Time-domain analysis of wave exciting forces on floating ...

Time-domain analysis of wave exciting forces
on floating bodies at zero forward speed

ROBERT F. BECK and BRADLEY KING*

University of Michigan, Department of Naval Architecture and Marine Engineering,
Ann Arbor, Michigan 48109, USA
*Presently at Bassin D'Essais des Carenes de Paris, Paris, France

1. I N T R O D U C T I O N simulations for body motions without great care regard-
ing the restrictions and assumptions of such a step. The
The problem of determining the exciting forces acting time-domain approach allows the direct calculation of
on a rigid floating body due to uni-directional waves has the transient solutions.
been extensively studied. The problem is usually for-
mulated in the frequency domain by assuming that the In this paper the methods developed in Beck and
incident waves are sinusoidal of fixed frequency. Results Liapis 2 and Liapis and Beck 3 will be extended to solve
for more general wave systems may then be found using the exciting force problem. The analytic basis for the
superposition and Fourier analysis. In this paper, the problem is presented in Section 2 followed by a discus-
problem will be formulated directly in the time domain. sion of the boundary conditions for the diffraction
Numerical techniques are developed for bodies of arb- problem in Section 3. Numerical methods and results
itrary shape. The solutions in the time domain and fre- are then presented in Sections 4 and 5, respectively.
quency domain are related through the use of Fourier
transforms. 2. MATHEMATICAL FORMULATION

Time-domain analysis has been used by several Consider an arbitrary three-dimensional body floating
authors to solve the radiation problem. For axisym- on the free surface of an incompressible ideal fluid. Let
metric bodies, Newman I determined the impulse Oxyz be a right-handed coordinate system with the x-y
response functions for right circular cylinders of various plane coincident with the calm water level. For ship
radius to draft ratios. Beck and Liapis z and Liapis and shapes the x-axis points toward the bow, the z-axis is
Beck3 use the technique to solve the radiation problem positive upwards, and the origin is placed at midship.
for arbitrary three-dimensional bodies at zero speed and Plane, sinusoidal waves are incident to the vessel. The
with forward speed respectively. waves are travelling in a direction which makes an angle
13 to the x-axis.
The use of time-domain analysis to solve the exciting
force problem has not been widely investigated. The incident wave amplitude at the origin as a func-
Wehausen 4 developed the analogue to the Haskind rela- tion of time is given by ~'o(t). The amplitude is assumed
tions for zero forward speed in the time domain. Using small so that a linear theory may be developed. The flow
these relations, the exciting force is determined as an is considered irrotational so there exists a velocity
integral over the body surface of combinations of the in- potential such that in the fluid domain
cident wave potential and the radiated wave potentials.
The solution of the diffraction problem remains of vEav= 0
interest because the Haskind integral relations cannot
be used to calculate local phenomena such as wave and (1)
elevations or hydrodynamic pressures. ,I, = CIo, (P, t) + dp7(P, t)

Direct solution of the diffracted wave problem in the where CIo, = incident wave potential
time domain is interesting for several reasons. Possibly, ,I,7 = diffracted wave potential
the most important reason is that the method used here P=(x,y,z)
may be extended to the cases of steady forward speed
and/or arbitrary maneuvers of the body without any For convenience the subscripts 0 and 7 are used to
serious modifications to the approach. To the contrary, denote the incident and diffracted waves respectively.
frequency domain methods involve very difficult Green The subscripts 1,2 ..... 6 are reserved for the radiation
function evaluations at steady forward speed and are potentials in the 6 degrees of freedom. This leads to an
not meaningful for general transient maneuvers. The efficient notation for numerical techniques when both
time domain approach is also very useful in verifying the radiation and exciting force problems are solved
and clarifying the relationship between frequency do- simultaneously.
main results and time domain results. Frequency do-
main solutions are often used to develop time domain The incident wave amplitude at the origin, ~'o(t), is
related to the incident wave potential by
Accepted March 1988. Discussion closes July 1989.
~'0(t) - 10 (2)
~o (0, t)

g at

re3 1989 Computational Mechanics Publications Applied Ocean Research, 1989, Vol. 11, No. 1 19

