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SECOND ORDER WAVES GENERATED BY SHIP MOTIONS 143
Second Order Waves Generated by Ship Motions
M.Ohkusu, M.Yasunaga (RIAM, Kyushu University, Japan)
ABSTRACT
We present measured contour maps of unsteady wave elevation generated by oscillatory motions of a ship navigating
at constant speed. A ship model used for this experiment is S175. The model is forced to pitch at two different amplitude.
Our computerized wave measuring system gives accurate wave contour map of the fundamental frequency and the second
harmonics component separately. Existence of the second order wave whose magnitude is 15 to 20 percent of the first
order's is confirmed. Pattern of the second order wave is discussed in the light of asymptotic analysis and a theoretical
method of predicting the second order wave is tested in its accuracy on the measured wave contour maps.
1. INTRODUCTION
Rational understanding of seakeeping and accurate prediction of ship behaviors in rough seas are required more than
ever. Nonlinear theory of three dimensional flow of ships, moving at forward speed in high waves, is indispensable for
these to be feasible. That is why various nonlinear theories and computational methods have been proposed and
implemented. However experimental evidence directly supporting or disproving those theoretically sophisticated methods
is scarce. This is what I argued at 22nd ONR Symposium as well (Ohkusu (1998)).
Examination of the accuracy of theories in predicting measured wave field generated by ship motions or the diffraction
of the incident wave will be more appropriate because the accuracy of the measurement is expected to be more reliable than
that of hydrodynamic pressure and the less integration effect of the wave field than hydrodynamic forces will be suitable
for proving or disproving nonlinear theories more vividly. Furthermore the accurate prediction of wave elevation close to
the hull surface is crucial for practical purpose such as accurate estimate of wave load on bow flare and deck wetness.
Wave field away from the ship is related to damping of the ship motion and added resistance in waves.
Hydrodynamics of waves generated by ship motions or the diffraction of the incident waves when the ship's advance
speed is not zero is a challenging and interesting subject for its own sake (for example Cao, Shultz and Beck (1994)).
Although several authors have presented their results on the computed wave field but experimental data obtained under the
exactly controlled condition is very rare (Ohkusu and Wen (1996)). Actually they are not visible clearly in the tank test
because other effect like the Kelvin wave pattern and the incident waves are mingled. Yet we wish to see experimentally
how those unsteady waves are generated, not at a spot but a whole area around the ship, and how the nonlinearity is
exhibited in the wave field.
In this paper an example of the second order effect manifesting itself in the structure of measured wave field generated
by a ship motion is presented. We investigate hydrodynamic characteristics of this nonlinear effect with asymptotic
analysis and the computation of the second order wave elevation by the so-called 2.5D theory.
2. MEASUREMENT OF THE SECOND ORDER WAVES
An objective of our study is to visualize nonlinear effect in the flow (unsteady free surface elevation) around a ship
moving at forward speed in sea waves. A ship model is undergoing an oscillatory motion and towed at a constant speed in
the water tank; the wave field, e.g. the contour map of instantaneous unsteady free surface elevation, around the ship model
is what we wish
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SECOND ORDER WAVES GENERATED BY SHIP MOTIONS 144
to obtain experimentally.
Accurate measurement of oscillatory wavefield around a ship model, which is doing oscillatory motions and running
at constant forward speed, is not straightforward. One of the present authors has already reported at many occasions our
technique to realize the measurement. General idea of this technique is briefly described below (refer Ohkusu and Wen
(1996) for the details).
Wave is recorded at several locations continuously during a run of a ship model; the locations are fixed to the water
tank and set on a line parallel to the ship model track. These records are transferred to time series of wave elevation at every
location in the reference frame moving with the ship model if the locations are on the line parallel to the ship model track.
The transformation is based on the fact that the wave probes reach to an identical location in the moving reference frame at
different time instants. The transformation requires some mathematical computation at every location; a computer must be
involved with the measuring system. The measurement is to be repeated at different lines of the wave probe locations so
that it covers some region around the ship model
Our wave measuring technique enables us to obtain ζ 0, ζ1 and ζ2 so on at every location (x, y) in the expression of the
wave field
(1)
where the coordinate system moves with the average position of the ship model. The x−y plane coincides with the calm
water surface and the positive z is vertically upward; the origin is at FP of the ship model and the x axis directs rearward; ω
is the frequency of the ship model's oscillatory motion.
