WAVE RADIATION AND DIFFRACTION
BY A FLOATING SLENDER BODY
by
JAMES
HARRY MAYS
B.S.E., Basic Engineering, Princeton University
(1966)
S.M., Division of Engineering and Applied Physics
Harvard University
(1973)
SUBMITTED IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE
DEGREE OF
DOCTOR OF PHILOSOPHY
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
June, 1978
Signature of Author.
Redacted
Signature
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.
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. fo., .
Redacted
Signature
Sup erv
h.
. . . . ......
t
Accepted by.
Thesis Supervisor
Signature Redacted
Chairman, Departmelta-l Commit
.
Certified by .
e on Graduate Students
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Department of Ocean Engineering, May 22, 1978
1978
-2WAVE RADIATION AND DIFFRACTION
BY A FLOATING SLENDER BODY
by
JAMES HARRY MAYS
Submitted to the Department of Ocean Engineering on May 22,
1978, in partial fulfillment of the requirements for the
degree of Doctor of Philosophy.
ABSTRACT
A linearized theory is developed for the oscillation
of a slender body floating on the free surface of an ideal
fluid in the presence of incident oblique plane progressive
The method of matched asymptotic expansions is
waves.
used to derive a unified slender-body velocity potential
valid for all frequencies that accounts for axial interactions at low frequencies (ordinary slender body theory)
and transverse sectional interactions at high frequencies
The source strengths and dipole moments
(strip theory).
of the inner and outer problems are related through an
integral equation the solution of which is used to express
the added mass, wave damping and exciting forces as the
sum of the two-dimensional hydrodynamic forces and an axial
Numerical computations are performed
interaction term.
Results for
for spheroids of several length/beam ratios.
added mass and wave damping in heave and pitch show very
good agreement with "exact" three-dimensional solutions.
Thesis Supervisor:
Title:
J. Nicholas Newman
Professor of Naval Architecture
-3ACKNOWLEDGEMENTS
Support during my graduate study has come primarily
from the Veteran's Administration
National Science Foundation
Traineeship).
and the
(G.I. Bill)
(Energy Related Graduate
Funds from the Office of Naval Research
Fluid Mechanics Program (Contract No. N00014-76-C-0365)
the National Science Foundation (Contract No.
7402576-A02-ENG)
are also gratefully acknowledged.
I am indebted to the criticism, encouragement, and
advice from members of my thesis committee:
Professors
J. H. Milgram, C. C. Mei, and Ronald W. Yeung.
For his
ideas, stimulation, patience, and enthusiasm it is to my
advisor, Professor J. N. Newman, that I direct my most
sincere thanks.
and
-4TABLE OF CONTENTS
Page
ABSTRACT
2
ACKNOWLEDGEMENTS
3
TABLE OF CONTENTS
4
LIST OF TABLES
5
LIST OF FIGURES
6
NOMENCLATURE
7
12
I.
INTRODUCTION
II.
RADIATION
BY
A
SYMMETRIC SLENDER BODY
19
III.
RADIATION
BY
AN ASYMMETRIC SLENDER BODY
69
IV.
DIFFRACTION BY AN ASYMMETRIC SLENDER BODY
85
V.
RESULTS AND CONCLUSIONS
121
148
REFERENCES
APPENDIX A:
The 2D Source Potential S(y,z)
152
APPENDIX B:
Summation of G* for Small
Ikir
156
APPENDIX C:
Behavior of G* at Low Frequency:
K= 0(k) = O(1)
165
Behavior of G* at High Frequency:
K>> Iki
167
Inverse Fourier Transform of the
Low Frequency Matched Wave Source
169
Application of Green's Theorem to
the Homogeneous Solution
173
Other Slender Body Theories
178
APPENDIX D:
APPENDIX E:
APPENDIX F:
APPENDIX G:
-5LIST OF TABLES
Page
I.2D Source Strength, Added Mass and Damping
131
of a Circular Cylinder
LIST OF FIGURES
Page
II-1 Matching Region and Cumulative Error
versus Wavenumber
68
V-1
Source Strength of 8:1 Spheroid at KB/2= .3
135
V-2
Sectional Added Mass of 8:1 Spheroid at
KB/2= .1
136
V-3
Sectional Damping of 8:1 Spheroid at KB/2= .1
137
V-4
Heave Added Mass of 16:1 Spheroid
138
V-5
Heave Damping of 16:1 Spheroid
139
V-6
Heave Added Mass of 8:1 Spheroid
140
V-7
Heave Damping of 8:1 Spheroid
141
V-8
Heave Added Mass of 4:1 Spheroid
142
V-9
Heave Damping of 4:1 Spheroid
143
V-10 Pitch Moment of Added Inertia of 8:1 Spheroid
144
V-ll Pitch Damping of 8:1 Spheroid
145
V-12
Pitch Moment of Added Inertia of 4:1 Spheroid
146
V-13
Pitch Damping of 4:1 Spheroid
147
A-l
Contours of Integration Il,
12 for S(y,z)
152
-6LIST OF FIGURES
(continued)
Page
A-2
Contours of Integration I3' 14 for S(y,z)
154
F-1
Contour of Integration for Application of
Green's Theorem
174
-7NOMENCLATURE
a (x)
-
2D conjugate source strength
a..
-
added mass
13
A2
A.
- 2D source strength
(x)
1
- 2D even wave-free multipole moments
b.
- wave damping
b. (x)
-
B
- beam
B
- body boundary condition of forced motion problem
(Equation 2.8)
BI
- body boundary condition of diffraction problem
(Equation 4.4)
B
(x)
-
2D conjugate dipole moment
2D dipole moment
B2n (x)
- 2D odd wave-free multipole moments
C
-
D(y,z)
- wave dipole =
1
D2n (y,z) -
Dh(y,z)
ey = 1.781...
S(y,z)
2D odd wave-free multipoles
- 2D Helmholtz dipole = 1
(Equation 4.7b)
K
S (Y'z)
D 2n (y,z) - 2D odd wave-free Helmholtz multipoles
h
(Equation 4.13)
E (u), Ei(u) - exponential integrals
- iwb..
f..
-
complex hydrodynamic force =
(Equation 2.56)
F
-
time dependent exciting force = Re{e iWtX
F.
iw
- time dependent hydrodynamic force = Re{eiwt s f
13
1
g
-JaJ
-
gravitational acceleration
(2a..
1J
1J
-8-
(Equation 2.6)
G(x,y,z)
-
H
(u)
- Hankel function of order 0
H
(u)
- Struve function of order 0
J (u)
0
(u)
3D wave-free multipoles
- modified Bessel function of lst kind of order n
- Bessel function of lst kind of order 0
K
- wave number = w 2 /g
K n(u)
- modified Bessel function of 2nd kind of order n
L
-
L
- Laplace equation
length
(L =
1)
L
(u)
-
slender body interaction kernal (Equation 2.38)
L
(u)
-
slender body free surface kernal (Equation 2.38)
L2(u)
K-v
- slender body free surface scattering kernal = L 1( K u)
cL(a(x))
-
(7 (x)) -
interaction operator (Equation 2.39)
scattering interaction operator (Equation 4.18)
OC 7 (x))Ct
7
x)
O7~ 7x)
cos
~logI2
4 7r
-l
+ cosh sics
+ine
1s
j
... (Equation 4.19)
-6(x-u)
(Ix - ui)-
(x
('rri+ log K + y)
+
1K
L0
(x-u)
+
L (x-u)
... (Equation 3.22)
n
-
3D body normal vector
N
-
2D transverse body normal vector
P2n (u)
-
Legendre polynomial of order 2n
-9-
sectional transverse radial coordinate = /Y2+ z2-
r
-
r (x)
- sectional transverse body radius
R
-
R
- radiation boundary condition (Equation 2.4)
R(x),
R
3D radial coordinate = /x2+ y2 +z2-
- reflection coefficients
sgn (u)
- signum function =
S (y, z)
- 2D Green function (wave source)
and 2.12)
S (y, z)
- 2D conjugate wave source (Equation 2.13)
1x>
(Equations 2.10
Z - 2D even wave-free multipoles (Equation 2.11)
SS2nn(y,z)
S
- linearized free surface boundary condition
(Equation 2.4)
Sh(yz)
- 2D Helmholtz wave source (Equations 4.7a, 4.9)
2n
5h
(y,z) - 2D Helmholtz wave-free multipoles
(Equation 4.13)
T(x), T
-
x,yz
- Cartesian coordinates with origin at body center
X.
- exciting force (Equations 4.42c, 4.44)
W..1]
0J
- weight factor (Equation 5.2)
Y0 (u)
- Bessel function of 2nd kind of order 0
transmission coefficients
2n
2n
^2n
^2n..
2n, 2n
a',
,2n ,rn
moments of strip solution of ith mode
-
- moments of generalized radiation potential
p2n
A'2n
rVA 'Ay
'A S A
2n I
~2n%'AA
O
%M 7 = gecr1-4i,--2n I 1 - snlti
aIf eve9='nh1%T
M 1mnt A&"4 .#s
_&
of eV~r
C, 2n ,ag 2n - moments of odd homogeneous solution
-
angle of incidence
-10-
- Euler's constant = log C = .5772...
Y
r
0
r0
-
(Equation 2.28)
-
(Equation 4.16)
- K/IkI
6 (x)
- Dirac delta function ={u
A
-
(1 + /1 - k 2 7K 2 ) /2
A
-
(1 + IsinSI)/2
E
-
slenderness parameter = B/L
C
-z
0
-
sectional transverse angular coordinate
X
-
wavelength
A
- even interaction coefficient =Ot(y()
A7
- ev n scattering interaction coefficient
=
0
+ iyl
= 2h (a7W
y
S
(x)
2n
(x)
V
-
3D unified dipole moment of ith mode
-
3D odd wave-free multipole moments
-
K cos
- complex amplitude of i th mode
p
-
fluid density
a . (x)
-
3D unified source strength of i th mode
a2n (x)
-
3D even wave-free multipole moments
P(x,y,z)
-
spatial velocity potential
9(x,y,z,t)
- time dependent velocity potential
= Re{
(x,y,z)elot} (Equation 2.2)
-11-
$. (x,y,z)
- outer velocity potential (Equation 2.4)
o.(y,z;x)
-
inner velocity potential (Equations 2.8,
2.14, 2.15)
0. (y,z;x,3) - generalized radiation potential
(Equation 4.50)
-
cosh 1 (K/IkI)
X
-
cos
$(n)
-
Psi function
w
-
radian frequency
Q
-
odd interaction coefficient =
07
-
odd scattering interaction coefficient =
*
- Fourier transform (Equation 2.16)
X
(Equation 2.23)
~1.
A
-
(Equation 2.23)
(K/Ikl)
(Equation B.4)
(overbar) complex conjugation
- non-dimensional
(p(x))
(y7 (x))
-12CHAPTER I
INTRODUCTION
The study of the motions of a floating body in the
presence of waves has intrigued scientists for decades.
It has only been in the last thirty years that the mechanics
of such motions have been understood to the extent that
Two-dimensional
reasonable motion predictions are possible.
problems may be solved with relative ease by a variety of
methods.
The extension to three dimensions is much more
difficult requiring considerable time and cost.
It is for
this reason that slender bodies have been almost exclusively
analyzed using a quasi-three-dimensional theory.
The problem at hand concerns a floating rigid body
with one length scale much larger than the others and along
which the body geometry varies gradually.
The body is
regarded as slender and assumptions simplifying the
governing equations can be made to advantage as described
in Chapters II and IV.
We shall only discuss in this
oscillatory body motions based on linear theory.
thesis
The
problem of steady forward motion by itself or in conjunction
with oscillatory motions is
not discussed.
The
application of the theory developed here to ships with
forward speed is outlined in Newman
(1978b).
Zero-speed
slender body motions, however, can be applied specifically
-13-
to the study of stationary ships or elongated offshore
structures.
By assuming "small" amplitude motions, we can
linearize the equations, decouple the excitation from the
response, and thus solve the diffraction and radiation
problems independently.
The separate analyses yield the
complex hydrodynamic forces (in terms of the added mass
or moment
of inertia, and wave damping) and the excitation
forces exerted on the body by the incident waves
respectively.
This information constitutes a transfer
function that determines the body motions as a function of
the "input" ambient waves.
Linearity also implies the
ability to analyze motions in "irregular" seas by spectral
decomposition (Price and Bishop, 1974).
To motivate the analysis to follow, we describe in
primitive form two complementary theories that have
produced much discussion in recent years.
"Strip" theory
is a method originally promoted by Korvin-Kroukovsky
(1955,1960) that assumes the flow around each section of
a slender body is essentially two-dimensional.
One can
intuitively understand why this approach indeed does work
especially at high frequencies of oscillation.
If we
consider a slender body in vertical oscillatory motion
(heave) such that the wave length of oscillation is small,
-14-
say on the order of the beam of the ship, then such waves
created instantaneously along the length will add to
produce in effect a wave system that appears to propagate
at right angles from the axis of the body.
As the wave length of radiation increases with decreasing frequency, the directivity of the radiated waves becomes
less focused (Ogilvie, 1977).
The components of the waves
that propagate along the length of the hull will not
necessarily cancel each other.
Thus, at lower frequencies,
we anticipate interactions along the length of the body.
This phenomenon was the basis for what we shall refer to
as the "ordinary" slender body theory of Ursell
Newman
(1964), and Newman and Tuck (1964).
(1962),
This theory,
described in greater detail in the sequel, presupposes
that the only interactions of any consequence occur along
the axis.
Strip theory, on the other hand, assumes inter-
actions only around transverse sections.
are mutually exclusive.
These two theories
Strip theory is not satisfactory
at low frequency and ordinary slender body theory is
invalid at all but low frequency.
Strip theory has gradually gained acceptance so that
today it provides the rationale for most seakeeping
programs
(e.g., Salveson, Tuck, and Faltinsen, 1970).
Strip theory computations are relatively inexpensive and
-15-
give quite good results over much of the frequency range
of interest.
Numerical three-dimensional solutions are
possible (Chang, 1977) but are too expensive for most
design applications.
Theoreticians were skeptical of strip theory for many
years because of its derivation by intuition rather than
rational mechanics.
Much of the doubt was removed by
the work of Ogilvie and Tuck (1969), who showed its justification at high frequency.
The optimism that accompanied
the early discussion of ordinary slender body theory
(Newman and Tuck, 1964) was shortlived as the theory was
not found to be of much utility in the range of frequencies
relevant for ship motions.
Grim (1960) attempted to synthesize a theory that had
the proper behavior at all frequencies by postulating
the existence of a "longitudinal wave" that would correct
the local strip theory to account for the three
dimensionality of the body.
Maruo (1970, 1978) developed
an "interpolation" theory that partly reconciled strip
theory and ordinary slender body theory.
His results
appear to demonstrate good correlation with experiment but
the theory has theoretical deficiencies as discussed in
Chapter V and Appendix G.
This thesis presents a "unified" slender body theory
uniformly valid for all frequencies.
The unified theory
-16-
is formulated in terms of separate "inner" and "outer"
problems.
In contrast to the strip theory approach
(Ogilvie, 1977), the inner problem has no specified radiation condition.
The method of matched asymptotic
expansions is then used to determine the three-dimensional
source strength of the outer problem in terms of the twodimensional source strength of the inner problem.
Expressions for the added mass and wave damping in sway,
heave, roll, pitch, and yaw (i= 2,3,4,5,6) for bodies of
general cross-sectional shape are derived.
They are shown
to consist of a strip theory term and an interaction term:
f..
=
dx f
(x)
+ 2p
dxA . (
((1.1)
The longitudinal axis is x; f.. is the three-dimensional
1J
2D
force given by the integration of f. (x), the sectional
1J
strip theory force, and the sectional interaction term 2ffp.
A. (x)oa. (x).
J
1
The two-dimensional source strength 27ra. is
multiplied by the integral transform A .(x) operating on the
J
three-dimensional wave source strength a. (x) defined in
J
Chapter II; i.e.,
A. (x) = A(a. (x)).
J
J
(1.2)
The source strength a. is determined by the solution of the
J
-17-
integral equation:
2T
= O. 3
.- (a /-a + 1)A.
7F
j
(1.3)
j
where a., a., and A. are all functions of x.
J
J
I
The oblique wave diffraction problem is solved as well
with expressions for the exciting forces X
dx
e-iKxcos
dxeX.
=
17
2D
(x;S)
1
d eiKxcos
+ 2p
+27rp
fdx
eA
;3A
7x..i(x;
)
X.(3)
given by:
where S is the angle of wave incidence and X
dimensional strip sectional exciting force.
2D
is the two-
As discussed
by Bolton and Ursell (1973), a. is a generalized source
strength that corresponds to the sinuous motion of the body
generating waves that propagate away from the body at an
The coefficient A7 (x; ) is a complicated integral
transform of the scattering source strength a 7 (x;
)
angle S.
discussed in Chapter IV.
The remarkable feature of both the unified radiation
and diffraction solutions is that the hydrodynamic forces
can be given in terms of the two-dimensional added mass and
-18-
damping (or excitation) of a two-dimensional strip solution
that is commonly computed today in seakeeping programs and
a term derived from the solution of an integral equation
involving two-dimensional source strengths.
Chapter V discusses the solution of the integral
equation (1.3) and presents the added mass and damping for
heave and pitch of slender ellipsoids of beam/length ratios
1/4, 1/8, and 1/16.
Comparisons with ordinary slender body
theory, strip theory, and a primitive composite slender
body theory are presented.
The unified theory is found
to be superior to these other methods and to compare quite
favorably with "exact" results obtained independently by
W.D. Kim (1975) and Yeung (Bai and Yeung, 1974) by
numerical procedures that treat the three-dimensional
problem explicitly with no slenderness approximations.
The encouraging results for zero speed suggest optimism
for application of unified slender body theory to ships
with forward speed in a seaway.
-19CHAPTER II
RADIATION OF A SYMMETRIC SLENDER BODY
IN HEAVE AND PITCH
Z
z
-L/2
L/2
y
--
'I
B-d
L
If1
E = B/L
Consider a body symmetric about y= 0 of length L, beam
B, and draft H whose offsets vary smoothly along its length.
The axial direction is taken as x, z is positive upwards with
respect to the free surface (z=0), y is orthogonal to x and z
in the horizontal transverse direction.
Polar coordinates
in the body section are defined by
z = -r cosO
y = r sine
The body section is described by a local radius r 0 = r0 (6;x).
-20-
The fluid is assumed inviscid and incompressible.
The fluid is considered irrotational, thus potential theory
can be used.
The body is allowed to heave and pitch with
small oscillations of radian frequency w on the surface of a
The "deep water" dispersion
fluid of infinite depth.
relation defines the wave number K in terms of the frequency w
and gravitational acceleration g:
2
K =
(2.1)
--
g
A time dependent velocity potential will be defined in terms
of a spatial velocity potential with simple harmonic time
dependence
(Newman, 1977):
O(x,y,z,t)
(2.2)
= Re{f(x,y,z)e it.
