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lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line PREDICTION OF WAVE PRESSURE AND LOADS ON ACTUAL SHIPS BY THE ENHANCED UNIFED THEORY 368 Prediction of Wave Pressure and Loads on Actual Ships by the Enhanced Unifed Theory M.Kashiwagi (Kyushu University, Japan) S.Mizokami, H.Yasukawa, Y.Fukushima (Mitsubishi Heavy Industries, Japan) ABSTRACT To establish a new practical calculation method in place of the conventional strip method, performance of the enhanced unified theory is investigated through the comparison of computed results with a large number of experiments conducted with VLCC tanker and container ship models. In this paper, compared are the ship motions, the pressure distribution, and the wave loads. The enhanced unified theory is essentially based on 2-D computations but takes account of 3-D and forward-speed effects. Furthermore the effects of wave diffraction from the bow part near the waterline are taken into account in a rational way. Despite these theoretical improvements, the results of comparison for the wave loads are not so good as expected. Since the pressure and wave loads are strongly influenced by the accuracy of ship motions, more improvement is needed for precise prediction of the ship motions particularly near the resonance of heave, roll, and pitch. INTRODUCTION In the design stage of actual ships, the strip theory is still in routine use for computing the ship motions, added resistance in waves, pressure distribution, and so on. Recently, 3-D computation methods based on the free-surface Rankine panel method have been developed, but they are time-consuming from a practical viewpoint, and validity for various ship shapes is not confirmed. On the other hand, the strip method is versatile and its prediction is relatively good, considering that the computation time is short and the theory is simple. However, several shortcomings in the strip method have been recognised; for instance, the pressure distribution near the ship bow and stern and the added resistance in short waves are not in good agreement with experiments. These shortcomings are related to improper treatment of 3-D and forward-speed effects. To account for these effects in the framework of slender-ship theory, many theoretical works have been made. Among them, the unified theory, originally developed by Newman (1978) and extended to the diffraction problem by Sclavounos (1984), is recognised as one of the successful slender-ship theories. The unified theory could bring in a certain amount of 3-D effects to a strip-theory type solution in a rational manner. However, it was still not satisfactory. For instance, the wave diffraction from the bow part near the waterline could not be taken into account, and thus the wave- exciting force in surge and the added resistance in short waves were usually underestimated. To incorporate the effects of the wave diffraction near the bow and other effects dismissed as higher order in the slender-ship theory, the original unified theory was enhanced by Kashiwagi (1995); in which the radiation problem of surge is solved in the same fashion as the heave and pitch modes, and the effects of wave diffraction from the bow part near the waterline are taken into account by retaining the x-component of the normal vector in the body boundary condition of the diffraction problem. Furthermore, 3-D and forward-speed effects on lateral modes of motion are incorporated as well. Validity of this enhanced unified theory (abbreviated as EUT in the present paper) has been confirmed only for mathematical ship models like a prolate spheroid and a modified Wigley model (Kashiwagi et al. 2000). The unified theory is essentially based on 2-D computations and thus the computation time is very short compared to that needed in 3-D Rankine panel codes; this feature is promising as a practical design tool in the design stage of actual ships. For the purpose of establishing a new practical calculation method in place of the strip method, we have investigated usefulness and applicability of the enhanced unified theory, through comparison of computed results with a large number of experi the authoritative version for attribution.

lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line PREDICTION OF WAVE PRESSURE AND LOADS ON ACTUAL SHIPS BY THE ENHANCED UNIFED THEORY 369 ments using actual ship models. In the present paper, some results of comparison are shown for models of a VLCC tanker and a recent container ship. The experiments were conducted in regular waves, and the incidence angle of the wave was changed rather densely. Although there are many experimental data, shown in this paper are the ship motions and pressure distribution of a VLCC tanker and the wave loads (vertical bending and torsional moments) of a container ship. We can see some noticeable improvements over the strip method particularly in the pressure distribution and wave loads, but predictions of the enhanced unified theory are still not perfect in some cases when compared closely with experiments. Discussion is made on possible reasons of disagreement with experiments. ENHANCED UNIFIED THEORY Mathematical formulation We consider a ship advancing with constant speed U and undergoing oscillatory motions with circular frequency Ï in deep water. The analyses will be performed using a Cartesian coordinate system, which moves steadily with the same constant speed as a ship. The x-axis is directed to the ship's bow and the z-axis is directed downward (see Fig. 1). Fig. 1 Coordinate system and notations Assuming the inviscid fluid with irrotational motion, the flow can be described with the velocity potential, which is expressed as (1) (2) (3) (4) where denotes the incident-wave potential; A, Ï0, k0, Ï are the amplitude, the circular frequency, the wavenumber, and the incidence angle of incoming wave, respectively; g is the gravitational acceleration. in (1) denotes the steady disturbance potential due to the forward motion of a ship. in (2) denotes the scattering potential and the radiation potential of the j-th mode with complex amplitude Xj, where in particular j=1 for surge, j=3 for heave, and j=3 for pitch. To obtain a solution for the purpose of practical calculations, the enhanced unified theory (hereafter abbreviated as EUT) is applied in this paper. In the subsections below, the outline of the theory will be given. For more details, we refer the readers to Kashiwagi (1995, 1997). Radiation problem In the inner region close to the ship hull, variation of the flow along the x-axis is small compared to that in the transverse section and the wave radiation at infinity is out of concern. Therefore, the velocity potential in the inner region satisfies (5) (6) (7) where K=Ï2/g. nj and mj in (7) denote the j-th component of the unit normal directing into the flu0id and of the so- called m-term representing interactions with the steady flow; these are considered on the contour (CH) of the transverse section at statio0n x along the ship's length. The general inner solution satisfying (5)â(7) takes the following form: (8) where Ïj and are the particular solutions, corresponding to the first and second terms on the right-hand side of (7), the authoritative version for attribution. respectively. denotes a homogeneous solution, which can be explicitly given by for the symmetric modes (j=1, 3, 5) and by for the antisymmetric modes (j=2, 4, 6), where the asterisk means the complex conjugate.

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lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line PREDICTION OF WAVE PRESSURE AND LOADS ON ACTUAL SHIPS BY THE ENHANCED UNIFED THEORY 371 Hydrodynamic and hydrostatic pressure Retaining only the first-order linear terms in Bernoulli's pressure equation, the spatial part of the unsteady pressure is given by (20) Here the first term on the right-hand side is the hydrodynamic part, with V defined as and the second term represents the change in the hydrostatic pressure due to ship motions from the equilibrium position. In accordance with neglect of in computing the m-term, an approximation of is employed in the present paper. Substituting (2) as in (20), the total oscillatory pressure can be divided into three components; those are written as (21) where pD, pR, and pS denote the diffraction pressure, the radiation pressure, and the change in the hydrostatic pressure, respectively. In the diffraction problem, differentiation with respect to x may be applied only to the rapidly-varying term, eiâx, and thus pD is given by (22) R S Likewise, p and p are given in the nondimensional form as (23) (24) The symmetric part of by EUT can be expressed by the homogeneous component (the second line) in (17). The same is true for the asymmetric part, although its explicit form is not shown here. The radiation potential by EUT is given by (8). Consistent with approximations for the m-term and the hydrodynamic forces (which will be explained next), differentiation with respect to x in (23) is performed only for j=5 and 6. The complex amplitude, Xj/A, of the j-th mode of motion will be given as a solution of the ship-motion equations. Hydrodynamic forces In the radiation problem, the force acting in the i-th direction is computed in terms of pR and the results can be summarised in the form (25) (26) where Aij and Bij are the added-mass and damping coefficients in the i-th direction due to the j-th mode of motion. In this paper, as shown in (9), mi and can be expressed with ni and Ïj. Therefore, all integrals in (26) along the contour (CH) of the transverse section at station x are evaluated using the following 2-D results: (27) The wave-exciting force in the i-th direction can be computed by integrating pD multiplied by ni over the ship hull. Using (22) and (17), the symmetric components (i=1, 3, 5) are expressed as (28) We note that hydrodynamic forces related to surge (i=1) are computed by EUT, with 3-D and forward-speed effects taken into account. Ship motions the authoritative version for attribution. In the linear theory, the symmetric modes (i=1, 3, 5) of motion can be computed independent of the antisymmetric modes (i=2, 4, 6) for a ship symmetric with respect to y=0. Therefore, the longitudinal motions (surge, heave and pitch) can be computed from the coupled motion equations: (29)