Wave exciting forces on floating bodies at zero forward speed: R. F. Beck and B. King

where g = acceleration of gravity. On the free surface, An integral equation which must be solved to deter-
both 40 and 47 must satisfy the free surface boundary mine (I)7on the body surface is obtained by applying the
condition Green theorem to the volume of fluid bounded by the
body surface, the free surface, and the surrounding sur-
024 04 (3) face at infinity and then integrating both sides with
Ot2 + g ~ = 0 on Z = 0 respect to r from - oo to t*. The integrals over the free
surface and the surfaces at infinity vanish because of the
The boundary condition on the body surface, So, is boundary conditions. Using the boundary condition on
the body (4) gives the final result:
04
--=0 It47(P, t) + - ~dr dSQ47(Q, 7")
On
so that

-oo So

0t~0 0(I)7 (4) 0 (8)
x-- G(P,Q,t- r)
- - on So
an an OnQ

where n = unit normal on the body surface out of the IS27r dr d S o G ( P , Q, t - r)
fluid domain. The incident wave potential is assumed to So
be zero at t = -oo. This implies that the initial condi- 0
tions on 41,7 are
x ~nQ 4 o ( Q , 7.) PeSo
47-'0 t~ -oo
a (5)
t ----~ - o o
Ot (I)7 ---) 0 The hydrodynamic forces acting on the body are found
by using the linearized Bernoulli's equation, and inte-
In addition, the radiation condition requires grating the pressure over the body surface as follows

V(I)7 ~ 0 a s r ~ oo Fj(t) = Fjo(t) + Fir(t)

An integral equation which must be solved to deter- = - 0 fl dsnj ~04- 0 - P fl dsni 004-7-7- (9)
mine the diffraction potential can be derived usipg the
Green theorem and a Green function for an impulsive where
source below the free surface. Beck and Liapis z give the
Green function, which was first derived by Finkelstein, 5 Fjo = Froude-Krylov exciting force in the jth
as direction

('r 1)G(P, Q, t - r) = - 7- ~(t - r) FiT=Diffraction exciting force in the jth
direction
+ H(t - r)G(P, Q, t - z)
j--(1,2,3) forces along the x,y,z axis
where (6) respectively

G ( P , Q, t - r) = 2 dk,,,'-gksin(,,gk (t - r)) j=(4,5,6) moments about the x,y,z axis
o respectively

X ek(:+ :c)Jo(kR) nj = generalized unit normal
(hi, n2, n3) = n
P=(x,y,z) (m, ns, n6) = r × n

r= (x,y,z)

Q = (~,'0, ~')

r 2=(x-~)2+(y-'0)2+(z- ~)2 3. LINEAR SYSTEM THEORY AND THE
DIFFRACTION BODY BOUNDARY CONDITION
r ' 2 = (X -- ~)2 + ( y -- ,0)2 + (Z + ~-)2
Because the hydrodynamic problem is linear, linear
R 2=(X-~)2+(y-'0)2 system theory may be used to put the expression for the
Froude-Krylov force and the diffraction exciting force
6(0 = delta function in the form of a convolution of the arbitrary incident
H(t) = unit step function wave amplitude with a kernel function which is indepen-
dent of the incident wave amplitude.
The Green function gives the potential at a point P
and time t due to an impulsive source at Q created at For the Froude-Krylov exciting force we seek a
time r. The (1/r- 1/r') 6(t- r) term represents the im- function, Kjo(P, t) such that the force on the body as a
pulsive source plus its negative image. Together they function of time can be written in the form:
satisfy a 4 = 0 boundary condition on the free surface.
The impulsive source generates a Cauchy-Poisson type S SSF~o(t) =
wave system which is represented by (~(P, Q, t - r). The -co dr~'o(t - r) ds~O(Q, r)nj
Green function satisfies the following problem:
s,, (lO)

VZG = -47r 5(P- Q) 6(t- r) = f~-oo dr~'o (t - r )Kjo(r)

