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OCR for page 465

Near-Field Nordinearities and Short Far-Field Ship Waves
F. Noblesse, D. Hendrix (David Taylor Research Center, USA)
ABSTRACT
The short divergent waves in the steady wave
pattern of a ship are analyzed on the basis of a
linear far-field flow representation and a nonlinear
near-field flow approximation. More precisely, the
far-field wave spectrum is determined in a simple
and practical manner by means of a waterline-
integral approximation obtained from a modified
Neumann-Kelvin integral representation; and the
nonlinear near-field flow along the ship waterline
is determined via a nonlinear correction defined by
a simple analytical expression. Numerical
calculations for the Wigley hull predict short
divergent waves too steep to exist in reality within
a significant sector in the vicinity of the ship track.
The predicted waves also exhibit a well-defined
peak at an angle from the ship track equal to about
1° to 2°.
INTRODUCTION
The ongoing search for explanations of the
features displayed by remote-sensing images of ship
wakes has prompted the formulation of various
alternative theoretical hypotheses. One such
hypothesis is that some features of ship wake radar
images might be attributable to characteristics of the
steady Kelvin wave pattern.
Efforts to determine whether the pattern of
steady far-field waves generated by a ship does in
fact exhibit any notable property capable of causing a
corresponding identifiable feature in remote-
sensing images have motivated a number of recent
studies of the Kelvin wake, including Scragg [1i,
Barnell and Noblesse [2i, Keramidas and Bauman
[3i, Milgram t4], and Trizna and Keramidas [5~. All
these numerical studies are based upon a highly-
simplified hydrodynamic model, namely the
zeroth-order slender-ship approximation proposed
in Noblesse [6~. It is shown in Baar t7i, Andrew,
Baar and Price t8i, Lindenmuth, Ratcliffe and Reed
[9], and Noblesse, Hendrix and Barnell t10] that the
first-order slender-ship approximation provides
fairly realistic predictions of the wave profile, the
465
near-field wave pattern and the long waves in the
wave spectrum. However, the zeroth-order
slender-ship approximation is a poor
approximation to the Neumann-Kelvin theory in
the short-wave limit, as is indicated by the
numerical results depicted in Figure 2a in [10] and is
confirmed by the results obtained further on in this
study. The wave calculations reported in [1]
through [5] therefore are unlikely to provide
realistic representations of the short waves in the
wave pattern of a ship.
Earlier numerical calculations of the wave
spectrum or the wave pattern of a ship may be
found in the literature, e.g. Sharma [11] and Tuck,
Collins and Wells [12~. However, these calculations
are also based upon a highly-simplified
hydrodynamic model, namely the Michell thin-
ship approximation. Similarly, the far-field wave
calculations of Ursell [13] correspond to an
elementary free-surface pressure singularity.
The steady wave pattern and wave spectrum of a
ship thus have been relatively little studied, and are
ill known. In particular, the asymptotic behavior of
the wave-spectrum function in the short-wave
limit is not known, and it is not known whether
this asymptotic behavior might explain the
common observation that the wake of a ship in the
vicinity of the track exhibits no short divergent
wave.
The short waves in the wave-spectrum function
and the far-field wave pattern of a ship are
examined in the present study within the context of
a somewhat more realistic hydrodynamic model
than the Michell thin-ship theory and the slender-
ship approximation used in the previously
mentioned numerical studies. More precisely, the
theoretical framework adopted in the present study
is that provided by the linearized Neumann-Kelvin
flow representation. This theory defines, via well-
known formulas, the steady wave pattern (and the
wave resistance) of a ship in terms of the wave-
spectrum function, which is defined in terms of the
flow at the ship mean wetted-hull surface.

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Accurate theoretical predictions of the steady wave
spectrum of a ship therefore are necessary for
obtaining reliable wave-signature (and wave-
resistance) predictions.
However, accurate numerical calculations of the
wave-spectrum function cannot readily be obtained
because this function is defined as the sum of two
integrals, namely a line integral around the ship
mean waterline and a surface integral over the ship
mean wetted-hull surface, which very nearly cancel
out in the manner recently shown in Noblesse, Lin
and Mellish t14~. Inaccuracies which inevitably
occur in the numerical evaluation of the waterline
and hull integrals cause imperfect cancellations
between these two integrals and correspondingly
large errors in their sum. The foregoing
cancellation phenomenon and the resulting
numerical inaccuracies are especially acute for the
short waves in the wave spectrum.
