Transactions on the Built Environment vol 5, © 1994 WIT Press, www.witpress.com, ISSN 1743-3509
Time domain analysis of vertical ship
motions
N. Fonseca & C. Guedes Soares
Instituto Superior Tecnico, Universidade Tecnica
de Lisboa, Av. Rovisco Pais, 1096 Lisboa, Portugal
Abstract
The time domain description of the vertical motions of a ship travelling at
constant forward speed is studied. The hydrodynamic forces are represented by
infinite frequency added mass, convolution integrals and restoring force
coefficients which are obtained from Fourier transforms of the frequency
domain solutions. The exciting forces are calculated in the frequency domain.
The hydrostatic forces are evaluated both around the mean equilibrium position
and over the instantaneous wetted surface in order to assess the effect of the
non-linearity of these forces. A strip theory is used to obtain a numerical
solution to the problem. Results of heave and pitch motions are presented for a
large containership and a tanker in head waves, and comparisons are made
between the solutions for linear and for non-linear hydrostatic forces.
1 Introduction
Time domain solutions of the ship motions are important to study various
phenomena. Of special importance are the different non-linear responses that
can be associated with the motions. A specific example are the stresses that
result from the impact of ships with the sea surface. Often in large ships the
motions can still be described adequately by linear theory but the forces
associated with the impact phenomena will be non-linear. For small ships or for
ships with large bow-flare the magnitude of the non-linear forces can be such as
to induce non-linear motions. Therefore, it is clear that at least for the problem
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mentioned it is important to be able to describe the ship motions in the time
domain so that non-linear forces can be calculated in an irregular sea state.
The interest for time domain representations of ship motions is relatively recent
as compared with the frequency domain formulations that were developed in the
late 60's and early 70's. After the development of a two-dimensional harmonic
flow solution by Ursell [1] for the heaving motion of a half-immersed circular
cylinder and the generalisation to arbitrary shapes using conformal mapping
techniques, these results were applied in frequency domain strip theories, firstly
by Korvin-Kroukovsky [2] and then by many other researchers who improved
the theory.
A more recent development in the linear frequency domain solutions of ship
motions was achieved by Chang [3] who represented the ship hull by a
distribution of singularities, and solved the fluid domain and free surface
equation using a panel discretization of the mean hull position. This is a very
acurate method to calculate the linear wave excitation load and response of
ships, but the numerical evaluation of the speed dependent Green function is
very difficult and it requires powerful computers.
The time domain simulations are often used to study non-linear ship motions
and two main approaches have been used. In one of them only the equations of
motion are solved in the time domain, and the hydrodynamic forces are
represented by frequency dependent coefficients. This is correct if the motion is
stricly sinusoidal in time, but not in the case of strongly non-linear motions or of
irregular motions.
Another approach solves the hydrodynamic problem in the time domain
together with the equations of motion. Finkelstein [4] and Stoker [5] were the
first authors who discussed the time-domain direct solution of hydrodynamic
problems. They applied the formulation to the problem of water waves. The use
of time domain analysis to solve the unsteady ship motion problem was initiated
by Cummins [6]. Since the instantaneous hydrodynamic forces depend of the
unknown body motion, Cummins decomposed the motion and the associated
velocity potential in a sequence of impulses. This decomposition allows the
hydrodynamic forces for arbitrary motions to be evaluated in terms of
convolution integrals of impulse response functions, which are independent of
the body motion. Ogilvie [7] studied the problem by the same approach, and
related the time domain motion equations with the classical frequency domain
equations using Fourier analysis. However, Cummins and Ogilvie did not obtain
a numerical solution to the time domain ship motion problem.
A linear time domain formulation for ship motions was developed and applied
by Chapman [8]. This theory is in some aspects similar to Chang's panel
method, but Chapman did not use Green's functions representing solutions of
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227
the free surface equation as Chang did. A panel discretization is used and the
solution is generated in terms of a source distribution representation of the body
and a spectral representation of the free-surface. The use of a time dependent
hull boundary to evaluate the hydrodynamic forces was proposed, dealing thus
with some effects of large-amplitude motions, but the computational effort of
such an extension is enormous.
