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OCR for page 567
Ship Internal Waves In a ShaBow Thermocline:
The Supersonic Case
M. Tulin (University of California, Santa Barbara, USA)
T. Miloh (Tel Aviv University, Israel)
ABSTRACT
We develop a general theory of ship internal waves in a
thermocline of moderate thickness below a well mixed
upper layer, when the ship is traveling faster than the
fastest internal waves. This theory provides both the
kinematical pattern earlier discussed by Keller and Munk,
and also the amplitude of the internal waves (the deflection
of the top of the thermocline) in terms of the spectral
amplitude function generated by the ship.
The wave far field consists of an inner and outer wake.
In the inner wake, near the track of the ship, each phase
line originates in a cusp, periodically spaced on the track,
with a time interval corresponding to the average Vaisalla-
Brunt period of the water in the thermocline. The inner
wake is created by the high frequency content of the
disturbance. In the outer wake, further downstream, each
phase line approaches the Mach angle, sin~l~llF), defined
by the densimetric Froude number corresponding to the
depth of the thermocline and the density jump across it.
The kinematical wave field near the limiting Mach line (the
outer field) is independent of the thermocline thickness.
The distance between the dominant wave crests in the outer
wake corresponds to (kin) of order one.
The spectral amplitude function is given in terms of
two factors, one depending on the thermocline but not the
ship, and another depending on the ship, Froude number,
and the thermocline. The latter is shown to be related to
the wave disturbance just behind the ship in its near field.
An asymptotic non-linear theory is developed for the
calculation of the near field around the ship. For
supersonic speeds the field equation is hyperbolic and can
be solved numerically by the method of characteristics.
Calculations show the development of a narrow wake
immediately behind the ship, consisting of three lobes, a
central lobe of elevation and two side lobes of depression.
567
The amplitude spectrum is concentrated in the region
0 ~ kh < 2-3. This coincides with the wavenumbers most
prevalent in the outer wake in the region of interest and
helps to explain why internal waves are so readily made by
ships traveling at supersonic speed in shallow
thermoclines.
IN1RODUCIION
At a previous Symposium on Naval Hydrodynamics
(Miloh and Tulin, 1988), we have presented a non-linear
theory of internal waves made by surface ships in the
transonic region, F = 0~1), with particular reference to
early studies of "deadwater" (Ekman, 1904~. Here F is the
ratio of ship speed to the speed of longest internal waves,
*
c .
Here we consider the case of internal waves made by
ships traveling over stratified water in the supersonic case,
F > 1. Since values of c* in nature lie in the range 20-70
cmlsec, while ship speeds are normally an order of
magnitude larger, we are especially interested in the
hypersonic case, F >> 1.
Our interest in this problem has been created originally
by the fact that ships at sea are known sometimes to leave
behind them narrow V wakes of great length (measured in
kilometers), detectable by remote sensing radar, (Hughes,
1986~. The circumstances of occurrence and the
hydrodynamic mechanism of their origin remains
unknown. The angles of the V wake are sufficiently small
(normally less than 10°), however, to be consistent with
the notion that they are surface manifestations of a pattern
of limiting thermoclinal waves propagating at speeds close
to the so-called Mach angle, a = sin-l~l/F). Such a
pattern, including waves internal to the V have been
theoretically predicted using ray kinematical considerations
by Keller and Munk (1970) and by Yih (1990~. An
OCR for page 568
adequate theory of such ship waves, connecting their
amplitude distributions with the parameters of the problem,
and taking adequate recognition of non-linear effects
remains absent, however.
Heightened interest in the problem is caused by recent
field research conducted by a team led by the Royal
Aircraft Establishment Space Division (Farnborough, UK)
recently reported at a Workshop held at Farnborough
under the aegis of the British Remote Sensing Society
(Workshop Proceedings, 1990~. The RAE team has
systematically measured in the water, internal wave wakes
produced by a number of ships traveling over shallow
thermoclines with a depth of the order of the ship draft as
well as above water, visual, and radar signatures due to the
internal waves. It is noteworthy that in these careful
scientific studies, no above water V-wake signatures have
yet been observed except those which originate with the
internal waves.
