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Wave Resistance and Squat of a Slender Ship Moving Near the Critical Speed in Restricted Water H. S. Choi Seoul National University Seoul, Korea C. C. Mel Massachusetts Institute of Technology Cambridge, USA Abstract The wave resistance and implied squat of a slender ship advancing near the critical speed in restricted water are studied. Employing matched asymptotic expansion techniques, it is shown that the response can be described by the homogeneous Kadomtsev - Petviashvili (KP) equation with flux conditions on boundaries, when the channel is wide compared to the ship length. Numerical results show the generation and radiation of straight-crested solitons in a periodic manner ahead of the ship, when it moves at transcritical speeds with moderate blockage. The solitons are initially three dimensional, which are followed by a depressed region and a train of complicated ship-bound waves in the wake. Hydrodynamic forces are computed by using slender body approximation, and the implied sinkage and trim are estimated based on hydrostatic relations. These quantities vary with time and strongly depend on the ship's speed and blockage. Near the critical speed, the wave resistance and the trim oscillate around mean values in phase with the emission of solitons, while the sinkage takes place out of phase. Ibe calculated results are in crude agreement with the measurements. 1. Introduction A couple of investigators have reported the fascinating phenomenon observed during shallow tank tests that a ship model towed near the critical speed ~ (g = acceleration due to gravity, h = water depth) radiates a succession of upstream-propagating waves in an almost periodic manner (Thews & Landweber 1935; Izubuchi & Nagasawa 1937; Graff 1962; Huang et al. 1982~. As a result, the ship experiences 439 considerable changes in resistance, trim and linkage, or better known as squat. One of most exciting aspects in this now identified phenomenon is that a three-dimensional disturbance, such as a ship, generates 2 dimensional waves propagating upstream in a tank of finite width. In addition to it, the propagation speed of the upstream waves, named solitons, is faster than the constant towing speed so that a steady state cannot be attained. Linear theories fail to predict the flow. Katsis and Akylas (1987) clarified it in the light of the linearized dispersion relation. Among all waves radiated from a disturbance advancing at a speed u, those which may remain stationary with the disturbance in the direction ~ must have the wave number k such that F kh cost = (kh ta~2~2' where F is the depth Froude number (=u/~. It means that at a transcritical velocity, F = ~ + O(kh)2, long waves must be in nearly the same direction as the moving disturbance, i.e. cost= ~ + of. Furthermore the group velocity tends to vanish in the moving frame. Consequently the long waves become almost nondispersive and the associated wave energy cannot be radiated. It implies, in order to deal with the problem, we have to include a balanced interplay by the nonlinear and dispersive effects to the leading-order wave equation, and to keep it in mind that transient waves evolve slowly. Wu and Wu(1982) were the first who calculated the generation and propagation of solitons for a moving disturbance spanning uniformly across the channel by using the generalized Boussinesq equation. Akylas (1984) also considered a two-dimensional pressure band travelling on the free surface but focused attention to the immediate neighborhood of the critical speed. He showed that the physics

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can simply be described by an inhomogeneous Kortewegde Vries (KdV) equation. In a joint theoretical and experimental study, Lee et al. (1989) found a broad agreement between the experiment and two theoretical models, the generalized Boussinesq and the forced KdV equations, for a moving two-dimensional bottom topography in a shallow water tank. All these works are concerning with 2 dimensional disturbances and thus one-dimensional wave fields. For a rectangular patch of surface pressure whose width is comparable with the tank width, Ertekin et al. (1986) solved Green - Nagdhi's directed - sheet model numerically. Their calculations have yielded two-dimensional flow upstream and three-dimensional flow downstream, in qualitative agreement with the measurements they carried out systematically (Ertekin, 1984 and Ertekin et al.,1984~. Wu and Wu (1987) obtained similar results using the generalized two- dimensional Boussinesq equation. Katsis and Akylas(1987) calculated some 3 dimensional long waves bounded by side walls using the Kadomtsev-Petviashvili (KP) equation. Quite recently, Lee and Grimshaw (1989) studied the three-dimensional slowly-varying evolution of wave fields ahead of a bottom topography in a honzontaDy unbounded fluid domain by using a forced KP equation. Meanwhile, Bai et al. (1989) applied finite element method to a vertical strut sliding in shallow water. 