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A Coupled Time and Frequency Approach for Nonlinear Wave Radiation P. Ferrant (Laboratoire d'Hydrodynamique Navale, France) ABSTRACT In this paper we report on computations using a semi nonlinear time domain formulation for the threedimensional wave radiation problem with a free surface. The body boundary condition is applied at the actual time-depending body surface, and the free surface conditions are linearized. An initial value problem is solved for the potential on the moving body, using a boundary elements method. The method allows the hydrodynamic forces on bodies of arbitrary geometry undergoing large amplitude forced motions in six degrees of freedom to be determined, as well as the unsteady wave field generated by the body motions. Two different applications are presented. The first one refers to the problem of a sphere undergoing large amplitude periodic motions below the free surface. A thorough parametrical study (amplitude - frequency) has been completed in this first case, and the influence of the body nonlinearity is clearly highlighted both on the hydrodynamic forces and on the structure of the radiated wave field, which is investigated using specific frequency domain Green functions associated to the body-nonlinear problem. Some results on a submerged spheroid starting from rest with a constant velocity parallel to the free surface are also presented, as an unsteady approach to the wave resistance problem. INTRODUCTION Time domain modelization of free-surface hydrodynamics is all but a new subject. Some time before the publication of the commonly quoted paper by Finkelstein (1957) on the time-domain Green function, Brard (1948) gave the expression of the time depending free-surface potential generated by a submerged source of arbitrary path and strength. The formulation of the integral equations of the linearized wave-body interaction problem in the time domain has been presented by many authors, including Stoker (1957) and Wehausen (1967), the latter with a clear presentation of the connection between frequency and time domain solutions for fully linearized problems. With the increasing power of computers, practical numerical solutions of linearized time domain formulations have become available, starting with 2D problems with for example Adachi & Ohmatsu (1980) or Young (1982) with boundary elements methods (BEM). Jami (1982) solved the 2D problem using a mixed formulation associating finite elements and integral representation, and formulated the solution of the 3D problem, giving practical expressions for (1) now with SIREHNA SA, 2 quad de la Joneliere. 44300 NANTES-FRANCE. 67 the computation of the 3D time domain Green function. Newman (1985-a) solved the problem of the impulsive heaving motion of a floating cylinder, using time depending ring sources. Some time later, the 3D problem was solved numerically, both for forced and free motions of a floating sphere by Jami & Pot ( 1985), whereas Liapis ( 1985) presented the first results of a time-domain BEM formulation for the radiation problem with forward speed. Specific algorithms for the computation of the 3D time domain Green function were given. Newman (1985-b) presented his own algorithms for the Green function computation. The solution of the forward speed problem was extended to diffraction by King & al (1988). The accurate numerical schemes for the computation of the time-domain Green function, developed at that time, were a prerequisite for the reliable solution of the time domain wave body interaction problem, and besides academic applications, computations on realistic bodies such as Tlp's became possible and demonstrated the interest of time domain methods for industrial applications, compared with more conventional frequency domain methods (Korsmeyer & al 1988). Nevertheless, the Cpu requirements of the model, mainly due to Green function evaluations, restricted the use of the corresponding codes to vector computers, with Cpu times much higher than for linear frequency domain analysis, on equivalent cases. A considerable speed-up was obtained by Ferrant ( 1988-a), using a tabulation- interpolation procedure for the evaluation of the time-domain Green function in infinite depth. The power of the method was demonstrated on the linear time domain analysis of the ISSC Tlp, discretized by 1200 panels. The results were obtained on a scalar computer (Vax8700) with moderate Cpu times. Besides its interest for linear time-domain analysis, the tabulation-interpolation procedure puts Cpu requirements at a sufficiently low level to allow the so-called body-nonlinear problem to be solved in the time domain. In such a formulation, a linearized free surface condition is maintained, but the body boundary condition is applied at the exact time-depending body surface. This leads to an integral formulation very similar to the fully linearized one, except for a line integral appearing in certain cases for surface-piercing bodies. The additional difficulty mainly lies in the numerical implementation, for the Green function terms in the convolution integrals have to be entirely recomputed at each time step, due to the changing position of the body surface on which the integral equations are solved. This results in O(Nt2) Cpu times, where Nt is the number of time steps, to be compared with O(Nt) when the body boundary condition is linearized.