Excepting the kinematical studies mentioned above,
almost all of the previously published work on ship
internal waves is concerned with the description of
singularities (Green s functions) in a two-layered fluid
[Stretensky (1959), Uspenskii (1959), Hudimac (1961),
Crapper (1967), Miles (1971~; Sabuncu (1961) applied
the source singularity to the derivation of a theory for the
interracial wave resistance of a thin ship, a la Michell, and
carried out some calculations. Unpublished work on
internal waves due to ships has been carried out and
presented in unclassified contractor reports: (Munk, et al.,
1968~; (Holliday, 1981~. These have resulted in
algorithms for computation.
Our interest is in providing an adequate mathematical
analysis and concurrent description of the wake. Our
ultimate practical interest is in predicting the long internal
waves propagating away from the track of the ship and
comprising a dominant pattern near the limiting Mach
angle.
THEORETICAL DEVELOPMENT: THE LINEAR FAR
FIELD
General Theory
We assume a mixed upper layer depth, h, and wish to
calculate the wavy displacement at that depth. The
approach of Havelock, introduced for the prediction of the
Kelvin wave pattern of a ship, is generalized. It involves
synthesizing the far field as a summation of waves
propagating in the ship direction and at all angles, 8, to that
direction within a sector, + ~t/2. The amplitude of the
individual waves is given by abode, and the wave
amplitude at any point (x,y;z = -h) is given by,
~x,y)=R{ ~ a*~)eik[XC°S8-YSill0)d8> (1)
-~12
The wave number, k, for each wave element is not
arbitrary, but corresponds to the phase velocity, c, which
on account for stationarity, is simply related to 8:
c = cO cost
(2)
where c0 is the ship speed; therefore k = kite. The
relationship between k and c follows from the dispersion
relation ~ = make, where c = o/k. These relations alone,
allow the determination of the asymptotic wave pattern due
to a steady disturbance propagating in the general medium
defined by Arks. If the disturbance is located at (x,y) = 0
and the coordinates (x,y) are replaced by polar co-
ordinates (r, 0), see Figure 1, then:
+~12 ~
q~x,13) = R ; able) exp [ix. g(8,0~] do ~(3)
-~l2 J
where
g(8,~) = k~cos~ - tang sin0] (4)
For large values of x, the stationary phase solution of (3)
. .
Is given by:
~x,,B) = Rt~ ~/2 eXptix. g(8s)
+ sgntg (8s)] ~ / 4~] + 0~1/ x) (5)
where dg/d~ = g (0s) = 0, the stationary phase condition.
This corresponds to:
cg /c0 . sin as
1-cico cost
(6)
where cg/co is evaluated at as and where cg, the group
velocity, is given by cg = dm/dk. This same result, (6),
can be readily obtained by geometrical construction, Figure
1. The shape of the phase lines follows from (6) and from
the relation (see Figure 1~:
dy/dx = cot Bs
568
(7)
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which allows the elimination of Us in (6) and (7). This
first order system requires an initial condition y(xn) = 0
which defines the nth phase line. This condition follows
from consideration of the wave along the track; using (1):
(X,0) = R{ ~ a*(~)eikXC°s~d8}
-rc/2
= Rig ~ a*(~)e( ° )d~j
-/2 1
(8)
where we have used c = co cost and co/k = c, and where
m* corresponds to the frequency of waves on the track of
the ship (,B = 0). We are entitled to shift the origin of this
far field result corresponding to a given value of 11(0,0).
Here we take q(0,0) = 0, since no internal waves are
generated ahead of the ship in the supersonic case. In
addition we expect the first wave at the ship to be given a
net initial downward impetus due to the early action of the
bow of the ship. Therefore we replace iei(0' X/co) by
it Deco) . The values of xn corresponding to crests in
rl(x,O) behind the ships are therefore given by:
Xn = co/~* [fen - ~/2] n=1,2,... (9)
Therefore phase lines originate at intervals of 2~co/,co
along the track.