2. Formulation We consider a ship advancing with a constant speed in a shallow channel. For simplicity, the ship is assumed to possess the lateral symmetry and to move amid the channel so that it is enough to take only the half of fluid domain into account. A rectangular coordinate system is introduced, which is fixed on the waterplane of the ship. The x-axis coincides with the longitudinal axis of the ship and the centerline of the channel (Fig.1~. Under the usual assumptions of potential theory, fluid motions are described in terms of the velocity potential Mix*, y*, z*, ,*y, which is the solution of the following initial - boundary value problem. The Laplace equation holds in the fluid region 92+ =0 ~ -it 5 Z S ~ ). (1) The kinematic and dynamic boundary conditions on the free surface at z = `* are ~ *=t,*+(u+l *it *++ *< *, (2) 9~*+~ +U~ +~/2 (vl*~2=o. (3' , X It is, however, as yet unclear how these methods with pressure distributions can be applied to three dimensional bodies such as a ship. In this respect, Mel and Choi (1987, hereinafter referred to as I) extended the theory of Mel (1986) to treat the transient forces on and responses of a slender ship. They found that the on the channelwall waves in the far field can be described by one- dimensional inhomogeneous KdV equation for a special class of channel width (=2w) and ship's slenderness parameter (= 8) as follows; w / ~ = 0(,u-m) with 0 5 m 5 1,2 and 8=o(,~25), where 2~ stands for the ship length and ~ = ~ / ~ = oLl) for the dispersion. It correctly predicts the upstream solitons, and the estimated sinkage and trim for a destroyer favourably compare with the time - averaged experimental values of Graff et al. (1964~. But the theory fails to render three-dimensional waves in the wake. In viewing the result of Katsis and Akylas (1987), it strongly suggests us to modify our theory in their direction. In fact, it was already pointed out in I that the KdV equation is to be replaced by the KP equation, when the canal is much wider, i.e. w, ~ = of. We have done it in this paper by following the same scheme described in I, but for a wider channel. 440 No net flux condition holds on the channel bottom ~ *= 0 (z =h ), **=0 6*=W), and also on the ship's hull r*=R*(x*,0) (4) (5) 4>~* (U++x*)R *[1+(Rg ~R*~2]-1,2 (6) where r* and ~ are polar coordinates on the cy*, z*) - plane. Under the assumption of slender body,the normal derivative on the ship surface has been approximated by that on the transverse plane at constant x* along the ship. Before the initial instant ,=o, there was no disturbance +*=o, t*=o `~*=oy. (7)

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In order to solve the problem approximately, we employ perturbation techniques by defining two smallness parameters e=A/h , ~=~2/L, and assume without loss of generality E= ~2 which implies that the nonlinearity and the dispersion are both important to the leading order solution of the problem. In eq.~), A means a typical wave amplitude. Vanables are normalized as below: ~ =AL,, ~ =(gAL/~, 2' =~- (9) Herein we rather focus our attention to the flow near the critical speed. Thus the Froude number is expanded F2=1 - 20~2 Wit/2 ~=0~1~. To cope with the slow variation of flow, we have to rescale time T= I/, The channel is assumed wide in comparison with the ship length W/L=~/ ~0 Wit/2 ~o=O(~. (12) As pointed out by Mel (1986), the blockage coefficient SB must have a magnitude of 0~4) 5~,RO2 / TWIT = 0 (~4), (13) where RO denotes the characteristic transverse radius of ship and the blockage coefficient is simply the area ratio of the midship section to the channel cross-section. Consequently the slenderness parameter must be b=Ro/L=O(~2~. (14) It implies that the nonlinearity arises from the slenderness of disturbance. Because of these vastly different length scales involved, it is relevant to divide the channel cross-section into three regions as: (i) the far field ; X* ~ < =, y* = 0 (W) = 0 (L/~), a: =o(~=(~L), (15) (ii) the intermediate field; x*=O(L), ~ ~ z )=O(h)=O(~L), (16) (a) (iii) the near field; X*=O(L), ~ ~ z '=0(Ro'=0(,~2L). (17) We shall first analyze each field separately and then conduct matching by following the line reported in I. Accordingly most parts must be very similar to those in I and it is already obvious that modifications which will appear are solely attributed to the different definition on the length scale of win. Neverthless steps are explicitly given here in order to keep this paper self- contained. 3. The Far Field In accordance with the scheme defined by eq. (15), the far field variables are made (11) dimensionless x =Lx,y =Wy,z =hz. (18) The normalization of other variables holds effective as defined in eqs.~9) - (11~. In this field, the banks and the bottom of the channel directly affect the propagation of waves. However, the boundary condition Qn the ship hull is of no meaning and the forcing agency is not known. The governing equation and the boundary conditions are rewritten in terms of the dimensionless farfield variables: (6Pxx+~ ~lo2di,yy)+4pzz=o (-l 5 Z 5 ~2~) (19) 2~1 _ 2 OCR for page 439