Beck and co-workers [9-11] formulated the time domain hydrodynamic
problem with the same basic assumptions as Chang [3]. The linear potential
flow theory and the linear free surface condition are used together with a panel
discretization of the mean hull boundary under the still water line. Again the
solution is obtained by applying Green's theorem to the fluid domain.
Comparing with the frequency domain Chang's panel method the proper time
domain Green functions are easier to derive and to compute regardless of the
ship speed. Following the work described, Beck and Magee [12] generalized the
method to include a time dependent hull shape, but numerical solutions for ships
travelling on the free surface were not available yet.
The main disadvantage of the time domain panel methods is the need of
powerful! computers to solve the complicated integro-differential equations at
each time step. Basically the idea of the present work is to develop and apply a
time domain method, where the hydrodynamic forces are treated with realism
for non linear and irregular motions, and on the other hand the computational
effort is reduced to a point where the simulations can be made by a desk top
computer in a few minutes.
In this work, the formulation developed in [13] is briefly described, and it is
applied to a containerhsip and a tanker in order to illustrate the different
characteristics of the response of these ships hulls. The linear time domain
solutions were found to reproduce almost exactly the frequency domain
solution. The influence of including a non-linear description of the hydrostatic
term was also investigated and these results were compared with the linear
ones.
2 Present Theory
A time domain strip theory is developed, based on the linear potential flow
formulation, and on the linearized free surface and body boundary conditions. In
order to linearize the velocity potential and the boundary conditions, the ship
must be slender, her forward speed low, and both the amplitudes of the
incoming waves and the oscillatory motions must be small. The radiation forces
are formulated in the time domain and they are related with the frequency
domain solutions by means of Fourier transforms. The exciting forces are
evaluated assuming sinusoidal incoming waves acting upon the restrained ship.
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The hydrostatic forces have a non-linear characteristic since they are evaluated
up to the instantaneous wetted surface.
The time domain method followed for the radiation forces is based on the
formulation presented by Cummins [6] where the basic assumption is the
linearity of the forces. The formula for the potential obtained is an
hydrodynamic analogue of the impulse response function, which Cummins
assumed that exists but he did not calculate it. The equations of motion are valid
for any excitation, as long as they result in small amplitude motions. The inertial
properties of the fluid were represented by coefficients which are independent of
the frequency, of the previous history of the motion and of the ship's forward
speed. Cummins called them "legitimate added masses". The effect of past
history is represented by a convolution integral over the oscillatory velocity.
The method used here applies a strip theory to obtain a solution to the
formulation proposed by Cummins. This method works with truly time domain
radiation forces, and on the other hand asks for much less computer effort than
the existing panel methods since the boundary value problem is solved only
once rather than at every time step. Only the final equations are indicated in the
text while the detailed derivation of the expressions can be found in Fonseca
and Guedes Soares [13].
A coordinate system fixed with respect to the mean position of the ship is
defined, X = (x , y , z), with z in the vertical upward direction and passing
through the centre of gravity of the ship, x along the longitudinal direction of
the ship, and y perpendicular to the latter and in the port direction. The origin is
in the plane of the undisturbed free-surface.
t» (h«ve)
S, (surge)
$4 (roll)
Figure 1: The coordinate system and six modes of ship motion
Considering a ship advancing in waves and oscillating as a unrestrained rigid
body, the oscillatory motions will consit of three translations and three
rotations. The translatory displacements in the x, y, and z directions are
respectively the surge (£, ), the sway (%% ), and the heave (^ ). The rotational
displacements about the x, y, and z axes are respectively the roll (£4), the pitch (
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229
£5 ), and the yaw (^ ). The coordinate system and the linear and angular
displacements are shown in figure 1.
2.1 Radiation Forces
In order to implement a time domain method all the forces in the equations of
motion must be represented in the time-domain. This raises no major difficulties
in the evaluation of the exciting and restoring forces since, for the present
linearized model, these forces do not have the time dependency of the previous
history of the fluid motion. However the radiation forces behave in a different
manner. The existence of radiated waves implies a complicated time dependence
of the fluid motion and hence of the pressure forces. Waves generated by the
body at time t will persist, in principle, for an infinite time thereafter, as well as
the associated pressure force on the body surface.