This completes the general theory. It remains to dx x
specify the dispersion relation.
For a shallow thermocline of moderate thickness in
very deep water, a useful approximation for the dispersion
relation has been given by Phillips (1977; pg. 213)
o2 = gk P {1 + k£ + coth(kh)) 1 (10)
see Figure 2 for definitions of h, £, and Ap/p. The
corresponding asymptotic limits are:
where the longest wave speed, c*, and the maximum
frequency, m*, are:
C*=(Apgh)
; lo* = (/`P g / £) (13)
Since the local Brunt-Vaisalla frequency is,
Rev = g P / , the average of mBV over the thermocline
is clot.
Note that the long wave limit is independent of the
thermocline thickness, £, while the short wave limit is
independent of the thermocline depth.
The Hypersonic Case: Wave Patterns
In the hypersonic case, c/co = costs << 1, so that
sine ~ 1. Then, (6) and (7) simplify to:
F >> 1: y/x ~ cg/co ; dy/dx ~ c/c0 (14)
Therefore the shortest and slowest waves are found near
the track of the ship (y/x ~ 0), the inner wake, and the
longest waves near the Mach line, y*/x = 1/F, the outer
wake. Using cg = cg(c), (12) and (11), the asymptotic
shape of the phase lines in the inner and outer wakes may
be calculated ( upperbranch ):
Outer wake
2d(y - y*) (y _ y*) x _
dy 1
dx x~oo F
where K is a constant depending on the phase.
Inner wake
dy *1/2
dx Lcog x
[O) E ] t~ _ :~2;
to C C C (C) Y > - I 11 ·A l
xn X~Xn 4L co AL xn J
(kin) ~ O c*k[l- kh] c*[l- kh] c*[l- kh] c = 2c - c*
(1 1)
(kin) ~ Oo ~ [l -1 / (ke)] ~ /k co*lek2
(15)
(16)
Therefore the phase lines are cusped at their origin
(xn), with a curvature, d2y/dx2 = [~*£/Co] / [2Xn1 , and
c = c /c ~are straight in the outer field with a slope, dy/dx ~ 1/F.
g According to these relations, it is only necessary to know
(12)
569
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Representative terms from entire chapter:
outer wake
cO, xn, y'~x ~ no); y"(xn), in order to determine the
oceanographic variables, £, h, and Ap/p. Therefore, in
cases where ship internal wave patterns are visible from
above, oceanographic surveys of shallow thermocline
characteristics may be made by remote observation.
We have calculated numerically the crest lines for a
range of F and Ah, using the eqns. (6), (7) and (10), see
Figure 3. These in general, closely resemble the
observations in Loch Linnie. We also show F * 2
corresponding values of (kh), Figure 4. · exp i: * ~ Y - Y _ ~
The crest lines provide a good kinematical description (2x / hJ h 4
of the wave field. For example, the wave lengths are
given by the normal distance between crest lines.
Hypersonic Case: Amplitude
It is useful to express the amplitude, (5), in terms of
the amplitude spectrum in wavenumber space, A*(k),
where:
acre) = A* dk/d~ = A*(k) [kco/(c - cg)] (17)
Then (5) can be put in the following useful form:
h {( h2 A. B(x / h;6 / h;n)
·exp~iLk~( c )+sg 4~l and,
where,
B(x/h;£/h;n) = [
(x/h~"|
Gt)"= d2(C')h / C*) / d~kh)2
and where we have used the following results in (5~:
(19)
d2g/dB2= co .4 t0+O(l/F3) (20)
(cg-c) dk
The amplitude at each point in the field is seen to be
given by the product of two factors, one involving A*
depends on the ship and must be separately calculated,
while the other, B/
l m n = 1/ X - Xn 3/2 (25
Xerox_' h I l ' 2F(2~n - ~/2) (h) ~ h ]
~~n
However, B decreases rapidly along the crest line leaving
the cusp, and for F = 0(10), £/h = 0(1), then B ~ 0(1)
when ((x-xn)/h) becomes larger than about 3. It then
decreases slowly with distance downstream, remaining
0(1) out to x/h = 600. In the outer wake, far downstream,
it decreases as (x/h)-~/2, see (21).