The boundary value problem for
P
is
given by the Laplace
equation
V2
0=
x +
y +
z = 0
(L)
throughout the fluid domain; the linearized free surface
boundary condition,
-21-
K 9
-)z
=
0
(S)
on the free surface z=0;
a kinematic body boundary
condition,
Re{fn iot} = velocity of body
(B)
on the body surface; and a radiation condition
stipulating
only outgoing waves far from the body,
lim
0
r'
ir
=
(R)
y2-.
X-2--+
The indicial notation and normalization of Ogilvie
(x,y,z)
where
3 and
=
( 3 1 3 (x,y,z) +
For this particular problem,
5
505 (x,y,z)
,
(1977) will be followed here.
5 are the complex heave amplitude and pitch
angle respectively.
The equations above describing the
complete boundary value problem pertain as well to
93 and
5 with the body boundary condition (B) given specifically
by:
-22-
Re
{_3n3
iwt
it
t
= Re{iwn3
(2.3)
Re
iwt
'05
Re
n
where n
Re{ion 5 eit
5
= nz and n 5 =-xn
w
+ zn ; n
and n
are longitudinal
and vertical components of the unit body normal n.
Define E as a slenderness parameter such that
= B/L = O(H/L).
If
the wetted radius is
ro (e;x) , ro = 0(eL)
.
E
Now assume the body is of unit length L= 1; then r0 = O(E).
The coordinate system is located at the midsection of the body;
pitch motions are defined accordingly.
If the body geometry
is such that there are only negligible or slowly varying
changes in the transverse dimensions along the x axis, then
D/Dx = O(1).
Transverse gradients, however, will be 0(1/s).
We warn that such is not the case in the immediate vicinity
of the ends where longitudinal gradients are significant.
Newman (1964) and Ursell (1962) simplified their slender
body results by assuming various degrees of taper near the
ends.
Methods of dealing with these local singularities
are discussed in Thwaites (1960) and Tuck (1964)
for flows in infinite fluids.
The recent paper by
Ogilvie (1978) discusses end effects for stationary vessels
-23-
as well as for vessels with forward speed on the free
surface.
The boundary value problem for fi(x,y,z) above is not
well posed when the limit E+O is approached.
The slender
body problem described when s is "small" is a singular
perturbation problem necessitating particular solution
techniques.
The method of matched asymptotic expansions has
been widely used in recent years to solve such problems and
will be applied to the problems discussed in this thesis.
The review article by Ogilvie (1977) discusses many of the
applications of this technique to singular perturbation
problems in ship hydrodynamics.
The method of matched asymptotic expansions will allow
us to derive a velocity potential 99 that is uniformly valid
throughout the fluid domain and across the radiation
frequency spectrum.
The "global" potential
0 will be
described by either an inner potential or an outer potential
depending upon the region under examination.
We shall
derive partial solutions for the boundary value problems in
these two separate regions and resolve the indeterminacy in
each solution by matching these solutions in a region common
to both to lowest order.
To motivate the procedure the following physical analog
is offered.
The outer problem is described by an observer
-24-
many body widths from the body.
He sees wave motion
caused by the body heaving and pitching.
From his panoramic
vantage point it appears as if the waves are generated from
a single line.
Because he is so far away the outer field
observer cannot distinguish any of the detail of the body
itself or of the nature of the fluid motion in the vicinity
of the body.
Contrast the
outer
observer with an inner observer
who sees all the detail of the body and the local flow but
because of the apparent great length of the body cannot see
the ends.
In fact, he has no depth perception along the
longitudinal axis as changes there are very slight.
considers his problem as essentially two dimensional.
He
The
technique of matched asymptotic expansions reconciles
mathematically these two disparate and incomplete perceptions
of the problem by a synthesis of both solutions.
The Outer Problem
In the outer region,
all length scales
considered 0(1).
/
--
-0
'g
shall be
-25-
Because of the transverse
symmetry of the body, this
observer will see a symmetrical wave pattern with respect to
the plane y=O.
Define the outer field potential f(x,y,z)
as the solution of the forced-motion boundary value problem:
(L)
xx +
-
x
yy
(2.4)
Z~r
-iKr
lim $~= OC( Kr,'
'
-
= 0
zz
= 0
z -K
(S)
+
r' =5
2
+ y2
-
(R)
We note that we have not specified a body boundary
condition
(B).
There is no unique solution to the problem
To ensure uniqueness we insist that it match
as posed above.
asymptotically with the solution of an inner problem posed
below in a region in which both the inner and outer solutions
are valid and hence overlap.
function G (x,y,z)
The 3D source potential or Green
representing a wave source at x= y= z= 0 that satisfies (L),
(S), and ()
is documented and discussed in Thorne
Wehausen and Laitone
(1960), Ogilvie
(1953),
(1977), and Newman (1978a,b).
satisfies a 3D Laplacian with 6(x)-6(y)-6(z)
To be precise, G
on the right hand side (L') rather than 0 as in
(L):
00
G
(x,y,z)
=
f
0
--A-K*
y
(p/ 2 +y2)ez .
(2.5)
-26-
Although we shall demonstrate that,to leading order,the
heave and pitch forced motion potentials in the outer problem
are described in terms of wave sources, we shall include here
additional contributions offering more hydrodynamical detail
by considering 3D wave-free multipoles
(Havelock, 1955).
These potentials satisfy
(L') and (S) but form non-propagating
waves that decay as O(R
) rather than outgoing progressive waves
with a decay factor O(R-1/2) prescribed by
P2n (-z/R)
2n
2
(-z/R)
2nR2
2n
R
where R = VIE 2 + y 2 + z
K P 2n-
(R).
and P2n (-z/R) are Legendre poly-
nomials.
The outer potentials will be described by a continuous
distribution of these 3D wave sources and wave-free multipoles along the centerline of the body at the free surface:
$i (x,y,z)
=
+
-
fdE[a i(E)G (x-Ey,z)
r2n (C)G2n (x- ,y,z)]
i=3,5
(2.7)
The 3D wave source and multipole strengths are given by
a
(x) and a
2n
(x) respectively.
The Einstein convention of
implicit summation over repeated indices will apply throughout
-27-
The integration is over the length of the body
this thesis.
to+-- as the source and multipole strengths are
from--
necessarily zero outside that interval.
The solution of the
2n
outer problem is thus the determination of a.I (x) and a 1
(x).
The Inner Problem
One could strain the coordinates and define a
perturbation scheme to show formally that the following
Because of the amount of work
problem is properly posed.
that has been done on this problem and because only leading
order behavior in e, the slenderness parameter,
The interested reader
we shall merely state the problem.
(1970) and Ogilvie
is referred to Newman
/ax
+
9.
iyy
(L")
-
(S)
(B)
-
0.
Z
a/Dy,
= O(1),
0
=
izz
S=
1
3D 3
3
D
a/az =
(1977).
O(1/E)
i
i=3,5
= EKO.
3n3
O(C2)
is desired,
-( nn
( n ,n
5
an
3D5
x
,nz) = iwnz
,nz)
z
=
iw(-xnz+ zn ).
z
x
-28-
Now the inner problem to leading order in e can be given
by dropping terms in (L"),
order.
(g), and (B) that are of higher
We will, however, keep both terms of the free
surface boundary condition
(S) in order to preserve the
existence of wave motion in the inner solution regardless of
the magnitude of K.
When K = 0(l)
the inclusion of both
terms is inconsistent with the rest of our slender body
simplification, however the inclusion of higher order
inconsistent terms should in no way affect the accuracy of
our solution to lowest order.
The original contribution
of this thesis is to solve for the global potential
92
over
a wide range of radiation frequencies by keeping both terms
of
(S) and hence preserving wave motion in the inner problem
for all K.
The boundary value problem for the inner potential
is now:
(L")
+
.
1
1
yy
(S)
-
.
= 0
D.
zz
K .=
0
z
(B)
V2D
3
-N
V 2 D 5 -N
iwnZ
=-iwxnz
(2.8)
-29-
where N is the 2D body section unit normal with components
n
y
and n.
z
The inner potential has now become only parametrically
We shall incorporate the sectional x
dependent upon x.
dependence in a "stripwise" fashion by weighting a distribution of two-dimensional solutions according to:
(.
(y,z;x) = A. (x)S(y,z) + a. (x)S(y,z)
+
A2
1
(x)
S 2 n (y,z).
(2.9)
2n
where S(y,z),
S(y,z), and S
(y,z) are the 2D wave source,
conjugate 2D wave source, and 2D wave-free multipoles
respectively.
As in the 3D case, we note that these
singularities are Green functions and as such satisfy a
Laplacian (with a delta function on the right hand side),
the free surface condition (a), and, by proper choice of
source strength A. (x), conjugate source strength a. (x) and
1
1
2n
wave-free multipole source strengths A.
1
o. (y,z;x) can be made to satisfy
(B).
(x),
We shall show that
the inclusion of S, the conjugate wave source, is essential
in the outer 3D solution.
-30-
We recognize the absence of a radiation condition in
the inner problem.
In fact,
in (2.9)
we have already
implied that we shall admit the existence of incoming waves
by allowing contributions from S, the complex conjugate of
the 2D Green
function S.
The "radiation condition" that
ensures solution uniqueness will be
achieved through matching
with the outer solution for which no body boundary condition
was specified.
The two dimensional boundary-value problem can be
solved by numerous methods all involving numerical analysis
to greater or lesser degrees.
Ursell
Pioneering the solution was
(1949) who calculated the radiation potential of a
heaving circular cylinder.
Various refinements through the years have been added
to include finite depth, motions other than heave, and
families of shapes other than circles by conformal mapping.
Integral
equation
techniques
utilizing wave sources of
unknown strength distributed over the wetted surface of the
body
have also been popular methods (Frank, 1967; Garrison,
1969).
Other methods that also solve this two dimensional
problem include the finite element/hybrid element approaches
of Bai
(1976)
and Chen and Mei
(1974) , and the singularity
source distribution technique of Yeung
(Bai and Yeung,
1974).
-31-
The classical two-dimensional solutions of the radiation
problem only admit to outgoing waves by virtue of the usual
radiation condition which was motivated by physical considerations to provide a unique solution.
In the inner
two-dimensional problem we can ignore intuitive physical
reasoning that specified only outgoing waves and instead
allow both incoming and outgoing waves, the relative amounts
of which will be determined through matching with the outer
Uniqueness is assured when matching.
solution.
by Ogilvie and Tuck (1969)
The analysis
"justifying" strip theory on
theoretical grounds imposes a radiation condition on the
inner problem.
That work, however, only claimed to be valid
for high frequency, K = O(l/s).
spectrum, K = 0(1), Ursell
and Tuck
t1964)
At the lower end of the
(1962), Newman (1964), and Newman
essentially approximated away the existence
of waves in the inner solution by imposing a rigid lid
@ = 0 on z=O.
The solutions 5. (y,z;x) of the inner problem will be
expressed in terms of 2D wave sources, 2D conjugate wave
sources, and 2D wave-free multipoles located along the
centerline of the body.
surrounded
by
a
circle
Suppose the 2D body is completely
in
the
y-z
plane.
(1949) has proven that the velocity potential
Ursell
-32-
can be expanded in terms of a wave source,
and
wave-free potentials outside the circumscribing circle.
This representation is an analytic continuation of the
potential that satisfies the boundary-value problem between
this circle and the body.
The outer problem will be described
by 3D wave sources, dipoles, and higher order wave-free
multipoles following Grim (1957,1960) and Ursell
The requisite 2D wave source is
d zcosy .
(2.10)
S(y,z)
=
-J.
(1962).
(Thorne, 1953):
P
p-K
fo
This potential, derived from the logarithmic singularity,
is symmetric about y=0, satisfies
(L") except at r=0, the
free surface boundary condition (S), and contributes only
outgoing waves as
y+I
The symmetric
S
2n
(y,z)
=
-.-
0
(even in y)
cos 2n6
2n
r
K cos (2n-1)0
~2n-l
(2n-1) r
such that r is a 2D radius, r =
with R =
/X2
y2+
2D wave-free multipoles are:
/ry2+ z 2 -,
(2.11)
2.1
not to be confused
2-
Now we can specify the solution of the 2D forced motion
problem resulting in outgoing waves as some combination of
-33-
S(y,z) and S2n(y,z) such that the body boundary condition
is satisfied.
Appendix A proves that S(y,z) can be written as the
sum of a wave term and a local term involving the exponential
integral E 1
S(y,z)
=
,rieK(z-ilyI)
= ffieK
-
Re[eK(z-iyJ)E 1(K(z-ilyi))]
- Re[eKCE(
(2.12)
where C = z+ ilyl
=
z- ily.
For small values of Kr the local term behaves as log Kr+ y
where y is Euler's constant, .577....
Far from the body
this term diminishes as O(1/K).
To describe the potential whose far field expansion
results in incoming waves, the contour of (2.10) is changed
from
-L.
to
-. v-..4 which can also be interpreted as taking
the complex conjugate of (2.12).
This modification is
reasonable as this would be the change if the time dependence
were e-
.
S(y,z)
Thus:
= -7ieKC -
Re[eKE 1 (KC)]
.
(2.13)
-34-
Expressed in terms of undetermined 2D source and
multipole strengths, the complete inner solution takes the
form
(x, y,z)
= A(x)S(y,z)
+ a(x)S(y,z)
+
A 2n
2n y'z)
.
= A(x)rrie
- Re[e KE
cos 2n
r 2n
-
a(x)7rie
(KC)] +
_ K cos
-
[A(x)
.
.(2.14)
+ a(x)]
A 2n(x)
(2n-l)0]
2n-1 r2n-1
(2.15)
where use has been made of the fact that
Re[eK E 1 (KC)] = Re[eKE 1 (K )]
Fourier Transform of the Inner Solution
The matching will be done in Fourier transform space.
Defining the Fourier transform pair by
-3500
dx eikxf(x)
=
f(x)
=
dx e-ikxf*(k)
,
f*(k)
00
-00
rf00
-
(2.16)
we see that the transform of the inner solution is trivial
involving only the local 2D source strengths A, a,
and A
2n
Thus,
* (k,y,z) = frie Kzcos Ky[A*(k)
+
re Kzsin KjyI [A*(k)
-
[A*(k)
+
-
a*(k)]
+ a* (k)]
+ a*(k)]Re[eK E
A 2n k) S2n
(Kg)]
i=3,5
(2.17)
.
For brevity, we shall not include the argument of terms when
their interpretation is obvious; e.g., we shall use S for
S (y, z) .
For now the mode symbol i will be suppressed.
Expansion of the Inner Solution at Low Frequency,
Kr =o(l)
The inner solution in terms of a distribution of 2D wave
sources and 2D wave-free multipoles is valid outside a circle
C of radius rc,
such that rc > r0 (e) where r (0) is the
-36-
sectional vector radius.
If r0 is O(e), then rc = 0(c) as
*(k,y,z)
we fit C just outside the body section.
expanded for small values of Kr.
will be
We seek a region where
the matching can legitimately take place, r= eE,
If K= E~q, then q< p is Kr is to be o(l)
0< p<1.
for the expansion
The inner expansion of the outer solution
of 0*(k,y,z).
will be effected in the same way.
We use the same
transverse variable r in both the inner and outer problems.
Because the wave-free multipoles are 0(r- 2n),
one
might expect a significant contribution from them in the
matching region.
However, when their strengths A
are
estimated by application of the body boundary condition
(B)
we see they are negligible in comparison to the source
strength A:
n
(B)
DN
N
-iw(
Z
~xn
VO-N = 0(l)
since w= /K
(2.18)
0(-1) N = 0(l).
Thus,
0 = O(E), but since the part of 0 that is composed of
multipoles is
A2n O(E-2np),
then by transposing the
last relation above we see that A2n =
(,2pn)
-37-
Applying the same argument to the wave source S, we arrive
at A = O(c).
The expansion of S(y,z) for small Kr is tedious but
straightforward.
Higher order terms than needed for matching
the source strengths of the inner and outer solutions will
be kept for illustration in a later section dealing with
asymmetric bodies and wave dipoles.
=z - iyj
=
e
-re
Expanding terms of (2.12):
z = -r cos e
y= r sin
-ie
2 2
+ K2
+ O(K r )
= 1 + K
E 1 (K?) = -y
-
log KC + K?
-
K
(2.19)
2 + O(K r3
Substituting these small parameter expressions into
(2.17) and ordering 0*(k,y,z)
O(log Kr):
0(l):
in terms of Kr:
A* + a*
ffi(A* -
0(Kr log Kr):
a*)
+ y(A* + a*)
-(A*
+ a*)cos 0
(2.20)
-38-
-
-7r(A*
0(Kr) :
O(K2 r 2log Kr):
a*)cos0
(A* + a*)
-
COS 2e
fi(A* - a*) COS 20 r):2
O2(Kr2 O(
) :
+ a*)((y - 1)cos0 - esine)
(A*
(A* + a*)((i
2 -
y)
cos 20 + 6 sin 20
2
2
Above the dotted line is the information we need to
The wave-free multipoles are not
match the wave sources.
considered in light of the discussion above.
Fourier Transform of the Outer Solution
There are a number of ways to represent the Fourier
transform of the 3D Green
is:
(Newman, 1978b)
z~k2 Tj+
G*(k,y,z)
An integral form
function.
2+ iyy
0 d
=
(2.21)
_ , _2A 42
Ursell
tion;
(1962) developed an infinite series representa-
the following result is the complex conjugate of
Ursell's series because of the different harmonic time
dependence;
coshmX
-(7Ti+ X)cothX
()mI(Iklr)cosme
(Iklr)+ 4
G*(k,y, z) =
(fr -
X)cot
m=l
1
cos mX
-39-
+ 2K
(kIr)+
4
f13I
sinh(mX)cothX
(-)m-
(IkIr)cosve
m=1
sin (mX)cot X
(2.22)
where
X and
X
are defined according to
X = cosh 1(K/Ikl)
for K >
jkj
(2.23)
X = cos~
(K/Ik)
for K <
Jkj.
The Fourier transform of the far field potential is
the transform of the convolution of G (x,y,z) with the
source strength a(x) as given by
(2.7).
By the convolution
theorem:
$*(k,y,z)
=
-
4 7r
*(k)G*(k,y,z).
(2.24)
The three-dimensional multipoles will be of higher order
throughout the outer region including the matching region.
For that reason, we shall neglect them from here on.
In the Fourier transform domain the slender body
assumption of slight longitudinal gradients can be interpreted as equivalent to k = 0(1).
Consider the far field
-40-
potential created by line distribution of wave sources
appropriate for a slender body.
We anticipate that the
magnitude of the Fourier transform of the source strength
a*(k) will be peaked at k=Q and will decay at least as
0(1/kE).
If the body were of a shape characterized by a constant
source strength over the length of the body,
a(x)
=
0
0 0 < x < 1
0, otherwise
then the Fourier transform of the "boxcar" function is the
well known result
a*(k) = .
sin k = 0(1/k).
The side bands of sin k/k are a result of the additional
harmonic content needed to generate the sharp corners of the
boxcar.
The smoother the axial gradients of the body, the more
the Fourier transform of the inner solution behaves like a
delta function as E-0.
