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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

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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[XCS8-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*(~)eikXCs~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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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/ OCR for page 567
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

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2 ~ ~ [C* - Co - 2Coto - 2COX]XX - [2Co(toy + Y)]XY + [C*2 _ ~ Joy _ (,~y)2]yy [ OXx]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

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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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a> n 1 1 -a l-w l - - ~ ~ 1 ~ ~ Q X a>\ 3 Xi; / ~ \ O/ T ~n \ \ \ ~C~ o C~ 10 \\N, 11 ~ \\\\ 1 , , I I I I , I I , , I I I I , Il _ _ ~ -. .. ". c' _ ~ ~ ~ ~ .. ~ t ~ ~ - o *011~ c~ o 4 - V - 1 4- o V 11 ~c oo V ~ ~: ~CQ V - 11 574 c C~ 3 C 0 c,OO V V 11 c, 11 ~Ix =1~ - ~v x ._ ce c a m c ._ C C) o o cn ~ Q 0) tn 1- ._ C o ._ C, - C o ._ a o ~_ C ~O a) 1 ._

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too - -~ ~F ' 5 . ~ / h 0~ 0 1 Go 200 300 ITS f00~ ~ 5 ~ ~ ~ h ~ ~ 0~ X / h ~0 7 6~00 10C ~ 501 10, (/ ~ ~ o" ~ DO 200 o 300 X / 40O 500 600 ~,~ 10, t/6 ~ 1 0 100 20n 300 IT 0~76~00 0~ X / h ~500 [~ internal Wave Crest ' I 6_ _- and Mach Line ~ddO)t)& TrU9h Lines ( dashed 575

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2.0 1 .S i 1'`\''\' ~\ `` \ `` \ F - 5, ~ / h - .25 2.0 1.S ~ Illit, '\~'~^ 0.5 0.0 4 , ~,' ,. ~.~ 1 00 200 300 "0 500 X / h ~ ~ it\''`\ I ~ ~ / h - 1. 1.0 0.5 0.0 O.S O.Q ~ rid ~ ~ I ~ ~ son o.o j _ F ~ 10, ~ / h - .25 _ . , ~, , , ~. 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

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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

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~-~ 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

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-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

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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

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- 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

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