The Shipform Amplitude Factor. A*(kh:F)
The shipform amplitude factor will determine the order
of magnitude of the interracial wave in a large part of the
wake. It can be related to the narrow interracial wave
disturbance in the region near, but behind the ship, Ant
before dispersion has created the kinematical patterns
previously discussed.
If we take x as the origin of the wave wake, then it
follows from (1) and (17~:
1l~x;y) , +, A (k) -i(kh)(Y/h)~,, ~(26)
J 2 e ~~ man)
_00 h
The Fourier Integral Theorem then allows determination of
A*(k) in terms of Six ;y):
h2 = 2~; 1l' h;Y)ei(kh)(y/h)d~y/h' (27)
where on account of symmetry the exponential function
can be replaced by the cosine.
The Nearf~eld Wave Disturbance, Alex ,y)
As the ship passes over the thermocline, it deforms
into a surface having the impression of the ship s form.
The thermoclinal water closest to the ship s track is
pressed largely downward under the ship at the bow and
back up again at the stern. Water sufficiently far from the
ship s track will be pressed largely to the side around the
ship as it slides forward. The relative effect of the
thermocline on the fluid displacement near the hull is
O[~Ap/p)(gL/c~], where L is the ship length. When this
quantity is much smaller then unify, then the flow field
about the shin may be represented to first order, neglecting
the presence of the thermocline, by a potential TO + cOx,
which can be approximated by the double hull potential
flow.
One of the consequences of the ship s motion is to
cause a pressure field in the thermocline, generated partly
by dynamic and mostly by hydrostatic effects due to
inclination of the isopycnics from the horizontal. The
component of pressure gradient, Vp0, normal to the
density gradient, Vp, in the thermocline, generates
vorticity there with its direction normal to both Vp0 and
Vp; Bjerknes law prevails. This vorticity generates a flow
field on either side of the thermocline (the Biot-Savart Law
prevails), which can be described by {(i', where i is an
index locating the flow above or below the thermocline.
The total potential in the flow field outside the thermocline
is therefore: 40 + Ax + tp. When ¢0 is the double model
disturbance, it is localized near the ship. Upstream ~
vanishes as the thermocline has not seen the effect of the
ship. Around the bow, at hypersonic speeds, outgoing
waves of depression propagate sidewards, and under the
stern, these are joined by outgoing waves of elevation.
These tend to cancel each other, except that they are
separated due to the time interval involved in their
generation. A residual signature results centered on the
track not far behind the ship, which has an upward lobe in
the center, and two lobes of depression on either side. The
entire wave wake downstream will find its origin in this
residual nearf~eld wake. Examples of the near field wave
patterns just described, are shown in Figures 6 and 7.
The most rigorous way to carry out the calculation of ~
is through strictly numerical means, without further
approximation; this is not simple and involves its own
problems.
As a simpler, and for us, more feasible alternative, we
have derived an asymptotic, non-linear theory (long waves
and sharp thermoclines) and we have solved the resulting
second order PDE numerically in the dispersion-free limit.
The justification for these approximations in the near field
is that our interest settles on long waves near the leading
characteristic (which are most visible in the experiments).
As these become dispersion free in the long wave limit, it
seems justifiable to neglect dispersion for their calculation
in the near field. They then propagate away from the ship
track as acoustic waves would.
In the hypersonic case, our PDE is normally
hyperbolic and may therefore be both accurately and
quickly solved numerically using the method of
characteristics (a particular forward marching procedure).