The first term on the right of (1) is the steady wave elevation the dominant part of which is the Kelvin wave pattern. It
will contain possibly the higher order effect resulting from the interaction of the oscillatory part. The second term is the
oscillatory part at the fundamental frequency and the third is of the second harmonics. Main part of the fundamental
frequency term will be the linear effect and the second harmonics part will be the second order effect.
The unsteady waves might be interacted with the steady part. It is however impossible to confirm it by experiment
because the unsteady waves without the steady part is not realistic to be compared with the one with it as long as the ship
has forward speed; the unsteady waves generated by the ship motions at no forward speed is physically other thing than
what we are concerned.
Consistency and repeatablity of the measured waves are almost perfect. It enables us to draw accurate wave contour
map with the results of the repeated measurement. The steady wave component is hardly affected by the existence of the
unsteady motion; the steady free surface elevation measured when the ship model is given oscillatory motion is in good
agreement with the one when it is towed on calm water at the same speed with the motion suppressed. One exception we
observed so far is a small steady depression of the free surface in very front of bluff bow produced by the effect of the wave
diffraction.
S175 model was forced to pitch at two different magnitudes expecting the larger nonlinearity with the larger
magnitude of motion. Our computerized wave measuring system enables us to obtain the instantaneous wave contour map
of the first harmonics ω component and the second harmonics 2ω component separately. The former is considered to
correspond to the first order effect and the latter the second order effect. Examples of the measured wave contour map are
shown in Figs.1 and 2.
Figure 1 is two snap shots of the contour map of the first order ζ1eiωt, the upper part of the figure represents the real
part of ζ 1 (cos component) and the lower part the negative of the imaginary part (sin component). Elevation is displayed
gradationally black-to-white as well as in the contour map. The elevation is normalized by the amplitude of vertical motion
at FP; this figure is the result for larger amplitude of the pitch, 0.00185 radian. This gives the vertical motion at the bow as
large as 70% of the draft.
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Fig. 1 Measured wave pattern (1st order)
SECOND ORDER WAVES GENERATED BY SHIP MOTIONS
Fig. 2 Measured wave pattern (2nd order)
145

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SECOND ORDER WAVES GENERATED BY SHIP MOTIONS 146
The ship speed is Fn =0.275 and the frequency ω of pitch normalized by the ship length L and the gravity constant g is
5.26. τ, the frequency normalized by the ship speed U and g, is 1.446.
With Fig. 1 and other result at smaller amplitude of pitch though not shown here, we may conclude that the measured
wave contour map at the fundamental frequency, when its altitude normalized by the pitch magnitude, is identical
regardless of difference in magnitude of the pitch motion. It suggests that Fig. 1 shows certainly a linear effect.
A wave system is noticeable in Fig. 1. The line connecting the peaks of the wave system makes about 21° with the x
axis. Wave pattern of Fig. 1 almost perfectly agrees with the predicted on linear assumptions as described later.
Figure 2 is the snapshots of the instantaneous free surface elevation of ζ2ei2ωt at the same condition of frequency of
pitch and the ship speed, and with the same normalization as Fig. 1. While the contour map can be drawn with this case, we
display the wave elevation only gradationally; it apparently exhibits the characteristics of the wave pattern more distinctly.
ζ2 measured at half the amplitude, not shown here, does not reveal so clearly the feature of the wave pattern. It suggests ζ 2
will be of the second order with respect to the amplitude of the motion.
Existence of the second order waves, whose magnitude is 15 to 20 percent of the first order's, was confirmed. Wave
pattern demonstrated in Fig. 2 is so distinct that it is certain that it has a clear physical meaning.
Two wave systems are seen in Fig. 2. One wave is almost like the wave system seen in Fig. 1; the line connecting the
peaks of the wave system makes about 21° with the x axis, though the wave length (the distance between two adjacent
peaks) is different from the one in Fig. 1. Other wave system has the narrower angle wake; the peak-line makes 13.5° with
the x axis.
We suppose that the second order waves have two components. The so-called bound component which has the same
phase speed as the first order wave (with an analogy of unidirectional 2D Stokes wave), and the free wave component
which satisfies the same dispersive relation as the first order wave.
Each effect of those two components in the measured waves is to be separated by utilizing the fact that their wave
lengths are different from each other. The free wave component will be mainly due to the nonlinearity in the ship body
condition and the bound component is a result of the nonlinearity in the free surface condition (again with an analogy with
Stokes wave). The former will correspond to the 13.5° wake in Fig. 2. The latter will be manifested as the 21° wake. This
conjecture will be confirmed more quantitatively in the next section.