The time domain formulation derived by Cummins [6] is used to represent the
radiation forces in terms of unknown velocity potentials. The basic assumption
is the linearity of the radiation forces. The equations representing the radiation
forces are given by:
where the dots over the symbols represent differentiation with respect to the
time variable, and the indices 'kj' are associated with forces in the k-direction
due to an oscillatory motion in the j-mode. In this study consideration is given
only to vertical motions i.e. k, j = 3,5.
In this expression Aj° represents the infinite frequency added mass coefficient,
which is dependent only on the ship geometry. The coefficient B^ is a constant
which depends on the ship geometry and on the forward speed. It was proven
by King [14] that B^ is zero. The coefficient C^ is also a constant which
depends on ship geometry and forward speed. The force proportional to this
coefficient is a "radiation restoring force", and it represents a correction to the
hydrodynamic steady forces acting on the ship due to the steady flow. The
correction arises because the steady force is evaluated assuming the ship in the
equilibrium position, and in fact the ship's position changes with time. In many
forward speed formulations this term does not appear, at least explicitly. This
constant can be obtained from the following expression:
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q, = <O*[A; - A,(*>)]-<»}[^(?)sin**]a
0
(2)
where A^(a>) are frequency dependent added mass, and the function K^(t) is
dependent on time, geometry, and forward speed. This quantity contains all the
memory of the fluid response. It is interesting to note that K^(t) is equivalent
to the impulse response function of any stable linear dynamic system. These
functions are obtained from the frequency dependent damping coefficients
BQ(G)) by means of Fourier transforms.
(3)
The last term in equation (1) is a convolution integral which involves the effects
of the whole past history of the motion, representing the memory effects due to
the radiated waves.
It is important to stress that none of the quantities described above (AJj°, B^,
Cjy, and Kq(t) is dependent of the past history of the unsteady motions. This
means that they need only to be calculated once for a given vessel, and then the
radiation forces can be evaluated for any arbitrary motion using equation (1).
The relations obtained for the radiation forces do not have coefficients
dependent on the frequency, and they are valid to evaluate the radiation forces
associated with non-sinusoidal motions, for example irregular motions. The only
condition necessary to apply this method is the linearity of the response, and this
means that the unsteady motions must be of small amplitude.
In spite of the fact that the forces obtained are truly time domain forces, they
are not evaluated directly in the time domain. This means that the boundary
value problem is not solved in the time domain, but instead infinite frequency
added masses, AJ°, and frequency dependent damping coefficients, B^.(co), are
used. A frequency domain strip theory very similar to the one presented by
Salvesen et al. [15] is used to obtain these hydrodynamic coefficients in terms of
sectional coefficients.
The sectional damping coefficients are computed by the Frank's close-fit
method [16], while the added masses corresponding to the infinite frequency
limit are computed by a modified Frank's method which takes into account the
zero potential condition on the free surface.
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231
2.2 Exciting Forces
The calculation of the wave exciting forces is treated as an independent problem
of a vessel restrained at its static equilibrium position and being subjected to
incoming waves. Again the strip theory approach is used to evaluate the total
force on the ship in terms of sectional forces but neglecting interferences
between adjacent cross sections.
The exciting force is divided in two components, F^ = F* +F°, where the first
part, the Froude-Krilov force, represents the effect of the wave incident
potential. The second part is the diffraction force which represents the effects of
the disturbance in the incident potential caused by the presence of the ship. This
last force is calculated with the formulation presented by Salvesen et al [15].