Substituting the expansions +~~(0)+ ~2~(V+ ~4~(4)+, . . (24) ~ t(0) + ~2~(2) + ~4~(4) + . . . (25) into eq.~19), we obtain +(o)=o, (I )= - ~(x), +(4)=_~(2)+~2~(0)) (26c) With help of no flux condition on the channel bottom, the general solution straightforwardly yields to - ilr'2~;(0)=o (Z=o). (3 ~ It is the homogeneous KP equation, which is the three-dimensional counterpart of the KdV equation, since it contains the transverse dispersion as well as the longitudinal dispersion. If t()=o, the KdV equation is recovered. An observer In the far field is too far away (26a) from the ship so that he just observes waves without knowing the details of wave generation, which can only be found through matching with (26b) the intermediate field. For this purpose, we need the inner expansion of the far field potential for small y =~(O)(X O. T)+y+()(X,O,7~+~ ~ l(V(X,O,T) _ ]rZ+~2~(0)~+or~3~ (31) .$~~=~3(x y ~) (27a) 4. The Near Field +(2)=~(2)(xyT)_~(z+~2~(0) (27b) +(4)= +(4)(X y T) _ ~ (Z + i)2~ (2)+ ~ 2 (0)) + y =Roy , z =Roz, In a fluid domain close to the ship, the characteristic length is Ro and thus let us nondimensionalize the near field variables + ~ (z+~4~0) (27c) r =ROr, R =RoR, (32) where +(n) are unknown functions to be determined later by matching with the intermediate field potential. The free surface elevation is derived from the dynamic condition, eq.~21) but retain the rest of the normalizations. The Laplace equation x < z s (~ /~L is now transformed for =_g,~o' (z=o), (28a) 82xx+~,+~--- (33) tt2' _~2~_~0~_ ~ ~0~_ ~ `~0~2 ~Z=O) `2gb) ~ the free surface at Z-=~3/~t, we have Utilizing the above relations, the kinematic condition of 0~4) on z=o turns out to be ~ (4) ~ (2)+ ~ (0)2a ~ (0)2~( )t ( ~ (Z = 0) . (29) Combining eqs.(27) - (29) and differentiating with respect to x, we finally get the leading order wave equation Liz= ~ 8~1 - 2~ Ace+ ~1 - 22 + + ~38xtx+ (~3/~, (34) (1 - 2~2~+ SIX'+ ~2~1 _ 2~ Aft+ + 1 ~2[<,2+(,,2+<,,2~82]=o. (35) On the ship hull r=R(x,0), the condition is tax)~`Xx) - _~(0)~2xx~ ~ (8 )2~1 - 2~2+~21 jR [1+(Re/R) ] . (36) 442

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As already discussed,the slenderness parameter is determined a sarnll quantity of o(~2) and thus we suppose the following expansions: ,4, A, (0)+ ,<~(1)+ ~2~(2)+ . . . (37) ~ `(0)+ ~;(1)+ 2~;(2)+ (38) Then the Laplace equation and the boundary conditions are readily expressed as below: ~,(n~)+~~,Ln)=0 ~x < z 5 O. n=0~l,2), (3 ~ chin)= o (Z = 0, n = 0,1 ,2), (40) {;(o)=_,-,() (z-=oy, (41) `1,(")=0 (r=R, n =0,1), (42a) ~>(2)=R ~~ + OR / R~2 ~-1,2 ~r=R~, (42b) From eqs.(39) and (40) it is clear that (~) may be absorbed by (3. The leading-order problem is of homogeneous Neumann type, of which solution formally takes the fonn ()=f()(X T) (43) The next-order solution contains a particular part owing to the inhomogeneous term in eq. (42b) `~(2)=f(2)`X At+ ~>~2~`` y A,. (44) The particular solution represents the disturbed flow due to the motion of the ship,of which outer approximation for large r will prove to be more meaningful in the viewpoint of matched asymptotic expansions. For large r, the ship shrinks to a line source and thus +` is expressed 4, (2)= _q (x ,T) In r + c (x ,7), (45) where q stands for the source strength and c may be regarded as a part off(2)(x,T). Applying the law of mass conservation to the fluid domain surrounded by the ship, the free surface and a control surface located far away from the ship, the source strength is readily 443 2 F2 q = 2 SBSXKX), To ~ (46) where SB jS referred to the blockage coefficient and six) to the longitudinal distribution of the cross-section area of the ship. It is reminded that the blockage coefficient has a magnitude of o(,u~4) near the critical speed, i.e. F = 1 + o(,u2~. For the sake of matching with the intermediate field solution, it is necessary to expand eq.(44) for large r ~ ~ ~f()~x Ty + p.,2tf(V(x ,T) +Lenr)] + (47) 1 5. The Intermediate Field Here the proper reference length for y* and r* is h, hence we introduce the dimensionless intermediate coordinates y = Ily^, r hr. (48) and keep all other normalized variables defined in eqs.(9)-(11) and (18). The nondimensional equations are ,u-2+Xx+ Fly+ fizz= 0 (- 1 s z s ,u-2~), (49) 4~/120t,u~ ~ + ,u~ (1 2a,u~ )`x + ~ ()xtx+ ~ (l~y~y ( = ~2~) (50) (1 - 2alt2)(` + ~X) + ~2~/1 - 2a~+ + _[~24,x2+~>2+~2]=0 (Z= 1~ A), (51) 4)z=0 (Z=-1), `~',t = (~/~)(1 - 2(Y. ~2+ ~2X) R. [1 + (R. / R )2 ] -~2 (52) (it =R ). (53) In anticipation of matching with the near field, we introduce expansions in the form of eqs.(37) and (38), with ~ and ~ replaced by ~ and Respectively, it then follows

