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A Mode] for the Generation and Evolution of an
~nerAngle Soliton In a Kelvin Wake
R. Hall, S. Buchsbaum
(Science Applications International Corporation, USA)
ABSTRACT
We develop a simple model for the
generation and evolution of an inner
angle soliton in a Kelvin wake. The
generation is modeled as an interference
maximum due to a sourcesink pair and
the evolution is modelled using the
nonlinear Schrodinger equation. The
model is used to explain the results of a
recent experiment. Although some of the
parameters in the model are fit to
experimental data, the values of the
parameters are physically reasonable,
thus we conclude that the model captures
the essential physics.
1. INTRODUCTION
It has long been observed that
rays appear inside the cusp line within
the diverging portion of the Kelvin wake
of a ship. The simplest explanation for
these rays is an interference pattern due
to the superposition of the wave fields
generated by the bow and the stern.
Interest in these rays has increased in
recent years because they are a possible
explanation for some of the long bright
lines observed in imagery of ship wakes
collected from space [1].
Since these rays can form as a
result of an interference pattern, they
will appear in any model for the Kelvin
wake of a ship as long at that model is
capable of calculating the farfield wake.
If the wave amplitude in the ray is small,
or if the Kelvin wake model uses a
linearized form of the free surface
453
boundary condition, then each ray
inside the cusp line diverges linearly as
it propagates aft: the width increases
linearly with distance aft and the
amplitude decreases as the inverse of the
square root of the distance aft (the cusp
line, of course, diverges slower).
In a recent experiment [2,3], a ray
in the Kelvin wake of a Coast Guard
cutter was studied. Figures 1 and 2 show
the near and farfield wakes,
respectively. The ship speed was 15
knots (the Froude number was 0.5) and
the ray appeared at an angle of 10.9
degrees from the wake cerIterline, which
is close to the angle predicted by the
interference model discussed above.
However, the farfield evolution of the
ray was not consistent with linear
theory. Although Figure 2 shows that the
width of the ray increased somewhat over
the first few ship lengths aft of the stern,
by a few hundred meters aft the width
did not increase and the ray was shown to
be an oblique nonlinear solitary wave
packet. Related theoretical work [4] on
the nonlinear evolution of a wave packet
showed similar behavior.
In this paper we will develop a
simple physical model for the generation
and evolution of the innerangle ray
observed in the experiment. In Section 2,
the generation of the ray will be modeled
using a simple linear theory in which the
bow of the cutter is represented by a
source and the stern is represented by a
sink. In Section 3, the evolution of the
ray into an oblique nonlinear solitary
wave packet will be modeled using a two
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::::::::::::::
I:
:::::::::::::::
: ~
Ad ::
::
::: 
Figure 1. The nearfield wake of the Coast Guard Cutter Point
grower. The ship speed is 15 knots. The lower photo
shows the port side cusp line and soliton. The soliton
intersects the lower left hand corner of the photo.
454
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dimensional nonlinear Schrodinger
equation for the complex packet
envelope. Compared to the current state
oftheart in modeling the Kelvin wakes
of ships, our model is crude, but it has
the virtue of capturing the essential
physics in a simple way. In the
conclusions we will review the
limitations of the model and will suggest
future improvements.
Included in the Appendix is brief
review of Kelvin wake kinematics,
including the geometry of the
interference rays generated by a source
sink pair, and a list of the parameters of
the soliton observed in the experiment.
2. LINEAR THEORY
Our model for the generation of
the ray consists of a sourcesink pair in
a potential flow with linearized free
surface boundary conditions. The
interior continuity equation is
V2~=SB +Ss,
the linearized dynamic boundary
conditions on z=0 are
A, Unix = P 877 (2)
and
P=O,
(3)
and the linearized kinematic boundary
condition on z=0 is
ql  U0x = Oz
(4)
The coordinate system is fixed to the
ship, which is moving with speed ~ i n
the positive x direction. y is positive to
port and z is positive upward. ~ is the
velocity potential for the perturbation to
the uniform oncoming flow, ~ is the free
surface elevation relative to the mean
ambient level z=0, and P is the surface
pressure divided by the density. The
source at the bow, SB, and the sink at the
stern, Ss, are given by point
singularities:
SB = UA §(X  a)~(y) 8(z  c) ~ 5 )
and
SS=SB, with a  a. (6)
The depth of the source is c and the
distance between the source and the sink
is 2a . Note that x = 0 is amidships;
hereinafter, when we refer to distance
aft, we mean distance aft of amidships, or
x. The volume flux emitted by the
source is UA. In the limit that a))~, A
is the crosssectional area amidships of
the region occupied by the fluid emitted
from the source. Since the source is
steady and the coordinate system is fixed
to the ship, the solution of Equations (1~
(6) will be steady.