The amplitudes of the Froude-Krilov force associated with the heave and pitch
motions due to sinusoidal incoming waves travelling on the negative xdirections are:
(4)
(5)
Here the integrations are performed over the ship's length L, £* is the incident
wave amplitude, co^ and k^ =co//g are the wave frequency and the wave
number, respectively and // is the sectional heave Froude-Krilov force due to
an unit amplitude incident wave which is given by:
(6)
where the integration is performed over the cross section contour C%, n, is the
vertical component of the two-dimensional unit vector contained in y-z planes
and normal to the hull surface at the mean position. q>*(z) is the amplitude of
the potential of an unit wave acting on a cross section, which in accordance
with the linear gravity wave theory is given by:
/W = ^-/°'
(7)
The amplitudes of the diffraction force, associated with the incident sinusoidal
waves, for the heave and pitch motions are:
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where the ff is the sectional heave diffraction force associated with an incident
wave of unit amplitude. This force can be evaluated using the Haskind-Newman
relations, which means that instead of solving directly the diffraction problem,
the solutions of the forced oscillations problem are used to represent the
diffraction force acting on the ship restrained at mean position. Haskind [17]
derived the relations for the ship's zero speed condition, and some years later
Newman [18] showed that an analogous result can be obtained for the case of a
ship with forward speed. Salvesen et al [15] used these relations together with
the strip theory simplifications to obtain the expressions for the sectional
diffraction forces. The sectional heave diffraction force becomes:
<;
(10)
C,
where ^¥^(y,z) is the two-dimensional radiation potential associated with the
forced heave oscillations of the cylinder with the cross section C% This
potential is calculated by the Frank closefitmethod.
Finally the time description of the total exciting forces is given by the real part
of the sum of incomming and difracted forces:
k = 3,5
(11)
2.3 Hydrostatic Forces
The hydrostatic forces are the result of the hydrostatic pressure acting on the
hull. The formulation for the hydrostatic forces used in the present model does
not impose any kind of restriction on the body shape or motion amplitudes. In
fact these forces are evaluated over the "exact" wetted surface at each time
instant, taking into account the free surface elevation due to the incident wave
It is assumed that the contribution to the free surface elevation from the
radiated and diffracted waves can be neglected. This hypothesis is reasonable
for wave lengths of the same order of the ship length, which is the wave length
range of interest.
The hydrostatic forces associated with the heave and pitch motions are given
by:
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F"(t] = Pg V*(0 - pg\ \(z ntyg dt
1C,
= -PS V*(/) x,(t) + pgj Jz(x n?)dg dt
LC,
233
(12)
(13)
where n^' is the vertical component of the two-dimensional unit vector normal
to the intersection of the free surface of the fluid with the hull at each time
instant. The instantaneous immersed volume under the intersection of the free
surface with the hull, V „,(/), and the coordinates of the centre this volume are
given by:
X, = —
M x d v , y,=-—\\\ydv
(15a,b)
The free surface elevation at any point represented on the reference system
advancing with the ship's speed is given by:
£ (x,t) = C cos[*o* cos(&)/)] - sin[*ojc sin(^)]
(16)
With the knowledge of the positions in space of the ship's surface and the fluid
free-surface, the intersection between both surfaces can be determined. It is
assumed that this intersection is given by a plane parallel to the x-axis, which
means that the hydrostatic force in the x-direction can be neglected.
2.4 Equations of Motion
The equations of motion for heave and pitch are obtained applying Newton's
second law:
(17)
+
-oo
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(18)
where x<,, y^ are the coordinates of the centre gravity and the coefficients with
indices 35 represent the coupling between the motions.
The method chosen to solve the problem is the fourth-order Runge-Kutta,
which processes the integration of the coupled first order ordinary differential
equations.
3 Numerical results
Results are presented for the motion of a containership and a tanker in order to
identify the different behaviour when the non-linear hydrostatic term is included
in the formulation. The non-linearity of this term is associated with non-vertical
ship sides which are more pronounced for the containership.
The containership hull is the one used in the experiments of Gie [19] and used
by Flokstra [20] to compare with calculated values from strip theories. The
main characteristics of the ships are listed in table 1.
Characteristics of ships
Length between perpendiculares(Lpp)
Breadth
Draught even keel
Displaced volume
Block coefficient
LCG relative to section 10 (midship)
Centre of gravity above base line
Longitudinal radius of gyration
Ship's speed
Contairicrship
270,0 m
32,2 m
10,85 m
56097 m'
0,598
10,12 maft.
13,49 m
24,8 %Lpp
24,5 Knots
Tanker
168,56 m
28,0 m
10,9 m
42596 m>
0,828
1,66 mfwd.
