-k4Xl
e-k3x2)
+
+
e-k4x2)}
(3.11)
Therefore the Greens function in equation (2.1) can be rewritten as follows:
a,b,c)
Gl(x,y,z;
R
R
f d0 0f F(0,k)dk
+
0
/2
+ 2g
f+
de (P.V.)
x /2
/2
i
,/I-4 cos
F( e,k)dk
{k (e
-klXl
+ e -k
x2)
+
y
-k
{k3(e
/2
J
e
f
0
3 Xl+ e
-k
3 x2)
k4(e
k2(e
-k x
4 + e
-k
-k4 x2
2 Xl+
e-k2x2)}
de
(3.12)
de}
or
+ G + G + G + iG
3
5
2
4
G
G
(3.13)
where
G1
G2
’
=2g f
R/2
Y
G3
de
0
f F(e,k)dk
0
/2
G5
f
+f
/2
,/1-4 cos O
,/I-4 cos e
{k
G
f + /2f e
4
f F(e,k)dk
(P.V.)
0
y
{kl(e-klXl+ e-klX2)
+
3(e-k3xl+ e-k3x2)
k
k2(e
-k
2 Xl+ e
-k2x 2 )}
4(e-k4xl+ e-k4x2)
de
de
(3.14)
There are two cases to be considered.
Case I
y= 0
if
and in this case G
0.
3
<
Therefore, equation (3.13) becomes
G
G
+ G
2
+ G
4
+ iG
5
(3.15)
584
M. RAHMAN
Case II
y
cos
-1
if
4T
%-
and in this case equation (3.13) becomes
G
+ G
G
2
+ G
3
+ G
4
+ iG
(3.16)
5
and G are highly oscillatory at large values of k
3
4
In order to calculate
because of the imaginary argument of the exponential function.
The double
in G
integrals
them numerically, at minimum computer cost, these integrals must be reduced to single
[5], and Inglis and Price [6]. We shall
treat Case I first and evaluate the Cauchy Principal Value (P.V.) integral in G
4.
integrals as suggested by Shen and Farell
The term G
4
72
G4
of the Green function can be written as
dO
y {l-4Icos 8
where
d0
7/2 I-4Tcos8
k-kl
0
13
(
for 7/2
k2exp (-kx
(P.V.)
dk,
k-k2
0
12
dk,
(P.V.)
0
14
(P’V’)
16
(P.V.)
18
(P.V.)
(15+16-17-18)
klexp(-kx 2)
k_kl
k exp(-kx
(P.V.)
I
for y
+
(II +12_13_14)
k exp -kx
2
2
0
k_k2
dk
(3.17)
(3.18)
dk
7/2
O
15
(P.V.)
17
(P.V.)
O
k2exp (-kx i)
k-k3
0
0
k4exp (-kx
k_k4
dk,
dk
k exp -kx
3
0
0
k_k3
k4exp(-kx 2
k_k4
dk
(3.19)
dk
,.
To obtain analytic expressions of these integrals, we consider a contour in the
K
k + ik’ plane as suggested by Smith et al [7] (see Figure 2) and later used by
Chen et al [8].
We impose the condition that
__[-KXll
1
on the
o
integration path 5 which makes an angle a with the real axis,
argument of the exponential can be made real along the ray.
Therefore, we get
I [-(k + ik’)
m
(Iz+cl-lw+)]
which simplifies to yield
and
a= tan
-I k’
tan
-I
w+
Z+C i-
0
so that the
THREE DIMENSIONAL GREEN’S FUNCTION FOR SHIP MOTION AT FORWARD SPEED
585
K--k+ ik
4
Figure 2" A closed contour for
Thus with this value of
,
+
w+> 0
2
w+
0
-kV
Also, we have
kV
K --k+ ik’
=V+c[-iw+
Integrating along the contour shown in Figure 2,
I
{klexp(-klXl)
2i
(i)
klexp(-klXl)
klexp(-kx)
klexp(-klX 1)
klexp(-kx
i
dk
dk
k-k
5
k-kl
5
Along the path 5
0
5
kl
exp(-kV)
(kV)
2
klXl
d(kV)
exp(-u)
k
0
u
k
Ix
du
exp(-klX 1) E l(-klx 1)
k
where
EI(_Z)
Therefore, for
w+ >
O,
I
k
e--{-
dt,
[arg(z) <
w
exponential integral.
-z
exp(-klx I) {El-(klX l)
It is to be noted here that for
+ ri}
w+< 0 the
contour will be as follows:
(3.20)
586
M. RAHMAN
K=k+
I
/4
k
Figure 3: Closed contour for
w, <0
Thus
I
Also, for
w+
k
exp(-klXl) {El(-klXl)
(3.21)
’rl}
0,
klexp(-klXl) {El(-klX I)
I
(3.22)
+ i}
-
which is obtained using the following definitions (see Abramowitz and Stegun [9] 1965,
p. 228).