A first experience on the solution of the body- and: nonlinear problem in the frequency domain was reported in Clement & Ferrant (1987), where partial results were given on a submerged sphere with forced heaving motion. Although results were successfully compared with the experiments of Dassonville (1987), the formulation was very heavy and the resulting code was not considered to be fully reliable, due to problems of convergence of the influence coefficients at large amplitude. Furthermore, the extension of the formulation to arbitrary motions was not possible. The only alternative was to solve the problem in the time-domain, but intensive computations with the basic code based on conventional schemes for the Green function were practically impossible, with about 12 Cpu hours on a Vax 8700 to reach steady-state with sufficiently fine time and space discretizations. This was in fact our main motivation for the development of the tabulation-interpolation procedure for the evaluation of the time-domain Green function. With simple arrangements in the convolution computations leading to O(Nt) Cpu times, the resulting code is considerably faster, and a typical run on the heaving sphere as presented in this paper requires now about 10 minutes. Intensive runs of the program being possible, a complete parametrical study (amplitude-frequency) in the case of a submerged heaving sphere has been undercome, the results of which are presented in this paper. Time-depending forces and wave elevation, as well as the results of the harmonic analysis of steady-state are given for various values of the amplitude, over the whole significant frequency range. A method for the fast analysis of the steady-state radiated wave field, based on frequency domain Green functions for the body-nonlinear problem is also presented. A few additional results are given on the time domain approach of the wave resistance problem, in order to demonstrate the versatility of the time-domain body nonlinear formulation, which is basically able to cope with any free-surface linear problem. TIME DOMAIN FORMULATION Basic Assumptions The fluid domain D(t) is bounded by a free surface Silt), the body surface Sb(t), and is unbounded in horizontal directions. The fluid depth is infinite. A fixed coordinate system is chosen so that the Taxis points upwards, and the origin lies in the mean free surface. An ideal fluid is assumed, with irrotational flow, so that the fluid velocity derives from a potential satisfying Laplace's equation: ^~)( x,y, z,t) = 0 in D(t) (1) U (x, y,z, t) = V4)(x,y,z, t) in D(t) (2) The body boundary condition is applied at the actual time depending body surface, while the perturbation at the free surface is assumed to remain sufficiently small for a linearized condition to be valid, so that: an - =V.n onSb(t) an (3) 68 at an +g =0 onSf(z=O) at az with: infinity: (4) n V unit normal on Sb(t) pointing out of the fluid domain D(t) local velocity of the body surface Additionally, the fluid velocity must vanish at spatial VO(x,y,z,~) - 0 for (x2+y2) moo or zoo (5) and the fluid is supposed to be initially at rest: A, Ot =0 fort=0 Integral Equation (6) The fluid problem being now completely defined, various integro-differential representations of the solution may be derived, either for the potential on the moving body (distribution of sources and normal dipoles), or for the source density on the body (sources only), using Green's theorem and the threedimensional time domain Green function G(M,P,t) (see Appendix 1). In the first case, we obtain after some transformations the following integral equation to be solved for the velocity potential O(M,t) on the moving body surface: fr Q(M)~(M,t)- | | O(P,t)aa Go(M,P)dSp Sb(t) = - Ji Go(M,P)-O(P) dSp Sit) P +: dl :: [me l )~F(P(I),M(t),t-l)-F(P, M,t-l)~(P,7 )] dSp O Sal) + J do | [~(P,7)-F(P(I),M(t),t-l)-F(P,M,t-l)-h(P,~)] (n Adl).V c(P,I) (7) a -onp u Cal) where Cb(t) is the the closed line defined by the intersection between the instantaneous body surface Sb(t) and the X-Y plane, Vc is the velocity of a point on Cb and Q(M) is the solid angle under which the fluid domain D(t) is seen from M. Go and F (see Appendix 1) are defined by: (8) G(M,P,t) = Go(M,P) . S(t) + H(t) . F(M,P,t) The actual occurence of the line integral in (7) is governed by Vc. For example, this term is zero for the linearized problem without forward speed (see e.g. Ferrant

1988), where Cb is time invariant. In the applications presented in this paper, the body is fully submerged, and the line integral obviously vanishes. Hydrodynamic Forces. The unsteady hydrodynamic pressure (without hydrostatics) in the fluid domain is given by: p(M,t) 3~(M,t) I (My) ~ (9) It is more convenient for the present study to introduce the total derivative of the potential on the body, yielding for M on Sb(t): P(-= - ~[~(M, t)] - ~ ~ VO(M,t) ~ + V. VO (10) The first term in the right hand side of (10) is directly obtained by finite-differencing in time the potential on the body. The two other terms require the computation of the fluid velocity on the body. The method used for this obtained from: computation depends on the space discretization scheme and will be discussed when describing the numerics. Force computations follows by simply integrating (10) over the discretized body surface. Free Surface Elevation According to the linearized