02 0 (7) The function/~(P, t) is effectively the time history of the
~Ot G + g Ozz G = O on z = O pressure at a point P caused by an incident wave whose
elevation is an impulse at the origin at time t = 0. The
G a n d - ~O-G= 0 for t - 7 . < 0 wave is similar to the wave system produced in the
Cauchy Poisson problem of Lamb 6 except that the

20 Applied Ocean Research, 1989, Vol. 11, No. 1

Wave exciting forces on floating bodies at zero forward speed: R. F. Beck and B. King

waves are travelling as opposed to the waves resulting Unlike a normal impulse response function it is not zero
from an initial disturbance. The elevation at the origin for t < 0. As with the incident wave kernel, this occurs
has the unique property of being zero for all time except because there is an induced velocity on the body surface
t = 0 where it is infinite. before the impulsive wave forms at the origin.

The function/~ is real and satisfies the condition that The diffracted wave potential may also be defined in
/~(P, t)--' 0 as t--, -oo. It does not satisfy the usual terms of an impulse response function
causality condition of being equal to zero for t less than
0. This does not imply that the system is anticipatory in Scb7(P, t) = dr~7(P, t - r)fo(r) (16)
nature. The condition results from the fact that when a
wave arrives at the origin it has already contacted the --oo
body (and exerted a pressure) at some earlier time. The
time delay increases as the wavelength decreases and where 4;7(P, t) is the impulse response function for the
hence there is no absolute limit except t = - o o . diffraction potential. @ ( P , t) is found by substituting
(15) and (16) into (8). After interchanging the orders of
To determine the impulse response function for the integration with respect to time, and several changes of
Froude-Krylov exciting force, ~'o(t) is assumed to be a variables the final result is
unit amplitude sinusoidal wave of the form
S If~7(P, t) + 1 dr dso~v (Q, r)

2"ff --~

~'o(t) = e i~t (1 1) So

For sinusoidal waves with an amplitude given by (11), 0 (17)
the velocity potential is × anO G(P, Q, t - r)

Cbo= ig ek z e_ik ~ ei~t (12) =1 ,dr If dSeG(P'Q't-r)

t.d So

where w = wave frequency X (n" K(Q, r))
k = 2[g
Equation (17) could be derived in an alternate manner
= x cos/3 + y sin 13 by assuming ~'o(t) is an impulsive wave such that
13= wave heading angle .~o(t) = 6(t). Equation (15) is then reduced to

An explicit expression for/~(P, t) is found by equating a,I,o = n . K ( P , t ) (18)
the forms of the Froude-Krylov exciting force given by On
(9) and (10) and substituting equations (11) and (12). An
inverse Fourier transform is then taken to obtain/~(P, t) Substituting (18) into (8) then results in (17). In this
as a function of time. The inverse Fourier transform is
defined by the transform pair: form, it is clear that q~7(P, t) is the diffracted potential

due to an impulsive wave.

F(~o) = dtf(t)e -i°'t To determine K(P,t) the known result for a

sinusoidal wave is used.

(13) Assuming ~'o(t) = e i'°t - oo < t < oo, then

1 0O%n - n_. f ~ d z K ( P , t - r)e i~ (19)

f ( t ) = ~-~ f ~ dwF(~0)e"~' --co

In order to take the inverse Fourier transform, negative OCI,o/cgn may also be found by taking the directional
values of the frequency range must be considered. Since derivative of the incident wave potential:
/~(P, t) is real, the continuation of (12) must be complex
conjugate symmetric with respect to w. The final result 0~o = n- [ (i cosl3 + j sin 13+ _ki)o~ ekz e -ig= e i°'t (20)
is: On

15(P, t ) = p g Re d~o e kz e i('°t-k~) (14) Equating (19) and (20) and taking the inverse Fourier
transform defined by (13) results in
71" 0

To solve for the diffraction potential, linear system K(P, t) = Re~/)' sin 1 dc0~o e kz e -ik~ ei°' (21)
theory is again used. First the normal velocity induced
on the body surface by the incident wave is put in the (Lk i a" o
form

_0 (15) AS with the Froude-Krylov exciting force, in taking the
inverse Fourier transform the function of frequency is
CbotP, t)=n. I~ drK(P,t-r)~'o(r) made complex conjugate symmetric in order to insure
that K(P, t) is real.
On - ~ --
For small values of z, equation (21) may be difficult to
where evaluate numerically because of the oscillatory nature of
the integrand. The problem may be avoided by using a
n_= unit normal to body surface non-impulsive input wave.