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Aft, (n)+ ~~, (n)= 0 (1 s z ~0, n = 0, i), (54a) ,:~(2)+~(2)=_,`> (0) (54b) From the Taylor expansions of the conditions on the free surface, we get i`> (n)= 0 (Z = 0, n = 0,1), (55a) `~(2)=~;(0) (z=o). (SSb) It is immediate to have +()=f()(X T) +(2~=f~2)(X7)_~(z+~)2f(0)+~(2) (56b) The particular solution (2) again corresponds to the response to a line source at the ship's centerline, hence takes the following form for small r <,(2)= ~ q^(x,T) in r=In (r). AT ~ ~ The solution has the inner expansion +~f (X,T)+ ~ ~ f(V(X,7)f (O)+ q O + ln( r)]+ (57) Matching with the outer expansions of the near field results in the relations q=q, f()(X T) =f( )(X IT) ~ `~()~ 2 qY (Yowl)' (61) so that the outer expansion of the intermediate field is, in terms of the far field van ables <~~f()`x'T)+ q +,2 Aft_ Size ~27(0)]+ . . .~62) to 2 Matching provides useful information: `,(~(X o :~=f(~(X,~=f()(x,~), (63) (56a) 2l~o 712 ~4 Differentiating eq.~64) with respect to x and recalling the relation of eq.~28a), we finally get the boundary condition for `(3 at y=o t( )(X,O,T)= 2 ~S=(X). (65) ~0 We realize that the result thus obtained is basically the same as those Katsis and Akylas (1987) derived. Once t;(3 is calculated, ,,(3=f~)=ff) is known. Since the leading-order dynamic pressure is linearly proportional to the surface elevation, the forces and moment acting on the ship can easily be evaluated by invoking the slender body approximation. The implied sinkage and trim (S8) may be estimated from the hydrostatic relations (Tuck, 1966), which are given in I. (59) (60a) x,=,_ if(O) (60b) The source strength measured in the intermediate field is indeed idendical with that in the near field. From mass conservation, the approximation for l(2) must be 444

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6. Numerical Results s(x)=l_x2 (1x1 5 1). To further investigate the wave field and resulting hydrodynamic forces, we have to rely on numerical computations for the homogeneous KP equation with proper boundary conditions. In contrast to the KdV equation, only a few literatures are available, in which numerical methods for the KP equation have been discussed. Katsis and Akylas (1987) applied an explicit finite-difference method for the standard KP equation and investigated the wave patterns due to normally distributed pressures in shallow water laterally bounded and unbounded. We adopted their scheme. To facilitate calculations, it is convenient to integrate eq.~30) with respect to x ((a)= aL()+ _~(o)~(o)+ _~(o>+ _~2i SIX (66) where use has been made that t(3 and its derivatives must vanish far upstream. If the integral term on the right hand side of the above equation is neglected, it becomes the standard KdV equation. In the discretization, simple forward differences are implemented for time derivatives, and central differences for spatial derivatives. But at the wall and the centerline of channel one-sided differences are used instead of the central difference, and the boundary conditions, eqs.~23) and (65), have been incorporated. Integrals are evaluated by trapezoidal rule. From diverse numerical tests, the scheme has proven to be stable for Ax = 0.1 , Ay = 0.1 , Ad= 0.00002 with the ship's length as 2.0 spanning from x= - Lo for bow to x= Lo for stem. These values have been used for all computations in this work. As Katsis and Akylas pointed out, reflected waves from the numerical boundaries far upstream and downstream seriously deteriorates the result. Several devices for radiation condition have been tried without success. Thus the computation domain is taken as large as possible and the portion is discarded, where numerically reflected waves are apparent. To save computing time, it is advised to begin with a small domain and to enlarge it continuously as time passes. Our primary concern is to examine the generation and propagation of solitons by a ship. For this purpose, let us consider a slender ship whose cross-sectional area varies parabolically 445 In Fig.2, the evolution of the wave field at the critical speed is illustrated with time interval T=0.2 Up to ~=~.o. The parameters are chosen as below: ~=h / L =0.25, W / L= 1.0 (~o=4~0~, SB=0.105 (~=25.6), F=1.0 (~=0.0~. The blockage coefficient seems somewhat too large for our theory to be valid. Neverthless this case is taken up, because it is possible to compare with the experimental and numerical results of Ertekin et al. (1984 and 1986~. They calculated Green-Nagdhi's model for a rectangular pressure patch,which should be roughly equivalent to a ship with blockage coefficient, SB=0 105, in the viewpoint of the hydrostatically displaced free surface. The figures are exaggerated vertically by 2.5 times. It is to note that 1 T corresponds to their nondimensional time UT,=. During this time span, the ship advances a distance of 8 times the ship's length. At T=0.2, three-dimensional