condition (4), the free surface elevation is given by linearized Bernoulli's equation: 1 am(~t) (it)=- g at (11) ~ may be computed by finite differencing in time the velocity potential at the free surface, or directly from an integral representation of Ot which for a submerged body (no line integral) and M on the free surface shrinks to: d ~(~ t) = at J= Adam o she T) [F,(P, M,t-~)~P(P,~) - O(P ~)a-n F. (P. TV-)] dSp (12) STEADY-STATE COMPONENTS FOR PERIODIC FORCED MOTIONS In the body-nonlin ear problem of the forced oscillations of a submerged body about a fixed mean position, the influence of the body boundary condition nonlinearity on forces and free-surface elevation is investigated. First, the time depending forces on the body are straightforwardly computed frorr' (10), after solution of the transient integral problem. These forces tend rapidly to a periodic steady-state which is Fourier-analyzed for a quantification of nonlinearities. On the contrary, the transient wave field, if computed at some distance from the body to eliminate near field components, needs a long simulation to reach the periodic steady-state, mainly because of the low group velocity of the higher harmonics. This point offers an opportunity to use special Green functions already developped for the solution of the body-nonlinear problem in the frequency domain (Clement & Ferrant 1985, 1987). These Green functions are shortly described in appendix 2. Thus, for a direct and economical computation of the steady state wave field, we first extract the harmonic components of the singularity distribution on the moving body, obtained from the time- domain solution after a few cycles of motion. That is, for example in the case of sources only and periodic forced motion: n2 o an(M) c06 n ~ + an ( M) sin n cat ( 13) for M on Sb(~) and t-- - so The steady-state potential in the fluid is then ~s(P t) = ~ Ii band G~n(M P t) + on (wG2n(~P t) ~ dsM sb (14) and the corresponding free-surface elevation, for P on the free-surface: (15) rls(P, I) = g ~ J: [c~n(M) ~Gln(M P t) + an (M) iG2,,(M,P,t) ] dSM sb For harmonic heave motion, the expressions of Gin are simplified, and for fixed points P on the free-surface, the harmonic components of Us and Us are directly calculated by eliminating the time variable from the expressions of Gin (A2.10), (A2.11). When only the far field is to be computed, which is sufficient to study the structure of the radiated wave field (amplitudes of harmonics, dispersion of energy on the components of A), asymptotic expressions of Gin are used. Again, the computation is drastically simplified in the present case of the heave motion, Gin reducing to the very simple expressions (A2. 12), (A2. 13). No numerical integration, but only computations of modified Bessel functions are involved. NUMERICAL IMPLEMENTATION The major part of the numerical results given in this paper have been obtained using a first version of the computer code completed in 1988. In this version, a very classical discretization scheme is used. The body surface is discretized into plane polygonal panels over which 69

singularity distributions are assumed to be constant. The impulsive part of the Green function (Go) is integrated analytically over the panels, and the memory part F is integrated using a variable order Gaussian quadrature. Numerical tests have proved that, at least for submerged bodies, one single point of integration per panel is sufficient for a good accuracy, the local flow being dominated by the singular part of Go. Thus, wave terms are treated as monopoles situated at panel centroids, which greatly reduces computational requirements. Note that such a mixed procedure for the space integration of a Green function as also been proved to be a valuable compromise for the solution of the steady wave resistance problem (Doctors & Beck, 1987). The time variable is discretized into constant time steps and the convolutions integrals are evaluated using a trapezoidal rule. At each time step the convolution terms at the right- hand side of the linear system of equations are actualized by computing and assembling the corresponding wave terms, and a new kernel is obtained by computing the motion- dependent part of Go. The linear system is then solved using a standard Gauss solver. Faster solvers are obviously available, but in fact the computing time is dominated by the evaluation of the convolution terms. Actually, two integral equations are solved at each time step. First a mixed distribution of sources and dipoles is used for a direct computation of the potential on the body. Then the integral equation for sources only is solved, and the result is used for the computation of the fluid velocity at panels centroids. Although increasing the computer time, the method allows the fluid velocities, and thus the full hydrodynamic pressure to be obtained without having to calculate the second spatial derivatives of the Green function. At the end of the simulation, the different terms in the hydrodynamic pressure are computed at panels centroids. Ot is obtained by finite differencing the time depending potential, and the quadratic terms are computed from the previously computed fluid velocity. Forces are then obtained by integrating the pressure which is assumed to be constant on each panel. The time depending wave elevation at prescribed points is also