A mathematical remedy to this fundamental
numerical difficulty is presented in [14~. The
remedy consists in an alternative mathematical
expression for the wave-spectrum function. This
alternative expression defines the wave spectrum
as the sum of modified waterline and hull
integrals. No significant cancellation occurs
between these modified integrals, which are of the
same order of magnitude as the wave-spectrum
function defined as their sum. The large
cancellations occurring between the waterline
integral and the hull integral in the usual
expression for the spectrum function thus are
automatically and exactly accounted for, via a
mathematical transformation, in the alternative
new expression given in [14~. This new expression
for the wave spectrum thus is considerably better
suited than the usual expression for accurate
numerical calculations, notably for the short waves
in the spectrum.
Nevertheless, the integral representation for the
wave spectrum given in [14] is not suitable for
evaluating the very short divergent waves. For
instance, the use of this (or indeed any similar)
integral representation for the numerical
calculation of ship waves with wavelength between
5 cm and 40 cm, corresponding to backscattering of
the electromagnetic waves in typical systems used
for remote sensing of ship wakes, would require an
exceedingly large number of extremely small panels
for representing the hull surface. In fact, such short
waves can only be evaluated analytically, via a
short-wave asymptotic approximation of the
integral representation of the spectrum function
given in [14~.
Reliable predictions of the short waves in a ship
wave spectrum are quite difficult to obtain because
of the significant numerical difficulties mentioned
in the foregoing, as well as for yet another reason.
This second source of difficulties stems from the
fact that short far-field ship waves are closely
related to the velocity distribution along the ship
mean waterline, especially the fluid velocity at the
ship bow and stern, as is shown further on in this
study. However, existing near-field-flow
calculation methods (so-called nonlinear methods
included) are unable to provide realistic velocity
predictions at a ship bow and stern because they
cannot model the strongly nonlinear flow in the
immediate vicinity of these points, as was recently
shown in Noblesse, Hendrix and Kahn [15~. The
nonlinear analytical/experimental and
analytical/numerical velocity distributions along
the mean waterline of the Wigley hull defined in
[15] are used in the present study for predicting the
short divergent waves generated by the Wigley
hull.
FOURIER REPRESENTATION OF THE WAVE
PATTERN
The wave potential owed at any point ~ = (t, it,
~ < 0) behind the stern of a ship advancing at
constant speed in calm water can be defined in
terms of a Fourier representation, as is well known.
Specifically, (20) in [14] yields
away = (2/11) .1; exp~v2(p2)cos~v2llpt)
Im exp~iv24p) K(t) aft, (1)
where the wave potential few and the coordinates
I,, 11, ~ are nondimensional with respect to the ship
length L and speed U. and v is the inverse of the
Froude number F; we thus have
v = 1/F with F=U/(gL)~/2, (2a,b)
where g is the acceleration of gravity. Furthermore,
p in (1) is related to the Fourier variable t as follows:
p = ~l+t2~/2. (3)
Finally the (nondimensional) function K(t) in (1) is
the wave-spectrum function. The wave potential is
defined by (1) in terms of a familiar Fourier
superposition of elementary plane waves
propagating at angles ~ from the ship track (the x-
axis ~ given by
. tang = t. (4)
The amplitudes of these elementary plane-wave
components are essentially given by the spectrum
function K(t), which thus contains essential
information directly relevant to a ship's signature
(and wave resistance). This study is concerned with
the numerical/analytical evaluation and the
behavior of the spectrum function K(t) in the short
wave limit t ~ so (0 ~ 90°~.
It is convenient and useful to express the
spectrum function K(t) as the sum of two terms as
follows:
K(t) = Knit) + Knit),
466
(5)

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where Ko represents the zeroth-order slender-ship
approximation and Kit the Neumann-Kelvin
correction term in the Neumann-Kelvin
approximation Ko+ Kit . More precisely, the
spectrum function Ko+ Kit corresponds to the
usual linearized Neumann-Kelvin approxima-
tion, in which the nonlinear terms in the free-
surface boundary condition are neglected. These
nonlinear terms result in an additional term in (5),
defined by an integral over the mean free surface
[6,14], which is ignored in this study. The slender-
ship approximation Ko is defined explicitly in
terms of the value of the Froude number and the
hull shape, whereas the Neumann-Kelvin
correction Kit also depends on the value of the
tangential fluid velocity at the hull [14~. For a ship
with port and starboard symmetry, as is considered
here, the slender-ship approximation Ko and the
Neumann-Kelvin correction Kit can be expressed
in the form
Ko = Ko + Ko,
Kq, = Kq,+ + K<,,
where the superscripts + and - correspond to the
contributions of the port and starboard sides of the
ship, respectively.
Modified mathematical expressions for the
functions Ko and K,~ are given in tip.
Approximate forms of these expressions valid in
the short-wave limit are now given.
