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The Numerical Solution of the Molions of a Ship Advancing in Waves G. X. Wu and R. Eatock Taylor University College London London, UK Abstract The hydrodynamic problem of a surface ship advancing in regular waves at constant forward speed is analysed using a three dimens tonal theory based on the linearized velocity potential. The potential is represented by a distribution of sources over the surface of the ship and its waterline. Various numerical schemes are introduced to overcome some of the maj or difficulties in this problem. Calculation is made for a submerged sphere. Results are compared with the analytical solution and very good agreement is found. Some preliminary calculations have also been made for a series 60 ship with block coefficient 0.7. 1. Introduction The wave induced motions of a ship have several implications for ship performance, increased resistance, deck wetting, slamming, vertical acceleration and propeller emergence, etc. While all of these aspects are important subj ects in ship hydrodynamics, the fundamental problem remains that of estimating the overall motion of the ship in waves. In order to predict ship motions in waves, the ship is usually regarded as a rigid floating body having six degrees of freedom, and the fluid loading is estimated from linearized potential flow theory. This theory assumes that the fluid is inviscid and incompressible, the flow is irrotational, and both incoming wave elevation and body oscillation are small. The velocity potential therefore satisfies the Laplace equation, and the corresponding boundary condition is imposed on the mean pos ition of the fluid boundary . Even after such drastic assumptions have been introduced, the solution of the resulting equation is still not easy to obtain. One of the maj or difficulties 529 arises from the complicated free surface condition. Further difficulty is associated with the fact that for a practical ship its shape is usually described by coordinates of discrete points rather than by a simple mathematical function. As a result, the solution can be only obtained numerically . Attempts to predict ship motions in waves can be traced much earlier, but a significant breakthrough was the work by Korvin-Kroukovsky and Jacobs (1957). Based on phys ical intuition rather than rigorous mathematics, they provided the early version of strip theory. Even though their theory was later found to be mathematically inconsistent, ( in particular it does not satisfy the Timman-Newman relation(l962) ), experimental data have shown that it nevertheless provides very good results in many cases. A number of modified versions of this strip theory have since been developed, of which, that proposed by Salvesen, Tuck and Faltinsen (1970) is widely used in ship des ign. Another very significant step was the work of Newman (1978). He overcame the limitation of the conventional theory to the region of high frequency, and proposed a "unified strip theory" which is valid throughout the whole frequency region. In particular this theory takes some account of wave interactions between different cross sections of the ship. Numerical results for heave and pitch (Sclavounos 1985) in infinite water depth have shown that the unified theory is superior to the conventional strip theory in such a case. Even though the s trip theory can provide satisfactory results in many cases, and has had a very important role in ship des ign, it has its inherent limitations. It requires the ship to be slender, and the magnitudes of forward speed and encounter frequency to be in appropriate ranges. Furthermore, while it