The equation is [here, c*2 = (Ap/p~g~h-rl)~:
571
2 ~ ~
[C* - Co - 2Coto - 2CO¢X]¢XX - [2Co(toy + °Y)]¢XY
+ [C*2 _ ~ Joy _ (,~y)2]¢yy
[ O¢°Xx]¢x [Cofo ]¢y-cohH(2)(V2¢
VPC a (p /p)
where the underlined term,
H(2)[ Vh ~XX] = 2~ 1 1 <( _ x')2 + (y _ y,)2
(28)
2 -
[ Vh (XX(X ,y )]
represents the effects of dispersion, and where the RHS is
the forcing term due to the ship's pressure field. The
disturbance field due to the ship without the thermocline,
40 = 40 - c0x, appears in the coefficients of the PDE and
represents convective effects of the ship's field, which
may not necessarily be neglected for small c*. In the field
near the ship, we assume that we may neglect dispersive
effects (the underlined term, [29]), as they are weak for
long waves and will require some time (distance aft) to be
effective. The potential, ¢, in (28) represents the flow
field in the upper mixed layer, and, from it, the elevation
of the top of the thermocline may be calculated. Some
examples of the calculated thermocline deflections due to
the passage of a semi-submerged spheroid (representing a
ship) are shown as Figure 6. Transverse cuts through the
wake reveal the emergence of the triple-lobe pattern at a
certain distance, x', behind the middle of the ship. This
pattern, rl(x',y) from which the amplitude function, A*(k),
may be calculated is shown as Figure 7 in a particular case.
Amplitude Results
The shipform amplitude function, A*(k), has been
calculated from Equation (27), using the triple-lobed wake
pattern q(x',y). Results for three Froude numbers are
shown, Figure 8, all for a semi-submerged spheroid,
whose draft is 90% of the mixed layer thickness, and 9%
of its own length.
The peaks of this spectral function are seen to vary
inversely with Froude number, and shift to higher values
of (kin) with increasing Froude number. The spectral
content is seen to become small for (kin) ~ 2-3. Since
B(kh) becomes very large for large values of kh (the inner
wake), the calculation of the wake wave amplitudes there
requires a separate calculation of A*(kh) for large
wavenumbers; we have not given it here.
The resulting wave amplitude, (~/h), along the crest
lines are exactly the product (A*lh2) ·B. The variation of
(~l/h) along these lines is shown as Figure 9. The peak
values are shown to vary only slowly from one crest line
to another; typical peak values are 5 x 10-2. The declines
for the smallest values of xlh are due to the corresponding
decline in A*. The region in the immediate vicinity of the
Mach line requires further study, as the method of
stationary phase used in the integration fails there.
DISCUSSION AND SUMMARY
The linear theory presented here is comprehensive in
that it provides for both the calculation of the kinematical
field and the wave elevations starting from a single
relation, eq. (1~. It then provides separate algorithms for
the calculation of the phase lines in space and for the wave
amplitudes on these lines. A necessary input is the
dispersion relation, m~k), which depends on the
thermocline shape. The determination of o~k) is a separate
problem, for which theory has long existed. In this paper
we took a simple model of the thermocline and an
appropriate approximate to its dispersion relation, due to
Phillips.
The amplitude is represented in this theory by the
product of two factors. One, B(kh), represents
kinematical effects and depends on to~k), but not on the
ship. The other, A* (kh), depends on both the
thermocline, and the ship; it may be determined from the
near field wave wake behind the ship. The latter requires a
separate theory and calculation, an example of which we
have provided.
The specific calculations shown here reveal that the
spectral content due to the ships disturbance (for the
thermocline depth h about equal to the ship draft, D), is
concentrated in wavenumbers, kh, between 0 and (2-3),
with a peak at about (kin) = 1, Figure 8. At the same time,
the wavenumbers on the first few phase lines in the outer
wake are also in this range, Figure 4. It is this particular
"coincidence" which result in measurable internal wave
patterns for large distances downstream under
circumstances which prevail at Loch Linnie, for example.
We do not believe that this "coincidence" depends critically
on the shape of the ship, or the shape of the thermocline,
or the hypersonic Froude number, although the non-
dimensional magnitude of the wave elevation will clearly
decay as the thermocline depth, h, increases beyond the
draft of the ship.