Because the Fourier transform of the far field potential
of the body is the product of a*(k) and G*(k,y,z), we can
ignore the behavior in G* that will be suppressed by the
source strength a* (k).
-41-
We assume the existence of a matching region where
r = O(EcP) , O< p< 1.
From above,
and we can sum (2.22)
IkI =O(1) , then IkIr= O(CE)
analytically.
For K > k, we show in
Appendix B that, with no upper bound on the magnitude of K,
(2.22) becomes:
G* (k,y, z)
K>IkI
--
k
2 --'-k2y
2Tricothx[e K co .,--_KAz
e
sin KAIyI]
+ (-coth
X) (log
2
- 2 Xcoth X[e Kzcos2 -k
-
Ikr
22
(/-2-
-
e KAzcos KAy]
2-
26cothX[e Kz sin K- :-kTI y
+ 2cothX
-
+ y)
-
Az
e Kzsin
KAy]
cose(log k2|r + y -
1)
6)
+ 2cothX- Re{eKA (z+iy) E 1 [KA(z+iy)]}
,
+ O(k 2 r 2 )
(2.25)
where
1 + 1/-k2-2
~
2*
To match with the inner source solution (2.20) we want
-42-
to show that in the matching region,
A*S + a*~S = -a
G*
(2.26)
The complexity of G* makes it impossible to algebraically
simplify (2.25) further.
matchings:
We shall perform two separate
one at low frequency Kr= o(l)
frequency K/kl
and one at high
>> 1.
Low Frequency Matching
When
Kr = o (l) , G* can be determined
to
leading order by expansion of either the series representation (2.22) or of Equation (2.25) above as is performed in
Appendix C for K > Iki.
G*(k,y,z)
Kr=o(l)
= -2log
For K < Ik
2
|kCrr
-2y
-
we use
2
(B.9)
giving:
(w+ X) coth X
(-T+ ^)cot
+ 2Kr log
+ 2Kr
2
cos O
(7i+ X)cothX
los - esine - cose (l-y))
(-7F+ X) Cot
2 2
+ O(K r logikir)
X
K 2
IkI.
(2.27)
-43-
For brevity, we shall introduce two new notations,
-(ri+ X)cothX
=
and
f
6 = K/IkI
As in the case of the 2D wave functions expanded in
(2.28)
(2.20),
we shall keep terms of higher order than are needed for
the source potential matching for use in a later section.
We can isolate log Kr terms by addition of log 26 terms in
(2.27).
The following expansion keeps terms through O(K2 r 2
G* (k,y,z) =
+
-2log Kr + 2F
0
2Kr log Kr cos
-
2y + 2log 26
e
+ 2Kr[- 0cos6+ (y-l)cosO- esine- 2log 26cosO]
- K 2r2 log Kr cos 20
+ K2 r 2
cos 20 - cos 20(y- 3/2)
+ Osin 20 + log 26cos 20
+ O(K3r 3 log Kr).
(2.29)
-44-
We now have all the information to match the Fourier
transforms of the inner and outer solutions.
solution is given in expanded form in
solution is -l/41
a*(k)
times
(2.29).
The inner
(2.20); the outer
The purpose of the
match is to relate the 2D source and conjugate source
strengths A(x) and a(x) with the far field 3D source
strength a(x).
We do the matching via Fourier transforms
because of the relative simplicity of the algebraic manipulations that are needed in the transform variable as opposed
to the more cumbersome manipulations with convolution
integrals in x-space.
We have introduced an extra degree of
freedom in the inner solution by allowing for the existence
of "incoming" waves.
Physically one might say that the
combination of incoming and outgoing waves in the inner
solution produce a standing wave and a net outgoing wave,
both of which are necessary to match with the outer solution.
At O(log Kr) and O(1), the match gives us what we need.
$*(k,y,z)
O(log Kr):
0(1)
:
--
D*(k,y,z)
(outer solution)
(inner solution)
a*/27T
A*+ a*
(F0 - y+ log 26)
Fi(A* - a*)
+ y(A*+ a*)
.
.
.(2.30)
-45-
The 2D and 3D source strengths are related by:
-
=
A* + a*
2irf
(2.31)
A* -
a*
A*=
-
(Fo + log 26)
-,y
or
(i
-
F
-
log 26)
(2.32)
a* = -
icy*
(fi + F
+ log 26)
The error of the inner and outer source strength match
22
2 2
at this point is 1 + O(k r log kr, K r log Kr,
2
e /r
2
The
).
last error is attributable to the neglect of the multipoles in
the matching region.
High Frequency Matching
In this case the matching is trivial.
For K/Ik
>> 1,
the inner solution is preserved with the outer solution
giving in the limit a term proportional to the 2D wave source.
The following result is derived in Appendix D:
Okr/K),
G*
K>>kl
=
-2S(y,z)
+
Kr = O
2
{O(l/K2 r 2),
(2.33)
Kr >>
l
-46-
The match then is:
cr*
A*=
(2.34)
a* = 0.
This result is expected as it implies that at high
frequency the far field source strength is proportional to
the 2D source strength.
The analysis of Ogilvie and Tuck
(1969) to provide a rational justification for strip theory
bounded this result by 0(1/Kr).
The above analysis shows,
however, that strip theory may be valid at a much lower
frequency, bounded by an error O(1/K 3r)
source.
in the wave
This improvement is because the exponential integral
term in the source strength (which behaves like 0(1/Kr)) was
preserved identically in both the inner and outer solutions.
High Frequency Inverse Transform
To leading order G* for K>>Ik
is independent of
Iki so
the inverse transform is simply
A
=.5
(2.35)
a = 0
-47-
Low Frequency Inverse Transform
We seek the inverse of (2.31) or alternatively
which,
rewritten with the equivalents of 1'
A* + a*
=r*
A*- a*
=-(7i+
x)coth x
2rL(Tr
-
X) cot 'X
Applying the inverse operator
for
+
log 2
(2.32)
and 6, is:
]
K>
Ik
(2.16) and substituting in
:
CO
A + a
2r
=
fdk
2
e ikx (y*(k)
-CO
00
if
A - a =
-
dk e -ikx
-
-00o
K
i - cosh
+ log 2
K
cos
(K/IkI)]
(K/IkI)1
(2.36)
The details of the inverse transform are recorded in
Appendix E; the results are:
A +A27aa =(x)
-481/2
jaix)
A - a =
log CK
- d- ')L(x1- 1/2
1/2
-
f
K
da(E)Ll(x- ()
(2.37)
-1/2
where log C = y = .577...,
L
(x) and L1 (x)
L
(x)
a' ()
= V/D
a(),
and the kernals
are defined by
= -log(21xI)sgn(x)
(2.38)
L (x)
H
= TT
[H(KIx)
+ Y 0(Klxi)
+ 2iJ
(Klxl)]
is the Struve function of order 0
and J
and Y
are
Bessel functions of the first and second kind (Abramowitz
and Stegun,
1964).
We shall define the integro-differential operator
(with operand a(x))
as:
1/2
t(a
Wx))
= 7T
(-ni - log CK)
+
1
-1/2
K
87r f
d~a (E) L 1(x -
-1/2
Then,
.
1/2
+
d~a' ( )L(x-()
(2.39)
-49-
A + a=
(2.40)
A-
+
a =--
2Tr
-((x)
Tr I
or
A
=
2 r +r
iL t(Y(x)
(2.41)
a =(
Y((x)
Type A and Type a Solutions
Equation (2.9) demonstrates how the inner solution can
be written in terms of incoming and outgoing waves.
refer to this formulation as a "Type A" solution.
We shall
An
equivalent approach, which we shall call "Type a", allows
the inner solution to be specified by a linear combination
of a particular and a homogeneous solution.
The particular
solution is precisely the 2D strip theory potential.
That is,
it satisfies the 2D Laplace equation (L"), the free surface
boundary condition (a), a body boundary condition
a radiation condition of outgoing waves.
of this strip potential
(B), and
The source strength
2D is given by a or,
-50-
2D (x,y,z) = a(x)S(yz)
(2.42)
The homogeneous solution satisfies a homogeneous condition on the body surface such that when it is added to the
particular solution, the appropriate body boundary condition
of the strip solution is still satisfied.
In the case of a
symmetric body in heave (pitch) the homogeneous solution
consists of the bi-diffraction potential resulting from
the scattering of two oppositely directed incident waves.
The potential of the bi-scattering solution will have a
source strength a A.
The amount of homogeneous solution
that is to be added to the particular solution will be
determined by the coefficient A.
Because of the symmetry of this problem, only wave
sources and even wave-free multipoles need be considered.
When we consider the slender body symmetric/asymmetric
diffraction and asymmetric radiation problems, not only will
wave dipoles and odd wave-free multipoles be included, but
an odd homogeneous bi-diffraction problem will also be
considered.
The inner Type a solution of the radiating symmetric
body in heave
=as
(pitch) is
+ a 2n 2n + A( 2 eKzcos Ky+ a S+ a2n
)2n
(2.43)
-51-
= 2D + A
(Type a Solution)
The source strengths of the 2D radiation problem and 2D
even bi-diffraction problem are a and a
respectively.
The
remaining wave-free multipole terms are defined accordingly.
Consistent with the analysis of (2.18), the contribution
of the multipoles will be neglected when the matching with
the outer solution takes place.
However, when discussing
the potential on the body itself as is necessary when
determining the hydrodynamic forces, the multipoles must
be kept.
The Type A solution including multipoles is:
= AS + a
+ A2n 2n
The 2D wave sources S,
part
W
(W
S = W
L
s
(Type A Solution)
S can be decomposed into a "wave"
) and a "local" part
+ L
s
= -Re[eK E 1 (K )]
= z + ilyl
L
S=W
Ws
(L
s
= L ):
+ L
s
K(z-ijy|)
(2.44)
-52-
The even bi-incident wave can be written as
Kz
W
- W
Ws
cos Ky
s
-
2e
With these terms the Type A solution can be rewritten:
A(W
+ L
) + a(W
+ L
) + A
2n 2n
S
(2.45
)
=
The Type a solution is then:
D = a(W
s
+ L )+
s
a2n 2n + A
s
7l
+ a2nS 2n
s + a (W + L)
A
s
s
)
(2.46
Equating the Type A and Type a solutions and isolating
the coefficients of Ws' Ws, L,
A
W;
A=a
+ -- + Aa
W:
w :
S
a =
A
A
Tr
Ls:
A + a = a + AaA
Tri
-
s
and S5,
A
(2.47)
-53-
S2n
A2n
_ 2n + Aa2n
A
which are clearly consistent.
These equations allow us to go back and forth between
the equivalent Type A and Type a representations.
The
matching of the wave sources between the inner and outer
solutions used the Type A model.
An Integral Equation for a(x)
Up to now A has been completely unspecified.
the results of the matching from (2.40);
A + a = a + Aa
and from
Recalling
(2.47):
=
2 Tr
A
(2.48)
A -
a = a + 2
TT).
+ Aa
A
-+
2Tr
7Ti
a x
cy (X))
,
Thus, the homogeneous solution weighting A is equivalent to
A = A(x) = c (a(x)) =
(-7i
-
log CK)
1/2
+ 1 C d~a'( )L
(x- ()
-1/2
1/2
+
dca()L (x- ()
-1/2
(2.49)
-54-
Writing out (2.48) provides an integral equation from which
a(x), the 3D slender body source strength, can be determined
from the source strengths a(x) and aA (x) of the 2D
radiation and bi-scattering
ax)
=
a +
(a(x))a,
problems:
or
=
a + aA
(-Tri - log CK)
+
1
d~a' (E)L (x-
)
1/2
-1/2
1/2
K
8Tr
d
/2
( )L
(2.50)
(x
-
+
Consider an even bi-incident wave eKz+ iKyI upon the
body , producing the scattered wave (R+ T- 1)e Kz-iKlyjI
Kz+iKI y
(R+T-l)eKz-iKIyI
(R+T-l)eKz-iky
The source strength of the even bi-scattered
by matching the outgoing waves
Kz+iKIyI
wave is provided
-55-
(R+T-l) eKz-iK I y
= s
A ieKz-iK y
2.51)
Thus,
R+ T- 1 = lffiaA
Now, rewriting the integral equation
(2.50) in terms of
R and T,
1/2
R+T+1
log CK +
+-
+
f
da
(x-)
' (E) L
-1/2
1/2
+
K
d(()
- L
(x - E)
(2.52)
-1/2
We recall that R, T, a, and a are all functions of x.
kernals L
and L
are defined in
The
(2.38).
The integral equation for the determination of the 3D
source strength from the 2D strip source strength was derived
from the low frequency matching.
The low and high frequency
solutions are consistent with each other as they were deter-
mined from the behavior at different limits of the same
function G*.
A question arises as A
is determined from the Fourier
-56-
inversion of the low frequency matched solution.
Is the
3D interaction specifically predicted by the function A
valid for high wavenumbers as well?
We shall show that
the behavior of A is asymptotically the same as the result
obtained from the separate high frequency analysis.
Reverting back to the Fourier domain, the inner solution
is written as:
4* = A*S + a*S
which, by matching with the outer solution, becomes
[-i+
T*
('ri+ X)cothX -
= -
-
2
log 26]S
[Tri- (Tri+ X)cothX + log 26]S
and
A* = -Tria*
cothX =
-
[Tii- (7ri+ X)cothX + log 26]
ki:
6
V62 _: l
+
+
For K >>
=
262
for K > ki:
-57-
X = cosh
-1 6 % log 26
-
4
and
e zcos Ky
log
a~
Kz
4*-:FT
+ 0()
where we have used the relation S-S = 27ieKZcos Ky.
*
[y
Thus,
+ 0(T)
log 2
The high frequency match produced a solution
G*S
27
If
+
*
k2
log jkr)
g
r = O(c ), K = 0 (C~q), and k = 0(1),
(2.33)
then the inner
solutions derived by asymptotic analysis of the low frequency and
high frequency match are respectively:
-
27r
S + O(a*e2
q
loge
(2.53)
C*S + O(c*6 2q
S
lor)
p
In the inner solution the sectional radius r cannot be
-58-
smaller than the body radius nor larger than the distance to
the matching region, thus 0 < p < 1.
The range of K for the
assumptions made above is K >> 1 implying q > 0.
The respective errors of
(2.33) and (2.53) are
O(a*qE2q
and
If O(p)
q)
0(a*p 2qlogEp)
= O(q), which is not very restrictive at all,
both errors are the same.
Thus, the high frequency behavior
of the inner solution determined from the low frequency
match is asymptotically the same as the inner solution
behavior determined from the high frequency match.
justified in using the interaction function
A
We are
for all K.
The error analysis performed in the last section of this
chapter describes the matching region r that minimizes the
cumulative error.
Added Mass and Damping of the Symmetric Radiation Problem
in Heave and Pitch
The determination of forces and moments on a body
requires a contour integration over the immersed surface of
the body.
To that end we shall use the following form of
the inner solution for heave and pitch (mode index= 3,5):
.
i =
i2D+ A i Geven
homo
i
=
3,5
(2.54)
-59-
From Newman (1977), we define the hydrodynamic force
(moment) as:
F3
Re
Re-e
t
ti 3 33 +
F5
where
3 f5 3
.
.35
(2.55)
5 55-
are body motions, and
1/2
f
+
5f
=-ipW
1/2
dx
dl Ni
i, j = 3,5
D.
.
(2.56)
-1/2 -1/2
The slender body assumption that derivatives with
respect to x are 0(E)
over most of the body allow us to
approximate the 3D normal n by the 2D sectional normal N,
n=
[n
,n y nz
(ynz-zny)
(-xnz+zn )
,
(xn -yn)]
(2.57)
'u [(0, n
, n
, 0,
-xn , xn )]
The complex force tensor f..
1J
=
N
is usually decomposed into
its real and imaginary parts, where a..
1J
is the contribution
to the added mass of the ith mode due to motion in the jth
mode and b.. is similarly defined as the wave
1s
damping which
is associated with the damping of body motions as a result
-60-
of radiated waves, thus,
2
f..=
a.. - iwb...
IJ
iJ
The contour CB of
(2.56) is around the wetted portion
of the two-dimensional section.
of
(2.58)
= 3,5
i,j
Writing out the components
(2.56):
1/2
.
IJ
= -iwp
dl N.
dx
-1/2
or
.
f.
CB
1/2
f3 3
= -iwp
f
dx
J
-1/2
f53 = -iWp
I
(2.59a)
dl N 3
CB
1/2
dx fdl N53
-1/2
CB
1/2
=
iWp
f
dx x
dl N 3
-1/2
3
=
35
CB
1/2
= -iwp
dx
-l/2
dl N 3
(2.59b)
5
CB
The identity between f35 and f53 is a special case of the
-61-
reciprocity relation f .. = f .. which is well known in ship
Ji
1J
1976)
(Newman,
.
hydrodynamics
Continuing the above, we arrive at:
1/2
f 5 5 = -iWp fdxf
-1/2
dl N 5 5
CB
1/2
= -iwp
x2
fx
-1/2
dl N 3 4 5
(2.59c)
.
CB
We can replace the normal N
inside the contour integral
(B).
by iwON by virtue of the body boundary condition
Substituting
(2.43) in for
:
1/2
f.
13
=
-p
fN
-1/2
j
H
)
(2.60)
21/2
fdx f d145D
CB
i
p fdxAj(x)
-1/2
dleve
CB
-
-1/2
i,j = 3,5
j
CB
1/2
= -P
(. + A. (x)
dl
dx
The left hand term of the last expression can be viewed as
the integral along the body length of the sectional 2D
(strip) hydrodynamical force
-62-
2D
2D
2D
= W 2 a. . (x) - i b. . (x)
. (x)
1J
J1
J1J
f.
(2.61)
.
Appendix F derives a theorem allowing us to express the
right hand term in (2.60) with respect to the 2D source
strength a. of the i th mode:
= -2ra. .
dl
(2.62)
CB
The respective components of the hydrodynamic force
tensor consist of the integrated 2D sectional forces and an
interaction term containing 2D source strength a3 and the
interaction coefficient A. (x) which we recall is a compli-
J
cated but straightforward convolution integral of the 3D
source strength along the body axis.
1/2
1/2
f..
+ 27rp
dx f2(x)
dxA (x)a (x)
-1/2
-1/2
1/2
1/2
dx f 2(x)
f33=
+
2i
f5
(2.63a)
dxA3 (x) a 3 (x)
p
(2.63b)
1/2
1/2
=
or,
-1/2
-1/2
f
The force tensor is simply
=
-
dx xf
-1/2
(x)
-
27p
dx xA
-1/2
3
(x)
3
x)
(2.63c)
-631/2
f
f
dx x2f2 (x) + 21TP
-1/2
1/2
dx x 2A 5 (x) a 3 (x)
.
(2.63d)
-1/2
The added mass and damping coefficients according to
this unified slender body theory follow directly from (2.58)
with due care taken to remember that both A.(x) and a 3 (x)
are complex.
The assertion that f 3 5 = f 5 3 made in (2.63c)
will be proved for the general case f.=1 f..