may provide good results for the total force on the ship, it usually gives a very poor prediction for the detailed hydrodynamic pressure distribution around the hull. Thus attempts to remove some of the limitations of strip theory, by using a three dimensional approach have been initiated by Chang (1977) and others (Inglis & Price 1981, Kobayashi 1981, Guevel & Bougis 1982). These investigations have all adopted the constant panel method (Hess & Smith 1964): the ship hull is represented by small panels on which the sources or dipoles are assumed to be constant. It has been found that these frequency domain three dimensional theories in general improve the results and provide better agreement with experimental data. However, it has been observed that the numerical solution is sensitive to the size of the panels, and high accuracy is not easy to obtain. This was also noticed in recent work by King, Beck & Magee (1988) using a three dimensional method in the time domain. The present work is part of an investigation which aims to obtain a stable and accurate solution of the linearized three dimensional problem, using the source distribution approach. Attention is focused here on certain numerical aspects. Firstly, we use quadratic isoparametric boundary elements instead of plane constant panels. As the associated wave resistance is known to be sensitive to the shape of the ship, it seems likely that isoparametric elements should enable us to model the ship hull with a higher degree of accuracy. They also provide a more convenient means of calculating the velocity of the fluid on the ship surface. Secondly, we impose the body surface condition by averaging over the body surface using the Galerkin method, rather than at discrete nodes. Experience has shown that this method usually gives more accurate results. In this particular problem, as the body surface condition on the waterline is averaged over the body surface, we can avoid the difficulty of both the source and field points being on the free surface when solving the integral equation. The use of the Galerkin method also avoids another serious numerical difficulty: the second order derivatives of the steady potential due to forward speed (which appear in the body surface condition on the unsteady potential due to the ship oscillation) can be reduced to first order derivatives, as in the coupled finite element method (Wu & Eatock Taylor 1987a) In the integral equation, we express the Green function in terms of the exponential integral (Wu & Eatock Taylor 1987b). Extensive tests have been carried out to try to achieve accurate evaluation of the Green function, and a technique has been introduced to remove the singularity in its integrand. We use a similar technique to that of Noblesse (1983) to reduce the order of the dipole singularity 1/r (where r is the distance between the source and the field points). We do not however need to evaluate the constant by integrating the Green function over the waterplane of the ship. Finally, to remove the singularity due to the source 1/r, we adopt triangular polar coordinates when calculating the contribution of an element to itself (Li, Han & Mang 1985). An alternative method for achieving this, by subdivision of the element, has also been investigated. These numerical procedures are found to be very effective for a submerged sphere. Compared with the analytical solution (Wu & Eatock Taylor 1988), the numerical method provides very accurate results when 12 elements for half of the sphere are used. Calculation are also made for a series 60 hull of block coefficient 0.7 at Froude number Fn~0.2. 2. Mathematical Formulation We define the right-handed coordinate system O-xyz so that x points in the direction of steady forward speed U of the ship and z upwards; the origin of the system is located on the undisturbed free surface and the middle section of the ship. The whole system is moving with the ship at the same forward speed. For a time-periodic incoming wave at a frequency w0, the total potential can be written as =-Ux+Ui (x , y , z ) +Re[j[0~7j jj (x,y,z)e ] (1) where ~ is the steady potential due to unit forward speed, ¢. ( j51, . . . ~ 6) are radiation potentials corresponding to the six degree of freedom oscillations of the body and ~.(j=1,...,6) are corresponding motion amplitudes; ¢0 and ¢7 are the potentials of the inch dent and diffracted waves respectively; and ~=~77 is the incoming wave amplitude. The encounter frequency ~ is given by ~> no ~ (~>O/g)U cost (2) where g is the gravitational acceleration and ,`3 is the incident angle of the incoming wave and ,B=0 indicates a following sea. Based on the assumptions of the linearized theory, we have for the steady potential v2¢ =o 530 (3)