THE WAVE SPECTRUM: APPROXIMATE
INTEGRAL REPRESENTATION
The expression for the slender-ship
approximation Ko defined by (30) in [14] involves
the exponential function exp(P2z), where p2 is given
by
p2 = v2p2 = (sec20~/F2 (7)
For negative values of the vertical coordinate z, the
exponential function exp(P2z) is negligibly small in
the short-wave limit p2 ~ 00 . We then have
Ko+ ~ iw E+ (nX2-u2) ty dl
-iv2u is exp(P2z)E+nzda (8)
asP2 moo
The expression for the Neumann-Kelvin
correction K'¢, defined by (73) and (74a,b) in [14] can
likewise be approximated by restricting the
integration over the hull surface h in (73) to the
hull side s. We then have
Kit+ ~ iw E+ Aw+ dl
+ iv2 is exp(P2z) E+ Ah+ da/ ~ t x s
as p2 ~ no, where the amplitude functions A
and Ah+ are defined as
(9)
Aw+ = (tXOt+sx~s~ty+u~v~u) (°t+~°s), (lOa)
Ah+ = [(V-CU)tx+(U+cv~ty-ictz] (¢S+£¢t)
- t~v-Cu~sx+(u+Cv~sy-iCsz] (Ot+£0S) . (lOb)
The functions E+ and E_ in (8) and (9) are the
trigonometric functions defined by (19) in [14], that
is we have
E+ = expi-iP2(ux+vy)], (11)
where u and v are given by
u = 1/p and v = t/p; (12a,b)
It may then be seen from (3) that we have 1 > u > 0
and O < v < 1 for O < t < so, with
u2+v2 = 1
Furthermore, w and s in (8) and (9) represent the
positive halves of the mean waterline and of the
mean wetted-hull side and dl and da the
differential elements of arc length of w and area of
s, respectively. Also, t = (tx, ty, tz) and s = (sx, sy, sz)
are unit vectors tangent to the hull side along
`6a' curves which roughly correspond to waterlines
`6b' and framelines, respectively. The vectors t end s
are roughly (but not necessarily exactly) orthogonal
and point toward the bow and the keel line,
respectively. At the mean free surface, the vector t
is tangent to the mean waterline w and we thus
have tz = 0. The normal vector n = (nx, ny, nz) to
the hull side is defined by
n = (txs)/|txs| .
The term ~ in (lOa,b) is defined as
and at and Us represent the components of the
velocity vector V) along the unit vectors t and s
tangent to the hull side; we thus have
Vo - nx n = at t + Us s, where the hull boundary
condition 3~/3n = nx was used. The components
of and as of Vo along the tangent vectors t and s
and the velocities Jo/Ot = V¢.t and aq,/Os = Voles are
related as follows:
34/3t = At + ENS,
3~/3s = US + cot,
at = (OO/&t - £3O/8s)/(l- £2), (14c)
Us = (30/3s - £~/3t)/~1- £2) (14d)
Finally, C in (lOb) is an arbitrary complex function
of t. Equations (9) and (lOa,b) thus define a one-
parameter family of mathematically-equivalent
expressions for the Neumann-Kelvin correction
K¢.
It may be seen from (30) and (73) in [14] that an
estimate of the error associated with the
approximate expressions (8) and (9) is provided by
the exponential function exp(P2z) where z is taken
equal to the negative of the ship draft d. This error
(13)
(14a)
(14b)
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estimate is smaller than a prescribed error £ for p ~
Pe, or equivalently for ~ ~ He = sec -lapel, where Pe
is given by pe(F; £, d) = F [ln~l/£~/d]~/2 . The values
of Pe and Be corresponding to values of the error £
and the ship draft d equal to 0.01 and 0.05,
respectively, and to five values of the Froude
number between 0.1 and 0.5 are listed in Table 1.
F 1 0.1 1 02 1 03 1 04 1 05
Pe 1 1.9 2.9 3.8 4.8
He 0 59o 70° 75o 78°
Table 1. Values of Pe and qe for £ = 0.01 and
d = 0.05.
The exponential function exp(P2z) in the
integrands of the integrals over the hull side s in (8)
and (9) decays rapidly with decreasing (negative)
values of z if p2 >> 1, that is for small values of the
Froude number and/or large values of t = tang.
The major contributions to the hull-side integrals
in (8) and (9) therefore stem from the upper part of
the hull side s in the vicinity of the waterline w.
These hull-side integrals can in fact be
approximated by single (one-fold) integrals along
the waterline. These waterline-integral
approximations for the spectrum functions Ko and
K`p are now given.