available. For simplicity, this computation is based on the source solution. In the case of a periodic forced motion, the solution of the time-domain problem is followed by the computation of the harmonic components of the steady state part of the response. For reasons initially related to an economical computation of the convolution terms, the time step is adjusted to obtain an integer number of steps per period, and the harmonics can be accurately computed by simple trapezoidal rule over one single period, typically the last, in order to provide the best approximation of the steady state. The harmonic analysis is applied first to the time depending forces, and then to the source solution. The latter results are used for the computation of the steady-state periodic wave field using the Gip functions described in appendix. In consistency with the method used for the solution of the time-depending problem, this computation is based on a monopole approximation of the source distribution on the body. In the case of the heave motion, the computation of the harmonic components of the far field wave system is straightforward and of negligible CPU cost. At last, for a quantification of the influence of the body boundary condition nonlinearity, the linear solution is systematically computed for comparison. REDUCTION OF COMPUTING TIME Apart from the use of frequency domain Green functions, the numerical implementation as described in the preceding paragraph is very classical. In fact, the main difficulty is related to the extensive Cpu and mass storage requirements of the body-nonlinear time-domain formulation. These requirements may be lowered first by accelerating the Green function computations, and secundly by reducing the number of Green function computations necessary to the evaluation of the convolutions. These two points have been addressed in the present study. Tabulation-Interpolation of the Green Function The well-known time domain Green function for a source of impusive strength in infinite depth is given in Appendix, with the following notation: G(M, Pi,) = 6(~) . Go(~P) + Ho) · F(~ Pit) (16) The memory part F of that Green function can be easily put under the following form: r°° gl/2r. 3/2 1 It2 21/2 _>, 1/2 F(M,P,t) = - 2~ J sin(x A) Jo[~(l~~ ) ] e ~ do o or: F(M P. t) = gl/2r 3t2~(~' 0 with ~ = -(z+z')/r' and ,B = t(g/r')V2 (17) (18) The first parameter is linked to the relative positions of points M and P. and varies from 0 to 1., whereas ~ is an essentially positive time parameter. Thus, the only non-trivial terms to be evaluated during the computation of the convolution integrals are reduced to the bivariate function F and its first derivatives. This fact can be exploited for deriving a very fast procedure for the evaluation of the wave terms. This procedure, based on very simple principles, has been already been described in Ferrant (1988). The 2D domain described by ,u and ~ is truncated at a large value Oman, and the remaining bounded domain is mapped by a discrete set of equispaced points for which F and its first derivatives are computed by numerical schemes very similar to the ones described by Newman( 1985) or King & al ( 1988). This computations are performed once for all, and the results are stored on permanent disk files. When a simulation has to be performed, the resulting evaluations of the memory part of the Green function are based on linear bivariate interpolations of the stored data. Note that the content of the file is read once for all at the beginning of the simulation, so that no disk access is necessary during the time-stepping procedure. The tabulated part of the (,u,0) domain is sufficiently extended to allow the use of simple large-time asymptotic expressions when ,B>pmax. In a very thin layer near 11=0 where the function presents large oscillations the ~ a, , precision of the interpolation may be unsuff;cient and we simply return to the original numerical schemes. However, 70

for the forced motions of a submerged body about a fixed mean position, computations never occur in this part of the domain and the interpolation is used throughout the simulation. The use of a regular tabulation grid allows for a very quick search in the tables, and the resulting code is very fast: less than 30 As are necessary for one evaluation of the Green function and its first space derivatives, on a Vax 8700 computer, from and back to physical variables. This is about the time needed for one evaluation of the sine function. The grid is composed of 200x1200 points in the rectangular domain defined by O < ~ < 1. and 0 < ~ < 30. For the Green function and its gradient, 3 tables have to be stored, resulting in about 720Kwords or 2.9 Mbytes on a 32 bit computer. Note that we do not pretend to have an optimized set of tabulation parameters. Such an optimization could be necessary to reduce memory requirements for a given accuracy, but the present size of the tabulation files is not a problem on the computer we use. When the problem to be treated leads to large horizontal displacements, as it happens with forward speed, a substantial number of computations may occur near ~ = 0, and a special scheme is needed to maintain both precision and low Cpu in this portion of the computational domain. Such a refined procedure has been subsequently derived by Magee & Beck (1989), exploiting an analytical approximation