8,956 m
27,2 %Lpp
14,0 Knots
Table 1 - Characteristics of the Ships Analysed
Before the time domain computations are performed it is necessary to compute
the retardation functions given by expression 3. These functions include the
effect of frequency dependent damping coefficients but they are frequency
independent. Figures 1 and 2 show the damping coefficients as a function of the
frequency for the tanker hull and figures 3 and 4 show the corresponding
retardation functions or the impulse response functions.
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0.000.0
1.1
i.b
235
,2:0'' *s.b'' '4/0
W(nod/s)
Figure 2: Heave (#33 / M^g I L^ \ and pitch (B^ I Mll^^gl L\
non-dimensional
damping coefficient for the tanker
-0.00-
0.75
-0.100,50
-0.20
853
-0.30
0.25
-0.40
W(rad/s)
W(md/m)
3.0
4.0
Figure 3: Heave (#351 ML^^gl L^ and pitch (B^ I ML^^gl L^ ) non-dimensional
coupling coefficients B% and ^3 for the tanker
1 00 •I
0 10 •
1
J0.50 •
8 0.05 -
0.00 • r
0 00 •r
*— ^H—
v/
A0.50^
V) >
—0 OSo.•0 10 0 20
0 'JO.0 " "40
' '20
.0" ' 40 o
0' ' 10.0Time
Time (sec)
&*
Figure 4: Non dimensional impulse response function for heave and pitch for the tanker
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).0
10.0 20.0 30.0
Time (sec)
400
10.0
20.0 30.0
Time (sec)
400
Figure 5: Non dimensional impulse response function for the coupling between heave and
pitch for the tanker
The corresponding results for the containership showed the same general aspect
but had different numerical values
An important conclusion from the observation of these retardations functions is
that they can assume significant values for thefirsttwo seconds and that they
vanish after 20 to 30 seconds. This shows the duration of the memory effects in
the response.
To verify the method of evaluating the radiation forces in the time domain, the
responses obtained from the frequency solution and from the time domain
response with linear restoring coefficients have been compared. The predictions
obtained for both motions were exactly the same, which was reassuring.
Several time histories have been obtained for the non-linear restoring forces and
a representative sample of them is shown in figures 6 to 12 for the tanker and
the containership. Since the motions are not initiated from the rest position of
the ship, the time histories of the motions before thefirstinstant (t=0) are given
by the frequency domain solutions. It can be observed that a transient phase
exists during the first few seconds of the run and afterwards the responses
become periodic.
8.00
4.00 A
A
A
0.00
S -4.00 -7 V
V
V
-8.00
0.0
10.0
20.0
30.0
40.0
50.0
Time (sec)
V
60.0
70.0
Figure 6: Linear and non-linear heave motion of the tanker in head waves (l^ IL^ - 2.4J
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10.0
"1
i 1 1 1 1 1 1 1
r
20.0 30.0
40.0 50.0 60.0
700
Time (sec)
= 2.4)
Figure 7: Linear and non-linear pitch motion of the tanker in head waves (z,^ /
6.00
^ 2.00 -
A
I -2.00
1
10.0
\j y
:
1
1
1
1
1—
20.0
30.0
40.0
Time (sec)
A
y
y
Figure 8: Linear and non-linear heave motion of the tanker in head waves (L^ I Lpp = 1.4)
"-6.0
0.0
10.0
1
20.0
1
1
1
r
30.0
40.0
Time (sec)
50.0
60.0
Figure 9: Linear and non-linear pitch motion of the tanker in head waves (L^ I Lpp - 1.4)
12.00
0.0
15.0
30.0
1
1
1 1 1 T
45.0
60.0
Time (sec)
75.0
90.0
Figure 10: Linear and non-linear heave motion of the containership in head waves
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9.0
sf
T3
0.0
-9.0
0.0
10.0 20.0 30.0 40.0 50.0 60.0 70.0 BO.O 90.0
Time (sec)
Figure 11: Linear and non-linear pitch motion of the containership in head waves
o.uu \w/
/
\ If
\ /
\j
V
V
V
10.0
20.0
30.0
40.0
50.0
60.0
Time (sec)
A
o
Q)
-C
J
_A nn 0.0
\
/
Figure 12: Linear and non-linear heave motion of the containership in head waves
8.0
0.0
A
I
Q.