]
J
in Chapter III.
-64-
Matching Regions
Throughout the analysis we have been accumulating
errors of different magnitudes due to various assumptions.
To examine these errors and their importance upon the
justification of the method of matched asymptotic analysis,
we refer to the method taken by Newman (1978b).
The slender body assumptions that allow the use of
the 2D Laplace equation in the near field give
22
of 1+ 0(k r ).
an error
The low frequency matching (2.32) has
22
22
errors 1+ O(k r , K r ).
Neglect of the multipoles
leads to errors 1+ O(E 2/r2 ).
We define the cumulative
error of the low frequency matching to be E 1
22
E 0 = 1 + O(K r
,
2 2
k r
,
2
E /r
2
where,
).
We define a similar quantity for the high frequency
match.
From (2.33) the difference between G* and S is
2
1+ O(k r/K)
for Kr= 0(l)
2 2
and 1+ 0(1/K r
)
for Kr >> 1.
2D multipoles also contribute, giving a cumulative high
frequency error:
1 + O(E 2 /r2 , k2r/K)
hi
1 + 0 (2/r 2, k2 r 2,
Kr = 0(l)
1/K2 r 2)
Kr >> 1
The
-65-
We have solved the matching problem in two frequency
regimes - primarily because of the intractability of the
3D Green function G.
If we can find a region where both
the inner and outer solutions overlap, then the matching
can be assumed valid (Van Dyke, 1975).
We shall determine
the matching requirements for such a region (r= EP)
to
exist for both low and high frequency matching problems
and show that there is a frequency overlap region common
to both the high and low problems.
spatial
Thus, we have a
(r) matching problem and a frequency overlap
problem that have to be satisfied for a uniformly valid
station.
We examine E o first.
22
2
At low frequency
2
O(k r ) and O(2 /r ) errors will dominate.
(K < 0(1)),
If we set
~q , k = 0(l), and choose as the optimum value
r=
, K=
of r
(p) where these two errors are equal, then 2p= 2- 2p
or p = 1/2 and E
= 1 + O(e).
Recall that 6 << r << 1
or 0 < p < 1 for the matching region to really exist.
22
2 2
At K = 1, O(K r ) will dominate O(k r ).
22
2
2
O(K r ) = O(E /r ),
or r = O(e 1 / 2 K-1 /2).
-2q + 2p = 2 -2p,
Setting
then p = 1/2 4 q/2
The error is now 1 + O(K 2 r 2) = 1+ O(K).
As we increase in frequency we shall have to transition
to the high frequency matching.
The terms of Ehi that
predominate for "low" values of K are O( 2/r ) and O(k 2r/K)
-66-
giving r = 0(c2/ 3 K 1 /3).
The transition will be effected
= O(Ks)
equals
or O(k 2/K 3r) = O(- 2K- 6).
Thus,
when the low frequency error E 1
Ehi = O(E 2/r 2)
= O( 2K-2/3 - 4 / 3 )
O(Ks)
Ktransition _
Or
-1/5
Going still higher in frequency we can define another
crossover where the high frequency error contributions
2
22
O(k r ) and O(k r/K)
22
O(k r
)
=
2
O(E /r
2
are equal.
Solving for r from
):
r = 0(El:/2).
This also specifies a new error:
E hi
1 + O(E).
This crossover is determined by setting the previous error
O(E2/3K 2/3)
equal to the new error O(c), giving K = E-1/2
Figure II-1 below delineates the respective frequency
regions, the magnitude of r (the overlap region of minimum
error), and the error incurred.
The actual magnitudes of
the ordinates in the graphs for E and r were determined
-67-
for 6 = 1/8.
Observe that as we sweep over the wavenumber range,
the contributions to the leading order cause the optimum
value of r to vary in different ways.
however, always remains between O(e 1/2
The value of r,
and O(E3/5
this choice of the transition wavenumber (K= Eerror is always bounded by 1 + O(E)
1 + O(C4/5
< Ecumulative
For
)
,
the
I
=
<
1 +0(e)
O(e
K < 1
K
<
2
0(e1/ K-1/
-1/5 < K < c-1/2
1 +0(Ke)
3
1 +0(C2/3K-2/3
1
0(e1/2
E-1/2 < K
)
1
2
)
3 1
0(e2/ K /
.5
Ii. 6
E
)
1
1
-
0 <
Cumulative Error
)
K
.6
1/8
Matching Region
r
11.5
-1
+O(c)
FIGURE II-1
1.4
.4
-1-11-141 -1:44411:1-1 -11111- 1 111
X]
r
i -1 I I I
.3
0 1,
11
,I
I I I I I1 1- -1 1 1-1 1 1 1 1- 1 1 -11 1
1-1
.3
tt j
.2
I
d 1-1-
E-- t [ff
--.00
1 V-
t
....
K
-t
~
1.0
-.
E
.......
14+ -14
-----------T
JiT
-ttt1111111VTTt1TFFflTPFFFFF
2.0
1.02.0
3.0
K.
.
1
Lip
I I
II
4.0
I
t
~~
-TT-
I I I I I I I I
I I I I I
I
1
-69CHAPTER III
RADIATION OF AN ASYMMETRIC
SLENDER BODY
The groundwork has been laid in Chapter II for the
extension of unified slender body radiation theory to bodies
not possessing symmetry about the plane y=O and for forced
motions other than heave and pitch.
The 2D wave dipole
D(y,z) will be defined as
_
D(y,z)
K
-
Thus,
a3 S(y
Dy
D
z)
Re[eKCE
= 'isgn(y)
K
eK(z-ijy|)
(KC)]
the potential for an arbitrary 2D body on the
free surface in the ith mode can be described by wave sources,
dipoles, and even and odd wave-free multipoles
2D
D = a S +
D + a
2n 2n
S2 +
2n 2n
nDn
(Ursell, 1949):
i=
2,...,6
The odd wave-free multipoles are:
D2n
K sin 2N_ 2N sin(2N+ 1)8
D
r2N
r2N+l
The inner solution can be expressed in "A" form as in
(2.14):
-70-
1
=
A.S + B.D + a.S + b.D + A2n 2n
1
1
1
+
B
1
D2 n
(3.1)
As in the earlier analysis for the case of symmetric heaving
and pitching, the multipoles will not contribute at leading
order to the matching.
They, however, will be retained
during the contour integration over the body when hydrodynamic
forces are computed.
The outer solution to leading order can be characterized as the integrated distribution of 3D wave sources
and
dipoles along the length of the body:
1/2
d[r
1
(1
G(x-Eyz)+p
(
!
G(x-E,y,z)].(3.2)
-1/2
The matching is performed with the Fourier transforms
of these solutions.
The inner solution transforms to:
1
=
A*S
+ B*D + a*S + b*D
1
1
1
1
(3.3)
while the convolution theorem provides the transformed outer
solution:
-71-
S
*=
(*G*
47
+ p*
1
a
i K ay
i
G*).
(3.4)
The matching can be performed exactly as in Chapter II
for both high and low frequencies.
Kr= o(l),
The low frequency match,
used the expansions for small Kr of (2.20) and
(2.29) for the inner and outer solutions respectively:
$ I(k,y,z)
(D#(k,y,z)
(outer solution)
(inner solution)
0(1/Kr):
-p*/2r
B# + b#
O(log Kr):
c
A# + a#
1
1
1
/27r
(3.5)
2
O():
3i
O(Kr):
-2
- a
A
(r+log 2
-*
127r
(F
0
+ log
2
T
B* -
1
b*
1
2 2
The source matching error we recall was O(K r
22
log Kr,k r logkr)
Note, however, the matchings determining A* and a* in terms
of a* were done at two adjacent orders, O(log Kr) and 0(1).
To find a corresponding relation between B*, b*, and p*
we have to correlate results at 0(1/Kr) and O(Kr), a difference in orders of magnitude of 6 2 .
Thus, we infer that
interactions at low frequency associated with the dipole
(i.e., non-vanishing b ) are existent at 0(6 2 ) which is
inconsistent with the leading order slender body assumptions.
-72-
We shall, inconsistently, carry the results of both the
source and dipole matchings through further analysis to
demonstrate the higher order coupling that is implied.
The
dipole interaction represented by b. will implicitly be
0(6 2 ).
assumed to be
The inverse transforms present no difficulty having
been performed for the symmetric problem, viz.
1
A.
1
2 T
-
A. + a.
-I a.1 =
(2.40).
2
+
21(a.) l
TT 7
-
(3.6)
vi
B.
S
b.
I
- 2Tr + 7Ti
-
(
.
-
B. + b. =
2r
I
or
A.
=
0i
+1_
2.
T'r r
a. =
(3.7)
b. = I
.
Tni
.
i
.
- +
r
I = 2n
B.
-73-
We now invoke the alternate representation of the
"a"
solution that the inner solution can be represented as the
sum of a particular 2D solution and a weighted homogeneous
solution.
Because of the asymmetry and the inclusion of
dipoles, we need an odd homogeneous solution as well as an
even homogeneous solution to complete the description.
The
mode symbol i will be implied in the following equations.
The symbols A and 0 weight the amount of even and odd
homogeneous solutions to be added.
S=: aS +
2nD2n
D + a 2n 2n +
+ A (2eKzcos Ky + a S +
+
Q(-2ie Kzsin Ky + aS +
+
Q2n 2n +
The A and a solutions
Thus,
AD + oA2n 2n +
A2nD2n
D
(3.8)
Q2nD2n
(3.1 and 3.8)
can be related by
decomposing the wave sources and dipoles into their wave
and local
(L) components:
S = Ws + Ls
D = Wd + Ld
5 = Wd + Ld
s + Ls
(W)
-74-
K(z-ijyj)
W=
Ls = -Re[e KE
(K)]
.Kz.
sin Ky =
= 1DRe[eKCE
L
(3.9)
(KC)]
s
=
2e Kzcos Ky
-2ie
Wd = fsgn y eK(z-ilyl)
wd
W
-
__
d ._d
Thus, we can associate the coefficients of the wave and
local terms as well as multipoles:
A= a
+
s
B=
W d:
+ A('A
A
T
-Tri
S : b
s
-Q
A+a = a
L:dL
B+b
S2N :
A 2N
(3.10)
-7r
=
+
L
1 + 6
a A
a =A
W:
s
d:
A ) + Q(a
A(--+
Tri
+
W
AaA
A
Qa
+ Q
+ A
=a
+
2N
2N
2N
+ AaA
+0
-75-
A
2N
+
2N + A2N +
B2N
D2N
These relations are self-consistent and can be correlated with the matching relations (3.6):
A+ a
a + AuA + Qa
=G=
(3.11)
B+ b =
"r=
+ AA
+ Qa
or in terms of the operator C(2.49),
o
Z
+
=
-
P7
=
+
(x~3
+
(3.12)
The source and dipole strengths aA, a
'
A, and S,
of
the even and odd bi-diffraction solutions can be related
to the reflection and transmission coefficients.
For even bi-diffraction:
eKz -iKy
(R-+T
)e Kz+iKy
(R
+T -e
Kz-iKy
K z+iKy
-76-
The incident wave from y= +oo
+
a reflected wave R e
Kz-
scatters off the body producing
i~yy
+
and a transmitted wave T e
Kz+iKy
The reflected and transmitted waves exist outside the body
in y
0 respectively.
side y= -c
is
The incident wave from the opposite
scattered
oppositely.
Now, if
we represent
the even bi-incident wave by 2e Kzcos Ky, then we have to
subtract off the component eKz iKy on the + side of the body
if waves are to be accounted for properly.
Thus, the waves
resulting from the diffraction of the even bi-incident
wave are, on
(
(R
+
y:
+
-1)e
Kz+iKy
(3.13
(.3
For odd bi-diffraction:
eKz-iKy
(R-
T+)e Kz+iKy
(-R++ T-)e Kz-iKy
_ Kz+iKy
The difference in this case is that the right hand
incident wave has a 180* phase shift from the even problem.
The same phase change occurs in R+ and T+.
Describing the
odd bi-incident wave by -2ie Kzsin Ky, the odd bi-diffracted
-77-
wave is:
-(
(+R-
T
T +
-
l )e
Kz+iKy
(3.14)
Now the source strength and dipole moment of the even
and odd bi-diffracted waves can be given:
aA =
+ 2T -
R+ + R-
2
2Tri
- R~
2T
R
A
(3.15)
- R
R
S=
2'ri
-R + -
R~ + 2T -
2
277
We have made use of the fact that T+ = T
(Newman, 1976).
The coupled integral equations whose solutions are the
3D source strength a and dipole moment p are:
a
2Tr
a +
(a)
(R+ +R
+i2T -
2) +
.d(R- R+
22 Tr)
-27Ti
(3.16)
P _)(R+
2'ir
-
+
- R)
27P
+
(-R+ - R~+ 2T - 2)
27T
-78-
At this point we must recognize that the dipole-dipole
0C(y)
was derived by a higher
order matching as noted earlier.
Inclusion of,(p) or Q
interaction represented by
is inconsistent with the slender body derivation that
ignored terms of 0(62) in posing the inner boundary value
problem.
The consistent equations are then:
-
a + C1 (G)
(R++ R
+ 2T- 2)
(3.17a)
2
2Y
The source
These equations are no longer coupled.
strength a can be solved for explicitly in (3.17a) and the
dipole moment p is given directly by
(3.17b).
(R+/ R~) there is a
Note that with an asymmetric body
source-dipole interaction.
Physically this suggests that an
asymmetric body in pure heave, for example, creates an
asymmetric fluid disturbance that is neither odd nor even
in y and thus needs both sources and dipoles.
The source-
like behavior then induces the interaction that contributes
to the dipole moment p.
For another illustration, consider
an asymmetric body in sway (yaw) or roll.
Except at low fre-
quency, the fluid disturbance will be asymmetrical and thus
described to leading order by both sources and dipoles.
existence
of
the
sources
creates
a
slender
body
The
-79in
interaction
sway
(yaw)
and
roll.
We recall that the variables in these equations are x
dependent as well as mode dependent;
i.e., a
The coefficients R~(x), T~(x), a(x), and
3(x)
(x),
i(x).
are themselves
parametrically x dependent.
Added Mass and Damping of an Asymmetrical Slender Body
The inner solution of the ith
mode radiation problem
is the sum of a particular 2D "strip theory" potential
D2D and an even homogeneous bi-diffraction solution
weighted by the interaction parameter A.
No further
reference will be made to the odd homogeneous solution as
that too was a consequence of higher order matching.
The inner potential is simplified to:
i
=D.2D + A4 even
n
i
ihomo
where
A.
1
[y.(x)
i
and
=even
2 eKZs
Dhomo =2'cos Ky + 0
DA is defined in
(F.2).
i= 2,..6
-80-
The hydrodynamic force tensor is;
1/2
f.
= -p
ij
f
dx
f
dl
DN
.
dx
~-p
-1/2 CB
dl
j
.
1/2
-1/2 CB
ji, j= 2,...,6
= W
2
a.. 13
(2.58)
iwb..
1)
where a. . and b. . are the added mass and damping tensors and
1J
1]
the 3D body normal n is approximated to a 2D body normal N by
[nn ,y,n, (yn Z-zn y) (-xn +zn ),(xny -yn X)]
n=
(3.18)
~
[0,
ny, nz, n , -xnz,
xn
=N
The roll normal n 4 is retained even though it is 0(c)
in
order to discuss the behavior of a slender body in this mode.
Surge, however, is not discussed.
Appendix F derives an important relationship between
the even homogeneous solution and the particular strip
solution:
dl 0even
homo
312D
=
i
-2_a.
(3.19)
aN
CB
Substituting this into f..
J and recalling the relationship
-81-
between the body normal N and the normal derivative of the
(2.18),
potential on the body D(./N,
1/2
1/2
=-iP-lJWPJ
dx
dl N.
'
2D + 27p
(x)
dx a.x
-1/2
-1/2 CB
1/2
1/2
dx f
.x)
+
2 p
dx a. (
A
.
f..
13
2r
2D
-1/2
-1/2
. (3.20)
Now f .
f 2D as can be easily
Ji
J
by invoking Green's
shown
theorem (Newman, 1976), but it remains to be proved that the
and damping coefficients are symmetric.
3D added mass
Examine the second term of (3.20).
The interaction
describes the 3D interaction in f...
coefficient
A is the result of the
It is this term that
operator on a.
0A
;(here operating upon a.)
Recall that
J
is:
a . (x)
(x)
-
+ 81
y= W
4Tr
fJ~
(ffi + log CK)
o(
+ K
d~ca
L~,~ (X
.
)
(a
. (3.21)
-82-
can also be regarded as the integral transas defined by
form with kernal
= fd(t
(lx-
J)
. (
)
(a. (x))
)
The operator d
where
CI)
+
+
('ri+ log CK)
=4~x- Tr
L
(x
(3.22)
KL L (x87rf
1 C-
When written out with the definitions of L
and L
,
(|x-
Ix- C|) becomes
-(x-
j)
4T
0)
(Tri + log CK)
-
1
+
K [H
+
2iJ 0(Klx- CI)]
log 21x -
Jsgn(x-
(Klx- CI)
+ Y
0
)
(Ix-
(KJx- CI)
(3.23)
use of the absolute value sign inside the kernalk
.
which is clearly an even operator about x= C justifying the
-83-
The integral equation that determines a
(3.17a)
and transposed to a. as a
I
can be rewritten in terms of a
function of ai:
-I
x- OI)a
aAfd I(
-
(3.24)
(
)
=
.
a..
Substituting this result into the right hand side of
(3.20)
we get:
2
p fdx a (x)A (x)
fdx{ c. (x)
2lff
= p2dx
2w
Cr )
f)}d-n t( I x --n 1 ) a ( n)
-
- aA (x)fd;(I x
=
2rp fdx
a
(lx
- ml)
x) dfdma i (C) a
= 2 p fdx a (x) A (x).
(T) X ( I x
I I
)
-
a (x) a. (Tn)
-
f
= P dx f d
(3.25)
Thus, the integrated slender body contribution to the
-84-
hydrodynamic forces is mode symmetric, confirming the
symmetry of the unified slender body added mass and damping
tensor.
We note that the strip theory contributions are
naturally sectionally symmetric but that the interaction
is only symmetric when integrated over the length of the
body.
-85CHAPTER IV
OBLIQUE WAVE DIFFRACTION
BY AN ASYMMETRIC SLENDER BODY
With the analysis of the radiation of an asymmetric
slender body, we have the basic tools to formulate the
solution for the general case of the diffraction of an
oblique incident wave by an asymmetric slender body.
The
simpler problems involving symmetric bodies or beam seas
are special cases.
We shall set up separate inner and
outer problems and correlate their solutions by matched
asymptotic expansions such that the two solutions will
collectively describe the potential throughout the entire
fluid region.
The Inner Problem
As discussed in Newman (1970) and more recently by
Troesch (1976) and Ogilvie
(1977), the inner diffraction
problem differs from that of the radiation problem in that
there now exists another length scale in addition to the
wavelength and beam (draft), viz., the projection of
the incident wavelength along the length of the body,
X/cos .