THE WAVE SPECTRUM: WATERLINE
INTEGRAL APPROXIMATION
The upper part of the hull side s can be defined by
the following parametric equations:
x = (~1) + xlkl~s + x2~1)s2/2 + ....
y = 11~1) + ye ills + y2(l~s2/2 + ....
-z = z] flus + z2(l~s2/2 + ....
where s 2 0 and the curve s = 0 corresponds to the
waterline w. The waterline is then defined by the
parametric equations
x = (~1) and y = Al), (16a,b)
where 1 is the arc length along w. The previously-
defined unit tangent vectors t and s to the hull side
s are given by
t = (tx' ty, tz) = 3~x/3l,
s = (sx' sy, sz) = IS
In particular, at the waterline w, we have
(tx, ty, tz) = (k,', it', 0), (17a)
(sx, sy, sz) = (xl, Y1' -Z1) ~(17b)
where the notation ~ )' denotes differentiation with
respect to the arc length 1 along w.
By using the foregoing representation of the
upper hull side s, we can approximate the integrals
on the hull side in (8) and (9) as integrals along the
mean waterline w. Details of this short-wave
asymptotic approximation are given in Noblesse
and Hendrix 1161, where the following waterline-
integral approximation to the spectrum function
K(t) is obtained:
K ~ [w {A+ expl-iP2(u4+v~)l
+ A_expl-iP2(ut-vll~l} dl .
The amplitude functions A+ are defined as
A+ [-sz+i~usx+vsy)] = Ao-+ S+¢s+ Thy, (19)
where Ao+ corresponds to the slender-ship
approximation and is given by
Ao+ = -szty~nX2-u2)
+ittynX2(usx+vsy)-syu2(utx+vty)~. (20)
The terms S+ and T+ in (19) are defined as
S+ = -sz~sXty + uv~sxtx + syty)]
-i [usz2ty - (utx + vty)(sxsy + uv)], (21a)
T+ = (tXty + uv) (-sz + ivsy)
+ iutx~utx + vty) (usy + VSx) (21b)
For sufficiently large values of p = secO, the
Neumann-Kelvin approximation K thus is defined
by the waterline-integral approximation (18), (19),
(20), and (21a,b).
These equations provide a simple and practical
basis for numerically evaluating the short
divergent waves in the steady wave spectrum of a
ship, given the value of the fluid velocity
components as and of at the waterline, by dividing
the waterline w into a large number of straight
segments within which the amplitude functions A+
are assumed to vary linearly.
Figures la,b,c depict the real and imaginary parts
(15a) of the Neumann-Kelvin approximation to the
`15b' spectrum function K(t) for the Wigley hull at three
values of the Froude number equal to 0.1, 0.25 and
(15c) 0.4. The dashed-line curves in these figures
correspond to the waterline-integral approximation
(18) obtained in this study; the solid-line curves
correspond to the exact mathematical expression for
the Neumann-Kelvin approximation to the
spectrum function given by (21), (22), (73) and
(74a,b) in [14~. The velocity potential on the Wigley
hull in these exact expressions, and at the waterline
in the waterline-integral approximation (18)
obtained in the present study, is taken as the (first
order) slender-ship potential defined in [6~. Figures
la,b,c show that the waterline-integral
approximation (18) does in fact become quite
accurate for sufficiently large values of t = tang.
The waterline-integral approximation (18) may
then be used henceforth in this study. Figures la,b,c
also show that the general numerical method based
on the exact expressions for the wave-spectrum
function given in [14] can provide reliable
predictions if a sufficiently large number of panels
is used for representing the ship hull form.
468

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WAVE-SPECTRUM FUNCTION
WIGLEY HULL AT F=0.10
Exact {98000 panels}
o ~Waterline integral
x ~
.O
0 lo
o
xr
~ Hi
WAVE-SPECTRUM FUNCTION
WIGLEY HULL AT F=0.40
Exact {16000 panels)
- - - Waterline integral
4 5 6
tanS
Fig. 1a. Real and imaginary parts of the wave-
spectrum function king.
WAVE-SPECTRUM FUNCTION
WIGLEY HULL AT F-0.25
Exact (32000 panels)
~--- Waterline integral
611111~1~14~
6 8 10
tang
12 14
Fig. 1b. Real and imaginary parts of the wave-
spectrum function king.
Y 0
a)
Cal
-1 , c~ !' i' r~> ,, me, . .
1
6 10 14 to
tend
Fig. 1c Real and imaginary parts of the wave
spectrum function kilo).