of the Green function in the vicinity of the p=0 axis, and a higher order interpolation scheme for the remaining part of the Green function on various subdomains. Although more precise for a given number of tabulated points, their method is certainly a bit less efficient in terms of computing time, but comparisons are delicate between runs on scalar and vector computers. The use of an analytical approximation is a very clever idea for reducing the oscillations of the function near ~ = 0, and we intend to adopt a similar scheme, but in our opinion, bilinear interpolation should be preferred for the interpolation of the remaining smooth function, whenever a very high precision is not necessary. We are not sure that the precision of 10-8 obtained by Magee & Beck is necessary when the overall accuracy of the computations is dominated by space & time discretization errors. A precision of about 10-5, easily obtained by a simple bilinear interpolation method in the major part of the computational domain is certainly sufficient for most applications. Reduction of the Number of Green Function Evaluations Basically, during one run of the program, Np2(It-1) computations of the Green function are required at time step It for the evaluation of the convolution. This results in a total of Np2.Nt(Nt-1) computations for a simulation over Nt time steps, if Np is the number of panels. A crude method for reducing this number is to truncate the convolution integrals, considering that the influence of the past history of the solution tends to zero for large time delays. However, such an approach is not safe, for it is very difficult to estimate the influence of the truncation on the final results. In many particular problems of forced motions, an adequate choice of the time step allows a substantial reduction of the number of Green function evaluations without affecting the numerical results. The idea is as follows: The Green function terms to be computed in the convolution depend essentially on MP and t-t, where M(t) is a point on the body at time t (field point), P(T)IS the position of a point on the body at time T (source point), and t-T.iS the time delay. The idea is to Just the time step At, i.e. the positions of the body at times iAt to reduce the number of combinations of MP and t-T to be considered in the computation of the discretized convolution. For clarity, let us illustrate the procedure on the case of forced motions of period T about a fixed mean position, for which the appropriate choice is At = T/Nper. Consider the discretized convolution to be computed at time step it, it 2 Nper. The Green function terms to be evaluated may be schematically written G[MP(it,iT),(it-iT)At], with IT < it. For iT 2 Nper, we have obviously: G [MP(it,i~),( it- iT)] = G [MP( it- Np~r ,i~-Nper),(it-Nper)-(*-Nper)] (19) The term at the right-hand side has already been necessary for the computation of the convolution at time step it-Nper, and thus has already been computed and stored. Thus the only new terms to be computed correspond to the influence at time step it of the Nper time steps of the first period. The total number of NpxNp sets of terms to be computed is now: N,(N,-1) NCaI! = 2 for N. ~ NPer N. (N. -1) N. In= per 2 per +(Nt-Npe,)Npe,. N. N. Nper(Nper~ 1) forN' 2 Nper (20) (21) The resulting total number of Green function evaluations is now a linear function of the number of time steps, as opposed to the quadratic behaviour obtained when At is arbitrary. The ratio of reduction, after a few periods of simulation is approximately: N. NT G 2 N per ~ (22) where NT is the number of simulated periods. For the computations presented in this paper, 4 to 8 periods are simulated, and the Green function computations are reduced by a factor 2 to 4 using this artifice. The procedure that we have illustrated in the case of periodic motions is also applicable when a constant forward speed is superimposed to the oscillations. In the case of a smooth starting of duration NStart, the gain is lower, but a linear law is maintained: NCal~ = N.' . (Now+ N ta ) (Nper NStart)(Nper + Nstart+ 1) (23) 2 For problems with constant forward speed without superimposed oscillations (unsteady approach of the wave resistance problem), simulations with an abrupt start lead to the same number of computation as for a linear code. If a smooth start is chosen, Ncall is given by (23), with Nper = 0. In this case, the introduction of a smooth start of duration N Start multiplies the number of Green function evaluations by the factor Nstart. 71

NUMERICAL RESULTS Submerged Heaving Sphere We give in this section the results obtained on the problem of a submerged heaving sphere, with a mean depth of submergence equal to the diameter (Zo/R = 2., Figure 1). Starting from rest, the body is given a purely harmonic motion, with frequency co. ~,~ Sc(t Figure 1 Time Domain Results Kl=~2R/g Z = Zo + A cos cot We first give in figures 2 to 9 a sample of the unsteady (vertical) force signals computed form the time domain formulation. The results of the fully linearized formulation (dashed line) have been systematically computed for comparison with the body-nonlinear formulation. The difference of the two results is also plotted. Results are given for two values of the amplitude, A/R = 0.50 and A/R = 0.70, and four values of the fundamental wavenumber K1 = ce2R/g=0.1, 