3. ASYMPTOTIC CHARACTERISTCS
We study asymptotic characteristics of the oscillatory wave pattern (Eggers (1957), Becker (1958)). ζ1(x, y) at a
distance relatively far to the wave length is given in the form of
(2)
where
The direction θ that does not give real k1,2 is to be excluded from the integration of (2).
When is large, method of stationary phase will give a good estimate of the wave pattern (2) except
at a caustic (Iwashita (1990)): Wave elevation at a location (x, y) is contributed by a discrete number of wave components
whose direction θ are given as the solutions of
(3)
Equation (3) is rewritten into
(4)
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SECOND ORDER WAVES GENERATED BY SHIP MOTIONS 147
(5)
|y/x|=tan φ must be less than a value φ1,2 if (4) or (5) has real solutions θ. φ1,2 is the angle of the outer bound of the
wave system outside of which the wave does not exist in the asymptotic sense. Crests and trough of the k2 wave system
computed by (5) at Fn=0.275, τ=1.446 is shown in Fig. 3. The outer bound at 21° is seen clearly. The k1 wave is limited
within much narrower region (about 10° to the x axis).
Fig. 3 Asymptotic Wave Pattern
The wave length on the line φ1,2 (the distance of two adjacent peaks of wave along the line) will be given
(6)
where θc is the angle of elementary waves contributing to the wave elevation along φ=φ1,2, which is the solution of (4)
or (5) at |y/x |= tanφ1,2
When we look at Fig. 1 in the light of the knowledge of asymptotic waves, the k2 wave is visible but the k1 wave is
not. Actually the pattern shown in Fig. 3 is just what we see in Fig. 2. φ2 of Fig. 3 is 21.1° and λ2/L 0.642. They are the
same as the angle shown in Fig. 1 and the distance between two marks. It confirms that the wave system we see in Fig. 1 is
the k2 wave system.
We study a wave system of the narrow angle of 13.5° we see in Fig. 2 in the light of the linear theory. The wave system
in Fig. 2 is of the second order, yet the linear theory can capture one feature of it. As explained in the previous section, the
wave field of the second order will be interpreted to be composed of the free wave component that satisfies the body
nonlinear but the free surface linear conditions and the bound component that satisfies the free surface nonlinear but the
body linear conditions. At radiation problem we may consider the former as a linear wave with the second order body
boundary condition at the frequency of 2ω. Naturally the body condition is decided by the first order flow but imposed on
the equilibrium position of the body as long as the body surface vertically intersects the free surface. Asymptotic
characteristics of this wave may be analyzed in the linear way described above.
Parameters determining this wave are Fn= 0.275 and τ=2.892. These parameters give φ2=13.5° and the wave length on
the angle is 0.268, which perfectly agree with the observed on a wave system of the smaller angle wake and prominent
behind the ship in Fig. 2. This will not be coincidence because there is no other possibility giving the angle and the wave
length exactly. We may conclude this wave is caused by the body nonlinearity but with the linear free surface condition.
Other wave system at the angle φ2=21° seen in Fig. 2 will probably be the bound component originated from the free
surface nonlinearity.
4. COMPARISON WITH THEORETICAL PREDICTION
Validation of theoretical methods is attempted in terms of accuracy in predicting the contour map of the first order and
the second order waves obtained in our experiment. Here we employ the 2.5D theory for this purpose. This theory is
understood to be practical in providing relatively stable results and requiring less computational burden as far as a ship is
slender and at moderate to high speed; fully nonlinear time domain computation is done often with this ap
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SECOND ORDER WAVES GENERATED BY SHIP MOTIONS 148
proach. If we are to be certain of the accuracy of these computations, experimental validation at the higher order will be
necessary, though it has never been attempted.
Theoretical approach we use here for the 2.5D theory is not a fully nonlinear time domain computation but a
perturbation analysis up to the second order in the frequency domain. A reason is to make the second order effect more
distinct; direct comparison of the fully nonlinear computational results in time domain and the measured wave elevation
might not necessarily provide a quantitative information on the higher order effect. We have to recognize that there is a
demerit in the perturbation analysis that it does not predict really large nonlinear effect but moderate nonlinearity.
and ζ(x, y, z, t) represent respectively the velocity potential of the unsteady flow and the unsteady
Let
wave elevation generated by the ship motion. Conditions of them to be satisfied at the free surface z=ζ are
(7)
(8)
Suppose the expansions
and ζ2 are of higher order than the first terms and ζ1 with respect to the
and assume that the second terms
magnitude δ of ship motions. It is because we simply assume that the steady flow around the ship is uniform flow U into
the x direction. Then one can find the first and second order free surface conditions by substituting the expansions into the
free surface conditions (7) and (8) to separate the first and the second order terms from each others.
and ζ1, and (11) and (12) are the second order condition for
Equations (9) and (10) are the first order condition for
and ζ2.