THE WAVE SPECTRUM: STATIONARY-PHASE
APPROXIMATION
An analytical approximation to the spectrum
function K(t) can in principle be obtained by
applying the method of stationary phase, since the
trigonometric functions expt-iP2(u4+vll)] in (18)
oscillate rapidly for large values Of p2 = v2p2 =
(sec20~/F2. This method shows that the major
contributions to the waterline integral (18) stem
from the end points of the integration range, that
is the ship bow and stern, and the points where
the phases of the trigonometric functions
expt-iP2(uE,+v~] are stationary. These points of
stationary phase are defined by the conditions
ud;+vd~ = 0, which yield the relations
utx+vty = 0, tx = v, ty = +-u . (22a,b,c)
by virtue of (17a), (13), and the identity tX2+ty2 = 1.
By using (12a,b) and (13) in (22b,c) we may obtain
tan) = ty/tX = Flit = -+cotanO,
where o is the angle between the x-axis and the unit
tangent vector t to the waterline. We thus have the
relation
1 0 1 = ~/2 - ~ , (23)
which shows that the very short divergent waves
in the steady wave spectrum of a ship primarily
stem from the central (midship) portion of the ship
469

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where the waterline is almost parallel to the
centerplane, as well as the ship bow and stern.
It is shown in [16] that application of the method
of stationary phase to (18) fails to provide a simple,
practically useful analytical approximation because
the second terms in the asymptotic expansions for
the contributions of both the end points (i.e., the
ship bow and stern) and the points of stationary
phase are of the same order of magnitude (in the
limit t ~ or) as the first terms, and thus cannot be
neglected. Unfortunately, the second terms in the
asymptotic expansions are extremely complex. It
can nevertheless be shown that we have
K ~ KB,S + KPhaSe,
where KB S and KphaSe correspond to the
contributions from the ship bow and stern and
from the interior plinths) of stationary phase,
respectively. Furthermore, we have
KB S ~ 1/t3 and Kphase ~ 1/t4 as t ~ ~ . (24)
NEAR-FIELD FLOW AND NONLINEARITIES
The tangential velocity components as and ¢~ in
(19) are merely taken equal to 0 in the zeroth-order
slender-ship approximation to the spectrum
function defined in [6~. More generally, the values
of these velocity components at the mean waterline
w may be predicted numerically using any near-
field-flow calculation method, including the
relatively simple first-order slender-ship
approximation defined in [6i, in the Neumann-
Kelvin approximation to the spectrum function.
However, it was already noted that existing near-
field-flow calculation methods, including the so-
called nonlinear methods, cannot provide accurate
predictions of the velocity components as and (~ in
the immediate vicinity of a ship bow and stern.
The velocity components as and of along the
wave profile of a ship can be defined in terms of the
nondimensional elevation e = Eg/U2 of the wave
profile and its slope en = dE(L)/dL in the direction of
the unit tangent vector t to the mean waterline by
means of analytical expressions given in [15~. More
precisely, (27) and (23) in t15] define the velocities
bo/3t = tx30/OX+ty3~/3y and
Jo/Os = sx34/3x+sy O¢/3y+szOo/3z at the wave
profile of a ship as follows:
bo/3t = tx - (l-2eyl/2/ [l+`l+~2ye~2~l/2 (25a)
arias = SX - t£+~1+~2)sze~(l-2e~l/2
/[l+~1+~2ye~2~1/2 (25b)
where £ and p2 are given by
£ = s · t = sxtX+Syty ~
p2 = n 2/~1_nz2) = (Sxty-Sytx)2/Sz2.
The tangential velocity components As and of can
then be determined from the velocities Jo/3s and
Jo/3t by means of (14c,d). The analytical
expressions (25a,b) can be used in conjunction with
either experimental measurements or numerical
predictions of the wave profile, corrected at the bow
and the stern in the manner specified by the
nonlinear local solution given in [15~.
In the special case of a wall-sided hull like the
Wigley hull, we have ~ = 0 since nz = 0 at the
waterline, and £ = 0 since we may choose the
tangent vector s to the hull at the waterline as s=
(0,0,-1~; (25a,b) thus become
a¢/at = tx- (l-2eyl/2/`l+et2yl/2 (26a)
30/3s = et (l-2eyl/2/`l+e~2yl/2 (26b)
It is shown in [15] that the steady wave profile at the
ship bow must be tangent to the stem, and likewise
at the stern. This tangency condition shows that we
have en = -or at the bow and the stern of the Wigley
hull. It then follows from (26a,b) that we have
Jo/3t = tx ~ 1 , (27a)
34/3s =-(l-2e~l/2~ 1 (27b)
at the bow and the stern of the Wigley hull.
The velocities 34/3t and JO/3s along the
horizontal and vertical tangent vectors t = (tX,ty,0)
and s = (0,0,-1) to the Wigley hull at the waterline
are depicted in Figs. 2a and 2b for a value of the
Froude number equal to 0.25. The dashed-line
curves in these two figures were determined using
the nonlinear expression (26a,b) in which the wave-
profile elevation e is taken as the experimental
profile obtained at the University of Tokyo and
corrected at the bow and stern in accordance with
the previously mentioned tangency condition [15~.