In the experiment discussed in
[ 1], the ship was a Point Class Coast
Guard cutter (Point grower) and the ship
speed was 15 knots. We obtain the
required parameters for our model from
Coast Guard Drawing No. 82(D) WPB
07001, "82 Foot Patrol Boat (Class D)
Docking Plan." The forward and aft (at
the stern) perpendiculars are defined
using a draft of 5'3" above the base of
the propeller. The distance between them
is 78' (23.8 meters), which is the
waterline length for this draft. The
crosssection nearest amidships shown
in the plan is a bulkhead located 36' aft
of the forward perpendicular. Using the
5'3" waterline, this crosssection has a
beam of 4.4 meters at the waterline, a
draft of 1.0 meters above the bottom of
the bare hull, and a crosssectional area
of 3.3 square meters below the waterline.
We do not know the actual static
waterline or the running sinkage and
trim for the experiment.
For our model, we will use U = ~ 5
knots and we will let A be 3.3 square
meters, which is the submerged cross
sectional area at the 36' bulkhead. The
depth of the source, c, will be half the
draft at the bulkhead, or 0.5 meters. We
could let the distance between the source
456
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and the sink, 2a, be the 23.8 meter
waterline length, but we chose a smaller
value of 22.2 meters instead. This value
is chosen so that the position of the ray
in the model is fit to the observed
position of the soliton in the experiment'
as shown in the Appendix. It is
reasonable that the value of 2a be less
than the waterline length because the
wave generation regions lie between the
forward and aft perpendiculars.
The solution of Equations ( 1)(6)
is given in terms of a superposition of
two of the Green functions defined in
Equation (7) of [5]. Figure 3a shows the
surface elevation out to a distance of 400
meters aft. Only the far field portion of
the solution, the portion given by
Equation (7b) of [5l, is shown. The first
ray inside the cusp line is the ray that
evolves into the soliton. The ray appears
at an angle of 10.9 degrees from the wake
centerline and the wave vector at the
peak of the ray is (kx, k,)=(0.426, l.Ol)m~i
(see the Appendix). Unfortunately, there
are two problems with this solution.
First, the wave slopes within the
diverging portion of the Kelvin wake are
unrealistically large. This can be seen
more clearly in Figure 4, which shows
the crosstrack slope ,7, at 100 meters
aft. Second, two additional rays, which
were not observed in the experiment,
appear inside the first ray. Both of these
problems can be corrected by replacing
the point sources at the bow and stern
with distributed sources. This is a
reasonable modification of the model
because the bow and stern regions are of
finite extent. For convenience, we chose
a Gaussian distribution in the horizontal
plane,
SB = 2 exp{ ( 2 ~ 2 Y ~(z  c) ~ 7 )
and
SS=SB, with a  a, (8)
because it reduces each spectral
component in the solution by the simple
factor
exp(~2k2 /2),
where k is the horizontal wavenumber.
Figure 3b shows the surface elevation for
the solution of Equations ( 1~~4), (7) and
(8), with a value of 1.2 meters for cr.
This particular value is chosen so that
the amplitude of the soliton obtained in
Section 2 agrees with the observations.
The value is reasonable because it yields
a value of 2.8 meters for the full width at
half height of the Gaussian, which is
consistent with the length scales in the
bow and stern. Figure 4 shows the cross
track slope at 100 meters aft for this
solution. Note that the values are now
physically reasonable.
1.5
LLI 1
53 0.5
~ O
Jim 0.5
1
1.5
 POINT 
 DISTRIBUTED 
X=100M
~ ~ ~ I ~ I ~
0 5 1 0 1 5 20 25 30 35 40
Y(M)
Figure 4. A comparison of the cross
track slope at 1 00m aft for the two types
of sources.
3. NONLINEAR THEORY
We model the evolution of the ray
into an oblique nonlinear solitary wave
packet by using a timedependent, two
dimensional nonlinear Schrodinger
equation for the complex packet
envelope. One procedure for doing this
is to take the crosstrack profile of the
ray at a couple of ship lengths aft and
457
OCR for page 453
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OCR for page 453
use it as an inflow boundary condition
for the nonlinear Schrodinger equation.