waves emerge ahead of the ship. A depressed region is built therebehind, which is followed by a train of complicated ship-bound waves in the wake and also reflected transverse waves far downstream. As time elapses, the upstream waves develop further and gradually become straight-crested as they are reflected from the wall. The first soliton almost completes its formation and becomes two- dimensional at T=0.4. The second soliton starts to take its shape at T=0.6, while the first soliton steadily propagates upstream and the depression is being elongated. At T= i.0, the second soliton is completed and the third one begins to appear. Such a trend can also be recognized in Ertekin et al., but the waves downstream here look more ship-bound. In Table 1, a comparison is made for the amplitude, propagation speed of the first soliton and the period of first two solitons. It is to note that the amplitude was taken at T=~.0 (UT/h=~), since it continuously increases during the developing phase. Let us first compare present results with those of Ertekin. The propagation speed agrees excellently,but significant deviations exist in amplitude and period. It is not probable that these are caused simply by numerical round-off errors. It was concluded in the works of Ertekin et al. that the amplitude increases and the period is shorter as the blockage is

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larger, and the details of the disturbance is less important. The blockage coefficient is identical in both cases, but the presumed ship of Ertekin is much fuller than our slender ship. It might act as a stronger forcing agency and result in higher amplitudes. If it is true, then the present period should have been longer in order not to contradict the measurements. But it is not the case and let us leave it as open question. Now we turn to Bai et al.~1989) who applied finite element method to a vertical strut sliding in a shallow channel. Their result is closer to the present for amplitude, but to E~tekin for penod. The propagation speed is slower than both. We may postulate that it is attributed to the different mathematical models. But no clear-cut conclusion can be drawn at this stage. Table 1 Soliton amplitude,propagation speed and period for ,s=2s.6 at a=O.O ,i A / h c / Jim | UTg / it I, Bai I Ertekin | Present I. 0.553 1.24 0.6248 1.280 n sr2s 1 1.281 30.0 29.6 28.8 Fig.3 shows the wave profiles along the centerline and the wall of the channel at ~=~.o. The completed first soliton assimilates each other closely. But there is a slight difference on the rear side of the second soliton, because a new soliton is just about to burst. A train of modulated wave packets follows a depressed region, which is directly reponsible for the sinkage and trim of ships. The downstream configurations are in general quite dissimilar. The time evolution of the wave resistance, sinkage and trim is depicted in Fig.4. The wave resistance and sinkage are normalized by the displacement and half length of the ship, respectively, while the trim is in degrees. Positive values indicate resistance, downward sinkage and trim by stern. The computed sinkage and trim are of qualitative meaning only, because no dynamic effects are included. Caution should be paid on three differently-scaled ordinates. All these quantities rise initially from zero to first maximum and then oscillate around mean values. The oscillation period is approximately ~g=0.45 in coincidence with that of soliton. It reflects the fact that hydrodynamic forces and their effects on the ship are dominated by the generation and radiation ofsolitons. The wave resistance and the trim fluctuate in phase with the periodic soliton emission, while the sinkage takes place out of phase. It seems unlikely that a steady state will be attained in time. To assess the effect of ship's speed, we consider the same slender ship as above but in a slightly deeper channel ,u=0.333, =3.0 (W /~=~.0), ~ = 5.0 (SB = 0.062) for five speeds: two subcritical speeds a=2.5 (F=0.667) and a=~.O (F=0.882), critical speed a=O.O (F=~.O), two supercritical speeds a=-~.O (F=~.106) and a=-2.5 (F=~.247~. The wave resistance,sinkage and trim are plotted in Fig.5,6 and 7 in this order. At transcritical speeds, the wave resistance indeed oscillates. The amplitude of oscillation reaches its maximum not at the critical but at a slightly faster speed, and the period becomes longer as speed increases, which supports the experimental findings. At the low subcritical speed,the wave resistance increases upto a certain threshhold with an intermediate step, and it arrives at a near - steady state. At the high supercritical speed,the wave resistance initially reaches a maximum and then diminishes with time to a small steady value. Similar trends are to be observed for the sinkage and trim in Fig.6 and 7. It is to note that the sinkage oscillates around zero at the critical speed and it becomes negative (lift up) at supercritical speeds. Generally speaking, the overall behaviour of the wave resistance, sinkage and trim is quite similar to two-dimensional cases described in I, as long as the channel width is not too wide and thus the flow around a ship is chiefly affected by upstream solitons. The variation of soliton amplitude, propagation speed and period according to ship's speed is listed in Table 2. The amplitude given here is referred to the computed value at ~=~.o for the first soliton. The numbers in parentheses designate the experimental counterparts. The difference in emission period is again remarkable. However, it can be said that theory provides at least crude predictions. 446