(9)
(10)
(11)
(12)
All the conditions are to be satisfied on z=0
We introduce here a crucial assumption so that we may proceed to the next stage of analysis: the magnitude of the
slenderness parameters ε is independent of that of δ i.e. we retain δ 2 terms that are of the lowest order with respect to ε
despite that practically ε>δ .
We may assume for any flow quantity f
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Yet we retain U∂f/∂x in the first and second order free surface conditions above because of high speed U of the ship.
and ζ2
From now on our analysis is concerned with the oscillatory part of the flow i.e. we understand hereafter that
represent the oscillatory part of the second order. Our analysis is in the frequency domain and everything of the first order
is oscillating at the frequency ω . Consequently it is straightforward to show
(13)

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SECOND ORDER WAVES GENERATED BY SHIP MOTIONS 149
(14)
(15)
(16)
Substitution of those expressions into the first and second order free surface conditions respectively will derive the
conditions for ψ1,2 and η1,2
(17)
(18)
(19)
(20)
In deriving equations (19) and (20) we have retained the lowest order terms of ε among the terms of δ2. The same
reasoning leads to
(21)
The body boundary conditions are derived in the same manner based on the same assumptions.
(22)
(23)
The body conditions (22) and (23) are both imposed on the surface of the ship at the equilibrium position. n is the
normal directed to the fluid on the ship's sectional contour and nz is the z component of the normal. x0 is the x coordinate of
the longitudinal center of mass around which pitch occurs. All the formulation above is under the condition that the ship
form is wall sided i.e. the hull surface intersects vertically the z=0 plane.
Equations (19), (20) and (23) suggest that ψ2 and η2 will be decomposed into tw parts, one part satisfying the
homogeneous free surface conditions corresponding to (21) and (20), and the body condition (22), and other part satisfying
the free surface condition (19) and (20) and the body condition (23) with the right hand side replaced by 0. The former will
be the free wave part due to the body nonlinearity and the latter the bound part due to the free surface nonlinearity.
Our computation of both parts revealed that the wave elevation by the former part is so small compared with the
latter's effect. While this is true only with the wave elevation close to the ship and the free wave part due to the body
nonlinearity is significant behind the ship model as shown in the previous section, we may ignore it in our computation of
the wave elevation relatively close to the ship. We concentrate hereafter our computation only on the bound term due to the
free surface nonlinearity. When we understand ψ2 and η2 stand for this term, then the body condition will be
(24)
The free surface condition (17) is integrated to march forward the value of ψ1 on z=0 in x direction. The condition (18)
updates the free surface elevation η1 at new x. Determination of ψ1z on z=0 and ψ1 on the ship' section at new x is by solving
the two dimensional boundary value problem for ψ1 with ψ1n on the ship section and ψ1 on z=0 given. Subscript denotes the
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differentiation into its direction.
Numerical implementation of this approach for the first order solution is straightforward (for example Wen (1997),
Faltinsen and Zhao (1991)). The 2nd order Runge-Kutta scheme was used to forward the solution from on section to next

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SECOND ORDER WAVES GENERATED BY SHIP MOTIONS 150
section; marching step ∆x in the x direction was chosen to be L/80. Two dimensional boundary value problem at each
sectional plane, when ψ1 is given on z=0 and ψ1n given on the sectional contour, was solved with segment size 4×103L on
z=0 on one side and every sectional contour divided into 30 segments. Behavior of ψ1 on z= 0 away from the body surface
(|y|>0.5L) is approximated by vertical dipole behavior (Wen (1997)). We employed rather classical initial condition
ψ1= η1=0 at x=0.
After ψ1 and η1 are obtained, we solve for ψ2 and η2 in almost similar manner. Equations (19) and (21) are integrated
to forward ψ2 and η2.