in the whole fluid domain R; Adz+ RIO (4) on the undisturbed free surface S. , where ~g/U ; F 3~/3n n (5) on the body surface SO, where n is the inward normal of the body surface and n is its component in the x direction; andX 3~/3n =0 (6) on the bottom SB of the fluid or z,-- in the present case of infinite water depth. To complete the boundary-value problem, we also need to include the radiation condition at infinity: it is usually assumed that there is no wave due to far in front of the ship but there are waves far behind the ship. The components of the radiation and diffraction potentials are assumed to satisfy the following equations (Newman 1978) in R; (7) ~jz+(, /~)¢jxx~2ir~ix~~¢i~o on SF (8) where r=wU/g and ~c~2/g; and where G is the Green function for a pulsating translating source, which is taken in the form derived by Wu & Eatock Taylor (1987b). From the above equation, we obtain ~n(P) 4~ HIP)+ It |S an~pjQ)a(Q)ds g JL an ~ P j ~ (Q) nX (Q) dY] ( 12 ) on S0, where a(P) is the inner subtended angle of the ship surface at point P and the integration excludes the point Q P. Substituting equation (9) into the above equation, we can obtain the corresponding boundary-integral equation for the source distribution. One difficulty in dealing with equation (12) is caused by the normal derivative of the Green function. It contains a second order singularity of the dipole when CAP. To avoid that we define F(P Q.) ~ 1 + l (13) where r is the distance between P and Q. and r is that between P and the mirror image 1of Q about the undisturbed free surface. Applying Green's second identity in the domain enclosed by the ship and its water plane where ~F/3n O. we have Bjj/3n -i~nj+Umj (j-1, ,6) (9a) akp) _ | ~F(P,Q) dS (14) 3¢j/3n .-~¢O/8n j=7 (9b) SO ~n(Q) on SO, where (n1,n2 ,n3) = (nX,ny,nZ) (n4,nS,n6) = X n U(ml,m2,m3) ~ -(n.V)W U(m4,m5,m6) - -(n.V)(X W) W = UV(~-x) . (lea) (lob) (lOc) (led) (lee) X is the position vector of a point on SO relative to the origin of the coordinates. The potentials ¢. alto satisfy the same condition on theJbott~m of the fluid as ¢. The radiation condition on ¢. states that the outgoing wave with itsJgroup velocity larger than forward speed travels far in front of the body; otherwise the waves propagate behind. Following the derivation of Brard (1972), the unknown potential can be represented by a source distribution over the body surface SO and water line L We have ¢(P) 4~ [ JS G(P,Q)a(Q)dS g JL (P Q)a(Q)nX(Q)dY] (11) By substitution of equation (14) into (12), the latter becomes n(P) 4~|Sot ~n`Pj ~(Q)~n`QjQ)a(p)]ds 4~ g JL ~n(Pj ~(Q)nx(Q)dy (15) It is easy to confirm that the order of the singularity in this equation has been reduced. 3. Numerical Discretisation We now discretise equation (15) using the shape function Nj. We write =.§ a.N. J=1 J J where n is the number of nodes. By use of the Galerkin method, equation (IS) can be written as [A][~]=[B] 531 (16) (17) where [A] is the square matrix with the coefficients ij 4~ iSOISo[ ~n`Pj ;(Q) ~F'QPjQ) Nj(P)]Ni(P)dSQdSp g iSo[iL ~ntP) Nj(Q)nx(Q)dy]Ni(p)ds }; (18a)

and [B] contains the body surface boundary condition and has the coefficients bij ~ JS ~ N (P) dS (lab) To obtain an accurate solution of the above equation, careful consideration must be given to several factors which have most significant effects. Firstly, the Green fucntion is expressed in an integral form with a complicated and highly oscillatory integrand. The numerical evaluation of such an integral requires a very small step and takes most of the computer time. In our analysis, we have perfomed certain transformations of variable to reduce the oscillation. To deal with the following singularity in the Green function (Wu & Eatock Taylor 1987b) I ~ Jo 7(4rcose 1) at ~ ~~acos(l/4~) when r>0.25, we introduce the following scheme I --1 {) sin~f(~)-sin~f(-r) do sin, J O J(4,cos8-1) sin, 2, 7(47-1) (19) A similar scheme is adopted in the range (,,~/2) and