0.25, 0.50, 2.0. A periodic steady-state is very rapidly obtained, and a significant influence of the body boundary condition nonlinearity is observed, mainly for K1 = 0.25, the difference between the two results being weaker at low and high frequency. This difference mainly appear in the form of higher harmonics in the response, the fundamental being apparently not much modified. At high and low frequency, the superharmonics tend to be in phase with the fundamental. The analysis will be easier on the harmonic components of the steady state response, given in a following section. Then, in figures 10 to 17, for the same two values of the amplitude, and for K1 = 0.2, 0.4, 0.7, 1.0, we give the transient wave elevation computed at a distance 20R from the sphere. Again, the linearized solution in dashed line is given for comparison. At low frequency, the influence of the nonlinearity is much stronger than on forces, but the difference between linear and body-nonlinear solutions tends to zero at high frequency, at least for the smaller amplitude, A/R = 0.50. For A/R = 0.7 and K1 = 1.0, a small second harmonic component remains sensible. At low frequency, the influence of the body boundary condition nonlinearity is already very sensible for a moderate amplitude, as demonstrated by figure 18 giving the wave elevations for A/R = 0.30. For such an amplitude, a linear analysis is commonly considered to give correct results. The present results prove that at least for wave generation, a linear analysis is inadequate, the body nonlinearity having a very strong influence on the structure of the radiated wave field. This phenomenon, already highlighted using a frequency domain analysis of the problem by Clement & Ferrant (1987), has been fully confirmed by the experiments of Dassonville (1987). Comparisons were made over a wide frequency range, but we simply give in figure 19 a comparison between experiments and time domain body- nonlinear analysis for A/R = 0.50, and K1 = 0.28, i.e. in the vicinity of the maximum nonlinear behaviour given by the numerical model. Due to experimental constraints, the distance from the sphere is here equal to 16.5R. A small difference in the starting transients is observed, due to the imposed smooth start of the experimental apparatus. During the steady-state part of the response, the wave elevation is slightly overestimated by the numerical model, but harmonic content as well as phases are perfectly recovered. At large time, the comparison is not significant, the experiments being affected by tank wall reflexions. Harmonic Components Concerning the hydrodynamic forces exerted on the moving body, the harmonic analysis is performed after obtention of the steady-state, and the results are given under the form: 00 F(t) = of, F cosn~t+ F. sonnet (24) pgKlR2A n=0 n n Full results may be found in table 1, where the harmonic components of the vertical force on the spheres from the constant term to the fifth harmonic, are given for three values of the amplitude, over the whole frequency range of interest, together with the purely harmonic frequency domain linear results. A part of these results is graphically represented in figures 20 to 27, for the two largest amplitudes, and up to the fourth harmonic. The results being scaled in the usual manner of frequency domain analysis, the first harmonic components tend to the usual linear added mass and damping coefficients when the amplitude tends to zero. In figures 22 and 23, the linear hydrodynamic coefficients are plotted together with the first harmonic components of the body-nonlinear results. The stronger relative difference lies in the damping coefficient, especially between 0.2 and 0.9, in the ascending part of the curve. At low and high frequency, F** components tend to zero. At high frequency, F* components tend to a constant. This behaviour cannot be observed at low frequency, the frequency range being insufficient. These observations agree with the asymptotic analysis of the problem, which gives predominant added mass effects at low and high frequency, where the steady-state flow is in phase with the body motion. The steady-state far-field wave elevation harmonic components are computed from equation (16), after an harmonic analysis of the singularity distribution on the body. This procedure avoids the computation of the transient wave field, and is of negligible Cpu cost. The results are given in the form of the distribution of the total radiated energy over the different harmonics of the wave field, in figures 28 and 29, for A/R = 0.50 and A/R = 0.70, as a function of frequency. The very strong deviation from the

non K1 = 0.25 KRIL3D+ -- Iineaire -0.6 - Differellce , 0 10 20 30 40 Figure 3 T*sqrt(g/R) A/R = 0 70 Zo/R = 2.00 A/R = 0.50 Zo /R = 2.00 o Figure 6 0.4 0.2- 0.0 -0.2- -0.4- fiO 60 70 0 Figure 7 ~A/ R = 0.50 Zo/R = 2.0C v." - 0.4- 0 0- -0.4 -0.8- 20 30 40 50 0 T*sqrt(g/R) Figure 8 -1.0 0 10 Figure 4 AL - - co * lo: * ~ o o \ _p . K1 = 2.00 KR~3D+ -- Idneaire - Difference -4 0 5 1 Figure 5 on 40 60 T*sqrt(g/R) A/R = 0.50 Zo/R = 2.00 . . . 10 20 30 40 T*sqrt(g/R) 10 20 30 T*sqrt(g/R) A/R= 0.70 Zo/R= 2.00 * 1 - * 0 - o ~ -1 - -2 - -3- . . 10 15 20 25 0 T*sqrt(g/R) Figure 73 O6 70 40 50 A R = 0.50 Zo/R = 2.00 . . 5 10 15 T*sqrt(g/R) - 20 25