The length of the body has been normalized to 1;
the slenderness parameter c is given as the ratio of beam
The angle of incidence measured
from the axis of the body is
.
(draft) to length (1).
-86-
The incident wave is given as:
= Re{(x,y,z)eiot
= Re
- eKz-iK(cos
= Re
- e(Kz-ivx-i/K1:)
such that v = Kcos ,
k2--2 = Ksin
x+ sinSy) + iWt
y +iot
.
O(x,y,z,t)
One might expect that the inner scattering potential
can be given in the form
iWt
Re D 7 (x,y,z)e-ivx +
where 0 7 (x,y,z) is assumed to vary slowly along the length
of the slender body.
The projected wave component is thus
mathematically separated out of the inner solution;
however, any equations that have derivatives with respect
to x are consequently modified.
Applying the Laplacian
to the spatial potential 4 7 (x,y,z)e-ivx
7-ivx
-
- 2i77-
2
-87-
+
2 +
(_
az2)
(4.2)
7 = 0.
Now, if we want a theory that is valid over a wide range
of wavenumbers
where 0 < q
K = O(C~)
then,
in general, the term v2 7 will be O(6 -2q7) except
The slender body assumption implies
for beam seas.
a/ay,
a/3z = O(1/c)
so that to leading order the governing
equation becomes a Helmholtz equation,
-
V22+
(4.3)
= 0
Newman (19.70) and Ogilvie
(1977) discuss this
derivation at greater length.
The free surface boundary condition (g) is invariant
under this change.
The diffraction body boundary condi-
tion is
(B')
() 7 e
recalling that P
)
a
(4 7 e
has explicit e
)
(4.4)
dependence
(4.1).
As
in the radiation problem, we shall not specify a radiation
-88-
condition in the inner problem but shall pose a general
solution that can be matched with the outer 3D solution.
The Outer Problem
The specifications remain the same as in the radiation
problem.
The velocity potential satisfies the 3D Laplace
equation,
(L), the free surface boundary condition,
(S),
and a radiation condition of outgoing waves at infinity,
(R).
As r-+ 0, the potential must match with the outer
expansion of the inner solution.
The far field potential
of wave sources
7 will be given as a line distribution
The Green functions for the source and
and dipoles.
dipole remain the same as in the radiation problem; the
factor e- iv
is inserted in the convolution integral to
provide the anticipated x-wise variation, thus,
$ 7 (x,y,z) =
-
f
1/2
d [c
+
7
7
K
(-G(x-y,z)e-iVE
-1/2
S.
.
(4.5a)
which has the Fourier transform
$* (k,y, z)
7
=
[a* (k-v)
4.
7+. 7(4
+
(k-v)1
y]G*(k,y,z)
.(4.5b)
-89-
The Inner Solution
As in the asymmetric radiation problem, the potential
will have both source (even with respect to y)
and dipole
(odd) behavior; the exception occurring for head seas
upon a symmetric body.
= 0 is a
The case of head seas
special problem by itself and will not be examined here.
Thus, we can specify that to leading order the potential
will have the form
07 = A 7 (x)Sh(yz)
+ B 7 (x)Dh(yz) + a 7 (x)
h(yz)
+ b 7 (x)Dh(y~z)
(4.6)
where overbar implies complex conjugation and the subscripts
7 and h refer respectively to the amplitude associated
with scattering and the wave singularity that satisfies
the Helmholtz Equation (4.3) in the near field.
From Ursell
Sh(y~z)
=
(1968),
dt cosh t evzcosh tcos(Vly~sinh t)
cosh t - K/v
0
(4.7a)
-90-
Dh(y,z)
h
=1
rdt
J
K
cosh t sinh t evzcosh tsin(vlylsinh t)
cosh t -K/v
0
. (4.7b)
This 2D Helmholtz source is similar mathematically to
the Fourier transform of the 3D wave source potential in
integral form discussed in Chapter II:
G*(k,y,z)
de
=
(2.21)
+
-jI
which becomes, after a change of variable,
= 2
dt cosh t e IkIz cosh tcos(k ly sinh t)
cosh t - K/IkI
0
Thus, Sh(y,z)
= -1/2 G*(v,y,z).
(4.8)
The Fourier transforms of
both the inner and outer solutions contain functions of
the form G*; however, the inner solution appears in terms
of G*(v,y,z) while the outer solution appears in terms of
G* (k,y, z) .
The two forms are
incommensurable
because of
the different arguments and no direct advantage can be
gained by the similarity.
-91-
Rewriting
(2.25), replacing
Iki by IvI = Kicos
and
performing some simplification:
Sh (Y Z) =
si
L sin|
[e Kzcos(KIsin 1y)- ieKzsin(KA |y|)
og Kr co
2
1
2 sinS|
+
sinI cosh~1
(cos
[eKzcos(Kysin|) - e
+
in
Kz
s-I) + y1
I)
KAz cos
(KAy)]
KAzs
sinf1 Krcose[log(Krlcos) + Y-]
2
2|sinS|
1 .
Isin I
where
Re [eKA (z+iy) E
e1
(KA^(Z+iy))]
(4.9)
-92-
A =
+
IsinI.
2
(4.10)
As 1j| approaches ff/2, all the terms are well behaved
except possibly for the term containing cosh~ 1(1/lcosfl).
However, formally taking the limit of this term with a
Taylor series about f/2 reveals that the behavior is
O[(- r/2) 2logjS- f/21]
At
and thus nonsingular at
=
f/2.
= ff/2,
Sh(yz) = 7i(e Kzcos Ky- ie Kzsin Klyl)
-
Re[e K(z+iy)E (K(z+iy))]
(4.11)
= S(y,z).
It should be borne in mind that the derivation of
(2.25) was based on the fact that k= O(l).
However, when
the argument is changed to v= Kcos , we have to be more
careful.
For beam seas, v= 0, so that the wavenumber K
can take on any value.
The problem in this case is in
almost exactly the same form as the case of asymmetric
radiation.
The more oblique the incident wave train
becomes, the more stringent the bounds on K have to be if
Kcos
is to be an 0(1) quantity.
If the second series of
-93-
(2.22) could be summed exactly, then this restriction on
the magnitude of K could be lifted.
As
Sh(y,z) becomes singular as O(1/ ).
+ O,
The
head sea diffraction problem cannot be modelled in a
consistent manner by the approach used here.
The behavior
of the potential downstream of the "bow" is clearly
dependent upon the nature of the upstream diffraction,
thus violating the relative sectional independence of our
Recall that interactions in the near field
inner solution.
were only allowed parametrically through the local 2D
source strengths and dipole moments.
A stronger mathe-
matical statement of the upstream dependence is clearly
in fact, a separate singular perturbation
necessary;
problem is indicated.
Faltinsen
(1971).
discuss this
This problem has been studied by
Recent papers by Ogilvie
point in some detail.
,
At low frequency,or more precisely,
Kcos
Sh(yz)
(1977, 1978)
= log(
+
-
Krj cosI3I
2
) + Y
. cosh -l (
1 - [rii+
si
gcossi
sin6I
Kr cosO log(
2rcs
1s
low values of
-94-
- Kr
s
-
(i+ cosh
IcosI
esinO - cose(l-
y)
22
+ O(K r log r)
(4.12)
Recall the inner solution (4.6) which kept only
leading order terms of the wave potential.
Among the
neglected terms was an infinite series of wave-free multipoles which, following Ursell (1968), satisfy the same
boundary conditions as Sh and Dh except for being
Their inclusion is essential when specifying
wave-free.
the detailed behavior of the potential near the body.
As
one goes farther away from the body into the matching
region where r= O(EY), 0 < p < 1, the same arguments as in
Chapter II can be used to show that the multipole terms
may be neglected during the matching procedure.
The wave-free Helmholtz multipoles are:
S2n = K
h
-
2n-2
(-vr)cos(2n-2)O+ K 2n (vr)cos 2n6
2 K K 2 n-(vr)cos(
2
n-l)
0
and
-95-
D2n
h
K
-
2n-1
K
V
(vr)sin(2n-l)6 + K
2 - Kn(vr)sin 2n
2n+l
(vr)sin (2n+1)0
6
2 nv~i
(4.13)
where K is the wavenumber and K 2n is the modified Bessel
function.
Taking the Fourier transform of the inner solution
(where here we have included the multipoles), we get:
7 = A*S
7 h + B*D
7 h + a*S
7!h + b*D
7h
+
An2n
+
*2n2n
(4
The Outer Solution
The same expression is used in the outer problem as
in the asymmetric radiation problem except that in the
present case, the Fourier inverse has a frequency shift of
-v as illustrated in (4.5) to account for the axial wave
component.
Matching of the Scattering Solution
The inner solution given in "A" form (4.14) is
expanded and ordered below as in the radiation problem.
-96-
Details are omitted, but the results are straight-forwardly
derived from Equation (4.2).
The expansion of the outer
solution is the same as in (3.5).
7 (k - v,y, z)
(outer solution)
$p*
<D* (k,y,z)
(inner solution)
0(1/Kr):
B* + b*
(4.15a)
0 (log Kr):
A* + a*
7
7
(4. 15b)
7
7
2ir2 r0 +log2K/I k I)
0(1) :
7
A7
2
(
o g'cosSI'
+i
a*
2
-(r0+log
2
.
-P*
0 (Kr)
:
2
i,
.
B *7
2
(Fo+log2K/Ikl)
7r 1
. (4.15c)
2
( -+log)cos
0
Io~
b*
7
0 W
2
*r 0 +log coS|
. . . (4.15d)
-97-
where
r0
and F
i+ cosh
(1/1Cos
(4.16)
is defined in (2.28).
The matching error is the same as derived in Section II:
22
22
O(k r log kr, K r log Kr) with an additional error of
22
2
O(K r cos 23 log Kr) which of course is bounded by
the previous error.
In Appendix B we noted that G* could
be derived using two complementary sets of assumptions:
viz, K= O(l), r<<l, or
k/K= O(l), Kr<<l.
The redimen-
sionalization affects the inner scattering solution
according to G*(K cos ,y,z) ++ G*(cos ,Ky,Kz) such that our
expression for Sh (4.12) is valid for cos = O(l), which it
always is, and Kr<< 1.
Thus, we can argue that Sh and
consequently the inner solution is valid so long as Kr<< 1.
The primary restriction on the inner scattering solution
lies in the singular behavior of
near 0 or ff.
In the radiation problem we were able to match
separately at high frequency K/k>>l because of the behavior
of G*(k,y,z) in this range.
The inner scattering solution,
as we have just seen, is valid for Kr<< 1, cos = O(l) or
alternatively, Kcos3= O(l), r<< 1.
The second case clearly
-98-
restricts the frequency to K < 0(1/cosS) while the first
case places greater demands on the matching region,
r <<
0(1/K).
Below, when we associate waves between the "A" and
"a" inner solution representations and the matching
results, we have to be careful not to attempt implicitly
to extend the analysis to regions of K, r, or cosS that
are unjustified according to the present discussion.
Inverse Transform of Oblique Scattering Solution
Again, relying upon earlier analysis
III), the inverse Fourier transform of
(Chapters II and
(4.15) is:
P 7 /2ff
= B 7 + b7
(4.17a)
CY 7/2fr
= A 7 + a7
(4.17b)
7
2+
2
Y7)
7
=
7
=-
DA
+ log
_
2
U
[
2
+ log-coCOW
a7
-i
7)
2iTr
(c
Tni
_7
7
Tr
2
r
+
r
-
l
(4.17c)
2_
Igo-s-
b 72
7Tn1
og
[P
I,(4.17d)
ICOW~+og
-99-
where
(a
=
-
a 7(x)
47)['ii+ logC(K-v)]
1/2
+
1/2
-d~a
(C)L0 (x-C) + (K
dr
-1/2
( )L2(X-0
-1/2
. .(4.18)
.0
and
L 2 (x)
2
=
L
1 (K-v
K x)
We can provide a more compact notation by combining
the terms on the right hand side of
(4.17c) and (4.17d).
Utilizing the results of the two preceding equations
and (4.17b), we
define
a7(x
47
7
a new operator
.
Lhi+
dh
2C(K-v)
log
IcosI
(4.17a)
cosh
+
(1/ 1cos)
sinS
J
1/2
d~ a
-
()(L(x-
) +
K
1/2
- dEa (E)L2
-1/2
(4.19)
*
+
-100-
The inverse transform of the matching relations
becomes:
(4.20a)
P7/2f = B 7 + b
a7 /2ff
27T
+
= A 7 + a7
2
i
(4.20b)
7a) = A
h 7~
7
- a
7
P7 2
*h
2
(P7)
= B7 - b
(4.20c)
(4.20d)
A
= i/2,
Jh = f= , and the matching
When
r
relations are identical to (3.6) found in the asymmetric
radiation problem.
Type A and Type a Solutions of the Diffraction Problem
As in Chapters II and III, the inner solution can be
expressed in two seemingly different but equivalent ways.
Type A solution consists of a certain combination of
Helmholtz sources and dipoles and their conjugates such
that matching to leading order with the outer solution can
occur.
The Type a representation is motivated somewhat
more physically by casting the solution in terms of a
particular strip solution and a homogeneous scattering
The
-101-
solution weighted by an interaction coefficient.
In fact,
we showed in the case of asymmetric radiation that two
homogeneous solutions appeared to need consideration:
an even bi-scattering solution and an odd bi-scattering
solution.
It seems plausible to expect that a similar
construct would be appropriate for this problem with
necessary modifications made to account for the oblique
angle of incidence.
The Type a solution has the following form, where A
7
is the weighting factor for the even homogeneous solution
and Q7 is the weighting factor for the odd homogeneous
solution:
0
7
=
Here, 02D,
2D + A Deven + Q 0
7 homo
7
7 homo
the 2D
(4.21)
.
oblique scattering potential,
is para-
metrically dependent upon x through the coefficients
modifying the Helmholtz singularities.
Written out, this
potential is:
a 72D
h +
7 Dh
+ a2n S2n +
2nD 2n.
(4.22)
The homogeneous potentials also bear close resemblance
to the asymmetric radiation problem except that they too are
-102-
expressed in terms of Helmholtz singularities rather than
the more simple 2D wave sources and dipoles of the
radiation problem.
The even and odd homogeneous solutions
are respectively:
even
= 2ezcos (Ksin~y)+ aA Sh +
homoAh
+ a2n 2n +
A Sh
ADh
Ah
(4.23)
2nD2n
Ah
+
(.3
and
odd
.Z
omo = -2ie Kzsin(K singy)+ a QSh +
homo,
+ a2n 2n +
+ h
0 Dh
(4.24)
.
2nD2n
h
We recall that Sh, Dh'
2n
D2n are
functions of y, z, and 5; the coefficients aA'
A, etc.
are themselves dependent upon x and S.
In light of the above discussion, the Type A solution is:
(D
7
=A S +aS
7h
7h
+ BDh+b
+ B 2nD 2n(4.25)
7 h
7h
7h
+
A2n S2n
7
h
-103-
In the radiation problems we demonstrated a method
for going back and forth between these two conceptualizations by examining the wave-like and local behavior of
the constituent parts of the wave singularities and
bi-incident waves.
This same method may be used in this
case except that the complicated expression (4.9) for the
Helmholtz source makes this decomposition more tedious.
We define a "wave" term,
Wh E
jeKz-iKisinSly,
(4. 26a)
a "standing wave" term,
KA
sin(KAy) +
sinI e
Ws
- [e
-l
.1
___KAz
Kz
cos (Ky lsinS| ) -e
KAz
+ jsin FKesin (Kyjsin|) - eKsin
and a "local" term,
Ls E
sinej
Isine|- 1l [
KrIcosSI
2
1
csin~I
Cosh
Icossi
+ Y
cos (KAy) ]
(KAy)
(4.26b)
-104-
+ y-lfl
Krcose (log Krcos
-
-
si
(4.26c)
1(KA (z+iy)).
yE
| Re[eKAz
Then the Helmholtz source and dipole are,
in terms of these
factors:
S
h
=
h
h +
2|sinr+
s
+ L
(4.27a)
s
and
Wh + Wh +
2i
+
Dh
a
y
s +
y
s
(4. 27b)
If we write out both the a and the A solutions in terms
of these factors, it will be observed that they are
equivalent representations if the following conditions are
met:
7 + A 7a
7 + A7
-
7
= b
+
7 a E=
A + Q72=
A7 + a 7
B7 + b
(4.28a)
(4.28b)
(4.28c)
-105-
sin A7
- i 7 =
(4.28d)
2
+ Q
at2n +
7 A7 An
2n 7= A 27n
(4.28e)
2n + Q
n = B2n
(4.28f)
2n + A
These relations are the diffraction analog to (3.10).
Note the inclusion of sine in the conjugate Helmholtz source
strength term a 7 .
These results can be immediately applied
to the matching conditions (4.20) giving the equations:
7a Q= a7/2c
(4.29a)
7 + A7 A + Q7Q = P7/27
(4.29b)
a7 + A7a
+
+ A 7 2 1 sin L+ .
2 )- +? h7
+
(4.29d)
7
A
7
+
A
7A
) + Q a
Oi h(7
(4.29c)
2
7T
=dh7
S 7=
+
-
lsin T
( i ()(4.29f)
2
2r
riJ
-7
(4.29e)
-106-
Solution of the Scattering Problem
Equations (4.29a) and
(4.29b) represent two coupled
integral equations in a7 and p 7 .
into a more familiar
To put these equations
form we seek to determine Helmholtz
source strengths and dipole moments of the even and odd
bi-scattering solutions in terms of the oblique reflection
and transmission coefficients.
The strategy will be the same as performed in earlier
sections.
We shall examine the fluid motion of the inner
solution as we go far from the body, yet still remain in
the region of validity of the inner solution.
We recall
that if we want the solution to apply over a broad range
of wavenumber K, we will have to make two matchings.
We shall assume that waves incident from y= -o refer
to
3= -ir/2; waves from +oo,
seas 0 <
<
= 7r/2.
In fact,
for non-beam
F corresponds to y> 0 and -r< fB< 0 to y< 0.
The labelling chosen here for the reflection and transmission coefficients refers to the angle of incidence of
the wave causing the disturbances.
Thus, R(
) is the 2D
reflection coefficient corresponding to the wave from
direction
.
Two-dimensional oblique wave diffraction
assumes that the body is infinitely long.
R( ) and T(O)
for the same body vary by both magnitude and phase as
changes.
-107-
The "single-sided" diffraction problem with incident
wave from 6 is:
,-iK
eiKIyI sin3
IyI sin
,-iK y sinj|S
The factor eKz-ivx is implied in the above wave terms.
The even and odd bi-scattering problems are set up
as in Chapter III, except that now we account for the
variable incidence.
Even bi-scattering:
(~N
e iK lyjsinj|
~
E|R~ ( ) + T+ (M I
-iKjyjsinjSj
[R+ ( ) + T_ ( ) ]I
.,1-iKjyjsinjSj
e-iKjyjsin
6
-108-
Odd bi-scattering:
eiKjyjsinSL
[R (S) -T
(r3)]
T(-)]
[-R()+
T+MI-eiKlysin
e -iKjy sinijI
.e-iKy sinj
where the factor eKz-ivx is implied in the above wave terms.