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Table 2 Variation of soliton amplitude, propagation speed and period for three ship's speeds A/h ! call .. .. . I a= - 1.0 ~x= 0.0 i a= 1.0 I _ 0.64 (0.60) 0.45 (0.49) 1.31 (1.26) 1.21 (1.20) 0.29 1.13 (0.26) (1.12) . . _ . _ _ .- UTg / h __ 50.8 (53.~) 40.5 (49.5) 31.0 (41.6) Next we examine the wall effect on the response at the critical speed. Three channel widths are chosen as w/~=o.s, Lo and 3.0, while the channel depth and the ship are kept unchanged (~ 0.333~. It implies that the corresponding blockage coefficients are 0.123, 0.062 and 0.021. Fig.8 a - c show the wave fields at T=~.O, when the ship moved a distance of 4.5 times the ship's length. The vertical scale is stretched by 3.2 times in comparison with those on the horizontal plane. Since the lateral coordinate is normalized by the half width of the channel in all cases,the banks are designated by -1.0 and 1.0. For a wider channel, it is necessary to reduce by for a better resolution. For w/~=o.s, there is an indication that the second upstream wave develops on the back of the front waves, whose crest line is spear- headed. The depressed region is relatively long and the downstream waves are pronounced. For w,~=3.0, there is no sign for upstream- propagating waves and diverging waves with large run angle prevail. The downstream waves are hardly two-dimensional. Katsis and Akylas (1987) suggested that the maximum canal width for which the downstream waves remain to a reasonable approximation two-dimensional depends on crudely the source characteristics. They found that the maximum channel width is about 20 h for an elongated pressure distribution. Since we are dealing with a slender ship, it may be possible that the downstream waves remain practically two dimensional upto a certain range of channel width. But we have not attempted to confirm it. The wave resistance, sinkage and trim are summarized in Table 3. Again the resistance and sinkage are made dimensionless with the displacement and the half length of the ship. Trim is in degrees. First two extremes are given with the nondimensional time in parenthesis at which they occur. For resistance and trim, the first 447 extreme corresponds to the first maximum and the second extreme to the first local minimum. The extremes and their deviations take greater values as the channel becomes narrower. For linkage, the first extreme represents lift up for w / r=o.s and 1.0, but downward sinkage for w/~=3.0. The ship sinks more In average in a wider channel. The time intervals between two extremes for all three quantities are consistently 0.4 and 1.7 for w/~=o.s and 1.0, respectively. But there is no such a correlation for w ,~=3.0. Hang S.Choi would like to thank the Korean Science & Engineering Foundation for financial support. He also wishes to thank I.H.Cho, a graduate student at Seoul National University, for his drawing pictures. References Akylas,T.B. 1984 On the excitation of long nonlinear water waves by a moving pressure distribution. J.Fluid Mech. 141,455-466. Bai,K.J., Kim,J.W. and Kim,Y.H. 1989 Numerical computations for a nonlinear free surface flow problem. to be presented at the 5th Inter''. Conf. on N~anerical Ship Hydrodyn. September, Hiroshima. Ertekin,R.C. 1984 Soliton generation by moving disturbances in shallow water : theory, computation and experiment. Ph.D. Thesis, Univ. Calif. Berkeley. Ertekin,R.C., Webster,W.C. & Wehausen,J.V. 1984 Ship - generated solitons. Proc. 15th Sym:p. Naval Hydrodyn. Hamburg, 347-364. Ertekin,R.C., Webster,W.C. & Wehausen,J.V. 1986 Waves caused by a moving disturbance in a shallow channel of finite width. J.Fluid Mech. 169,275-292. Graff,W. 1962 Untersuchungen ueber die Ausbildung des Wellenwiderstandes im Bereich der Stauwellengeschwindigkeit im flachem, seitlich beschraenktem Fahrwasser. Schifftech}~ik, Bd.9,Heft 47,110-122. Graff,W., Kracht,A. & Weinblum,G. 1964 Some extension of D.W.Taylor's standard series. Tra~zs.Soc.Naval Arch. & Marine E'~gnrs. 72, 374- 401. Huang,D.-B.,Sibul,O.J. & Wehausen,J.V. 1982 Ships in very shallow water. Festkolloquium zur Emeritierung von Karl Wieghardt, Institut fuer Schiffbau, Hamburg Univ. Bericht Nr.427, 29- 49.