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lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line PREDICTION OF WAVE PRESSURE AND LOADS ON ACTUAL SHIPS BY THE ENHANCED UNIFED THEORY 373 as (36) This transformation is consistent with the treatment in computing the added-mass and damping coefficients in pitch. With these taken into account, the nondimensional calculation formula for the vertical bending moment takes the form (37) where S(x) denotes the area of the transverse section and âB(x) is the z-coordinate of the center of transverse-section area. The expressions for other components of the wave loads can be obtained in an analogous manner. For the subsequent comparison with experiments, let us describe the expression for the torsional moment. Defining the torsional moment acting counterclockwise about the x-axis to be positive (see Fig. 2), the torsional moment on the transverse section at x=x0 is computed by (38) where and is the distribution of the moment of inertia in roll. As is the same as the vertical shearing force, we note that in the radiation pressure may be discarded from a viewpoint of consistency with the computation of the added-mass and damping coefficients in roll. Furthermore, since only the antisymmetric components of the pressure contribute to (38), the nondimensional calculation formula for the torsional moment is given as follows: (39) where is the transverse metacentric height and is the gyrational radius of roll in the transverse section; both are nondimensionalized in terms of b. It should be noted that the 3-D and forward-speed effects are taken into account in EUT even for the antisymmetric part of and the lateral modes of the radiation potential (Kashiwagi, 1997). RESULTS AND DISCUSSION Outline of the strip method In this paper, the results of the strip method established by Salvesen, Tuck and Faltinsen (1970) (which is abbreviated hereafter as STFM) are shown and compared with the results of EUT and model experiments. In STFM, the contour of the transverse section is approximated by the Lewis form, and 2-D hydrodynamic computations are implemented by Ursell-Tasai's method. Surge is treated as an independent mode, with only the Froude- Krylov force and the inertia force due to the ship's mass taken into account. The computer code used in this study solves the diffraction problem directly, in which the freesurface condition of (15) is satisfied; that is, the wavenumber in the free- surface condition is not K but k0. Wave-induced ship motions The experimental data of ship motions and the pressure distribution used for comparison in this paper are for a VLCC tanker model. The principal particulars of this tanker model are shown in Table 1. The experiments were carried out at Ship Research Institute and their results were reported by Tanizawa et al. (1993). Although many experimental data exist, only the amplitudes of heave and pitch are shown in Fig. 3 for various angles of the wave incidence, together with corresponding results by EUT and STFM. The Froude number was set equal to Fn= 0.131. EUT takes account of 3-D and forward-speed effects in the radiation and diffraction forces. the authoritative version for attribution.

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lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line PREDICTION OF WAVE PRESSURE AND LOADS ON ACTUAL SHIPS BY THE ENHANCED UNIFED THEORY 375 at Fn=0.131. The pressure was measured at some points on the contour of the transverse section at S.S. 4.22 (0.1 m ahead of S.S. 4). The abscissa of each figure is the position along the contour and Î¸=â90Â°, 0Â°, and 90Â° correspond to the weather side, the centerline, and the lee side, respectively. the authoritative version for attribution. Fig. 4 Pressure distribution at S.S. 4.22 of a VLCC tanker (Î»/Lpp=0.3, Fn=0.131) Figure 4 shows the results of Î»/Lpp=0.3, at which the ship motions are almost zero except for the roll motion around Ï=30Â°. Therefore the pressure distribution in Fig. 4 may be regarded as the pressure induced by the wave diffraction only. The overall agreement between computed results by