The sign of the argument
in R( ) and T(S) uniquely
defines R and T; e.g., S> 0 will always refer to reflection
in y> 0 due to an incident wave from y> 0.
We shall label
R and T redundantly by superscripts + and - when the value
of 3 is not specified.
replacing
by
jSWsgn
This convention is simpler than
in the argument.
existence of waves at y=
co
Although the
for the 2D oblique diffraction
problem is expected, it is not supported by Equation (4.9)
which is valid strictly only for Kr<< 1.
Asymptotic
analysis of (4.7a) and (4.7b) by Ursell (1968) confirms
the existence of these waves which, of course, are
suggested by the first terms of (4.9).
Far from the body
the Helmholtz wave source and dipole potentials are:
S h UKz-iKjylsin|SI
IsinI e
(4.32a)
-109-
y+->-
Dh % Trsgn(y)eKz-iKIyjsinj
(4.32b)
.
With the above observation we can associate the
scattered waves of the even and odd bi-scattering problems
with their respective Helmholtz source strengths and
dipole moments.
From the even bi-scattering (4.30):
a
T- -1
y +++
7rA
+
(4.33)
y
A
Tr
R- + T + -
+-O
and from the odd bi-scattering (4.31):
-R++ T~+1 =sin
y++o
+
(4.34)
-1T
.
R-T-a
y+--1
Transposing and recognizing that
we find:
A
Ag
sin
2Ri
1
297r
(R++ R~+ 2T - 2)
+R-)
-R)
T+
T-
(Newman,
1976),
-110-
S= IsinSI
2'rri
(-++
(-e +R-)
- R
(-R
=
(4.35)
+ 2T- 2).
Substituting these results into the integral equations
(4.29a) and (4.29b):
7 + jsin | A 7 (R++ R~+ 2T- 2)
7
27ri
7
2= Tr
2,ffi
P7+A
2 -
7
+
7
+ 2
(4.36a)
0 (-R++ R~)
|sin
(R+-R) +7
~7
+
-
+
(-R --R2+ 2T- 2).
(4.36b)
The reflection and transmission coefficients of the
waves scattered by the body due to a single incident wave,
say,
from y= o with incidence
to a 7 and
7:
S= Isin
7
can of course be related
2 Tri
(R + T -1)
(4.37)
R -
7
T + 1
2r
-111-
Equivalently, the integral Equations (4.36a) and
(4.36b) are:
CY7
_
21T
+ T - 1+ A (R+R+2T-2) + 0 (-R
+R)}
7
7
2ri
.
P
2Tr
(4.38a)
.
.
.
.(4.38b)
{R - T+ 1+ A(R+-R~) + 07 (-R+-R+2T-2)}
=fT
.
The coefficients of these equations are of course
dependent upon x and f.
coupled as A
Recall that these equations are
= /h(Y7 (x)) and Q7 =
(7
(x))'
We make the same observations at this point of our
derivation of the integral equations as we did in the
radiation problem.
The matching that yielded the dipole-
dipole interaction term Q
=
(p) is not consistent with
the degree of approximation that, for example, allowed us
to use the Helmholtz equation in the inner problem.
Equations
sini
2R i
r
2=
23
(4.38a) and (4.38b) then properly become:
7
1[R2TfT
[R+ T - 1+ A
+
7
T+ 1+ A
7
(R-
7
(R++ R~+ 2T - 2)]
R-)]
(4.39a)
(4.39b)
-112-
Again we notice how the asymmetry of the body induces
a source-coupling into the dipole term.
For a symmetric
= R~, and:
body R
7
_
2,f
7
2,f
lsinS|
(R+ T - 1) (l+ A )(4.40a)
7
2,ffi
(R-T+1).
1
2,f
The source interaction term A7
(4.40b)
(4.29e) is defined with
a factor 1/jsinS| so that the source strength does not
vanish as
-*Q as it might appear above (4.40a).
Exciting Forces and Moments in Oblique Waves
The procedure for evaluating the exciting forces on a
slender body in oblique waves is similar to the derivation
of the hydrodynamic forces in the radiation problem.
inner solution of the unified scattering potential,
including the axial wave component e~
separated out in
e-ivx
7
_
that had been
(4.2), is:
-ivx [2D + , even(
7homo
7
We recall that 12D and 0homonare each dependent upon B
and x.
The
-113-
(1977) , the exciting
Adopting Newman's conventions
forces and moments are defined:
F
iWAeiwt
J
n2 )
rxn
0
+ 4 e
(4.42a)
VX)ds
7
)
-pRe
M
SB
(4. 42b)
Re{AeiwtX
eA
=
X.
-iwp ,
((D
o
+ +
.
F ex
(4.42c)
S7 )n.ds.
SB
n.i
n.=
I (rxn)
i-3
i= 2,3
i= 4,5,6
We make the slender body assumption that the normal
(nx , n
rxn =
%
, n )
(yn -zn
, (0, n
, n
)
=
-xn z+znx , xn -yn
(N1 , N 2 , N
(4.43)
)
(yn -zn , -xn , xn ) = (N 4 , N 5 , N 6
)
n =
)
derivative on the body can be approximated according to
-114-
where we retain n 4
=
even though it is O(E)
N4
in order to
be able to calculate a roll exciting moment; however, as
in the radiation problem, surge forces will be neglected.
Substituting (4.41) into
(4.42c):
+ 4)
dl(U2D
X. = iWp dx eivx fJ
7
0
J
I
CB
dx e
iP
Vx A7 dl
7)C
CB
-X
(4.44)
even N..
homo
The exciting forces are now expressed by the sum of
two terms, the first of which is the integral along the
length of the body of the oblique 2D sectional exciting
force,
e
7
edl(12D
+ 1)
0
-e
X2D
(4.45)
i
CB
Note that this integral term contains a scattering
due to 0c
force due to 42D
70 and a component
commonly
referred to as the Froude-Krylov exciting force.
The
force X2D has been calculated by different techniques by
Bolton and Ursell (1973), Bai
(1975), Choo (1975), and
-115-
Troesch (1976).
All of these authors properly treat the
Helmholtz equation.
The second term of (4.44) represents the contribution
to the slender body exciting force from the interaction
of adjacent body sections as weighted by A
N.
The normal
can be related to the normal derivative of the potential
(D according to
i= 2,3,4,5,6.
iwN.
=
1
1
(4.46)
satisfies all the conditions of the
The potential $
radiation potential (. (3.17) except that it is governed
by the Helmholtz equation rather than the Laplace equation.
In this sense it is comparable to the potential created by
Bolton and Ursell (1973) in their "generalized heaving
For the case of a circular cylindrical body
problem".
section,
their potential can be identified with $3 which
creates a flexural wave travelling along the length of the
=/2,
03 30
body.
For
f
We examine the integral
fdl[i
CB
where (H =
N
H
H
even as in Appendix F.
(4.47)
By Green's theorem,
-116-
the above equals
ffds[iV2H
-
(4.48)
HV2i].
A
Adding and subtracting -v 2 D
H to this integral, we see
that the same result obtains as when the governing
equation was the Laplace equation:
Jds[.(V2
ffd[Di
_-2
H
H -VH
2
2
8.)]
=
(4.49)
(449
0.
0.
A
Thus,
(4.47) = 0.
As in the radiation problem, assume there are source
and dipole strengths modifying the Helmholtz wave source
and dipole Sh, Dh such that
a.iSh +
(.=
1
(4.50)
iDh + wave-free multipoles.
i
We shall refer to 1.
as the generalized radiation
potential.
The integration of (4.44) proceeds in the manner of
Appendix F.
at z= --
(CO)
The integrals on the free surface (fC) and
vanish as in the radiation case.
Helmholtz multipoles tend to zero at C
and Cr.
The
The wave
-117-
components of Sh, Dh, etc. are described by (4.32).
Equation (4.47), which we have shown to be equal to
zero, now becomes
+
D
S
dl [
(2e
Kz
cos(Kjylsin)+ a Sy+
D
C L+C
(2 eKzcos(Kjysin ) + a Sh +
* (&.S
(tiSh,y
f
dl H
+
+i
A.
-W
ADh)
Dh,y)
)
-
i
where Shiy
h, etc.
(4.51)
CB
The reduction of this integral is exactly the same as
in the radiation case except for the inclusion of a
Isinfl factor on the dipole term.
As in the earlier
problem, the dipole contribution is even in y which when
integrated with respect to the normal a/ay vanishes
leaving us with a result analogous to (F.5),
f
dl
CB
evenA
A
I=
-27rct.
(4.52)
-118-
The final expressions for the exciting forces and
1 1
are:
moments in terms of integrals over the body length (-gg)
,
-iKxcos
2D
X 2 = fdx e - iKxcos X 2
+ 27Tp fdx e
A 7 a2
X=
fdx e
-iKxcos
^
iKxcos
2D
X 3 + 2Tp fdx ei
A 7 a3
M= X4 = fdx e-iKxc
X4D + 21rp fdx e-iKxcos
-iKxcos X
-fdx
M2 = X5
S
=
M3 = X6 =
x e
2
3
A7
4
- iKxcos
D
-2p
dx x e
^
A733
eiKxcos
fdx xe-iKxcos X2D + 2-rrpfdx x
xe
+7"'d
xxeX2
.
.
a
y20
.(4.53)
The slender body exciting forces of a symmetric body
in beam seas
X
1 1 = ff/2 become:
= fdx X2D
-119-
1
X
=
x x2D + 2p
dx A7 a3
0
M
= X
=
fdx X2D
M2 = X
= - dx x X D
M 3 =X6
=
2p
dx x A7 a3
(4.54)
dx x X2D
We observe that a symmetric body in oblique waves
has inner generalized radiation inner potentials consisting
of both Helmholtz sources and dipoles
(plus even and odd
Helmholtz multipoles):
=
Sh +
i Dh + wave-free multipoles
which of course are neither even nor odd in y.
Thus, what
is purely an odd mode in 2D radiation (e.g., sway or roll)
as expressed by dipoles and odd multipoles now acquires
a source-like component due to the asymmetry forced upon
4. by oblique incidence.
The source-like component induces
-120-
slender body interaction through A7 , thus contributing
to the exciting forces and moments.
-121CHAPTER V.
RESULTS AND CONCLUSIONS
It has long been realized that for a symmetric body
the only motions that demonstrate a 3D interaction to
leading order are those that are "source-like".
In
Chapter II we verified that, for a symmetric slender body,
heave and pitch are such motions.
We have excluded surge
from consideration in this study.
For sway and yaw, there
are no such interactions and the strip theory suffices.
The archetypal slender body is the spheroid which in
the folklore of marine hydrodynamics has been called
"God's gift to the Naval Architect".
We shall compute the
added mass and damping in heave and pitch for spheroids
of several length to beam ratios
(c).
Before presenting the results of the unified theory,
we shall describe briefly other slender body approaches
that have been used to predict body response in waves.
Appendix G describes in greater detail these other
methods.
Strip theory assumes that there are no longitudinal
interactions regardless of frequency.
This is the assump-
tion generally used in practice which leads to errors at
low frequency.
The added mass and damping of the 3D body
are given by the integrated distribution of the sectional
-122-
added mass and damping weighted appropriately.
Thus,
.st
st
st
= W 2 a..
f..
- iwb..
1]
dx (W 2a.
=
where a..
Ij
IJ)
and b..
J
(x)
-
(5.1)
iwb. .(x))w.. x)
iJ
iJ
are added mass and damping coefficients
and the weight factor is:
i,j
1
w.
=1
<4
-x
i or
x
2
i or
x
i,j
i,j
j
j
=
5
(5.2)
=6
= 5 or
= 6
Ordinary slender body theory as mentioned in Chapter I
is
The inner
relevant only at quite low frequency.
solution here assumes that the free surface boundary
condition can be approximated by 4 z= 0, essentially
suppressing waves.
heave
We reproduce the expressions for the
(pitch) added mass and damping of a body of revolution
of local beam B(x) derived in Appendix G:
1
P fdx B 2 (x)
-
x
r
2rx
2
3
(log 2B (x)-
1
1
x
Jd
B()
L 0 (X-0
+
=
)
f osb
-123-
+ KB()
(5.3)
L1(x-
The integrand of the first term represents the added
mass
(moment of inertia) of a circular cylinder in the
zero frequency heave (pitch) mode.
The second term is the
slender body correction term while the third term contributes the slender body free surface effect.
We can construct a composite solution to cover the
high and low frequency range by adding the strip potential
Dst and the ordinary slender body potential dosb and
subtracting their common limit.
As derived in Appendix G,
this potential is
@cp
=
+
where a(x)
D st
+
-d~c(E)L
dca()L (x- )
(x-)
-
(log K+ y+ wi)
(5.4)
= -2iwB(x).
Appendix G also gives the hydrodynamic forces fcp
ii
for a body of revolution in terms of the local beam B.
The composite solution then has the correct asymptotic
behavior at high and low frequencies.
At intermediate
-124-
frequencies it has an error no larger than the error of
its constituents (Van Dyke, 1975).
The composite solution
is not unique and the primitive additive solution we
construct is not guaranteed to model the solution well at
intermediate frequencies.
An "interpolation" theory suggested by Maruo
demonstrates good agreement with experiment
Takura, 1978).
(1970)
(Maruo and
Maruo's theory is similar in many respects
to the unified theory; however, it only retains the low
frequency leading behavior of the proper homogeneous solution.
We can compare Maruo's potential:
Maruo
0
st
with the unified potential:
i
st
i
.un
+ A.(2eKzcos
I
Ky+ a S+
A
2n 2).
A
(2.43)
Maruo develops an integral equation for the determination of ac; however, the solution will vary from the unified
source strength because of the difference between
(2.43)
and
The interaction of the hydrodynamic forces given by Maruo,
2ip fdxA.(x)[B(x)+ KS(x)],
(S(x)= sectional area)
.
.
.(5.6)
(5.5).
-125-
can be compared to the unified interaction:
(5.7)
2pu fdxA i(x)a i(x).
For the unified slender body theory, a computer
program has been written to solve the following integral
equation (2.50) for the 3D source strength a
Y. iCx)
2
Yix
iT .-
(x):
x tA()4T (-Ti- y - log K)
1/2
+
1
(x-)
'(C)L
-1/2
1/2
+
K
da
( )L
i=
(X-)
3,5
(2.50)
-1/2
where we recall:
L0 (x)
L
=
(x) =
-log (2 Ix 1) sgn (x)
(Koex)
svr[H + Y
(K
and
+ 2iJ
nxa) (Kbjxea
We shall make several observations about the behavior
-126-
of this equation.
At very low frequency,
K<< 1,
the second
integral vanishes; the logarithmic singularity of
the Y0
Bessel function is dominated by the linear factor K.
The
major contribution will come from the log K term outside
the integrand.
More importantly, though, is the behavior
of the source strength a A of the homogeneous solution
which can be written,
R + T -1
(5.8)
TTl-
as previously noted in (2.51).
From Newman (1976) we also know that there is a
relationship between the scattered waves and the radiated
waves, in this case caused by heave of a symmetric body,
a3 + a 3 (R+ T) = 0.
(5.9)
R+T-l
wi Tr i 33+1) -(a5/5+1)
(5.10) A
because
=
1
-
Then we can say,
r
5= -x at3 .
However, the 2D source strength
at very low frequency is proportional to the local beam
times the heave velocity (Newman and Tuck, 1964) and
-127-
consequently is imaginary.
The ratio a3
3 in this limit
is zero so that aA vanishes at K= 0.
The solution of the integral equation at very low
frequency is:
a.
~ 27a.
(5.11)
i= 3,5.
If we substitute this low frequency source strength
into the inner solution, the result is equivalent to the
ordinary slender body theory described above.
At high frequency, K>> 1, Equation (2.50)
is most
easily examined in the Fourier transform domain.
From
Chapter II,
=- 1
(7i + r 0 + log
2K/Iki)
k 2 -1/2
_l
where
P
=
-(1
-2)
K
(ri+ cosh
K/IkI).
Expanding all terms for k/K<< 1:
*
=
O(a
k
-).
K
(5.12)
-128-
If we assume "reasonable" behavior of a* as was done
3
by Ogilvie and Tuck (1969):
a*
3
,
B = max beam
(5.13)
k
then with these assumptions:
* = 0(1/K5 ).
(5.14)
Thus, at high frequency the 3D source strength again
is given by the 2D source strength:
a.1
=
2fa.1
(5.15)
which, upon substitution into the outer solution, recovers
the zero speed strip theory as examined by Ogilvie and
Tuck (1969).
Even though the 3D source strength a. is given by the
2D source strength 27a. at both high and low frequencies,
the behavior of a. for intermediate frequencies must be
derived from the integral equation (2.50).
-129-
Although there are a number of numerical approaches to
this problem, we have chosen an iterative method of
solution because of our confidence in the zeroth order
approximation, 27a ..
There is no proof that such an
approach converges to a unique solution.
This technique
is sometimes associated with the method of Picard used in
the solution of differential equations
(Ralston, 1965).
The unified source strength a. was approximated by
the method of successive iterations.
The starting
solution was given by the 2D source strength.
With the
source strength and its derivative approximated by linear
piecewise-continuous functions, the logarithmic singularities in L
and summed.
and L
(due to Y ) were integrated analytically
The remaining integrations were performed by
Simpson quadrature.
expansions, Equations
"Economized" Chebychev series
(9.41, 9.42)
of Abramowitz and Stegun
(1964) were used for small arguments of the Bessel
functions J
and Y .
Struve function H .
-o
A Taylor series was used for the
For larger arguments, the formula
(9.43) of Abramowitz and Stegun was used to calculate J 0
and Y .
For argument greater than 5, the Struve function
-130-
was determined by a six term rational approximation of the
asymptotic expansion (Luke, 1969).
The values of the complex non-dimensional source
2D
2D
and damping b3 3 of a circular
strength C, added mass a
cylinder of unit radius were provided by Yeung (personal
communication).
These data (Table
I) were computed by
the hybrid integral equation method (Yeung, 1975) for
finite depth, where the minimum wavelength to depth ratio
was 1.33.
At very low frequency
(KB/2
,
.075) asymptotic
solutions (5.17a, 5.17b) were substituted for Yeung's
results as indicated.
The heave (pitch) source strength in dimensional form
is given by:
2wa. (x) = iwB(x)C(x)
i= 3,5
(5.16)
-x
The added mass (moment of inertia)
is non-
dimensionalized by the mass (second moment of inertia) of the
displaced fluid.
The damping terms are defined similarly
except divided by the radian frequency.
Interpolation and evaluation of x-derivatives were
performed when necessary by use of a cubic spline curve
routine that is well behaved for smoothly varying functions.