is found to be effective. The second important factor which significantly affects the accuracy is the method of discretisation of the ship. Initially, a coarse mesh can be refined by using more of the coordinates of the ship hull provided by the offsets; but this process is limited by the number of coordinates available. When a still finer mesh is needed, the commonly adopted procedure is to interpolate using the shape functions. Consequently this may refine the representation of the source distribution but it does not improve the representation of the ship hull. This leads to the problem that different shape functions will give different hulls. It may not be important when these ship hulls are close to each other; but a problem can arise when even then the results do not converge. Ultimately, different shape functions may lead to different converged solutions when the above subdivision procedure is adopted. When this happens, subdivision of elements must be based on measuring the nodal coordinates on the lines drawing of the ship. The third factor is the integration over the body surface in equation (18). After numerous tests and careful consideration of accuracy and efficiency, we have chosen the four point Gaussian scheme. To avoid the singularity when UP, we have investigated two methods: that proposed by Li, Han & Mang (l9,85) using a triangular polar local coordinate system; and a method based on subdividing the element when the integration is performed. We have found that both schemes give very similar results and the latter has been chosen in the main computer program. To improve efficiency, we have also used the fact that components of G(P,Q) are either symmetric or antisymmetric. This reduces the computer time by almost a half. Finally, to avoid the difficulty of calculating the second order derivatives in equations (lOc) and (led), we can perform the integration in equation (lab) by parts. This reduces the derivatives to first order (Wu & Eatock Taylor 1987a). After the solution has been found, the added masses A.. and damping coefficients Aij can be obtained from (Newman 1978) 2 - 1~> j P |so(i° fj+ w.v¢; )nidS P iso(iu)ni~Umi)¢j dS+PUtL¢j~znidL(20) where the second term has been transformed using the relation derived by Ogilvie and Tuck (1969). In general, the second form of this equation has no apparent advantage over the first. In fact the second order derivative in mi makes the calculation even more difficult. However, when the steady potential ~ can be neglected, such as for sufficiently slender ships, the latter form has the advantage of not requiring calculation of the derivatives of the unsteady potential. Thus for a slender ship we have 2 ~ . . ~ ~ . . - 1~> . . 1J 1J 1J P |So(i~)ni Umi)4q~ [ |sG(P,Q)a(Q)ds g |LG(P,Q)a(Q)nx(Q)dy] dS P 4~ k~lak k where a corresponds to +. and k |SO(i(~)ni-umi) [ |S G(P,Q)Nk(Q)dS (21) g |LG(P,Q)Nk(Q)nx(Q)dy] dS (22) Ck can of course be calculated when matrix [A] is assembled rather than after the solution has been found; otherwise the computer time would almost be doubled. 532

4. Numerical Results In the following calculated example we have taken ¢~0. This is in fact consistent with the linearized free surface condition given in equation (8). The presence of the steady disturbance potential in the body surface condition alone does not appear to provide a consistent improvement to the accuracy of the boundary-value problem for a surface piercing body. In general, when ~ can not be regarded as a small quantity, its contribution should be included in the free surface condition (Newman 1978) as well as in the body surface condition. If the incoming wave is of large amplitude, a fully nonlinear mathematical model should be used. Tables la and lb give the added mass and damping coefficients for a sphere of radius a undergoing forced oscillations. Table 2 gives the exciting force on the sphere in an incoming wave with incident angle p~0.75~. The sphere is submerged at hula (h being distance between the centre of the sphere and the mean free surface) and translates at a Froude number Fn=U/~(ga)~0.4. The