A second procedure, which is suitable
when the generation model is linear, is to
bandpass the source term for the narrow
band of spectral components that
generate the ray, and then include this
narrowband source term in the
nonlinear Schrodinger equation. We
adopt this second procedure.
We begin by returning to the
generation model and separating the
velocity potential ~ into a double body
term (DB and a wave term
4' = (DB + ¢.
(10)
The doublebody flow satisfies the
interior equation
V2$DB =SB +SS (1 1)
and the bondary conditions
{DBZ = 0
and
PDB = U¢DBX
on z=O. The wave field satisfies the
interior equation
v2¢ = o
and the boundary conditions
¢,U¢x =PDB grl (15)
and
1,  UNIX = {z
(16)
on z=O. We solve Equations (11~(13) for
PDB and bandpass the result so that it
only contains the narrow band of spectral
components PDB that generate the ray.
The rigorous procedure for
bandpassing PDB involves finding the
wavevectors of the two nodes in the
Kelvin wake that bound the ray, then
using a top hat filter to pass all of the
spectral components along the Kelvin
wake dispersion curve in between these
nodes. For simplicity, we adopt an
alternate approximate procedure: we
replace the top hat filter with a Gaussian
filter, we approximate the spectrum of
PDB in the vicinity of the filter by a
uniform value for the source and another
uniform for the sink, And we set each
uniform value equal the value at the
spectral component corresponding to the
peak of the ray. The resulting
expression for PDB is
PDB = Re tH exp[i (ken + kyy)~l, ( 17)
where
H=HB+HS'
(12) HE= k UAexp[ 2 kci (kxa)] FB,
( 13) and
(18)
(19)
Hs=HB, with a  a, (20)
(14) and where FB is given by
CThCTt,, ~ CT,` 2(Xa)2+~t,,2y2 1
B 2~ ~ ~
(21)
(kx, ky)=(0.426, l.Ol)m~t is the wavevector
of the Kelvin wave at the peak of the ray,
and k = l.lOm~ and ~ = 3.23rad / s are the
wavenumber and frequency, respectively
(see the Appendix). We choose a value of
~, = k' /2 for the standard deviation of
the Gaussian filter in the ky direction.
This is the most important direction
because it is roughly parallel to the
459
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Kelvin wake dispersion curve in the _ iamb I',12 .~2
(kX kit) plane. This value of a~ does a
reasonable job of passing the spectral
components that lie between the two
nodes that bound the ray, as well as
excluding the spectral components
beyond the nodes (see the Appendix).
The value of a~ must be large enough to
pass the spectral components that lie
between the nodes. We let it equal cry,
for convenience. Although this is a few
times larger than necessary, the
additional filter width is normal to the
Kelvin wake dispersion curve and does
not contribute to the farfield wake.
As suming that the surface 0.15
elevation has the narrowband form
O 0.05~
0.05
LL
o. 1 5
,7 = Re {BeXp[i (fix+ kyy)]~' (22)
we use standard procedures (see [6]) on
Equations ( 14~~17) to obtain the linear
Schrodinger equation for the complex
packet envelope:
B.  fJBx =  28 (kxBx + kyBy )
+ 8~ 3 t(2k~2~k~2)Bxx
6kyk'Bxu, + (2kx2  k'2 )Bn
+i~H
2g
(23)
The solution of this equation is shown in
Figure 5a. We note that the ray in Figure
3b appears somewhat larger that the ray
in Figure 5a only because the former ray
is superimposed on the transverse wave
(note the different gray scales in the
figures). The actual wave amplitudes are
comparable.
The nonlinear Schrodinger
equation is obtained by using the
nonlinear versions of Equations (15) and
(16) in the derivation of Equation (23)
(see [6]~. The wellknown result is to add
the term
460
(24)
to the right hand side of Equation (23~.
This term represents the increase in the
phase speed of the wave due to
nonlinearity. The solution of Equation
(23), with (24), is shown in Figure Sb.
The nonlinear term arrests the
dispersion of the ray yielding an oblique
nonlinear solitary wave packet. Figures
6a and 6b compare the linear and
nonlinear Schrodinger equation results
at 1 OOm and 400m aft, respectively.
X=4noM
. . .
50 60 70 80 90 100 1 10
Y(M)
0 25
0.15


0 0.05
~ 0.05
us
X=1 OOM
0 26 ~, ,
0 1 0 20 30 40 50 60
Y(M)
 NONLINEAR 
 ~ LINEAR 
Figures 6a (bottom) and 6b (top). A
comparison of the solutions of the linear
and nonlinear Schrodinger equations.