E
such that for
I
w+
e
(P V
l(-x+lO)
-E--dt
i,
E1(-x-lO)
0
klexp(-k2Xl) {El(-klXl)+
z’l]
Similarly, we can calculate the other integrals.
we get for
w+
I
and for
w+ <
)
0 or I
m
k
exp(-klX I)
0 or I (-k x
m
I
k
(-klX)
I) <
)
(P.V.)
dt +
J wle
Thus summing up the situation,
0
{E l(-klx I) + i}
0
lexp(-k Ix I)
{Z l(-k
Ix I)
-1)
Similarly,
Im(-klX2) > 0
klexp(-klX2) [El(-klX2) I], for Im(-klX2) < 0
13 k2exp(-k2x I)[E l(-k2x I) I], for Im(-k2x I) > 0
k2exp(-k2x I)[E l(-k2x I) + I], for Im(-k2x I) < 0
12 klexp(-klX2) [El(-klX2)
+ I], for
(3.23)
THREE DIMENSIONAL GREEN’S FUNCTION FOR SHIP MOTION AT FORWARD SPEED
14 k2exp(-k2x2)[El(-k2x2)
k2exp(-k2x 2) [El(-k2x 2)
k3exp(-k3x I) [E l(-k2x I)
k3exp(-k3x I) [E l(-k3x I)
k3exp(-k3x 2) [E l(-k3x2)
k3exp(-k3x 2) [El(-k3x 2)
k4exp(-k4x I) [E l(-k4x I)
k4exp(-k4x I) [E l(-k4x I)
k4exp(-k4x2) [E l(-k4x2)
k4exp(-k4x 4) [El(-k4x 2)
15
16
17
18
in G
Now adding the terms
4
and G
5
Im(-kzX2)
for Im(-k2x2)
for Im(-k3Xl
for Im(-k3Xl
for Im(-k3x2)
for Im(-k3x2)
for Im(-k4Xl
for Im(-k4Xl
for Im(-k4x2)
for Im(-k4x2)
i], for
+ i],
+ i],
i],
+ i],
i],
+ i],
i],
+ i],
i],
given by
the
587
0
)
<
0
0
<
0
)
0
<
0
)
0
<
0
)
0
<
0
equations
(3.24)
(3.17) and (3.14),
respectively, we obtain
12
G
4
{I +
+ iG
5
+ .I_w
i
/2 ,/I-4cos8
[(I 5 +
+ i
12 13 14
16 17 18
+ i
(Iii
+
(131 + 132
112
+
+
121
122)}d8
(3.25)
142)}d8
141
where
lij
k
i
exp(-kixj)
j
1,2,3,4; j
1,2.
Thus, if we combine the corresponding integrands of G 4 +
First Integral
1
J2
k lexp(-k
Ix I)
2 klexp(-klX
I__
2
2 klexp(-klX 2)
(
__l
-
E
,/1-4 cos8
k exp(-k x
Second Integral
we obtain
for I
,/1-4 cos 8
1
-f
[E (-k x )+2 ri ]d 8,
iGs,
2
,/I-4 zcos8
,/1-4 zcos 8
(-klX
)d 8,
[E (-k Ix 2 )+2 z-i ]d e,
E
(-klX2)d 8,
m
_(-klx I)
for I (-k x
m
for I
m
I) <
__(-klX
2)
for I (-k x
m
> 0
0
)0
2) <
0
588
M. RAHMAN
_I
Third Integral
2 k2exp(-k2x I) (-k2x
J2, k2exp(-k2x I) El(-k2Xl)d
--[-E
,/I-4
Tcos
)+2 ri ]d O,
for
Im(-k2x I)
)
0
0,
for
Im(-k2x I) <
0
(-k2x2)+2 ]d 0,
for
Im(-k2x2) <
0
(-k2x 2)d O,
for
Im(-k2x2) <
0
0
I
--
,/I-4
Tcos
/2 k 2exp(-k 2x 2)
Fourth Integral
_..1
Fifth Integral
1
Sixth Integral
z’2 k2exp(-k2x2)
i
/2
’
k exp -k x
3
3
[E
1
w
1
(-k3x
,/1-4 cos0
i/2 k3exp(-k3x2)
k 3exp(-k 3x
2)
[E
Tcos 8
i
k3exp(-k3x2)
i
k4exp(-k4Xl
E
1
for
Im(-k3x 2) <
0
for
Im(-k3x 2) <
0
(-k4x
)-2 rl ]d 0,
[E
i/2 k4exp(-k4x2
[-E (-k x )-2 ri ]d O,
4 2
Tcos
)0
(-k3x 2 )+2 i ]d 0,
i/2 k4exp(-k4x4)
,/1-4
(-k3x)
0
[-E
k4exp (-k4x2)
m
Im(-k3x I) <
/2 ,/I-4 zcosO
,/I-4 cos O
for I
for
(-k3x2)d O,
(-k4x
,/1-4 zcos8
Eighth Integral
)+2 ri ]d O,
)d 0,
(-k3x
E
,/I-4 zcos0
/2 ,/I-4 zcos0
Seventh Integral=
E
,/1-4 cos 0
/2 ,/I-4
1
[-E
,/1-4 cos O
y
)d 0,
(-k4x2)d 8,
E
O
for I
__(-k4x
I)
for I
m
__(-k4x
I) <
m
(-k4x 2)
for I
for
)0
0
)0
Im(-k4x 2) <
0
(3.26)
Therefore, for Case I, we can evaluate the Green’s function given in equation
(3.15).