hydrodynamic coefficients are nondimensionalized as ~ij/p~a ~ and the exciting forcees as F./pg~a uq0. In the tables, ~ and u0 correspond to the values of ~ and w0 al the critical point r,0.25. We have omitted r3 from table 1, since it is Observed That the Timman-Newman (1962) relation is very well satisfied at this forward speed. The results from the numerical method (designated N in the tables) are based on an idealization using only 12 elements (45 nodes) on one half of the surface of the sphere. These are seen to agree reasonably with results from the analytical solution (Wu & Eatock Taylor 1988), designated A in the tables. It should be noted, however, that these results are not the same as those in Wu & Eatock Taylor (1988), since the latter included the effect of the steady disturbance potential on the body surface boundary condition. We have observed that excluding this term has one marked effect: namely it leads to a non-zero rotational moment about the centre of the sphere. This should not occur, and it highlights the importance of including in this case. Nevertheless, the comparisons shown in the table provide evidence of the reliability of the numerical procedures adopted to solve the boundary value problem by equations (7) and (9). ~11 ~22 ~33 ~13 ma A N A N A N A N 0.1 0.7009 0.6948 0.6939 0.6996 0.7291 0.7251 -0.0028 -0.0029 0.2 0.7200 0.7141 0.7055 0.7116 0.7607 0.7577 -0.0259 -0.0264 0.3 0.7128 0.7064 0.7075 0.7137 0.7529 0.7501 -0.0758 -0.0773 0.4 0.6077 0.5987 0.6936 0.6996 0.6259 0.6153 -0.0799 -0.0828 0.5 0.6290 0.6224 0.6722 0.6779 0.6315 0.6244 -0.0198 -0.0211 0.6 0.6374 0.6311 0.6579 0.6629 0.6271 0.6201 -0.0007 -0.0011 0.7 0.6420 0.6356 0.6482 0.6532 0.6226 0.6156 0.0102 0.0103 0.8 0.6443 0.6379 0.6416 0.6465 0.6186 0.6115 0.0170 0.0174 0.9 0.6452 0.6387 0.6371 0.6423 0.6153 0.6081 0.0212 0.0215 1.0 0.6452 0.6388 0.6341 0.6388 0.6125 0.6051 0.0236 0.0240 Table la. Comparison of added mass coefficients for sphere (h=2a, Fn=U/~(ga)=0.4, ~ a=0.3906) >11 a submerged >22 N A N A N A N 0.1 0.0035 0.0035 0.0025 0.0023 0.0062 0.0063 0.0181 0.0184 0.2 0.0275 0.0279 0.0155 0.0153 0.0450 0.0460 0.0360 0.0367 0.3 0.0722 0.0781 0.0376 0.0380 0.1205 0.1237 0.0249 0.0253 0.4 0.0871 0.0884 0.0595 0.0605 0.1454 0.1513 -0.0891 -0.0937 0.5 0.0371 0.0380 0.0631 0.0644 0.0975 0.1013 -0.0695 -0.0712 0.6 0.0275 0.0280 0.0599 0.0609 0.0850 0.0879 -0.0592 -0.0604 0.7 0.0242 0.0245 0.0548 0.0557 0.0769 0.0793 -0.0502 -0.0511 0.8 0.0230 0.0233 0.0492 0.5000 0.0703 0.0725 -0.0420 -0.0428 0.9 0.0225 0.0228 0.0438 0.0444 0.0645 0.0665 -0.0348 -0.0355 1.0 0.0222 0.0225 0.0388 0.0393 0.0593 0.0612 -0.0285 -0.0291 Table lb. Comparison of damping coefficients for a sphere (h=2a, Fn=U/~(ga)~0.4,~ a=0.3906) 533 submerged

I F1 l A N A N A N 1F31 0.1 1.0669 1.0633 1.0789 1.0859 1.5441 1.5411 0.2 0.8434 0.8402 0.8601 0.8626 1.2345 1.2342 0.3 0.6957 0.6950 0.6828 0.6846 0.9640 0.9610 0.4 0.5560 0.5548 0.5395 0.5407 0.7678 0.7661 0.5 0.4449 0.4437 0.4299 0.4306 0.6119 0.6107 0.6 0.3559 0.3548 0.3444 0.3449 0.4889 0.4881 0.7 0.2851 0.2840 0.2768 0.2772 0.3916 0.3911 0.8 0.2286 0.2278 0.2232 0.2233 0.3145 0.3143 0.9 0.1837 0.1828 0.1801 0.1802 0.2530 0.2530 1.0 0.1478 0.1470 0.1457 0.1456 0.2039 0.2042 Table 2. Comparison of exciting forces on a submerged sphere (h'2a, Fn.U/~(ga)~0.4,~0ca~0.2937,§~0.75~) As an application of this analysis to a surface ship, we have calculated results for a series 60 hull with block coefficient 0.7. We first investigated convergenece by assuming a rigid free surface condition,and using two meshes on one half of the ship hull: with 168 elements and 567 nodes and with 280 elements and 935 nodes, as shown in figure 1. The second of these meshes is substantially finer than those used by others in earlier published work. he found that the former provides results within 3.1% and 5.3% of the finer mesh results for added mass in heave and pitch respectively, while these two meshes give virtually identical area and volume fqr the ship. Next we calculated the hydrodynamic coefficients by using the translating pulsating source Green's function in equation (22), with the source strength in equation (21) based on the rigid free surface calculation. This has the advantage of providing a much more rapid calculation of source strength, and is related to the approximate method used by Newman (1961) for a submerged ellipsoid. The results from the coarse mesh are shown in figures 2 and 3. (a) Coarse mesh (567 nodes) (b) Fine mesh (935 nodes) Figure 1. Mesh for Series 60 hull 534