We note that we have performed
additional calculations with the model
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OCR for page 453
using reduced forcing. If the peak slope
of the soliton just aft of the sourcesink
region drops to 0.1, then a soliton does
not form by 400 m aft. The required
conditions on the initial packet for
soliton formation are discussed in [3].
4. CONCLUSIONS
We have developed a simple model
for the generation and evolution of the
innerangle ray observed in the
experiment. The generation is modeled
as an interference maximum due to a
sourcesink pair and the evolution is
modelled using the nonlinear
Schrodinger equation. Although some of
the parameters in the model are fit to
experimental data, the values of the
parameters are physically reasonable,
thus we conclude that the model captures
the essential physics.
We have three recommendations
for future research. First, the generation
model should be improved using one of
the stateoftheart Kelvin wake models,
with the objective of replacing our fit
with a true prediction. This is
particularly important at low Froude
number, where there are a larger number
of interference rays and a model with a
detailed representaion of the hull form
and the nearfield flow is essential.
Second, the output from the generation
model should be used as an inflow
condition for the farfield equations,
rather than using the doublebody
pressure forcing, as in our evolution
model. The doublebody pressure
forcing is only suitable when the
generation model is linear. Third, the
farfield model should be upgraded to
include a higher order expansion [6] or a
fullynonlinear equation. This will
improve the approximation of the linear
dispersion relation near the nodes and
will add a nonlinear correction to the
group velocity.
ACKNOWLEDGEMENTS
We thank Carl Scragg for the use
of some of his linear Kelvin wake codes.
REFERENCES
Munk, W.H., ScullyPower, P., and
Zachariasen, F., " Ships from space, "
Proceedings of the Royal Society of
London, Series A, Vol. 412, 1987, pp.
23 1 254.
2. Brown, E.D., Buchsbaum, S.B.,
Hall, R.E., Penhune, J.P., Schmitt, K.F.,
Watson, K.M., and Wyatt, D.C.,
"Observations of a nonlinear solitary
wave packet in the Kelvin wake of a
ship," Journal of Fluid Mechanics, Vol.
204, 1989, pp. 263293.
3. Buchsbaum, S.B., "Ship Wakes and
Solitons," Ph D Thesis, University of
California, San Diego, 1990.
4. Akylas, T.R., Kung, T.J., and
Hall, R.E., "Nonlinear Groups in Ship
Wakes," Proceedings, 1 7th ONR
Symposium on Naval Hydrodynamics, The
Hague, the Netherlands, National
Academy Press, Washington, D.C., 1988.
5. Noblesse, F., "Alternative
integral representations for the Green
function of the theory of ship wave
resistance," Journal of E n g i n e e r i n g
Mathematics, Vol. 15, No. 4, 1981, pp.
24 1  265.
6. Dysthe, K.B., "Note on a
modification to the nonlinear
Schrodinger equation for application to
deep water waves," [~2ceedings of the
Royal Society of London, Series A, Vol.
369, 1979, pp. 1051 14.
APPENDIX
In this appendix we review Kelvin
wake geometry. The condition that a wave
is steady with respect to the ship is
k=U,
(Al)
where the dispersion relation for the
wave iS
m~>0 k_4kx2+k'2 >0. (A2)
462
OCR for page 453
The group velocity of the wave
determines its position in the wake,
given by the angle a with respect to the
wake centerline:
8~/ Ok
U  de i at
(A3)
The condition for an interference
maximum in a sourcesink pair model is
2akx ens, n odd.
(A4)
The condition for an interference
minimum is Equation (A4) with n even.
The observed frequency of the
soliton in the experiment was 3.28
rad/sec [21. Using a ship speed of 15
knots (7.7 m/s), the above formula imply
that (kx, k')=(0.426, l.Ol)m~~ and a =10.9
deg. Equation (A4) then yields a value of
2a=22.2m for the distance between the
source and sink that yields a ray at the
observed angle. Using this distance,
Equations (A4) and (A3) yield a value of
15.9 degrees and 8.3 degrees for the
nodes on either side of the ray peals. The
k, wavenumbers of these nodes are 0.393
m~l and 1.854 ma, respectively.
Based upon an average of 24 runs
beyond 0.5 km aft [2], the spatial width
of the feature was 8.9m (measured at 1/e
of the peak) and the peak wave amplitude
was 15.1 cm (this was 1.1 times the
theoretical soliton value calculated from
the other parameters). An average of the
runs at 0.5 km aft yielded a peak
amplitude of 20 cm.
463
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