The solid-line curves in Fig. 2a correspond to
numerical predictions obtained using the slender-
ship approximation defined in ted. Figure 2a shows
that discrepancies between these linear numerical
predictions and the corresponding nonlinear
analytical/experimental predictions are quite large
in the vicinity of the bow and the stern, where
nonlinear effects indeed are important. The solid-
line curves in Fig. 2b were obtained from the
nonlinear expression (26a,b) in which the wave-
profile elevation e is taken as the profile predicted
numerically using the slender-ship approximation
[6], corrected at the bow and stern in the manner
specified in [15] and used also for determining the
analytical/experimental dashed-line curves in Figs.
2a and 2b. Thus, both the solid-line curves in Fig.
2b and the dashed-line curves in Figs. 2a and 2b
correspond to the nonlinear analytical expression
(26a,b). The discrepancies between these nonlinear
analytical/experimental and analytical /numerical
predictions clearly are much smaller than the
discrepancies corresponding to the linear numerical
predictions shown as solid-line curves in Fig. 2a.
The oscillations in the latter curves correspond to
the divergent waves in the wave pattern. These
470

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VELOCITY AT WATERLINE
WIGLEY HULL AT F=0.25
slender-ship
approximation
-- nonlinear analytical/
experimental prediction
lo
ANEW.'
to _
0
1 1
- (~,,'\
, . . . . . . . . .
stern bow
Fig. 2a. Velocity components Abet and 3~/3s.
oo _
o
_
d.
~o _
no> _
o
oh
\ ~
~ 0
VELOCITY AT WATER LINE
WIGLEY HULL AT F=0.25
slender-ship
approximation with
nonlinear correction
nonlinear analytical/
experimental prediction
~1
stern bow
Fig. 2b. Velocity components Jo/3t and Jo/3s.
Oscillations do not appear in the corresponding
numerical predictions corrected for nonlinear
effects depicted in Fig. 2b because the number of
numerical data points used for defining the curves
in this figure is fairly small (only 26 data points are
used for both the experimental and the numerical
results).
THE WAVE SPECTRUM CORRESPONDING TO
FOUR NEAR-FIELD-FLOW APPROXIMATIONS
The modulus, ~ K I, of the wave-spectrum
function K(t) of the Wigley hull is represented in
Figs. 3a,b and 4a,b for values of t = tang in the range
7 ~ t < 19, which approximately corresponds to
values of ~ in the range 82° < ~ < 87°. Figures 3a
and 4a correspond to a value of the Froude number
F equal to 0.25, while Figs. 3b and 4b correspond to
F = 0.4.
The dashed-line curves in Figs. 3a and 3b
correspond to the zeroth-order slender-ship
approximation, so that the velocity components of
and as in (19) are merely taken equal to 0. The
solid-line curves in these two figures correspond to
the Neumann-Kelvin approximation defined by
(18), with the near-field velocity components ¢~ and
(PS in (19) determined from the first-order slender
ship potential defined in ted. The numerical results
depicted in Figs. 3a,b show that the predictions
corresponding to the Neumann-Kelvin approxi-
mation (first-order slender-ship approximation) are
significantly larger than those corresponding to the
zeroth-order slender-ship approximation. The
latter approximation thus appears unlikely to
provide realistic predictions of the short waves in
the wave spectrum of a ship, as was already noted.
The predictions corresponding to the foregoing
Neumann-Kelvin approximation are also depicted
in Figs. 4a and 4b. These predictions correspond to
the thick solid-line curves located much below the
other two sets of curves represented in Figs. 4a and
4b. The latter two sets of curves correspond to the
Neumann-Kelvin approximation defined by (18),
with the near-field velocity components ¢~ and as in
(19) determined from the nonlinear expression
(26a,b). These two sets of curves thus correspond to
nonlinear near-field-flow predictions, whereas the
thick solid-line curves in Figs. 4a,b correspond to
linear near-field-flow predictions. More precisely,
the thin dashed-line and solid-line curves in Figs.
4a,b correspond to the nonlinear experimental and
numerical, respectively, near-field-flow predictions
depicted in Fig. 2b. It may be seen from Figs. 4a,b
that discrepancies between these predictions of the
wave spectrum corresponding to the nonlinear
experimental and numerical near-field-flow
predictions are relatively small, whereas the
prediction of the wave spectrum corresponding to
471

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~ ----
cv~ 1A 1 1
WAVE SPECTRUM OF
WAVE SPECTRUM OF WIGLEY HULL AT F=0.25 0 WIGLEY HULL AT F=0.40
x
Cal
o
-- Zeroth-order slender-ship approximation
- Neumann-Kelvin theory
(first-order slender-ship approximation)
7 10 13
tang
(a)
o
16 19
Fig. 3. Modulus of the wave spectrum function king.