TABLE
KB/2
Re C
0.000
0.025
0.050
0.075
0.100
0.150
0.200
0.300
0.400
0.500
0.700
0.900
1.100
1.300
1.500
1.700
1.900
2.100
2.300
0.000
0.100
0.200
0.300
0.086
0.156
0.217
0.270
0.359
0.431
0.541
0.617
0.670
0.724
0.728
0.698
0.645
0.578
0.501
0.419
0.336
0.255
Im C
-1.847
-1.759
-1.684
-1.618
-1.501
-1.396
-1.212
-1.050
-0.904
-0.646
-0.428
-0.244
-0.092
+0.030
+0.123
+0.198
+0.249
+0.280
Im C (1)
-2.000
-1.858
-1.760
-1.679
a3
(0)
3
(1)
a3 3
b 3
-
(o)
00
2.483
1.922
1.615
1.411
1.152
0.988
0.801
0.700
0.642
0.597
0.597
0.615
0.640
0.668
0.694
0.719
0.741
0.762
2.522
1.960
1.632
2.187
1.992
1.841
1.717
1.518
1.360
1.122
0.944
0.806
0.599
0.453
0.347
0.270
0.212
0.169
0.136
0.111
0.091
2.177
1.985
1.836
1.713
1.516
1.358
1.122
0.944
0.806
0.600
0.454
0.348
0.271
0.213
0.170
0.137
0.111
0.091
33(1)
2.547
2.204
1.998
1.852
refers to Yeung's computations,
are asymptotic calculations according to:
C
-)u
2 -2
a33
(log KB/2 (log KB/2 + y)
7i + y + 2log2
3
-~
-
(o)
(1)
Re C(1)
I
r
8~
(5.17a)
(5.17b)
^ indicates damping computed by conservation of energy (Newman, 1976):
b33
3|Ck7
= 2
(5.17c)
H
LA.)
H
-132-
The average relative error is defined as the absolute
value of the average relative error between successive
approximations for a :
Average Relative
Error Between
mth and (m+l)th
iterations
Ia
(xk
i
(m))(m+)
k
a(m)
-
k
(xk
i= 3,5
.
.
.(5.18)
where N is the number of stations along the body length.
Iterations were continued until the Average Relative
Error was less than .001 or the number of iterations
exceeded a preset limit, usually fifteen.
On the IBM 370/168 each iteration took approximately
one second with the body divided into 24 intervals.
At low
frequency convergence was rapid, usually three or four
iterations.
For example, after five iterations at KB/2= .4
the Average Relative Error for a 3 was .002.
No appreciable
advantage was gained by going to more intervals except in
the high frequency range when the oscillatory kernal L
required it
to preserve accuracy.
-133-
For illustration, the heave source strength along the
length of the spheroid e = 1/8 for three different theories
are depicted in Figure V-1 for KB/2 = 0.3.
Figures V-2
and V-3
=
0.1 are plotted in
.
heave added mass and damping at KB/2
The sectional
The efficacy of the unified slender body theory lies
in its ability to compute the integrated quantities of
added mass and damping.
Figures V-4 through V-9 show the
unified added mass and damping for different slenderness
ratios E= 1/6, 1/8, and 1/4.
Other curves represent the
results of ordinary slender body theory, strip theory, a
composite slender body theory, and the numerical results
by W.D. Kim
(1965) and Yeung
(Bai and Yeung,
1974).
Kim
solved the 3D problem in an "exact" numerical fashion by determining the source strength over the surface of the body by
solution of a large set of integral equations.
Yeung's results
by the fundamental source singularity distribution are shown.
We observe the goodness of fit of the unified theory
to Kim's and Yeung's results.
The frequency range over which the
computations were performed can be given in terms of
wavelength X to body length L=l and X to beam as given
below.
The observation that strip theory is valid when
X= O(B)
is clearly validated here.
We note, however, that
ordinary slender body theory is inadequate except for very
low frequencies.
-134Range of
E:
KB/2
X/B
_/L
1/4
0.
-
.8
o
-
.98
c
-
3.93
1/8
0.
-
.7
o
-
.56
o
-
4.49
1/16
0.
-
.5
o
-
.39
c
-
6.28
The primitive composite solution generally fails worse
than either of the two theories that comprise it.
In fact,
it even shows a negative damping at very low frequencies
because of the O(K log K) behavior of the second convolution
integral in (5.4).
The added moment of inertia and damping for pitch are
shown in Figures V-10-13.
Pitch in the strip theory
sense can be regarded as heave motion linearly weighted
along the body length from the midsection.
Although
'
the 2D pitch source strength is given by 27a = -2rxa
5
3
the unified pitch source strength a5 has to be determined
by a separate solution of the integral equation.
quently, a 5 (x) /
Conse-
-xa 3 (x).
We note that where the composite solution gave poor
results for heave,
for pitch.
it gives totally meaningless results
SOURCE STRENGTH
0.5
Real
Source
Strength
-
K8/2 =0.3
0
Strip Theory
--
Unified Slender Body Theory
-0.5
-
4b
ft " " - -
- - - 0-
/
"ft
-
Imaginary
SourceStrength
Ordinary Slender Body Theory
-
K
-1.5
-2
I
0
I
0.2
7
I
I
I
I
0.4
II
0.6
x
FIGURE V-1
I
I
II
0.8
I
I
I
H
L&J
I,
EPS E 1/8
SECTIONAL ADDED MASS
KB/2 = .16
0.009 T
Heave
0.008 t
0.007 t
0.006
2DW
(x)
0.005 t
0.004 t
Re(2'r-A
0.003 t
s
=
7
2
0.002t
A
-0.001 ' -
0
--
0.1
--
0.2
- -
0.3
-
--
0.4
0.5
x
FIGURE V-2
-
a3 3
a
0.8
0.6
0.7
0.9
H
(..J
SECTIONAL DAMPING EPS
0.01
I
S1/8
KB/2 = .1
Heave
0.008
0.006
Tr
2
0.004
b2DW
b33
0.002 1
b 38S
0%
I
-0.002
I
-0+004
4.
Im (2TrA
WA
S043
-0.006
0
0.2
0.1
0.4
0.3
0.6
0.5
x
FIGURE V-3
0.8
0.9
C-.-)
-.1
HEAVE ADDED MASS
e =1/16
4
Unified Slender Body Theory
Strip Theory
Ordinary Slender Body Theory
---
3
033
~\
e-
0 33
Heave Added Mass
pV
0
I
.2
I
I
I
.4
KB/2
.6
-
0
-
Displacement
I
.8
I
FIGURE V-4
11-110"PMW!
OP.
HEAVE DAMPING
E=
4
1/16
--
Unified Slender Body
Theory
Strip Theory
Ordinary Slender Body
Theory
3
b33
2
Heave Damping
I
D33
ispm
pV= Displacemnent
0
I
0
I
.2
I
.4
KB/2
I
.6
I
I
.8
I
FIGURE V-5
1
-I '111 1 1, 1
011 a ON
1
ww.
HEAVE ADDED MASS
E= 1/8
4
Unified Slender Body Theory
Strip Theory
-
3 -r
033
Ordinary Slender Body Theory
--
Composite Slender Body Theory
r K Nmnr ica IS +ti
+- Kim 3D
A
+
Yeung 3D Nume rical Solution
1+
2
+
%^\Heave
Added Mass
\
= Displacement
+pV
44t
0
0
.2
.4
K8/2
FIGURE V-6
.6
.8
H
HEAVE DAMPING
e =1/8
4
Unified
Slender Body
Theory
Strip Theory
Ordinary
---
Slender Body
Theory
Composite
Slender Body
Theory
+ Kim 3D
Numerical
Solution
A Yeung 3D
Numerical
Solution
/ /
3
b3 3
'I
'I
2
'I
/
~
+
I
b33
1
Heave Damping
wp V
p-V= Displacement
I
.5 ...W...Mm ..MM."
0
.2
.4
KB/2
FIGURE V-7
.6
.8
I
HEAVE ADDED MASS
e = 1/4.
4
Unified Slender Body Theory
Strip Theory
Ordinary Slender Body Theory
3 T
+
I
a3
Kim 3D Numerical Solution
3
Heave Added Mass
2
033:
= Displacement
+
0
I
0
I
.2
.4I
1.1
.6
KB/2
FIGURE V-8
.8I
I
I
II
I
H
HEAVE DAMPING
4 -1I/4
Unified Slender Body
Theory
--
--
Strip Theory
--
Ordinary Slender Body
Theory
LA
/+
b3 3
Kim 3D Numerical
Solution
Heave Damping
b 33
WV
pVz Displacement
0
0
I
.2
I
.4
I
.6
I
.8
I
KB/2
FIGURE V-9
I
PITCH MOMENTOFADDED INERTIA
E = 1/8
4
+
3
055
W
Unified Slender Body Theory
Strip Theory
Ordinary Slender Body Theory
Composite Slender Body Theory
Kim 3D Numerical Solution
Yeung 3D Numerical Solution
/Pitch
Moment )f Added Inertia
55
2
I
I =2 nd Moment o f Displaced
+
Fluid
'~
ift
0
I
0
I
.2
. I
I
.4
KB/2
FIGURE V-10
I
I
.6
I
I
.8
I-J
PITCH DAMPING
E 1/8
4
-
--
Ordinary
--
/ /
N
Slender Body
Theory
Composite
Slender Body
Theory
+ Kim 3D
Numerical
Solution
A Yeung 3D
Numerical
Solution
K
--
/
3
2
-
/
'4
4%
4%
4%
'4
41%
0000, IN
.9'
9%
/ %%bj
/
4.
9,
~
+
+
Pitch Damping
9
b555 =
+I0+
0
+
Unified
Slender Body
Theory
Strip Theory
me
I1a2nd Moment of
Displaced Fluid
04-
/
0%
8
I
U,
KB/2
FIGURE V-1l
PITCH MOMENT OF ADDED INERTIA
4
E 1/4
Unified Slender Body
Theory
Strip Theory
4
3
--
Ordinary Slender Body
Theory
I
055
+ Kim 3D Numerical Solution
I
2
I
/
I
a5 5
Pitch Moment of Added Inertia
I
2nd Moment of Displaced
Fluid
1
)
+K
C
.2
.4
.6
KB/2
FIGURE V-12
.8
1.0
I
1.2
H
I
PITCH DAMPING
Unified Slender
--
6= 1/4
4 r-
Body Theory
-
Ordinary
Slender Body
Theory
/
3
Strip Theory
+
/
2
Kim 3D
Numerical
Solution
b5 5 =
Pitch Damping
wI
I
+_.+________--_
0
ft's I
WWWWWW1910
.2
.4
.6
KB/2
.8
I= 2nd Moment of
Displaced Fluid
I
1.0
1.2
FIGURE V-13
H
-148REFERENCES
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(1972). A variational method in potential flows
with a free surface.
Ph.D. Dissertation,
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Bai, K.J.
(1975). Diffraction of oblique waves by an
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Proc. Int.
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Massachusetts Institute of Technology, Cambridge.
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Frank, W.
(1967). Oscillation of cylinders in or below
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Grim, 0.
(1960). A method for a more precise computation
of heaving and pitching motions both in calm
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Havelock, Sir Thomas (1955). Waves due to a floating sphere
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Korvin-Kroukovsky, B.V. (1955).
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(1970). An improvement of the slender body theory
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Maruo, H., and Tokura, J. (1978). Prediction of hydrodynamic forces and moments acting on ships in
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(1977). Numerical methods in water-wave diffraction and radiation. Annu. Rev. Fluid Mech. 10,
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Newman, J.N. (1970). Applications of slender-body theory
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-152APPENDIX A
2D SOURCE POTENTIAL
The 2D source potential given in
S(yz)
(2.10) can be divided
into the sum of two separate integrals over respective
contours I
and 12:
00
S(y,z)
00
deP(Z+iIyI)
= dpeP zcos py =1
=
-K
0
0
lf dpeP(Z-ilyI)
2
(A.1)
'p-K
0
-~*
2
Im y
Im P
for
z < 0
R
Rep
K
K
I2
FIGURE A-l
Re p
-153-
dpepJ(z iy)y1
p-K
l,2
2
r+
'2
0
1.
ds eis(z ijy)
f1
is-K
+00
0
(A.2)
7
ieK(z-iy)
Thus, on the free surface z=0,
0
Tr ie -iK |y
}
+00O
1
I
dse+sI y
,1
1, 2 1 2f
s+iK
00
1
dse-s Iy
s+iK
-1f
1,2
+
2
(A.3)
0
which suggests that far from the body,
1
-Ie -i+ 0y as Iy I-IC
2
(A.4)
-154-
Now looking back at the decomposition in
(A.1), the
integrals can be simplified by a change of variable to the
following where r = z+i y
and T = z-ijyj,
_dyePzcos py _ _1 [eK
p-K
0
dse-s + eKC fdse~]
s
s
Kr
(A.5)
K3
I3
The contours I3 and I4 are sketched in Figure
4
(A-2) below.
These expressions are almost in the form of exponential
integrals except that we have to take special note of the
contour and the branch cut that specifies the exponential
integral function E 1 (z) which,
Stegun
following Abramowitz and
(1964), is:
00
E
(z)
sds
=
(argjzj < fT)6
z
Im S
KC
I3
1R
4
K
Figure A-2
-155-
Going from s = K
to w for I3 in the upper half plane
is straightforward and -A-
is satisfied automatically.
The
integral of 14, however, demands we go above the pole s=0
but this means crossing the branch cut.
The branch of
is associated with log z so the toll for crossing the
E 1 (z)
next Riemann sheet is -2rri.
=E
1
= E 1 (K) -
(K
)
I3
2fi
Hence,
00
S(y,z)
=f_
de
Cos Py
eKCE
(KC)
0
+ eKC [E(K ) -2Tri]
=
rieK(z-ilyl)
-
Re[eKCE1 (KC)]
(A.6)
-156APPENDIX B
SUMMATION OF G* FOR SMALL
Ikir
function is
The Fourier transform G* of the 3D Green
(1962) in the form of an infinite series:
given by Ursell
-2(Tri+ X)COthX
=
G*(k,y,z)
I
(jkjr)
-2(-T+ X)cot Xo
+ 21 (-1)mI (Ikir)cos mecos
Cos mx
M=
X
+ 2K0 (ikjr) + 4 1 (-1)m-1
m=l
sinh mX cothX
-2-
I
( klr)cos
(B.1)
vol
cot X
sin mX
where
{E
1
(K/IkI)
log(6
cos~1 (K/Iki)
-ilog(6
cosh
and
+ /62
1)
(B.2)
+ i/l- 62)
6 E K/IkI.
Taking the derivative with respect to order in the
second series generates two series,
the first of which is
similar to the first series of (B.1).
Application of Graf's
-157-
(Abramowitz and
Addition theorem for Bessel functions
Stegun, 1964)
allows the summation of these series in
Im(Iklr):
=
Kz
e- cos(
2
-
k2
+
-2(i+ x)cothX
G* (k,y, z)
-2(-7r+ ^Xcot ^X
+ 2K
cothX
+
(B.3)
(IkIr)
26
Kz
ez siA
2 -
k2 y
cot X
coth X
oo
ml
sinh mX
I kr)
I
cos mO
(-1)
+ 4
sin mX
cot X m=1
v=m
So far the summation is exact.
In the matching region,
r= O(cP), while the slender body assumption in the Fourier
domain suggests that k= 0(1).
The derivative
with respect to order
of the modified
Bessel function can be described in terms of the following
infinite series (Abramowitz and Stegun, 1964):
(-)m
a I (kr)= (-)m[lok
3v V
2
-
O(m+1)]I
m
(kr)
v=m
+
where
i(m+l)
= -Y
+
1
j=lj
(-l)
(1 m+I (m+2 i) Im+2i (Ikjr)
i=l
i (m+i)
(B.4)
-158-
For small argument, Im (Ikr) can be expanded
Im(Iklr)
=
(B.5)
(Ikr)m/m! + O(Ik m+2rm+2)
which, upon substitution into the series of
klk0 r
cothX
* (-1)iCos me sinh mX
k
( 2
2
sin m^(
o
cot Xt M=l
(B.3), is:
m
+ $ (m+1)).
(-log
.
.
. (B.6)
The terms in X and 'Rcan be expressed as follows:
sinh mX=
(eMX_ e-MX)
=
+ V2---)
1
= I [ (6 +
sin
mX =
[ (6 + i/l- 62)m
-
2 ---i
(6 - i/ -6
We treat first the case K < Iki.
m
-
6+
-
(6 /_
2
-im
Ikir
only
Adding this to
the rest of the series for G* where the modified Bessel
function K 0 (IkIr)
has been expanded,
) -m I
(B.7)
2 )m]
For small
the first term of the series need be taken.
VI62 ----
-159-
G* (ky, z)
= -2 (-ir+ cos
K
K
-1
K<Iki
-
K
2
V
r + y.)
2 (logi
-
Kz
e cos /E-k2T
K2 .- k 2 T
Kz i
+ y-
+ 2Krcos a (log'
But since K< Iki = 0(1),
2
sink-k
e
-
k2
Kr = 0 ( P)
TY
1),
(B.8)
and the above is
further expanded to
K
G*(k,y, z) -- 2
K<IkI
K2 -7k 2 T
2OKy
-
-
+
-2(log2
(1 + Kz) (-Tr + Cos~
) ))
y)
2Kz (log
2
+ y -
+ 0(r 2log r).
For the case K>
(
1)
(B.9)
k I, we have to evaluate the two series
C
-2coth Xlog k 2r
(-1) mcos me
( Ikir)m
m=1
[ (6 + /62-Z-1-
I
-_
( 6 /6-2-Z1-)
m]
B.10
(B.
)
-
of :
-160-
+2cothx
(_l)mcos mOep(m+l)
m=l
m!
[(
6
2_-)m -
+
,ftfry
(/_
m
2
m]
2-4
(B.11)
The first series is summable by inspection with the
value:
-2cothX
log
r/2(5+
r Re e-
-
The second series
2
-- 1)e
IkIr/2(6 -
2)--
(B.ll) is more involved.
1
(B.12)
We use the
relation taken from Hansen (1975):
(-1) mum$p(m+l)
_ e-u (log u -
Ei(u))
+ y
m!
m=l
where
Ei(u)
e dt
=
- uo
and u is real and positive.