xt~1 ~1 20. to. Q 10 S. , , 2 3 t ~ Legal) (a) heave x,~2 ~1 5 -10. -t5 20- -25' t 5 2 6J /~) (c) heave/pitch xlr2 1K to to to. 8. ~ ._ x'0-2 16. 11. . 12. 10. a Q - ; ~ 5 ~ '~q~L) (b) pitch/heave . . . . 3 t S Legal) (d) pitch Figure 2. The added masses of the series 60 (Cb=0.7) at Fn=0.2 535

xt~l X18-l 8. 25. ~ ~ _' ~ J tS. to, Is ~ Ethyl) , . 2 3 ~ Mogul) (a) heave (b) pitch/heave xt.-l Xt. j, 6, ~ t1. 3. 7 1~ _ ~ ~ to. ~ , 0- 1 . o. ~ 8 >3 6. 1, , , , , ~ ~ 2. . J 1 2 3 t S 2 3 ~ AL) ~ ', GEL) (c) heave/pitch (d) pitch Figure 3. The damping coefficients of the series 60 (Cb=0.7) at Fn=0.2 536 ~ _z J _ 4 5 7 - J ~ 5

5. ConcIusions The hydrodynamic problem of a surface ship advancing in regular waves at constant forward speed is analysed based on the linearized velocity potential theory and using the boundary integral technique. The numerical techniques introduced have been found to be effective in overcoming some difficulties encountered previously. A major remaining difficulty is the evaluation of the Green function when both source and field points are near the free surface, which appears to be the direction towards which further work in this area should be directed. References Brard, R. "The representation of a given ship form by singularity distributions when the boundary condition on the free surface is linearized", J. Ship Res., Vol. 16, pp.79-92, (1972) Chang, M.S. "Computation of three-dimensional ship-motions with forward speech, 2nd Int. Conf. on Num. Ship Hydrodyn., pp.l24-135, University of California, Berkeley, (1977) Guevel, P. and Bougis, J. "Ship-motions with forward speed in infinite depth", Int. Shipbuilding Prog., Vol. 29, pp.1-3-117, (1982) Inglis, R.B. and Price, W.G. nA three dimensional ship motion theory-comparison between theoretical predictions and experimental data of the hydrodynamic coefficients with forward speed", Trans. R.I.N.A., Vol. 124, pp.l41-157, (1981) King, B.K., Beck, R.F. and Magee, A.R. "Seakeeping calculations with forward speed using time-domain analysis", 17th Symp. on Naval Hydrodyn., The Hague, The Netherlands, (1988? Kobayashi, ~ God the hydrodynamic forces and moments acting on an arbitrary body with a constant forward speed", J.S.N.A. Japan, Vol. 150, pp.61-72, (1981) Korvin-Kroukovsky, B.V., and Jacobs, W.R. "Pitching and heaving motions of a ship in regular waves n, Trans. SNAME, Vol. 65, pp. 590-632, (1952) Li, H.B., Han, G.M. and Mang, H.A. "A new method for evaluating singular integral in stress analsysis of solids by the direct boundary element method", Int. J. Num. Meth. Eng., Vol. 21, pp.2071-2098, (1985) Newman, J.N. "The damping of an oscillating ellipsoid near a free surface", J. Ship Res. Vol. 5, pp.44-58 (1961) Newman, J.N. "The theory of ship motions", Adv. Appl. Mech., Vol. 18, pp. 221-283, (1978) Noblesse, F. "Integral identities of potential theory of radiation and diffraction of regular waterwaves by a body", J Eng. Math., Vol.17, pp.l-13, (1983) Ogilvie, T.F. and Tuck, E. O. "A rational strip theory for ship motions n Rep. no. 013, Dept. of Naval Architecture and Marine Eng., University of Michigan, (1969) Salvesen, N., Tuck, E.O. and Faltinsen, O.M. "Ship motions and sea loads", Trans. SNAME, VO1. 78, pp.250-287, (1970) Sclavounos, P.D "The unified slender-boa] theory: ship motions in waves", 15th Symp. Naval Hydrodyn., O.N.R., Washington, (1985) Timman, R. and Newman, J.N. "The coupled damping coefficients of a symmetric ship", J. Ship Res., Vol.5, pp.1-7, (1962) Wu, G.X. and Eatock Taylor, R. "Hydrodynamic forces on submerged oscillating cylinders at forward speed", Proc. Roy. Soc. London, Vol. A414, pp.149-170, (1987a) Wu, G.X. and Eatock Taylor, R. "A Green's function form for ship motions at forward speed", Int. Shipbuilding Prog. Vol. 34, pp. 189-196, (1987b) Wu, G.X. and Eatock Taylor, R. "Radiation and diffraction of water waves by a submerged sphere at forward speed", Proc. Roy. Soc. London, Vol. A417, pp.433-461, (1988) 537