= (nl, n2, n3) Assume the input wave has the form
~'o(r) = arbitrary incident wave amplitude at the
u~
origin

The vector function, K(P, t), is effectively the impulse ~'o(t) = a / ~ g e - at2 - oo < t < ~ (22)
response function for the incident wave velocity. When
it is convolved with the arbitrary incident wave where L is some characteristic length. This form of the
amplitude, it gives the induced velocity on the body sur- input wave is very convenient because all derivatives of
face. K is a real function which goes to zero as t --, - ~ .

Applied Ocean Research, 1989, Vol. 11, No. 1 21

Wave exciting forces on floating bodies at zero forward speed: R. F. Beck and B. King

~'0(t) are finite and continuous. Increasing the size of the where Kj7(t) = - p dsnj ~ q~7(P, t) (27b)

constant a will adjust the input to as close to an So

impulsive wave as desired.

In addition, the Fourier transform of (22) has the The value of gj7(t) may be easily found by interchang-

simple form ing the order of integration over the body surface with
the time differentiation and recalling that differentiation
F(£o (t)) = ,./-[L]g] e- '°Zl4a (23) with respect to time is equivalent to multiplying the
Fourier transform by ico. If the diffracted wave exciting
where F(...) denotes the Fourier transform defined by
(13). force due to a wave amplitude given by

The velocity potential for this input is found by a~gge_at,-

solving (8) directly. The input normal velocity, 8~o/On,

is determined from (15) as follows:

O¢'°=n- 'f:0n ~ d r K ( P ' t - r ) ~ g is defined as the time derivative of the function

= n • Vffo (24) gj(t) = -p It dsnjO(P, t) (28)

where K(P, t - r) is given in (21). Equation (24) may be St)

evaluated using (21), (20) and the fact that the Fourier it can then be shown using (27b) and (26) that:
transform of a convolution is the product of the in-
dividual Fourier transforms. The real part of the inverse KiT(t) = F-'( icoF(gj(t)! ~ (29)
Fourier transform of the product is then taken to give: ,/[ L/g]e ~ /4a]

V*o(P,t) 1 ~ ZgRe ~([|i)scions B131| In numerically evaluating (29) care must be taken
--~ because at large cothe quotient approaches 0 over 0. To
determine the proper limit, it is noted that the
LL_ J numerator simply represents the Fourier transform of
the exciting force due to the incident wave given by (22).
o dcocoe kz e -':'/4" e ik,~eio:l (25) Thus, (29) may be rewritten as

, {xj (co),,'i Lig] e z]4a~ (30)
-

Kj7(t) = F- \ ,/~L/g]e_~:14" j

Note that as opposed to (21) the integrand in (25) con- where X;(co) represents the traditional complex

tains the factor e -'~214a. This exponential decay makes frequency-domain diffraction force response at fre-
quency co. While not proven here, it is reasonable to
the integral easy to evaluate even when z is small. expect Xj(co) to be absolutely integrable and the inverse
Physically, this choice of ~'o(t) corresponds to a wave Fourier transform to be well defined. Numerically, for

system with exponentially small short wave content. large values of co, the term icoF(gj(t)) in (29) cannot be
Using (25), the solution to integral equation (8) is the
diffraction potential for an incident wave amplitude of expected to have exactly the same exponential decay
the form as the denominator. To obtain reasonable numerical
results, a cut-off frequency is used and the value of the
~'0(t) = ja_a~L e-at'-. arbitrary constant a must be chosen sufficiently large

To find ~7(P, t), the impulse response function for the that the numerator (io~F(gj(t))) becomes small before
diffracted wave potential, recall that for a linear system
the Fourier transform of the output is equal to the the exponential in the denominator. It has been found
Fourier transform of the input times the Fourier
transform of the impulse response function. Thus, it that for reasonable values the parameter a has no effect
can be determined that on the results. Specifically, a value for a was chosen
such that the expression e ~/4o was no smaller than 0.1
~p~(p,t)= F-l(F(O(P't-))~ (26)
\ F(~o(t)) ] for values of co within the range of interest. Typical

= F-'(... F(4~(P,t)) "~ values of aL/g were around 1.0.