WAVE SPECTRUM OF WIGLEY HULL AT F=0.25
| A
7
Linear Neumann-Kelvin theory
Nonlinear analytical/numerical
Nonlinear analytical/experimental
^) -. ~
13 16
tang
(a)
Fig. 4. Modulus of the wave spectrum function kite).
the linear near-field-flow calculations are much
smaller. These numerical results indicate that the
major contributions to the waterline integral (18)
clearly stem from the ship bow and stern, and that
the simple nonlinear correction of the linear
numerical predictions of the wave-profile elevation
presented in [15] and depicted in Fig. 2b thus can be
used effectively for predicting the short-wave tail of
the wave-spectrum function.
THE FAR-FIELD DIVERGENT WAVES
It is appropriate to analyze the far-field wave
pattern of a ship in terms of the nondimensional
far-field coordinates (x,y,z) = v2~t,q,() = (X,Y,Z)g/U2,
where (X,Y,Z) are dimensional and ((,ll,() =
(X,Y,Z)/L are the nondimensional near-field
coordinates used in (1), the nondimensional
10
13 16 19
tend
(b)
WAVE SPECTRUM OF
0 WIGLEY HULL AT F=0.40
x
0
co
Y 0
-cot
0 .
~ .
to
19 7 10 13
tang
(b)
16 19
potential ~ = v2q, = ~g/U3 and the wave-spectrum
function k = v2K. By using these far-field variables
in (1) we may obtain the equivalent alternative
expression
(x,y,z) = Im J; [E+(t; x,y,z)
+ E_(t; x,y,z)] kits aft, (28)
where the functions E+(t; x,y,z) are defined as
E+(t; x,y,z) = exp~zp2+i~x+yt)p]
with p given by (3~.
The nondimensional free-surface elevation
e~x,y) = Eg/U2, where E is dimensional, is given by
e = 3~/Ox, where the function 3~/Ox is evaluated at
the mean free-surface plane z = 0.
By differentiating (28) we may obtain
~e~x,y) = Re [e+(x,y)+e_(x,y)], (29)
472

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\/(211) = 1/p2, cosO = sin I 0 I = 1/p, (48a,b)
tang = (p2 1)l/2/(2p2-l) (48c)
In summary, the far-field pattern of divergent
waves is defined by (40), and the waves' amplitude,
wavelength, and steepness by (41), (42), and (43).
These expressions involve the value of the wave
spectrum function k - v2K at the stationary point
t = ~(c') defined by (34) and (35). The corresponding
values of p - (l+t2)~/2, the wavelength X, the wave
propagation angle 0, the wake ray angle or, and the
waterline angle ~ ~ ~ are related by (48a,b,c). A
practical approximate expression for the spectrum
function K(t) is given by the waterline
approximation (18). This approximation is valid in
the limit p ~ or, although Figures la,b,c indicate
that it may be used in practice for moderate values
of p. The near-field flow-velocity components ¢
and as at the mean waterline in (19) can be
determined from the wave-profile elevation e by
using the analytical expressions (25a,b) obtained in
[15~. The foregoing expressions provide an
approximate but complete theoretical basis for
predicting the short waves in the steady wave
spectrum of a ship.
Figure 5 depicts the steepness (-x)~/2s, defined by
(43), of the divergent waves in the steady wave
pattern of the Wigley hull for values of the wave °
angle or in the range 0 < or < 4°, which
approximately corresponds to values of t in the
range 7 ~ t < to for which the waterline
approximation (18) was previously shown to
provide fairly accurate predictions. The top and
bottom parts of Figure 5 correspond to values of the
Froude number F equal to 0.25 and 0.4, respectively.
The solid-line and dashed-line curves in this figure
correspond to the nonlinear near-field-flow
predictions defined by (26a,b), where the wave-
profile elevation e is determined from
experimental measurements and numerical
calculations, respectively, in the manner explained
previously. These analytical/ experimental and
analytical/numerical steepness predictions are in
reasonable agreement with one another. Both
predictions exhibit a fairly well-defined peak at
values of of approximately equal to 1.9° and 1.4° for
F = 0.25 and 0.40, respectively. Let ~ denote the
value of (_x)~/2 s. We thus have s = 6/(-x)~/2 =
6F/(-X/L)~/2 since x = Xg/U2. We then have s =
F6/10 at a distance of 100 ship lengths behind the
ship. It may be seen from Fig. 5 that the predicted
peak values of the wave steepness at 100 ship
lengths thus are approximately equal to 0.3 and 0.6
for F = 0.25 and 0.40, respectively. These predicted
peak values are extremely large and the
corresponding waves could not exist in reality.