DISCUSSION by R. Huijsmans The authors are to be congratulated on their treatment of the full forward speed diffraction problem. We at MARIN have the experience that using the exact Green's function for the translating oscillating source as was studied by eg Bougis, Inglis needs a very careful treatment of the panel sizes. In the above mentioned studies only very coarse grids were used. In recent calculations at our Institute, we calculated added mass and damping of a series 60 ship with the number of panels increasing up to 856. Apart from the very large computational burden the results for the added mass and damping did not seem to converge really with increasing panel sizes. This was especially the case when the pressure distribution was examined. Our opinion is that there is conflicting requirement regarding the stationary and the oscillating part of the Green's function with respect to the panel sizes, especially when the forward speed is not very large. 1) Does the authors have the some experience regarding the statement made above? 2) Can the authors give some indication of the type of computer they used and computer time they have used for the non-zero speed case? Especially when comparing with the zero speed case. 3) Does the authors have some idea/indication how the local hydrodynamic quantities, like pressure and velocities behave when increasing the number of panels? Author's Reply We thank Dr. Huijsmans for his interesting comments. From our experience using quadratic boundary elements for the problem, we are certainly not surprised that using 850 or so constant elements to represent the series 60 hull did not always provide satisfactory results. Our own coarse mesh used 1085 nodes for the submerged hull (the numbers in Fig.1 referring to one half of the hull), and the finer mesh 1789 nodes. Like Dr. Huijsmans we have been looking at various ways of overcoming the conflicting requirements at small forward speed, and some of our thoughts on this are to be published elsewhere[A1]. We do not yet have an efficient algorithm for the Green function, and the computing time therefore still quite long. For the series 60 results given in Figs.2 and 3, they range from about 2000 to 7000 seconds per frequency on a CRY 1, with the longer runs corresponding to results at higher frequencies. We also run these programs on a Microvax II, and have to wait a few days for results at one frequency! We have not evaluated pressures and local kinematics for the case of the body with forward speed. But we would expect to draw similar conclusions to those given in Eatock Taylor and Sincock[A2]. [Al] Wu, G.X. and Eatock Taylor, R.: The Hydrodynamic Force on an Oscillatory ship with Low Forward Speed, J. of Fluid Mech. to appear 1990). [A2] Eatock Taylor, R. and Sincock, P.: Wave Upwelling Effects in TLP andSemi- submersibleStructures,Ocean Engineering 16, pp.281-306 (1989). DISCUSSION by G. Jensen Isoparametric elements are associated with numerical integration. Could you please give some more details about the computation of the velocities and may be higher derivatives on the body? Author's Reply The first order derivatives of the velocity potential on the body surface (and hence the fluid velocities at any point on the surface) can be obtained from the nodal solutions, using the shape functions, together with the known normal derivatives. Thus uses ax = acax + away + adz ad axar cyan azar an apex away adz _ _ ~ _ + an- Dxarl ayarl azar, an an an an On axnX + Dyne + aznZ to solve for the derivatives of ~ in the x,y and z directions. This approach can not be used directly to obtain the higher order derivatives. As discussed in the paper, however, it may only be the integrated effect of such derivatives that is required on the body surface, and in some circumstances this can be obtained by alternative means. 538