572
Peak wave amplitudes, Ah, of the order 5 x 10-2 are
typical for the cases considered, and these values are
reached at substantial distances behind the ship, see Figure
9. The peak values do not vary much from crest to crest.
This theory does not take any account of non-linear
effects. We have studied these separately and derived an
evolution equation for the outer wake, allowing for soliton
generation. Although solitons of depression traveling
ahead of the leading Mach line are, in principal, possible,
they require a significantly large initial disturbance; their
generation is enhanced by very shallow thermoclines (h <<
D). It is unlikely that they can be generated under
conditions which prevail at Loch Linnie.
It would be highly desirable to have available
systematic model scale measurements of internal wave
wakes for validation of theory such as we have presented
here.
ACKNOWLEDGEMENTS
The first author is grateful to Dr. Dennis Holliday of
RDA and to Dr. Brian Barber of RAE for many useful
discussions. The authors are also very grateful to Mrs. Pei
Wang Yao and Mr. Yi Tao Yao of the Ocean Engineering
Laboratory at UCSB for their invaluable help, especially in
carrying out numerical calculations.
This work was partially supported by the Office of
Naval Research Ocean Technology Division and that
support is very gratefully acknowledged.
REFERENCES
Crapper, G.D., "Ship Waves in a Stratified Ocean," J.
Fluid Mech., Vol. 29 (1967), p. 667.
Ekman, V.W., "On Dead Water: The Norwegian
North Polar Expedition 1893-1896," Vol. V, Ch. XV
(1904), Christiania.
Holliday, D., "Internal Wave Wake of a Ship," RDA-
TR-118100-001, R&D Associates, 1981.
Hudimac, A.A., "Ship Waves in Stratified Ocean," J.
Fluid Mech., Vol. 10 (1961), p. 229.
Hughes, B.A., "Surface Wave Wakes and Internal
Wave Wakes Produced by Surface Ships," Proceedings of
the 15th Symposium on Naval Hydrodynamics, National
Academy Press, 1986.
Keller, J.B. and Munk, W.H., "Internal Wave Wakes
of a Body in a Stratified Fluid," Physics of Fluids, Vol. 13
(1970), p. 1425.
Miles, J.W., "Internal Waves Generated by a
Horizontally Moving Source," J. of Geophys. Fluid
Mech., Vol. 2 (1971), p. 63.
Miloh, T. and Tulin, M.P., "A Theory of Dead Water
Phemonena," Proceedings of the 17th Symposium on
Naval Hydrodynamics, National Academy Press, 1988.
Munk, W.H., et. al., "Generation and Airborne
Detection of Internal Waves from an Object Moving
through a Stratified Ocean," JASON Study S-334, IDA,
1969.
Phillips, O.M., The Dynamics of the Upper Ocean,
Cambridge University Press, 1977.
Sabuncu, T., "The Theoretical Wave Resistance of a
Ship Travelling Under Interfacial Wave Conditions,"
Norwegian Ship-Model Experiment Tank," Trondheim
Pub. No. 63 (1961~.
Stretenski, L.N., "Wave Resistance of a Ship in the
Presence of Internal Waves," Izv. Akad Nauk SSSR, Otd.
Tekhn. Nauk. Mekh., Mashinostr. (1959), p. 56.
Uspenski, P.N., "On the Wave Resistance of a Ship in
the Presence of Internal Waves Under Conditions of Finite
Depth," Trudy. Morsk Gidrofiz. Inst., Vol. 18 (1959), p.
68.
Yih, C.S., "Patterns of Ship Waves," Engineering
Science. Fluid Dynamics, World Scientific Publishers,
1990.
573
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O 1 00 200 300 400 SOO 600
X / h
- 111 'a"
0~
. . , ( , , , ,~
0 100 200 300 400 BOO 600
X / h
. , ~, , , ~.
O t 00 200 300 400 500
X / h
0.0
~0
1~5
-
~ 1.0
0~5
0.0
. _ .
_ ~e~
O 100 200 300 400 500 600
X / h
~ ~-
.