-8.0
0.0
10.0
A
20.0
A
40.0
30.0
Time (sec)
A
50.0
600
Figure 13: Linear and non-linear pitch motion of the containership in head waves
The wave steepness defined as the ratio of wave height to ship length (2£* / L)
was kept constant and equal to 1/40.
For waves with length equal to the ships length both responses were basically
linear and they therefore are not shown The non-linear time domain response
when compared with the linear one, shows different positive and negative
amplitudes. For the heave motion the positive amplitudes are very similar for
both models but this is not the case for the negative ones.
To summarise the response as a function of the ratio between the length of the
wave and the ship, linear transfer functions are shown in Fig. 14 for the tanker
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and 15 for the container ship. In view of the asymetry of the non-linear
responses as concerns the positive and negative peaks of the response, pseudotransfer functions have been determined for the non-linear responses as shown.
1.35 i
1.60 -.
1.20 -
0.90 -
oQ.80 -
***** linear
•nin linear
linear ((—+
0.45 0.00
0.00
1.00
2.00
Lw/Lpp
3.00
0.00
0.00
1.00'
' ' 2.60 ' ' 3.00
Lw/Lpp
Figure 14: Heave and pitch transfer functions for the tanker
2.00 -i
1.60 -|
***** linear
non
non linear
linear (•+
(—
0.00
i .60' ' ' 2.60
Lw/Lpp
0.00
1.00'
' ' 2.60
Lw/Lp
3.00
Figure 15. Heave and pitch transfer function for the containership
Observation of those figures makes it clear that the non-linear results show
difference with respect to the linear responses and the method differenciates
between the amplitudes of the positive and negative peaks. Furthermore, the
results also show that the differences are much larger for the containership than
for the tanker, as expected.
These results demonstrate that the present theory is capable of predicting the
effects of the non-linearities associated with the non-linear description of the
restoring forces. Although it is felt that this effect is one of the major
contributors to the non-linearity of the response [21,22] it is still necessary to
investigate the influence of the non-linear formulation of other terms of the
motion equations.
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4 Conclusions
The comparisons of time domain solutions with frequency solutions for linear
hydrostatic forces has shown that this method of calculating the radiation forces
is accurate.
The time domain model shows that the depicted non-linear effects, although
relatively small for the tanker they are large for the containership. The nonlinear behaviour is more pronounced in pitch than in heave.
Although the present results concern the response to regular seas, it is believed
that the main advantages of the time domain solution is on calculating the
motion response to irregular seas. However this can be done whenever the
responses to regular seas have been validated.
5 Acknowledgements
This work has been made within the scope of the research project "Wave
Climate and Ship Behaviour in the Portuguese Exclusive Economic Zone",
which has been financed by JNICT: Junta Nacional de Investigate Cientifica e
Tecnologica under contract PMCT/MAR/931/90.
6 References
1.
Ursell, F. "On the heaving motion of a circular cylinder on the surface of a
fluid", Q. J. Mech. Appl. Math., Vol. 2, (1949), pp. 218-231.
2. Korvin-Kroukovsky, B V., "Investigation of ship motions in regular
waves", Trans., Soc. NavalArchit. Mar. Eng., Vol. 63, 1955, pp. 385-435.
3.
Chang, M.S., "Computations of three-dimensional ship motions with
forward speed", Proc. 2nd. Int. Numer. Ship Hydrodyn, University of
California, Berkeley, 1977, pp. 124-135.
4. Finkelstein, A. B., "The initial value problem for transient water waves",
Commun. on Pure andAplliedMathem, Vol. 10, 1957, pp. 511-522.
5.
Stoker, J. J. Water waves, Interscience Publishers, Inc., New York, 1957.
6. Cummins, W. E, "The impulse response function and ship motions",
Schiffstechnik, Vol. 9, 1962, pp. 101-109.
7. Ogilvie, T F., "Recent progress toward the understanding and prediction of
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