[L//g]e- '~ ]4a/ To summarize, the total exciting force in the jth
direction acting on the vessel at zero forward speed for
arbitrary long crested seas is given by

where F - ~(...) = Inverse Fourier transform F,(t) = f~ dr~o(t-r)Kjo(r) (31)
~b(P, t) = solution of (8) using (25) to determine
the induced normal velocities. + I~ dT~o(t-~)Kj7(r)

The exciting force due to the diffracted waves is Kjo(r) is found by direct integration of the function
found by substituting (16) into (9) which results in /~(P, t) (eqn. (14)) over the body surface as given in (10).

F : ( t ) = - O dsnj ~O -oo drq57(P, t - r ) f o ( r ) Kj7(r) is determined by evaluating the Fourier

So transform given in (29). The time derivative of the func-

(27a) tion gj(t) is the non-impulsive exciting force in the jth

= f7~o drKj7(r)~o(t-r) direction due to an incident wave amplitude of the form

a~g e_at2

22 Applied Ocean Research, 1989, Vol. 11, No. 1

Wave exciting forces on floating bodies at zero forward speed: R. F. Beck and B. King

from direction ¢3. This force is found by evaluating (28)
where q~(P, t) is the solution to the integral equation (8)
with the induced velocity on the body surface given by
(25).

4. NUMERICAL METHODS / ' -" ISwoY ....... [

It has been shown that the diffracted wave potential and q-ff- '-*J ** -s!** -J!.. - , ! . -~!~o .~* 3,~* , ~0 J~* ,.~, ,*!~
resulting forces due to an arbitrary incident wave on an
arbitrary body may be found using relations (16), (26) Figure 1. The Froude-Krylov impulse response func-
and (27). These relations require that the potential tion for a sphere in heave and sway
q~(P, t) due to an incident wave of the form
i
f~
•/• ,.g* ,.~* *3** i.l,, ,.!u
~'o(t) = ~jra-~Lge-atZ
Figure 2. The diffraction impulse response function for
be found. This potential may be computed using a a sphere in heave and sway
numerical discretization of equation (8). The surface in-
tegrals are performed by dividing the body surface into Krylov force exhibits a very large hump and is even
M plane panels which have constant potential strengths. about t = 0. The function n2 is odd in space so that the
The convolution integrals in time are performed by a sway impulse response function is also odd and displays
trapezoidal rule from to to tN. The discretized integral many humps. The area under the Kjo curve is propor-
equation takes the form of a system of algebraic equa- tional to the zero frequency Froude-Krylov exciting
tions given as force. For this reason, the integral of K2o(t) with respect
to time is zero and the heave curve integrates to 1
~(Pm'tN)"]-1271i~="1 ~)(Qi,tN) ffOSQ~QO(1_ 1) because of the nondimensionalization.

S~ The diffraction exciting force impulse response func-
tions shown in Figure 2 do not display the same nice
I!x ('r l)=1 dSQ -~- n.V~o(Q, tu) symmetry properties as the Froude-Krylov forces.
2r i=1 i Because of the memory effects of the diffracted wave
system, the results are neither odd or even. In addition,
(32) the curves are more oscillatory due to interference
effects.
+At ~ dSQG(PmQ, , tN-n)n
Figures 3 and 4 present the amplitude and phase of
n=l i the total exciting force versus kR for the sphere in sway
and heave respectively. Three sets of results are shown
• WI,o(Q, t . ) - ~b(Q~, t . ) 1t dSo in the figures. The solid lines are frequency-domain
results obtained from the time-domain results by taking
Si the Fourier transform of equation (30). The plus signs
are the results of CohenS using a multipole expansion.
0 G(Pm, Q, tN-n)]l m= 1,2, ...,M The circles were computed by Breit 9 using a higher
order panel method. As can be seen the results agree
where n • VOo is given by (24). The end points of the very closely.
trapezo-[dal rule summation do not appear because of
the choice of initial time at which q~(Q, to) = 0 and the The results of Figures 3 and 4 are somewhat mislead-
condition that G(P, Q, t)= 0 for t < 0. The Rankine ing because all three methods predict the Froude-Krylov
source terms are calculated using analytic expressions part of the exciting force with equivalent accuracy.
following Hess and Smith. The integrals involving the Since this component of the force dominates for about
wave terms of the Green function are performed using half the kR range shown, the results compare extremely
2 x 2 Gaussian quadrature. The numerical methods well. Figures 5 and 6 show only the diffraction exciting
used to evaluate G(P, Q, t) are discussed in detail in force component. In these figures Breit's results are not
Beck and Liapis 2 and King 7.