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Izubuchi,T & Nagasawa,S. 1937 Experimental investigation on the influence of water depth upon the resistance of ships. (in Japanese) Japan Soc. Naval Architects, 61, 165-206. Katsis,C. & Akylas,T.R. 1987 On the excitation of long nonlinear water waves by a moving pressure distubution.Part 2.Three-dimensional effects. J.Fluid Mech. 177, 49-65. Lee,S.-J.,Yates,G.T. and Wu,T.Y. 1989 Experiments and analyses of upstream-advancing solitary waves generated by moving disturbances. J.Fluid Mech. 199, 569-593. Lee,S.-J. and Grimshaw,R.H.J. 1989 Upstream- advancing waves generated by three-dimensional moving disturbances. to appear in Physics of Fluid. Mei,C.C. 1986 Radiation of solitons by slender bodies advancing in a shallow channel. J.Fluid Mech. 162,53-67. Mei,C.C. & Choi,H.S. 1987 Forces on a slender ship advancing near the critical speed in a canal. J.Fluid Mech. 179,59-76. Thews,J.G. & Landweber,L. 1935 The influence of shallow water on the resistance of a cruiser model. US Exp. Model Basin, Navy Yard Rep.408. Tuck,E.O. 1966 Shallow-water flows past slender bodies. J.FIuidMech.26,81-95. Wu,D.M. & Wu.T.Y. 1982 Three-dimensional nonlinear long waves due to moving surface pressure. Proc. 14th Symp. Naval Hydrodyn. Ann Arbor,103-129. Wu,D.M. & Wu,T.Y. 1987 Precursor solitons generated by three-dimensional disturbance moving in a channel. Proc. IUTAM Symp. on No''li''ear Water Waves, Tokyo,69-76. Table 3 First two extreme values of wave resistance sinkage & trim for a slender ship in channels with different width ~ ~ = 0.0, `~0.333) W/L l Rw s aT 2w .~_ 1 ~1 1 2L -1 Fig.1 Definition sketch of a slender ship advancing in a channel 448 . 0.5 0.140 (0.7) 0.101 (1.0) -0.008 (0.5) 0.013 (0.9) 12.082 (0.6) Q ADA {1 ~N 1.0 0.105 (1.1) 0.074 (1-8) -0.005 (0.8) 0.014 (1.5) 8.827 (1.0) A.__ `~.VJ 6.456 (1-7) 1 3.0 0.085 (2.1) 0.082 (2.8) 0.017 (0.5) 0.011 (1.9) 7.900 (1-7) 5.894 (3-9)

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(a) T = 0' Fig.2 Evolution of wave field generated by a slender ship 449 (b) ~ = 0.4 {~1 ~ = Or (d) ~ = 0.E (e1T= 1.0

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~ o o - a, 8- c~ , 1 o o 0- l l -40.00 -24.00 1 1 1 -8.00 8.00 24.00 I ~ I I I ~ 40.00 56.00 72.00 X (a)y = 0 ~ 8- ~D 8- o o o m_ -40.00 -24.00 I I I I I I 1 1 1 1 1 1 -8.00 6.00 24.00 40.00 56.00 72.00 X (b)y = W Fig.3 Wave profiles along the centerline & the wall of channel at ~ = 1.0 RW - O O KI ~ . O l l 0.00 0.20 AT S ~ mi F ~ , 8- on 0 . , ~ ~ ~ I I I ~ I T 0. 40 0.60 0.80 1 .00 1 .20 1 . 40 Fig.4 Evolution of wave resistance, sinkage and trim for a slender shin ~ a = 0.0, ~ = 1!6-5, TO = 4.0, ,u~ = 0~25) 450 1

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~1 o- oooo ~ 1 1 ~ T (degrees ) 8 ~ 1 o of Rw (U] o: _ /< ~ ~ ~o,o 8- ~ g: /, , 1 , o `,.00 0.40 0.60 1.20 Fig.S Evolution of wave resistance on a slender ship for five different speeds (,s = 5.0, ~0 = 3.t), `~ = 0.333) O o s o_ 0 ~ 1 1 1.60 2.00 2.40 2.80 1 \a = -2.5 1- 1 1 1 1 1 1 1 1 1 1 1 1 '1 0.40 0.~30 1.20 1.60 2.00 Fig.6 Evolution of sinkage for a slender ship for five different sneeds ( ,s = 5.0, ~0 = 3.t, ,~ = 0.333) or=-1. O ~ \~, a =1. 0 ~ \ 2.40 2.~30 `,~ _ _ ~ _ ~ _~ a =-2 . 5 8 1 ~ = 0 - I I , 1 1 1 1 1 1 1 , 1 I--- i 1 O.CO 0.40 0.80 1.20 1.60 2.00 2.40 2.BO ~ Fig.7 Evolution of trim for a slender ship for five different soeeds ( ~ = 5.0, ~0 = 3.~), ,~ = 0.333) 451

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-it (a)W7L=0.5 I it. it, it' 'by\ (C) W/L = 3.0 -I ,~ o , ,~ Fig.S Wave pattern generated by a slender ship for three different channel widths at ~ = 1.0 ( ~ = 0.0, ,u = 0.333) 452 ,~_