We need to evaluate the forcing terms due to ψ1 and η1 on the right hand side of these equations. We must rely on
numerical differentiation for evaluating ψ1zx as ψ1z is given on z=0. We evaluated ψ1zz by solving a new boundary value
problem for ψ1z when ψ1z are prescribed on both the ship section and the free surface.
The condition imposed at large |y| is determined as follows. The forcing terms behavior at large |y| is already known
because ψ1's was assumed of vertical dipole at the first order computation. For ψ2 we assume the slowest attenuation
conceivable at large |y| i.e. we assume the same behavior as ψ1.
The singularity of ψ1 at the intersection of the body surface and z=0 will be Z 2 log Z where Z is the complex coordinate
with the origin at the intersection (Cointe, Molin and Nays (1988)). It leads to the singularity log Z of ψ1zz. We avoided the
difficulty in solving the boundary value problem in each crosssectional plane by collocating not at the intersection but at
locations very close to it.
Figures 4 to 13 are the comparison of the measured and the computed wave elevation ζ1eiωt in cross sectional planes at
several different x/L. The measured is taken from the contour map shown in Fig. 1. Figures at the left side of each page
show the wave elevation at the instant of ωt=0 (cosine component) and the right figures the wave elevation at the instant of
ωt=π/2 (sine component). In those figures Theoretical (3D) depicts the computed by three dimensional and desingularized
Rankine panel method (Sclavounos (1996)) using the double-model flow as the basic steady flow.
First of all we see that the wave elevation normalized by the amplitude of the ship motion at x=0 is almost identical
despite the difference in the amplitude (white and black circles). It implies that the measured must be accounted for by
linear theories. While both the theoretical 2.5D and 3D capture the general features of the wave elevation, either of the
theoretical does not succeed in predicting accurately the height of wave elevation; the measured is higher in the crest and
deeper in the trough than the theoretical predictions. It is not supposed to be caused by the 2.5D's incorrect initial condition
at x=0 because even the 3D not suffering from this defect fails to predict them.
Figure 14 is the contour map predicted by the 3D computation. Once again we see that the whole feature of the
theoretical agrees perfectly with that of the measured shown in Fig. 1.
In Fig. 15 to 23 we compare the theoretical prediction of the second order wave ζ2ei2ωt given by the 2.5D theory with
the measured corresponding to the result shown in Fig. 2. The format for the comparison is the same as for the first order
except that the sine component represent the wave elevaation at 2ωt=π/2. It appears that the general feature of the wave
elevation is predicted by the 2.5D theory. It means that the second order wave elevation observed close to the ship might be
the bound component caused by the free surface nonlinearity as suggested in the previous section.
The relatively large wave elevation of the second order is not predicted by our theoretica method. It may not be
surprising because the prediction of the first order wave is not so correct in the magnitude of the wave.
The theoretical result on the second order wave elevation far behind the ship is not available (the 2.5D theory is not
capable of computing the wave field behind the ship's stern). We can not confirm our statement on the narrow angle wake
we see in Fig. 2.
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SECOND ORDER WAVES GENERATED BY SHIP MOTIONS 151
5. CONCLUDING REMARKS
We presented instantaneous contour maps of the first order and the second order waves generated by a ship motion.
They are not visible on usual experiment at water tank. Those contour maps were made available by our computerized
technique for wave measurement.
Validation of a theoretical method was attempted in terms of accuracy of predicting those contour maps of the first
order and the second order waves.
Theoretical prediction of the first order wave is not entirely accurate despite the slender hull form of the ship model.
While pattern of the wave contour is accurately predicted, discrepancy exhibits itself in the predicted wave height being
much smaller than the measured wave height near the hull surface. The discrepancy suggests that our theoretical model is to
be improved since we believe the measured first order wave is of “the first order”.
Two components are clearly distinguished in the second order measured wave pattern, one satisfying the linear
dispersive relation and the second order body condition, and other satisfying the nonlinear free surface condition and the
homogeneous body condition. The latter apparently is dominant in the wave elevation at the location close to the hull
surface.
A theoretical approach to compute the second order wave contour was proposed. The prediction of general feature of
the wave contour does not seem so inaccurate. However the magnitude of the measured wave is larger than the predicted.
This will be inevitable when we consider inaccuracy in the prediction of the first order; the second order wave was
computed utilizing the first order prediction in our theoretical approach.
Practical implication of our result to sea-keeping study is to be studied in future.
ACKNOWLEDGEMENT
The authors acknowledge the help of Prof. H.Iwashita, Hiroshima University, in providing his computational result by
Rankine panel method.