5. RESULTS

The technique was tested on a floating hemisphere since
accurate results are available from other methods for
comparison. Taking advantage of the body symmetry,
the integral equation (32) was solved with panels on 1/4
of the submerged hemisphere's surface.

Figures 1 and 2 show the Froude-Krylov and diffrac-
tion force impulse response functions for a sphere of
radius R, respectively. The two modes of motion of
heave and sway are shown. In Figure 1, the Froude-
Krylov exciting force is even about t = 0 for heave and
it is odd for sway. This is the result of the impulsive
wave pressure which is an even function in time about
t = 0. Since n3 is always positive, the heave Froude-

Applied Ocean Research, 1989, VoL 11, No. 1 23

Wave exciting forces on floating bodies at zero forward speed: R. F. Beck and B. King

*e--I

'°-~ Time Domo i n --

Cohen +

°° ~° ,% =% =.~o ,.~°
kR

,ii Cohln +
Holkind 0

I.......... ]
COhen +

a,ell o o

kR Figure 5. The amplitude and phase o f the diffraction
sway exciting force acting on a sphere versus nondimen-
Figure 3. The amplitude and phase o f the total sway ex- sional wave frequency (kR). The solid line was obtained
citing force acting on a sphere versus nondimensional by Fourier transform of the impulse response function
wave frequency (kR). The solid line was obtained by shown in Figure 2. Cohen "sresults were calculated using
Fourier transform of the impulse response function. multipole expansions. The circles were computed using
Cohen's results were calculated using multipole the Haskind relations in the time domain
expansions. Breit's results were computed using a
higher-order panel method in the frequency domain

.,,M

Time Domoin
Cohen
Breit +
0

JiI ..

kR kR
~o

Coh=~ + 2:;'7.,

kR Figure 6. The amplitude and phase of the diffraction
heave exciting force acting on a sphere versus non-
Figure 4. The amplitude and phase of the total heave dimensional wave frequency (kR). The solid line was
exciting force acting on a sphere versus nondimensional obtained by Fourier transform of the impulse response
wave frequency (RR). The solid line was obtained by function shown in Figure 2. Cohen's results were
Fourier transform of the impulse response function. calculated using multipole expansions. The circles were
Cohen "s results were calculated using multipole computed using the Haskind relations in the time
expansions. Breit's results were computed using a domain
higher-order panel method in the frequency domain
Furthermore, there is a slight hump in the heave time-
shown since he only computed the total exciting force. domain curve around kR = 2.2. The reason for this
They are replaced by results using the Haskind relations hump is not known.
developed by Wehausen. 4 The Haskind relations deter-
mine the diffraction exciting force in terms of the inci- To investigate the accuracy of the convolution in-
dent wave potential and the solution to the forced tegral for the exciting forces (equation (31)), the exciting
oscillation problem. The results presented in Figures 5 force in a random sea was computed using equation (31)
and 6 were computed by Liapis using the solution to and a frequency-domain addition. The pseudo-random
forced heave or sway given in Beck and Liapis 2 (1987). sea shown in Figure 7 was produced by summing six
The agreement is again good. At high frequencies unit sine curves of the wavelengths shown on the figure.
differences begin to appear in the sway amplitude.