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DISCUSSION . by R.C. Ertekin I would like to make a few comments before asking some questions on the points that are not clear to me. Precursor soliton generation is not restricted to the critical speed. These waves have been observed and reported for Froude numbers as low as 0.2. The theory you used may be restricted to critical speeds but the phenomenon is lot. Soliton speeds are not necessarily faster than the towing speed always. As Fr -1.3 solitons form a bore attached to the bow. Around critical speed if the soliton amplitude is very low it is also possible that the soliton and model speeds are almost the same. For all subcritical speeds it is not clear that steady state cannot be reached, since soliton amplitudes (if 2nd, 3rd, etc. solitons) decrease as the ship continues to move forward. In some cases, solitons would not have been generated if we had a model tank which is very long. I am surprised that Sommerfeld's condition did not work in your case. We have had no problem so far at the open boundaries. Perhaps something went wrong in the implementation process. I am curious why you have not compared your resistance results with the experimental data. Could you please comment on this? Your theory seems to be valid at Fr=l. However, most practical ship speeds are well below that. Can you extend the theory for subcritical speeds which may be low? If you can, your method will be much more efficient than a numerical method which is valid for all Froude numbers. Author's Reply Prof. Ertekin's discussions are highly appreciated. Since we have no experimental experience, your comments on the phenomena observed during tank tests will be much helpful for our further research. It is not to mention that the Boussinesq equations are effective over a wide range of speed in shallow water, once the corresponding Ursell number is close to unity. It is also true that the expressions are rather complicated and an immense computation is required. For a simplification, we need an additional assumption that the depth Froude number is expandable near the critical value. We have done it to obtain the two-dimensional Kdv equation or KP equation. Consequently it is obvious that our theory is valid for transcritical speeds. In reply to the question about the comparison of wave resistance with experimental data, we tried to calculate for a destroyer model. But still more computations are necessary before we are able to arrive at a conclusion. DISCUSSION by T. Inui This morning's Session (Session 7) reminded me my undergraduate diploma thesis (1943) on "Restricted Water Effect on Ship is Wake"(1943). We measured the wake at two lengthwise positions, i.e. at midship and at the propeller position, and we also traversed in beamwise direction at midship. For the midship wake, which is approximately "potential" wake, we found an unexpectedly good agreement between our measurements and Kreitner's simple 1-D theory. The three different flow stages, i.e. subsonic, tran- sonic, and supersonic, were clearly obtained. Couple years later, I applied linear wave- making theory to this phenomena (1946), and found that i) For purely shallow water dRW/dV is discontinuous at Fh=1, and ii) For restricted water Rw is discontinuous at Fh 1. However, naturally, I could not succeed to get theoretically the transonic region. Since then it was my dream to bridge this gap by CFD, because it is essential to take into account the bodily sinkage and squat for the hull boundary condition. The authors already obtained the first approximation for this. Then you may have a sufficient possibility by applying iteration. The authors' comments are highly appreciated. Author's Reply We thank Professor Inui for the comments on his own research. In response to it, we would like to emphasize once again that both the nonlinearity and the dispersion are important to the leading-order solution for the flow caused by a moving disturbance near 453

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the critical speed in shallow water, and that a steady state can hardly be attained. To the question, direct approaches such as the works Prof. Bai and Prof. Ertekin presented at this conference may give a better answer by taking the instantaneous hull- boundary condition exactly into their numerical schemes at the expense of a rapidly increased amount of computing ti me. In our slender-body approximation, it can be done only indirectly, if we include the temporal variation of the longitudinal distribution of ship's cross section in terms of source strength . 454