*10 -3 12 . RSL = 20*R A/R = 0.50 Zo/R= 2.00 -12 ~ ~ n - . K1 = 0.20 -- -sol. Iineeire KRll:3D+ . l l l l l 0 20 40 60 80 100 120 140 160 180 200 Figure 10 T*SQRT(g/R) RSL = 20*R A/R = 0.50 Zo/R = 2.00 -3.0 *10 -2 K1 = 0.70 -- - sol. linea~re -ER~D+ K1 = 0.40 1 -- - sol. linea~re | .KRI~D+ | 0 20 40 60 80 Figure 11 T*SQRT(g/R) RSL= 20*R A,/R = 0.50 Zo/R = 2.00 100 120 140 . . 0 20 Figure 12 $10 -2 F.n RSL = 20*R A'R = 0.50 Zc,/R = 2nn -5.0- 20- RSL= 20*R A/R = 0.70 Zo/R = 2.00 5 0 ~: 5 E3 O -5 -10 -15 - . 1 1 1 1 1 1 1 0 20 40 60 80 100 120 140 160 180 200 Figure 14 T*SQRT(g/R) *10 -2 RSL = 20*R A/R = 0.70 Z0/R = 2.00 . . . . 0 20 40 60 80 Figure 15 T*SQRT(g/R) - 6- RSL= 20*R A/R = 0.70 Zo/R = 2.00 5_ 2 1 c,3 0~ -1 -2 -3 _-4 -5 - 40 60 80 100 120 T*SQRT(g/R) . . . 100 120 140 K1 = 1.00 || . -- -sol. linesire KRE.3D+ _ I I ~ 0 20 40 BO T*SQRT(g/R) Figure 13 6- . 4 a, ~ ~ I -3 -4 -5 .~.. 80 100 ° 20 Figure 17 7~1 1 1 1 · . . ~20 40 60 80 100 120 *lo_2 Figure 16 T*SQRT(g/R) RSL= 20*R A/R = 0.70 Zo/R = 2.00 11: 11 K1 = 1.00 v I -- -sol. 1iIleeire Y KRIL3D+ 40 60 80 100 T*SQRT(g/R) 120

*10 -3 al 6 4 Ad; 2 EM o -2 -4 RSL = 20*R A/R = 0.30 Zo/R = 2.00 i, 0 40 Figure 18 80 120 160 200 T*SQRT(g/R) linear analysis already observed on time domain results is confirmed, with a maximum about co.sqrt(R/g) = 0.2 (K1 = 0.4) where for A/R = 0.50 only 60% of the total energy input by the moving body is recovered on the first harmonic of the wave field. This phenomenon appears approximately in the same frequency range as the maximum deviation from linear results of the damping coefficient F1**. For confirmation, we give in figures 30 and 31 the damping coefficient computed from the body nonlinear formulation, together with linear results. The difference is also plotted, as a function of frequency. Results on the sphere have been obtained with a discretization of 200 panels on the entire body. Time Domain Wave Resistance of a Submerged Ellipsoid. Results are now given on the problem of a submerged ellipsoid of beam/length ratio B/L = 0.2, and submergence Zo/L = 0.16. Two different discretizations are used, with respectively 60 and 168 panels on the half-body, and the simulations were run over 150 time steps. Starting from rest, the body is abruptly given a constant velocity in the positive X direction, parallel to the free surface. We give in figures 32 to 34 the transient horizontal force on the body, for three values of the Froude number, Fr = 0.4, 0.45, 0.5. After some oscillations, the force tends to a constant, which actually is the wave resistance of the body. We give in table 2 the Cw wave resistance coefficient estimated from the ultimate value of the transient force, compared with the results of a conventional Neumann-Kelvin wave resistance code (Delhommeau 1987), obtained with a 192 panels discretization. The results of a semi-analytical formulation, deduced from curves given in Farell (1973), are also presented. With the finest discretization, Npan=168, the agreement between time-domain analysis and steady-state results is correct. The influence of the time-step has not been investigated. A smaller time step would certainly lead to a better agreement, especially for larger Froude numbers. Fr Cw (time domain) (Np=60) (Np=168) 0.01226 0.01267 0.01175 0.45 0.01592 0.01667 0.01734 0.50 0.01619 0.01671 0.01775 Table 2 Cw (N-K) Cw (Farell) (Np=192) 0.01320 0.01713 0.01822 Note that the oscillations are greater for smaller Froude numbers. At Fr = 0.8 (figure 35), only a small *10 ~3 RSL=16.5*R A/R = 0.50 z0m = 2.00 0 10 TO 30 40 50 60 T*sqrt(g/R) Figure 19 70 80 90 100 overshoot is observed during the transients, a constant resistance being very rapidly obtained. Similar results on the transient approach of the wave resistance problem have already been published by Jami & Gelebart (i987), and more recently by Magee & Beck (1990), with comments on the oscillatory behaviour of the time depending wave resistance and its connection with the critical parameter ~ = 0.25 of the forward speed seakeeping problem. Present computations simply intend to demonstrate the versatility of the body-nonlinear time domain formulation. CONCLUSION A systematic attention on computational efficiency has led to a time domain body-nonlinear code with Cpu requirements sufficiently low for intensive computations to become possible, as demonstrated on the problem of the heaving sphere. Strong nonlinear effects, both on forces and wave field, already detected using a frequency domain approach (Clement & Ferrant 1987) have been confirmed and their validity extended to the whole frequency range and to larger amplitudes. Furthermore, the time domain formulation can cope with arbitrary motions and is absolutely robust. Future computations will be undercome with a refined version of the code based on a linear representation of the velocity potential on the body. Other researchers are also working on the time domain body-nonlinear problem, and for example a considerable experience has been gained on the time domain seakeeping problem with forward speed by Professor Beck's team. The variety of possible applications will certainly motivate many interesting studies in the future. The extension to finite depth depends on the development of fast algorithms for the computation of the corresponding Green function, on which preliminary results have been obtained by Newman (1990). 75