24 Applied Ocean Research, 1989, Vol. 11, No. 1

Wave exciting forces on floating bodies at zero forward speed: R. F. Beck and B. King

The incident elevation is given as may be noted that for the examples shown, the sphere
had 65 panels on 1/4 of the submerged surface. With
6 t), regard to convergence of the numerical scheme it was
found that the results followed closely the findings of
g'o(t) = ~ sin([~] Beck and Liapis 2 for the problem of wave radiation by
the same body.
i=1
CONCLUSIONS
where k~ is the wave n u m b e r o f the j t h c o m p o n e n t .
The heave exciting force as a function of time acting Solving the exciting force problem directly in the time
domain is a viable alternative to the more conventional
on the sphere due to the pseudo-random excitation of frequency-domain approach. At zero forward speed the
Figure 7 is shown in Figure 8. The data labeled as fre- frequency-domain approach is probably computation-
quency domain were computed by summing (with ally faster because of the convolution integrals and time
proper amplitude and phase) the exciting forces due to stepping which is required in the time domain. The next
each of the 6 individual sine wave components. The time step is to extend the theory to include forward speed.
domain line was obtained using (31) and the impulse
response functions for heave presented in Figures 1 and ACKNOWLEDGEMENTS
2. The two sets of results agree except near t = 0, where
there is a starting transient in the time domain. It should This research was sponsored by an Accelerated
be noted that by the assumptions used to compute the Research Initiative of the Office of Naval Research,
frequency domain results, the excitation continues Contract No. NI4-85-K-0118. Computations were made
indefinitely from t - - -Qo to t = oo. The time domain in part using a Cray Grant, University Research and
approach correctly predicts the starting transient. It Development Program at the San Diego Supercomputer
Center.
v vv vVV
REFERENCES
-..1o-'° s le 91h ....,..|
*,!,::'2; 1 Newman, J. N. Transient axisymmetric motion of a floating
:i: ....... ,.... ,..... ,..... , cylinder. Journal of Fluid Mechanics, 1985, 9, 190-199

Figure 7. Irregular wave amplitude versus nondimen- 2 Beck, R., and Liapis, S. Transient motions of floating bodies at
sional time. The wave amplitude is formed by summing zero forward speed. Journal of Ship Research, 1987, 31, 3,
six sine waves of the frequencies shown. The starting 164-176
phases were all set equal to zero
3 Liapis, S., and Beck, R. Seakeeping computations using time-
...*-4 m0,0mon A,°q0 ....• domain analysis. In Proceedingsof the fourth international con-
ference on numerical ship hydrodynamics, 34-54. Washington,
A::: D.C.: National Academy of Sciences, 1985.

-A A 4 Wehausen, J. V. Initial-value problem for the motion in an
i v V, tvV V v_, "~ undulating sea of a body with fixed equilibrium position. Journal
of EngineeringMathematics, 1967, 1, 1-19
. * ,.~.o ,o?x s.~.l ..~°° .°~...?~ , o % . . 0 % 0 ..~°° , ° . , ~ , , . ' . ° ,~¢.** ,~.'**
5 Finkelstein, A. B. The initial value problem for transient water
Figure 8. Irregular exciting force acting on the sphere waves. Communicationson Pureand Applied Mathematics, 1957,
due to the wave amplitude shown in Figure 7. The solid 10, 511-522
line is obtained by convolution integral in the time do-
main. The dots are obtained by summing the exciting 6 Lamb, H. Hydrodynamics, Cambridge University Press, 1932.
forces due to the six sine waves in the frequency domain 7 King, B. K. Time-domain analysis of wave exciting forces on ships

and bodies. Ph.D. Diss., The University of Michigan, 1987.
8 Cohen, S. 13. A time domain approach to three-dimensional free

surface hydrodynamic interactions in narrow basins. Ph.D. Diss.,
The University of Michigan, 1986.
9 Breit, S. R. A higher-order panel method for surface wave radia-
tion and diffraction by a spheroid. In Proceedingsof the fourth
international conference on numerical ship hydrodynamics,
200-215. Washington, D.C.: National Academy of Sciences,
1985.

Applied Ocean Research, 1989, Vol. 11, No. 1 25