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lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line PREDICTION OF WAVE PRESSURE AND LOADS ON ACTUAL SHIPS BY THE ENHANCED UNIFED THEORY 377 Fig. 6 Pressure distribution at S.S. 4.22 of a VLCC tanker (Î»/Lpp=1.25, Fn=0.131) We can see that the wave pressure at Î»/Lpp= 1.25 shown in Fig. 6 is relatively small in amplitude except for Ï=90Â°. In the present case, the roll motion becomes large around Ï=90Â° due to the roll resonance. In fact, the change in the the authoritative version for attribution. hydrostatic pressure due to the roll motion, the second term on the right-hand side of (24), becomes dominant near the roll resonance. Therefore precise prediction of the roll motion is crucial in estimating the pressure. For that purpose, as shown in (31), inclusion of the nonlinear viscous damping force in the ship-motion equation is very important. It should be noted that

lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line PREDICTION OF WAVE PRESSURE AND LOADS ON ACTUAL SHIPS BY THE ENHANCED UNIFED THEORY 378 the same modification for the roll damping coefficient using (31) is made both in EUT and STFM. (Unless the nonlinear viscous damping is taken into account in roll, the pressure at Ï=90Â° in Fig. 6 becomes tremendously large.) Fig. 7 Vertical bending moment at S.S. 5 of a container ship (Fn=0.215) Wave loads In order to make a thorough investigation on the wave loads, measurements of the vertical bending moment and the torsional moment have been carried out using a container ship model at Nagasaki R&D Center of Mitsubishi Heavy Industries. The experiments were conducted for various angles of the wave incidence and at five different Froude numbers. Furthermore, the wave loads were measured at seven stations along the ship's length. The length-to-breadth ratio, L/B, and the block coefficient, Cb, of the tested ship model are 6.45, and 0.59, the authoritative version for attribution. respectively. The nondimensional metacentric height in roll, was set to 0.03 and the gyrational radius in pitch, Îºyy/L, was 0.25 in the experimental setup. Figures 7 and 8 show the vertical bending moment and the torsional moment, respectively. These

lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line PREDICTION OF WAVE PRESSURE AND LOADS ON ACTUAL SHIPS BY THE ENHANCED UNIFED THEORY 379 were measured at S.S. 5 and Fn=0.215. Looking at closely, computed results of EUT differ from the measured results in some respects. Firstly for longer wavelengths in following and quartering waves, EUT tends to overestimate the vertical bending moment, and secondly in beam wave (Ï=90Â°), EUT is different from the measured results even in the variation tendency. These discrepancies may be attributed to imperfect agreement of ship motions, because the wave loads are strongly dependent on the results of ship motions. Fig. 8 Torsional moment at S.S. 5 of a container ship (Fn=0.215) Regarding the torsional moment at S.S. 5 (Fig. 8), the overall agreement between computed and measured results is favarable, considering the value itself is small compared to the vertical bending moment. However, we can see a difference in the variation tendency for oblique waves (Ï=60Â° and 120Â°). Figures 9 and 10 show the same items of the vertical bending and torsional moments, respectively, but the measured section along the ship's length is S.S. 7. Compared to the results at S.S. 5 (Fig. 7), the amplitude of the vertical bending moment decreases but the variation tendency with respect to Î»/L and Ï is more or less the same. Looking at the torsional moment at S.S. 7 (Fig. 10), a noticeable improvement by EUT over STFM can be seen in the shorter wavelength region for the case of Ï=30Â°. However, in other angles of the wave incidence, there are still discrepancies between computed and measured results. Since nonlinear and forward-speed effects on the damping force are important in the roll mode, those effects might be a reason of discrepancy in the torsional moment. Figure 11 shows the dependence of the Froude number on the vertical bending moment at S.S. 5 in head waves (Ï=180Â°). We can see that the value increases slightly as the ship's speed increases, but the overall variation tendency with respect to Î»/L is the same. Regarding the degree of agreement, we can point out that EUT tends to underestimate around Î»/L=0.7 and this tendency becomes prominent the authoritative version for attribution.

lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line PREDICTION OF WAVE PRESSURE AND LOADS ON ACTUAL SHIPS BY THE ENHANCED UNIFED THEORY 380 with increasing the Froude number. In fact, the forward-speed effects are not properly taken into account in EUT especially for higher Froude numbers. To improve in this respect, the forward-speed terms must be incorporated into the free-surface condition even in the inner problem of the slender-ship theory. Fig. 9 Vertical bending moment at S.S. 7 of a container ship (Fn=0.215) CONCLUDING REMARKS The enhanced unified theory (EUT) encompasses the strip method and takes account of the 3-D and forward-speed effects in a rational way. Moreover, EUT can compute the surge-related hydrodynamic forces in the same manner as for heave and pitch, and thus the longitudinal ship motions (surge, heave and pitch) are computed from fully coupled motion equations among these three modes. In the diffraction problem, EUT can also account for the wave diffraction from the the authoritative version for attribution. bow part near the free surface, because contributions of the n1-term are retained in the body boundary condition. We expected that these theoretical improvements over the strip method would result in good prediction of the distribution of wave pressure and wave loads even for actual ships.

lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line PREDICTION OF WAVE PRESSURE AND LOADS ON ACTUAL SHIPS BY THE ENHANCED UNIFED THEORY 381 Fig. 10 Torsional moment at S.S. 7 of a container ship (Fn=0.215) With this expectation, EUT was applied to a VLCC tanker model and a recent container ship model, and computed results were compared with experiments to confirm applicability of the theory. However, the results were not so good as expected before starting this study. One important reason of this is that the prediction of ship motions is not improved so much when compared to the results of STFM, since the ship motions are influential in the prediction of the wave pressure and resulting wave loads. Of course, the effects of nonlinearity and three-dimensionality of the flow may be important particularly around the bow part, and viscous effects are also important near the stern. These effects must be accounted for by more sophisticated 3-D computation methods. However, EUT is still advantageous from a practical viewpoint, because it is efficient in computations. Further study is needed for precise prediction of the ship motions, for which the forward-speed effects on the free-surface condition must be taken into account in a more rigorous way. ACKNOWLEDGMENTS The authors would like to thank Mr. H.Sueoka and Dr. T.Kuroiwa of Mitsubishi Heavy Industries for their help in the course of the present study. Mr. Y.Tozaki in assisting numerical computations is also greatly acknowledged. REFERENCES Kashiwagi, M., âPrediction of Surge and Its Effects on Added Resistance by Means of the Enhanced Unified Theory,â Transactions of West-Japan Society of Naval Architects, No. 89, 1995, pp. 77â89. Kashiwagi, M., âNumerical Seakeeping Calculations Based on the Slender Ship Theory,â Ship Technology Research, Vol. 4, No. 4, 1997, pp. 167â192. Kashiwagi, M., Kawasoe, K. and Inada, M., âA Study on Ship Motion and Added Resistance in Waves (in Japanese),â Journal of Kansai Society of N. A., Japan, No. 234, 2000, in press. the authoritative version for attribution.

lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line PREDICTION OF WAVE PRESSURE AND LOADS ON ACTUAL SHIPS BY THE ENHANCED UNIFED THEORY 382 Newman, J.N., âThe Theory of Ship Motions,â Advances of Applied Mechanics, Vol. 18, 1978, pp. 221â283. Salvesen, N., Tuck, E.O. and Faltinsen, O.M., âShip Motions and Sea Loads,â Transactions of SNAME, Vol. 78, 1970, pp. 1â30. Sclavounos, P.D., âThe Diffraction of Free-Surface Waves by a Slender Ship,â Journal of Ship Research, Vol. 28, No. 1, 1984, pp. 29â47. Sclavounos, P.D., âThe Unified Slender-Body Theory: Ship Motions in Waves,â Proceedings of the 15th Symposium on Naval Hydrodynamics, Hamburg, 1985, pp. 177â192. Tanizawa, K., Taguchi, H.Saruta, T. and Watanabe, I., âExperimental Study of Wave Pressure on VLCC Running in Short Waves (in Japanese),â Journal of Society of Naval Architects, Japan, No. 174, 1993, pp. 233â242. Fig. 11 Vertical bending moment at S.S. 5 of a container ship (Ï=180Â°) the authoritative version for attribution.

lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line PREDICTION OF WAVE PRESSURE AND LOADS ON ACTUAL SHIPS BY THE ENHANCED UNIFED THEORY 383 DISCUSSION P.Sclavounos Massachusetts Institute of Technology, USA I would like to congratulate the authors for a series of thorough studies, including the present one, refining the unified slender body theory for the prediction of motions and wave induced loads of realistic ship hulls. I would like to commend the authors' diligent refinement of the theory and its validation against careful experimental measurements. It is the experience the authors are reporting that has led me to switch my energies since the mid 80's to the development of the 3D Rankine Panel Method SWAN. I am however very pleased to see that several of the shortcomings of the unified slender body theory with forward speed have been removed by the authors. I have a few comments and questions to which I welcome the authors' response: The good correlation of STFM and EUT for the motions of the tanker is probably due to her low Froude number. It would be interesting to see how both methods perform at U=0. We have seen some very good performance of unified theory in that limit which seems to be corroborated by the results the authors present. In our experience with SWAN, one of the most important forward speed effect arises from the careful and complete treatment of the m-terms on the body boundary condition. These terms may actually be computed from the double body flow without the need to model surface wave disturbances. Their evaluation would require the use of a 3D panel method, yet their values can be input into EUT as forcing terms in the body boundary condition. It would be interesting to see the effect of this âexperimentâ on the performance on the EUT. It has been our experience that the m-terms in the body boundary condition contribute much more significant forward speed effects than their counterpart on the free surface condition, as the authors appear to suggest. What is the behavior of the EUT near the Ï=1/4 singularity in quartering waves. It is my recollection that the unified theory solution developed in the late 70's and early 80's by Nick and myself was predicting a singularity in the motion predictions at Ï=1/4. We have since seen that this singularity is not really present in SWAN, yet it remains difficult to resolve. How does EUT deal with this delicate regime when it arises in your computations and experiments? AUTHOR'S REPLY 1) Concerning good performance of EUT in the limiting case of U=0, we have shown many supporting results not only for a single body in open sea but also the tank-wall interference and catamaran problems (see reference [A1]). From theoretical viewpoint, the unified theory is a perfect one in the framework of slender- body theory. 2) Regarding the effect of the steady disturbance in the body boundary condition, our recognition seems to be different from that of discussor. Firstly, our purpose is to develop a practical calculation method which must be easy to implement but reliable in the design stage. Secondly, the results of the radiation forces (especially the diagonal coefficients) by the present method are in good agreement with experiments. Recent results for modified Wigley model with L/B=6.67 are shown in reference [A2]. The degree of agreement by means of EUT is almost perfect in A33, B33, and A55. The results of High-Speed Slender-Ship Theory (HSSST) are also shown in reference [A2] and HSSST gives obvious improvement in B55. This tendency is much more prominent in the cross-coupling terms. HSSST uses also the uniform flow assumption in the body boundary condition, but the forward-speed effects in the free-surface condition are fully taken into account. In fact, we have learned from Rational Strip Theory (RST) of Ogilvie & Tuck that the forward-speed effect in the free- surface boundary condition is important in the cross-coupling terms, which improves clearly over the strip theory. HSSST encompasses RST, and thus it is natural to see good agreement with experiments. From experiences mentioned above, we suggest that the forward-speed effects in the free-surface condition may be more significant than the effects of disturbance potential in the the authoritative version for attribution.

lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line PREDICTION OF WAVE PRESSURE AND LOADS ON ACTUAL SHIPS BY THE ENHANCED UNIFED THEORY 384 body boundary condition. 3) As shown in some figures in references [A1] and [A2], EUT shows singularity at the frequency equal to Ï=1/4, and experimental data also show rapid variation near Ï=1/4. I suppose the results of SWAN are imperfect near Ï=1/4, and we cannot discuss this sensitive behavior with questionable numerical results. [A1] M.Kashiwagi (1997); Numerical Seakeeping Calculations Based on the Slender Ship Theory, Ship Technology Research (Schiffstechnik), Vol. 4, No. 4, pp. 167â192. [A2] M.Kashiwagi (2000); The State of the Art on Slender-Ship Theories of Seakeeping, Proceedings of the 4th Osaka Colloquium on Seakeeping Performance of Ships, October, 2000. the authoritative version for attribution.