To evaluate the Green’s function for Case II given in equation (3.16), we need to
express the G
3
term in exponential integrals as given below:
k
G
3
0
0
ek[(z+c)
gk
+
i(x-a)cosO]cos[k(y_b)sln 8 dk
(w + kUcosS)
THREE DIMENSIONAL GREEN’S FUNCTION FOR SHIP MOTION AT FORWARD SPEED
i
where
/
0
kl
Jl
k
and k
and k
2
+
(J
exp(-kx l)dk,
I-
(3.27)
J3- J4 )d@
J2
4 cos 0-I
kl
i
J2
exp(-kx 2)dk
-Z-f
k
2
589
2
(3.28)
are the complex roots of
(m + kUcos0)
gk
2
0.
Using the contour in Figure 2, it can be easily shown that
0
2ff]
Im(-klXl)
Im(-klXl) <
0
2i]
Im(-klX2)
Im(-klX2) <
Im(-k2Xl) )
Im(-k2Xl <
0
Jl klexp(-klXl)El(-klXl
klexp(-klXl) [El(-klXl)
J2 klexp(-klX2) El(-klX2)
klexp(-klx2)
[E 1(-klx2)
J3 k2exp(-k2Xl) El(-k2Xl)
k2exp(-k2Xl)
[E l(-k2xl) -2ri]
J4 k2exp(-k2x2) [El(-k2x2)
-k x
+ 2J.
Im(-k2x2)
0
0
0
> 0
E (-k x
Thus, with this information, we can evaluate the Green’s function for Case II
from equation (3.16).
4.
RESULTS AND CONCLUSIONS.
The present form of Green’s function is equivalent to that used by Wu and Taylor,
but in a different form.
Chen et al [8].
However,
The terms GI, G and G are all identical to those used by
3
2
and G terms
in the present study, we have combined the G
to correspond with the form of Wu and Taylor.
5
4
It appears that our studies are quite
similar to those of Wu and Taylor, and Chen et al.
The
double
integral
arising
in
the
evaluation
of
Green’s function has been
replaced by a single integral with the use of complex exponential integrals.
The
present work has provided an alternative form but similar to that of Wu and Taylor,
and has been found to be efficient for the analysis of the three dimensional potential
problem of ship motion with forward speed.
M. RAHMAN
590
ACKNOWLEDGEMENTS.
This work has been performed under Contract No. OSC87-00549-(010)
with the Defence Research Establishment Atlantic while the author was on Sabbatical
Leave from the Technical University of Nova Scotia.
REFERENCES
I.
2.
3.
HASKIND, M.D., The Hydrodynamic Theory of Ship Oscillations in Rolling and
Pitching, Prik. Mat. Mekh. I0 (1946), 33-66.
HAVELOCK, T.H., The Effect of Speed of Advance Upon the Damping of Heave and
Pitch, Trans. Inst. Naval Architect, I00, (1958), 131-135.
WU, G.X. and TAYLOR, R.E., A Green’s Function Form for Ship Motions at Forward
Speed, International Shi p
P.ogress 34 (1987), 189-196.
WEHAUSEN, J.V. and LAITONE, E.V. Surface Waves, Handbuch der Physik 9 (1960),
446-778.
5. SHEN, H.T. and FARELL, C., Numerical Calculation of the Wave Integrals in the
Linearized Theory of Water Waves, Journal of Ship Research 21 (1977).
6., INGLIS, R.B. and PRICE, W.G., Calculation of the Velocity Potential of a
Translating, Pulsating Source, Transaction, RINA 198..
7. SMITH, A.M.D., GIESING, J.P. and HESS, J.L., Calculation of Waves and Wave
4.
Resistance for Bodies Moving on or Beneath the Surface of the
Sea, Douglas
Aircraft Company Report 31488A, 1963.
8.
CHEN,
H.H., TORNG,
J.Mo
and
SHIN,
Y.S.,
Formulation,
Method of
Solution
and
Procedures for Hydrodynamic Pressure Project, Technical Report RD-85026, November
1985.
9.
ABRAMOWITZ, M. and STEGUN, A., Handbook of Mathemat.ical Function, Dover
Publications, 1965.