474
WAVE STEEPNESS (WIGLEY HULL)
--- Analytical/numerical
Analytical/experimental
F=0.25
1_
0 1
F=~.40
2 3 4
~ (degrees)
Fig. 5. Steepness of the short divergent waves.
CONCLUSION
A simple practical analytical/numerical method
for calculating the short divergent waves in the
steady wave spectrum of a ship has been presented.
The method is based on a waterline-integral
approximation obtained from a modified
Neumann-Kelvin integral representation of the
wave-spectrum function. A comparison of
numerical predictions obtained using the exact
integral representation and the waterline-integral
approximation showed excellent agreement for
sufficiently short waves. This agreement
demonstrates both the validity of the waterline-
integral approximation and the robustness of the
more general numerical method based on the exact
integral representation. The waterline-integral
approximation to the spectrum function shows that
the short divergent waves in the wave pattern of a
ship are defined in terms of the near-field flow
along the ship waterline, and may indeed be
regarded as an image of the flow along the ship
waterline.

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The foregoing linear far-field flow representation
has been used in numerical experiments seeking to
determine the sensitivity of the far-field waves to
the near-field flow along the ship waterline. The
simplest approximation is the trivial
approximation corresponding to the zeroth-order
slender-ship approximation, in which the
disturbance potential is merely ignored. The next
level of approximation for the near-field flow is
that corresponding to the first-order slender-ship
approximation. Significant discrepancies were
found between the wave-spectrum predictions
corresponding to these two near-field-flow
approximations. The other two near-field-flow
approximations were determined by applying a
simple analytical expression defining the fluid
velocity along the waterline in terms of the
elevation of the wave profile, corrected at the bow
and the stern according to the nonlinear local
solution given in t15~; both wave profiles measured
experimentally and predicted numerically using the
slender-ship approximation were used.
Discrepancies between the wave-spectrum
predictions corresponding to the latter
analytical/experimental and analytical /numerical
near-field-flow approximations were found to be
much smaller than the discrepancies between the
wave-spectrum predictions corresponding to these
nonlinear near-field-flow approximations, on one
hand, and to the linear near-field flow prediction
given by the slender-ship approximation on the
other hand. This result indicates that the short
divergent waves in the wave pattern of a ship stem
mostly from the ship bow and stern; a result that is
not surprising but points to the fundamental
difficulty of predicting the short divergent waves
since the flow at a ship bow and stern is strongly
nonlinear and quite difficult to compute.
Numerical calculations for the Wigley hull
showed that the steepness of the short divergent
waves predicted on the basis of the foregoing linear
far-field/nonlinear near-field flow analysis is too
large for the waves to exist in reality within a sector
of several degrees in the vicinity of the ship track.
The wave steepness was also found to exhibit a
well-defined peak at an angle of approximately 1° to
2° from the ship track. It is unclear whether or not
these numerical predictions have real physical
implications. On the one hand, the physically-
unrealistic predicted wave steepness might be
regarded as a theoretical indication for the common
observation that the wake of a ship in the vicinity
of the track does not contain divergent waves. On
the other hand, these physically-unrealistic
numerical predictions might be regarded as an
indication that the assumptions of linear far-field
waves and of steady nonlinear near-field flow
underlying the mathematical model and the
numerical results may not be correct. In particular,
assumptions about the flow in the immediate
vicinity of the ship bow and stern clearly are crucial
for the prediction of the short divergent waves in
the wave pattern. Indeed, the present study
underscores the need for a better understanding of
the flow at a ship bow and stern.
It should be recognized that the assumptions
underlying the short-wave analysis presented in
this study are also adopted for numerical
predictions of the longer waves in the steady wave
pattern of a ship. In this respect, numerical
experiments analogous to those performed in this
study for the short divergent waves also seem
useful for determining the sensitivity of the longer
waves in the wave pattern to the assumptions used
in the calculation of the near-field flow at a ship
hull. Such numerical experiments will be
presented in [16~.
ACKNOWLEDGMENTS
This study was performed at the David Taylor
Research Center with support from the Surface
Ship Wake Signature Task, the Independent
Research Program, and the Applied Hydrodynamics
Research Program funded by the Office of Naval
Research. The authors also wish to thank
Dr. Robert Hall at SAIC and Dr. Patrick Purtell at
DTRC for discussing the paper.
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476