We can extend this series to the complex plane by
(B.13)
-161-
analytic continuation of Ei(u).
u real and positive.
argument E 1 (-w)
Ei(u)
is valid only for
The exponential integral of complex
(w=u+iv)
is analytic everywhere except for
Connectivity between Ei(u)
Re(w) > 0, the domain of Ei(u).
and E 1 (-w) along the branch v=O, u> 0 according to
Abramowitz and Stegun (1964) is
E 1 (-u iO)
Thus,
= -Ei(u)
+
iT
(B.12) can be extended to
)
$^(m+1) =-we
(log w + E
(-w) + i'rsgnv) + y
m=l
. (B.14)
allowing us to sum the second series (B.10):
2cothX Re exp[-Ik r-
[log
+ E [kr
2
6 (6+
e -i6
+
e-i
-/2
2f)
2
_)
l)]
+ iTrsgn 0]+ y
-162-
exp[_
[log
+
[-
6 (6 _
ie
r
r
2 r
-i6
_5
2
-
_
2
Combining this s eries with
2xcothX-e-IkIr/2
+ e-IkIr/2
V2_)
2
(5+
(5-C
2
_ 1)
1) ] + inrsgn6]
-
-
(B .15)
y
(B.12) and collecting terms:
-l)cos6
:
-l)cos6 C
cos[
k2r(6+ /6 2
os[ k2r ( -v
=1)
sine]
2
-1) sine]
}
+2 (6 - 7Tsgn6) cothX
e{ek I r/2
(6 + VT2 l)cosG
(6 2---l) COS sin
[sinIk
[ k2
2
Ir (
52
+ 1
/6 2 ii
II
:T) sine]
}
-163-
+2cothX Ree kr2e-
- E [-
2
-i6
+
Now
(6+/32-1)
=
0(6),
2
6 (6
ie
but
that by expanding terms in
(B.16)
- /62
)
Ik r
/-62::r)
/62j)
- IkIr/2 e~ i(6-
-E[-
(6+
(6_ -
2
(6 _
2-1),
T)
= 0(6
) for
K> k so
(B.16) now becomes:
2Xcothxe KAzcos KAy + 2(6- isgn)cothXe KAzsin KAy
+ 2cothXRe eKA (z+iy) E
+ 2cothX
y + logIk
(log k2
-
+ y
-
Ik
1)]
(KA (z+iy))
2 Ir
cosO (6-
/2-l)
(B.17)
where
1 +
~E
k2/K2
2*
Finally, we can combine the results of this summation
-164-
to the remaining parts of G* from (B.3).
The equation
below places no restriction on the magnitude of K;
assume, however, that k = 0(1).
The error given is
attributable to the approximation
(B.5):
=
-27cothX eKzcos (VK2 - k 2Y)
-
ieKAzsin(KAIyI)
+ (1- cothX) (logIkIr
+
G*(k,y,z)
-
2XcothX eKzcos
-
2ecothXe Kzsin/K2 - k 2 y
kr
+ 2cothX
+
22
_ eKAz cos KAy7
2-k2
cos6(log
2cothX Re eKA(z+iy) E
+ O(k r
it does
-
2
eKAz sin KAY]
+ y -
1) (/2--7- 6)
[KA(z+iy)]
)
(B.18)
An alternative approach is to assume that Kr<<l and
IkI/K= 0(l).
Then,
Ikir
=
K
Kr<< 1 so the expressions
derived in this appendix apply equally well to the two sets
of assumptions:
Kr<<l,
K=
(l)
or r<<l,
Iki = O(l).
-165APPENDIX C
BEHAVIOR OF G* AT LOW FREQUENCY
K = O(k) = 0(l)
The terms in
are expanded in Taylor series:
'\, -27TicothX[ (l+ Kz) - i(l+ KAz) (KAIy )I
y) (1-cothX)
-
2XcothX[(l+ Kz)
-
28cothX[(l+ Kz) /
+ 2cothX
-
(f
2--I)
-
(1+ KAz)]
2-- 2
y
(1+ KAz)KAy]
2|Ir cose (logIkjr + y
-
-
2
-
k2(log
-
+
6)
+ 2cothX[-logk2r + X -
Ikir
y +
- cos6 (6+ V2l---')logIk2r
+
|kr
-
(e - TrsgnO)
cose (6 + /-
2
2
---
j)
X
sine (6 +
V
2
----
)
G* (k,y,z)
K= O(k)
(B.18)
1)
-166-
-
=
(1- Y)
krCOS
-2log k2r -
2y -
]
(6 + V-21)
2(Tri+ X)cothX
+ 2Krcos 0 10g32
+ 2Kr [
-
(Tri+ X)cothXcose
cosO (1 - y) ]
-
OsinG
+ O (K2r 2log r)
(C. l)
This last equation can be obtained, of course, by
expanding to leading order the original series representation
of G*
(B.2) as was done by Ursell (1962).
-167APPENDIX D
BEHAVIOR OF G* AT HIGH FREQUENCY K
>
Iki
The various terms of G* given in (B.18) have the
>> 1:
following behavior for K/IkI
sin VI
2
2
-
k 2 y % cos Ky
- k 2 y % sin Ky
cos KAy
b cos
sin KAlyI
Ky +
K sin ky I
2K
k 2 y co
2K
4K
sin Ky
-. sin KjyI
eKGE
-
4K
(KG) (1 -
Ky
,
cos
k 2z
eKz
eKAz
+
1-4K 2
ky
1 +
k/2K
-
A
-
,
cothX =
X
lg 2K
X *109k
k2
4Kz
cos Ky , and
Kr= 0(1)
k)
4K
eKAE 1 (KAC) =
(D.l)
)
(
+
+(K2r2)
where
I
|
Kr> >1l
= r
Substituting these expressions into G*, retaining
terms of O(r)
and less,
-168-
-27rie K(z-iIy I)
k2
(sin
K2
+2
2
1 +
K)
-
2(log
Y)
+
%
G*(k,y,z)
2
k
2K
g FZT
k 2r
4K
-2)
- eKz (sin e sin Ky - cos e cos Ky)
+
2
Kz
k r
e
(sin 0 cos Ky +K=6
k2r
(19
2K
+ 2(1+
k+-r
2
cos 0 sin
Ky)
+y-1
k2)Re eK(z+iy)E
(D.2)
[K(z+iy) I}
TK-2k1
log k2r
2
(1 +
k
2K2
2(1 +
)
-
-27rie K(z-i yI
+
%
0 (k 2r/K)
-2S(y,z)
-
+
IkI = 0(1),
0(1/K r
Kr>>l
(D.3)
/K 2
(1/K r)
Kr>>l
where S (y, z) is the 2D Green function.
region for
Kr=O (1)
2 2
Kr=0(1)
0 (k 2r/K)
-
+
)
eK (z+iy) E 1 [K (z+iy) I
Thus, in the matching
the Fourier transform of the 3D Green
function approaches the 2D source:
G(x,y,z)
K>>kl
~ -2S(y,z).
-169APPENDIX E
THE INVERSE TRANSFORM OF THE LOW FREQUENCY
MATCHED WAVE SOURCE
(2.36) results in:
Rearranging
00
=
8
-
a
dk e
8TrT
(k)
+i
sgn (K-k
si
1-k2-K2-
-00
2K-1
+ [log
Ursell
-ikx*
)
I
log
Tk
- K
(E.l)
-
A
(1962) determined the contribution of the second
term in parentheses to be:
1/2
+
I + y)
d~a ' (E) sgn (x - E) (log 2K x-
7
-I
-
-1T3
1/2
fdc
d
K jx-Ej
' (2 )sgn (x-
fd[H
()
) log 2 1x-(
-
T
)
-
0
-1/2
2 (log K + y)
+
+
d'( )sgn
-1 /2
(x -
(x)
- Y
0
(h)]
}
-1701/2
d~ac(C)[H
-
(Kjx-Cj) -
Y
(Kjx-E|)]
(E.2)
-1/2
cy(0) =a(1) = 0
if
a' (x)
=
and
a (x)
H (r) is the Struve function of order 0 (Abramowitz and
Stegun, 1964).
1
dk e-ikx*(k)
sgn(K-Ik
k2-K
8 -00l
O-
Jd1a(E)jdk
e-ik(x-) 1
fd~~a(C)
-1/2
where
1
k
continuing:
2iK
ds cos Ks(x0
+2K
-d k 2-2
01
/T = +i;
[2wa(x)
sgn(K-Ik
1/2
ds Cos Ks (x-)
f 1/2 1-s 2
)
1/2
)
1
)
The first term in parentheses of (E.1) can be analyzed:
-171-
S
27r
Adding
(E.2)
x) + K'Ii[Y (KIx-EI) + iJ
(KIx- 1)]
}
(E.3)
and (E.3),
A.
{-i2xra (x)
,-22c
8i
a
2a(x)log CK
1/2
fda'()log
2jx-Cjsgn(x-C)
-1/2
+
f
dca(E) [H
(KIx-El )
+ Y0 (K IX-C
)
1/2
-1/2
+ 2iJ
(Kx-CI)]
}
where log C = y
(E.4)
Define
L
(x) = -log(21x Ign(x)
(E.5)
L 1 (x)
- .[H (KIxI) + Y
Then in more compact form,
(KIxI) + 2iJ
(Kixi)].
-172-
a =
(x)
a
Ta(x)log
1T 4f 2 f dcy (~Lo(X-C)
-
CK
d
)
1/2
y
A
-1/2
1/2
iKd~a (E) L
(x-0)
(E. 6)
-1/2
or expressed as the sum and difference of the above:
A + a
=
W
27r
(E.7)
1/2
-
a
ia(x)log
27r 2 CK
-
4'rr 2 f da'(c
-1/2
1/2
iK
d c (C) L
(X-)
.
A
-1/2
) L 0 (x-)
-173APPENDIX F
APPLICATION OF GREEN'S THEOREM
TO THE HOMOGENEOUS SOLUTION
Green's Theorem is widely used in ship hydrodynamics
to prove many useful relations.
Newman
(1976) catalogs
most of the presently known applications.
A common
form of Green's Theorem applied here to 0 1 and 0H is
([c.V2,
JJ
A
i
H
_
V
H
2
[
5.]dS =
i
.- @
H
i
iDN
-
-5
H3N
G.i]dt= Q
C
(F.1)
where (D. is a radiation potential
1
(i= 2,3,4)
and
H
H
0
oeven
homo
is an even homogeneous potential consisting of waves
incident from y=
<o
waves 0
The zero equality of
of V 2 ,iH
and 0~
and their corresponding
= 0 in the fluid.
(F.l) is a consequence
The discussion in this section
is concerned exclusively with the 2D case.
the bi-scattering
A A 7+
scattered
We shall define
solution
7-
aAS
A +
+A AD +
a2n 2n + a 2 nD 2 n
AD
(F.2)
where S and D are the 2D wave source and dipole, S2n and
D2n are even and odd wave-free multipoles.
These singular-
ities and their coefficients are discussed in Chapter II.
-174-
The incident waves that contribute to
S+-
even
H sum to
Kz
cosKy.
0=2e
Figure F-1 depicts A and C of
(F.1).
includes the body (Cb) , the free surface
right boundaries
(C1 , Cr)
The contour C
(Cf),
left and
far from the body, and a bottom
boundary (C o) at z=-w.
C
C
f
f
CB
A
kC
r
C 11
C =C
1
+ C
r
+ C
f
+ C
B
+ C
0
Coo
Figure F-1
The normal N is taken with respect to the contour of integration, not the fluid.
We assume a counter-clockwise
contour with the normal directed at right angles out of
the
fluid
region
A.
Thus,
/DN
at
C
and
Cr
-175-
are both +3/Dy.
C
is over the free surface where the linearized free
surface boundary condition (S) holds, hence fC
Because of the eKz dependence in all potentials,
cally zero.
fC
is identi-
is also zero.
Integration over C , Cr, and CB will
give us the results we seek. Writing out (F.1) in detail,
fdl
(a S +
D + a2n 2n +
2nD2n
C +C r+CB
a 2e
aN
+
2n2n 2n 2n
cos Ky + a S +
A N
-
(2eKzcos Ky + aS +
-
(ctS
i N + .D
i N + a2n
1
N2n +
A
~A
D +
A N
a
A S
2nD2n
N)
A
S2n
2n +
N
A
D2nD2n
N
+ ~AD
= 0.
(F.3)
Summation is implied over the multipoles; SN,DN imply normal
derivatives on S and D, etc.
On C
and Cr only the wave-
like components of S and D will play a role, the multipoles
can also be neglected.
S
Similarly:
% K'sgn y eK ziKyl
Dy . -iKre
iKjy
Transposing the integral over the body and simplifying the
-176-
integrand of the remainder:
dl(-2ffKa. sgn y e 2KZ + 2iiK.e2Kz
C 1 +C r
+ 45
[D
=
+ (AD
N
(F.4)
dk.
CB
The integral over the body has been simplified because of
+
the diffraction body boundary condition 3/N(O ++
A)
=
Recalling the convention of the normal in integrating the
left hand side:
0
-00
2'rrK(ax
i i)fdz(+)e2Kz + 2TrK(-a
+
+ i) fdz(+)e 2Kz
-00
0
22 (a+ is ) + 2(TK + i
)
= 27K-
i
i
2
12W
=
.
Thus,
(D
CB
+ 4~
+
A)
=
Id kH
B
N
= -2ra
. (F.5)
-177-
= odd , the homogeneous diffraction
homo
H
potential corresponding to incident waves of opposite phase
We now let o
converging on the body such that
odd
0]
-
S+
[0 +
=
2 ie Kz sin
Ky
The analysis is the same as for the odd
wave except that t he result is -2Tr
The results m ay be summarized:
dl 0even
f homo
i
3N
= -21Tc.
CB
dl0
=
homo
CB
aN
-2OD2.
bi-incident
i instead of -2Wa.
-178APPENDIX G
OTHER SLENDER BODY THEORIES
Strip Theory
Strip theory is the name given to the method of
representing the hydrodynamic forces on the body strictly by superposition of sectional 2D solutions.
Thus, the added mass
of the sectional 2D added mass and damping coefficients
-
and damping of the 3D body is the integrated distribution
normalized in a consistent fashion.
The hydrodynamic force tensor is defined in
Equation (2.56) and for strip theory is:
st
2
st
.
st
I
1J
LJ
=
= x(22D
(2a. .
fdx
-
.2D
iwb..)w...
(G.1)
where w.. is a weighting function,
1J
F
-x
x
ij
<4
i or j = 5
i or j = 6
i,j = 5 or_
i,j
= 6
Assume we are given the following 2D added mass and
damping coefficients non-dimensionalized
(-)
as indicated:
-1792D
13
b
-D
a. .
a.. = a2b
-2D
pS
1J
2D
b.
(G.2)
pSc
such that p is the fluid density, and S= S(x)
sectional area of the immersed body.
is the local
The quantities a..
1J
and b. . then refer to the added mass and damping of a body
JJ
of unit radius at a given frequency.
We shall normalize
with fdxS(x)w.
.
(x)
the 3D terms similarly except
replacing S in the denominator.
Thus, the 3D strip added mass and damping in terms of
~2D
f iare:
~s
f.
3
2D
dx S (x) f - (x) w. -(x)
(x)
fdx S(x)w. . (x)
(G.3)
Ordinary Slender Body Theory
"Ordinary" slender body theory
(OSB) refers to the
approach taken by Ursell (1962), Newman
(1964), and Newman
and Tuck (1964) briefly alluded to in Chapter V.
We
rewrite the equation for the inner OSB potential:
1/2
osb =0
+
fda
1/2
(E) Lo (x-E) +
-1/2
di
(E) L
(x-)
-1/2
i=3,5
i.
. .(G.4)
-180-
where () is the potential of the 2D infinite-fluid doublebody heave
(pitch) problem at zero frequency.
This
potential satisfies a Neumann boundary condition
=
.
0
z
on the free surface and is most easily interpreted
as a vertical pulsing as shown below.
z= 0
As this potential is strictly only valid at K= 0, to
leading order D. may be represented by a. (x)/2Tr log r.
Using a conservation of flux argument, it can be shown
that for small wavenumber the heave
(pitch) source strength
can be represented by
a
(x)
=
-2iwB(x)
1
(G.5)
where B(x) is the local beam (width) of the body at the
free surface.
(G.4) yields:
Substitution of this source strength into
-181-
Iosb = -iwB(x)
o
r
7rg
-
dE
BdB(E)
L
dCB (E)
i
L
(
-
(x- )
(G.6)
All integrals are over the body length from -1/2 to 1/2.
This potential describes the leading order behavior of the
fluid motion.
It does not, however, provide enough
information for the evaluation of the hydrodynamic force
on the surface of the body where additional detail in the
form of the neglected multipoles is needed.
Thus, we need
a more complete description of 1)m to be able to determine
the added mass and damping.
For a body of revolution of local beam B(x), Newman
(1964) gives the complete ordinary slender body potential:
(-l)n
n=1 n(4n -1)
-
-x
iwB(x)
Jl~-x
Bx)/2_
[ B(x)2
r
dC[B(C)
-
1
cos 2n0
] L (x-)
-
=x)~
log r
1
iB (x)
osb
1
J
-182-
fd B(
L
)
- r
The hydrodynamic heave
(x-).
i= 3,5
(G.7)
(pitch) force for a body of
revolution with fore and aft symmetry is given by
STr/2
S
1
0
osb
dO8 1sbcose.
dx B (x)
-
f..
(G.8)
Tr/2
i= 3,5
Substituting (G.7) into this equation and performing
the reduction, we arrive at:
11
-
L
TT
f1
-
P
f dx
dx B (x)
2
2 (3
(Ltog 2B (x)-)
(
f
B(x)
L(X-
-x
+ KB()
L
(x-K)).
(G. 9)
-183-
A Primitive Composite Slender Body Theory
(1966) and Newman
Tuck
(1978a) noted that the low
frequency limit of the strip theory potential was equal to
the high frequency limit of the ordinary slender body
potential:
lim Dosb = lim
where 4
0+
1
ist
(x) (logK +y +Tri)
(G.10)
K+O
K+o
is the 2D solution to the Neumann problem discussed
above.
A composite solution is created by summing the strip
theory potential and the ordinary slender body potential
minus the limit (G.10).
For heave
(pitch) this potential
is:
st +
K
+
-
dol ( )L
(x-)
i
-da
(E)L
(X-0
27r
(log K+y+Tri).
i= 3,5
The heave
(G.11)
(pitch) source strength used is the same as
derived for ordinary slender body theory (G.5).
-18 4-
The hydrodynamic heave force (pitch moment)
for a body
of revolution with fore and aft symmetry is:
f9
ii
=
f St
ii
dx B(x)
W
-
2Tr
Sw 2 Kp
fdx
f
B(x)
27r-x
+
IL2
(log K + y + Ti)
dC B(C)
-
J
L
0
(x-)
L 1(x-()
dCB(C)[
3.
dx B 2(x)
i= 3,5
(G.12)
Maruo's Interpolation Theory
Maruo
(1970) and Maruo and Takura
(1978) have developed
an "interpolation" theory which is similar in some respects
to the unified slender body theory.
Maruo's potential
takes the form:
Maruo
Daru=
1
2D
D 2D + 2A. (l+ Kz)
1
i= 3,5
(G.13)
which by comparison with (2.43) is seen only to retain the
leading order behavior at low frequency of the unified theory.
-185-
Maruo's first approximation for the source strength,
a
(x), and thus
2wfa i(x).
for A. (x) as well,
is the 2D source strength
The second approximation is obtained by
iteration of an integral equation determined from the body
boundary condition.
Because
(G.13) is only strictly valid
at low frequency, the solution derived from the integral
equation will not be uniformly valid.
The second
approximation of the source strength a (x) is used to
determine the interaction A (x)
in the expression for the
determination of the added mass and damping
(cf., 2.63a)
according to:
2ip fdxA(x)[B(x)
+ K S(x)].
(G.14)
The added mass and damping results presented by Maruo
and Takura
(1978) for a Series 60 ship, however, do match
experimental results quite well and qualitatively resemble
the results of the unified theory.