. ~,,,,,,,,,,,,e,,,,,,,,,,,,,,,,,,,,,,,,,,,, 5, ]
0 100 200 300 4)0 SOO 600
X / h
Figure 4. Wave Numbers Along Crest Lines ( solid ) & Trough Lines ( dashed )
576
6 ~
~ -
5)
c
IL
13 3
m
~ _
2.8 - 1
". _ ~
q 6 it\
q~42 \
F = 5, £ / h = 0.25
\
\
F=10, S/h=0.25
c
2- \
\ *e
._ ~o*~~~
300
_ \ F = 5, £ / h = 1 ! I \ F = 10, £ / h = 1
~1.2 ~ \ 1
05 4~ n ~ ~= ~
0.24
1.9
1
1.7
1.6
1.5
1.4
c 1~
1~2
1.1
1
O
_ OJ
O.,
0.8
0~
o
5 ' \
_I \
U.4~
c
hi
1
mAl \
7
200
X / h
lot i
lL all
, ,\
13 7- \
~ 6- \
m 6
800
\
300 340 3eO
Figure 5. Waveform Amplitude Factor
577
i
l
. , ,
Chin 3~0 450
X / h
F = 1 0 , £ / h = 2
1
l
UO 600 640 see
~-~
INTERNAL WAVE ( F-5, D/h-0. 45, L~hS5. ~ ( MIN. ) Ah-- . 077S, ~1 ( MAX. ) Ohm. 03633 )
~ ~.~ ~,,~
I_, -
`, 'a
- ~a,
IFITE - AL WAVE ~ FY10, D/h - 0. 45, LO-5, 1( t1IN.) Ah-.0~1113, fit t1AX.) ~h'.025E; ~
Figure 6(a). Thermocline Displacement Patterns ( Near Field )
578
-a
-
~o ~'
At'
~ -Io
~If.
INTERNAL WAVE ( Fee, D'h=0. 9, L/h~10. ~ ( MIN. ) Ohm- . 3574, ~ ( ~X. ) Oh - . 1800 )
It's
me-- ~-
~~ en
- ~486
~ To
At;
~ 1,%10
~- - .
INTE~L WAVE ( F-10, Do. 9, L/h - 1 B. ~ t ~IN. ) ho- . = - , 7( ~X. ) Ah.. 1070 )
Figure 6{b). Thermocline Displacement Patterns ( Near Field
579
oes - ,
0.4 -
0.3
0.2
o.1 -
o
~\
1. F = 2.5, x' / hFl/2 - 3.26
2. F = 5.0, x'/ hFl/2 = 3.45
3. F= 10., x'/hFl/2=3.43
-O.1 ~
-0.2
-0.3 -
-0.4
0.3
0.2 -
O.t -
_3~/
>0~
-0~5 1 , , ~, , , 1 , , .
-1 o -8 - 6 - 4 - 2 o 2 4
, , , , i
1' 1 1 1 1
6 8 10
y/ h
Figure 7. Near Field Wake Patterns
1. F= 2.5, x'/hFl~ = 3.26
2. F = 5.0, x' / hFl~ = 3.45
3. F= 10., x'/hFl~=3.43
.,
-0.2 -
l
, I
-0.3 -
o
2
Kh
Figure 8. Shipform Amplitude Factor
580
-
TTr ' ' ~ 1 1 1 1 1 | o.oo
1 00 200 300 400 500 60C
X / h
- 1
1111~1
0 1 00 200 300 400 500 600
X / h
._!
~\x W_~. ~
J I ~
0 1 00 200 300 400 500 600
X / h
0.1
3 0.10
SAC
a_
O.OS
0.00
_'
I m o.lo-
_
-
~: ~
I - o.os ~
o.oo -
1
F ' 10, ~ / h - 1. 1
1
:; _
0 1 00 200 300 400 SOO 600
X / h
0.15 ~
~ _ 5, ~ ~ h - 2.
b~ ig -a ;~
0 1 00 200 300 400 500 600 0 1 00 200 300 400 5.00 600
X /
F - 10, ~ / h - 2.
Figure 9. Crestline Wave Amplitudes
581