REFERENCES
Becker E (1958) Das Wellenbild einer unter der Oberfläche eines Stromes Schwerer Flüssigkeit Pulsierenden Quelle, Z.Angewandte, Mathematik und
Mechanik, Bd38
Cao YS, Schultz W and Beck R (1994) Inner-angle Wavepackets in an Unsteady Wake, Proc. 19th Symposium on Naval Hydrodynamics, Seoul
Cointe R, Molin B and Nays P (1988) Nonlinear and second order transient waves in a rectangular tank, proc. BOSS'88, Trondheim
Eggers K (1957) Über das Wellenbild einer Pulsierenden Störung in Translation, Schiff und Hafen, Heft 2
Faltinsen O M and Zhao R (1991) Numerical predictions of ship motions at high forward speed, Phil. Trans. Royal Soc. Lond. A 334
Iwasita H (1990) Green function method for ship motions at forward speed, PhD Thesis, Kyushu University
Ohkusu M (1998) Validation of Theoretical Methods for Ship Motions by Means of Experiment, Proc. 22nd Symposium on Naval Hydrodynamics,
Washington DC
Ohkusu M and Wen G (1996) Radiation and Diffraction Waves of a Ship at Forward Speed, Proc. 21st Symposium on Naval Hydrodynamics, Trondheim
Sclavounos P (1996) Computation of wave ship interaction, Advances in Marine Hydrodynamics edited by M Ohkusu, Computational Mechanics
Publication UK
Wen G C (1997) Theoretical prediction of sea-keeping of high speed ships, PhD Thesis, Kyushu University
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Fig. 8 Wave elevation (1st order, Cos)
Fig. 6 Wave elevation (1st order, Cos)
Fig. 4 Wave elevation (1st order, Cos)
SECOND ORDER WAVES GENERATED BY SHIP MOTIONS
Fig. 9 Wave elevation (1st order, Sin)
Fig. 7 Wave elevation (1st order, Sin)
Fig. 5 Wave elevation (1st order, Sin)
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Fig. 12 Wave elevation (1st order, Cos)
Fig. 10 Wave elevation (1st order, Cos)
fig. 14 3D-Computed wave pattern (1st oder)
SECOND ORDER WAVES GENERATED BY SHIP MOTIONS
Fig. 13 Wave elevation (1st order, Sin)
Fig. 11 Wave elevation (1st order, Sin)
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Fig. 19 Wave elevation (2nd order, Cos)
Fig. 17 Wave elevation (2nd order, Cos)
Fig. 15 Wave elevation (2nd order, Cos)
SECOND ORDER WAVES GENERATED BY SHIP MOTIONS
Fig. 20 Wave elevation (2nd order, Sin)
Fig. 18 Wave elevation (2nd order, Sin)
Fig. 16 Wave elevation (2nd order, Sin)
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Fig. 23 Wave elevation (2nd order, Cos)
Fig. 21 Wave elevation (2nd order, Cos)
SECOND ORDER WAVES GENERATED BY SHIP MOTIONS
Fig. 24 Wave elevation (2nd order, Sin)
Fig. 22 Wave elevation (2nd order, Sin)
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SECOND ORDER WAVES GENERATED BY SHIP MOTIONS 156
DISCUSSION
X.-B.Chen
Bureau Veritas, France
I am interested in the steady part of ship waves you measured. From the second-order theory, there is a contribution of
second-order to the steady part, which maybe the difference of wave pattern between that in calm water and the total steady
part in the unsteady measurement.
Is it true? If so, have you any information (results) on the steady part of measurement in unsteady tests?
AUTHOR'S REPLY
We are too interested in the second order interaction of the steady and unsteady ship waves. First of all we must know
that the effect of the unsteady waves upon the steady ones will be observable but not vice versa. Reason is that the unsteady
waves not under the influence of the steady waves is not realistic for the case of our concern i.e. when a ship has forward
speed..
While we have not reported the result here but certainly we measured the steady ship wave without the ship motion or
incident waves, and compared it with the steady wave component of the wave motion when a ship is forced to oscillate or
in the incident waves. What we have found is that the steady wave is hardly affected by the unsteady wave at least for the
magnitude of the oscillation we adopted in our measurement In other words the steady component of the wave motion when
the ship is oscillating agrees with the steady wave at no ship motion.
One exception is that the steady wave in front of very blunt bow is depressed slightly when the ship is in incident
waves.
the authoritative version for attribution.