Harmonic Components K1- 0.100 1 0~= 0.316 Fo I A=O. 00 1 K1- 0.025 1 AS0.30 1 0.0104 on. 0. 15B I A-O.5O 1 0.0194 1 A-O.70 1 0.0324 1 A-0. 00 1 K1- O.OSO 1 A-0.30 1 0.012S 0 ~0.224 1 A~0.50 1 0.0237 1 A-0.70 1 0.0411 I A~0.00 1 1 A.0.30 1 0.01Si7 1 A~O .50 1 0.0305 1 A=O.70 1 O .0554 I AsO. 00 K1= 0.2001 A~0.30 on. 0.4471 AeO. 50 1 A=O.70 I A- O . 00 A-O. 30 A=O. SO 1 A=O. 70 I AsO. 00 A-O. 30 A=O. SO 1 A=O. 70 I A~0.00 1 A=O.30 1 A. O . 50 1 A=O. 70 I A-O.00 K1- 0.600 1 A-0.30 0~= 0.775 1 Ar O. SO I A-0.7a I A=O.OG 1 Aa0.30 I A-O. 50 1 A=0.70 I A=O. oa A-O.30 A=O. SO 1 A-O.70 I A=O.00 1 A-O. 30 I A=O. SO 1 A-0.70 I A-O. 00 1 A=O.30 I A ~ O . SO 1 A-0.70 I A ~ O . 00 1 AsO. 30 I A=O. SO 1 A-O. 70 I A=O. 00 1 1 Aa0. 30 1-0.0312 1 A=O. 50 1-0.0535 1 A=O . 70 1 -O . 077S I A=O. 00 1 1 Au0.30 1-0.0327 1 AeO.SO l-O.OS70 1 A=0.70 1-0.0843 I A-O. 00 1 K 1 = 2. 000 1 A'0. 30 1-0 .0323 on ~1. 414 1 As O. SO 1-0. OSb8 1 A-0.70 1-0.08S6 I A=O. 00 1 K 1- 2. SOO 1 A-O.30 1-0 .0279 0~= 1.581 1 A-O.SO 1-0.0500 1 A*O . 70 1 -O .0778 I A-O. 00 1 K1. 3.000 1 A=0.30 1-0.023b 0~. 1.732 1 A-O.SO 1-0.042b 1 A-O. 70 1-0.0674 K1- 0.300 1 0~2 O. 548 1 K1. 0.400 1 0 ~O. 632 1 Kl= 0.500 1 0~- O.707 1 Kl - 0.700 0~- O. 837 K1- 0.800 1 0~= 0.894 1 K1- 1.000 1 0~= 1.000 1 K1 ~1.200 1 OM. 1.095 1 K1- 1.400 1 0~= 1.183 1 K1- l.bOO I 0~= 1.26S 1 K1C 1.800 1 0~- 1 . 34 2 1 1 0.0223 1 0.0433 1 0.0770 1 0.0281 1 0.0512 1 0.0829 1 0.0310 1 O.0530 1 0.0781 1 0.0305 I O . OSOO 1 0.0686 1 O . 026q 1 O. 043S 1 O .0576 1 0.0208 1 0.0337 1 0.0444 1 0.0131 1 0.0215 1 0.0287 I -O.0034 1 -O "0 0- 2 1 -O .0066 1 -0.0172 1-0 . 02as 1 -0.0391 1 -O .0264 1 -O . 044S I-O . Ob30 F1.F1#. 2.12900.0003 2. 13SS0.0003 2.14890.0003 2.17390.0004 2.1478 2. 1S68 2. 17S2 2.21 1S 2.1807 2.1941 2.2222 2.2782 2.2319 2. 246S 2.2757 2.3286 2.2533 O. 1 624 2. 2S87 O. 1 830 2.2696 0.2234 2.2888 O. 29~0 2.2287 O. 2570 2.2263 0.27 44 2.22 1 9 O.3084 2.2147 0.367S 2. ~ S67 O.3296 2.1520 0.3411 2. 1 422 0.3640 2.1240 0.4046 2.0784 O.3730 2.0750 O. 3805 2.0666 O. 39S7 2.0486 O. 4227 1. 99?6 O. 38b5 1. 9959 0.3925 ~ .9905 O.404 1 1.9769 0.4241 1.9247 O.3763 1.9238 0.3821 1.9204 O.3930 1.9106 0.4107 1.8190 0.3148 1.8168 0.3214 1.8121 0.3333 1.8033 O.3~20 1. 76S9 O. 2361 1.7604 0.2429 1. 750S 0. 2S53 1.7356 0.2749 1.7503 O. 1 648 1.741S 0.1711 1.7257 O. 1326 1.7023 0.20 12 O. 1 092 O. ll4S 0.1245 O. 1 4 07 O.0694 O.0737 0.0818 O. O9S 1 1.7563 1.7449 1.7244 1.6933 1.7723 1. 7S94 1.7357 1. 69eq 1.7914 0.0427 1.7777 0.0460 1.7524 0.0522 1.7121 0.0626 1.8333 1.8202 1.7950 1.7533 1.8603 1.8487 1.8261 1. 787S F2t O .0364 0 .0005 O . Ob68 0 . OOO' 0.1093 0.001S 0.0020 0.0022 ' O .0524 O .0026 0 .097 3 0.0039 O. 1631 0.0130 0.0156 0.0217 O.0364 o .07 1 1 0.0857 O . 1 1 S? O. 1747 0.0110 O . 0 1 2S -O .0508 O . 0 1 53 -O .09 1 B 0.0201 -0.1443 0.002 1 O. 0026 O.0037 0.0056 O .074 9 0 .037 2 O. 1383 0.0717 O .228 1 0 . 1 292 O .0354 0 . 1 062 O .0606 0 . 1 94 8 0.0853 0.3186 -O . 02b 7 0 . 102 ~ -0.0490 O. 1857 -O .082 1 0 .2939 -O.OS12 0.0701 -O .0927 0 . 1 287 -O . 1 477 0 .2060 -O .0489 0.0444 -0.0906 0.0823 -0.1471 0.1334 -O .0397 0 .0350 -O . 07S3 0 . Ob33 -O. 12S7 O. 1002 -O .0332 0 .034 7 -O .063 1 0.0604 -O. 1062 0.0912 -O .03 1 4 0.0376 -O . OS83 0 .0642 -O . O9S8 0. 093B -O .0370 0.0420 -O .0650 0 .07 1 7 -0.0991 0.1040 -O .0460 0.0403 -O . 079S O .0697 -0.1172 0.1033 -O .053 1 0 .034 2 -0.0918 O.ObO2 -O. 13S2 0.0917 -O .0568 0.0266 -O .0990 0 .0477 -0.1475 0.0748 -0.0576 0.0194 -O. IO1S 0.035S -O . 1 S33 0 . OS73 -O.OS65 0.0134 -O . 1 OOS O .0252 -0.1539 0.0419 -O .0454 0 .00 1 1 -O .0826 0. 0026 -O. 1316 0.00SS Table 1 F2. ~f3. 0.0052 0.0129 0.0100 0.0399 O .0 1 82 0.0936 O .004 S -O.0075 O .009 1 -0.0223 O .0 1 65 -O .04 8 1 76 F3~. F45 O .008 1O .0004 0.02490.0013 O.0575O.003 1 0.01220.01S8 O . 03bB0.0497 0.07880.1209 -0.00290.0137 -0.01070.056S -0.0325O. 1254 -O . 00350.0 1 53 -O. 026b0.0447 -O.0624O.0933 -0.01230.01 1S -0.0363O.0336 -O .07 aoo . 069q -0.01270.0062 -0.03730.0188 -0.07880.041D -O . 0 1 1 50. 0030 -O.0343O. OO9S -O.07 370.0224 -0.00960.0014 -0.0291O.0045 -0.06390.0114 -O .00 800.0009 -0.02420.0029 -O. OS39O. Q070 -O.0062O.00 17 -O .0 1 85O .0046 -O. 0408O.0090 -O. OObO0.0026 -O . 0 1 7S0.0072 -O .0370O .0 1 45 -0.0065O. 0029 -O .0 1 870. 008S -0.03BS0.0179 -O . 007 10.0028 -O.02040.0084 -0.04190.0183 -O. 007S0.0024 -O . 02 1 8O.0073 -0.04500.0165 -O.0077O. 00 19 -O.02250.0058 -0.04710.0137 -O.0070O.000 1 -0.02 1 1O. ooOB -O. 0462O.0026 o .00370.002 1 0.01180.0109 O.0306O. O3S3 O.0007-0.0008 O. 002b-O .0041 O .0068-O .0 1 2 1 FSt F5~' 0.00140.0002 O . 007SO .00 1 2 O . 024BO .0044 0.0013 O .0073 O .0270 O .00080 .003 ~ 0.00330.0164 O .00520.0549 -0.00160.0021 -0.00840.0102 -O .02930.0286 -0.001S0.0010 -0.00790.0048 -0.02470.0118 -O .00 1 40 .0009 -0.00680.0040 -O.Ol960.0101 -0.00130.0005 -O. OObOO .0027 -0.01660.0076 -0.00130.0003 -0.00630.0018 -0.01770.006t -0.00120.0001 -0.00610.0009 -0.01770.0038 -O .00 1 10 .0000 -O . O OSSO .0002 -0.01670.0017 -O .0009 -O .000 1 -0.0043 -0.0002 -O .0 1 35 -O .000 1 -O .0007O .0000 -O .00360.000 1 -0.01100.0004 -O .0007O .000 1 -0.00330.0006 -o.ooq?0.0017 -O .00070.0002 -O .00340.0009 -O.00990.0027 -O .00080.0002 -0.00360.0010 -0.01040.0032 -O .0 0080 .0002 -O .0038O .0009 -0.01 1 10.0031 O.0000 O .0005 O .0020 -O .0008-O .000 1 -O .0040O .000 1 -0.01210.0008 O .0002O .000 1 0.0019Q.0007 O.0090O.0040 0.0002O. 0003 0.00190.00~8 O .0074O.0 1 49 -O .000 10. 0004 -O.00 1 10.0036 . ooe40. 01 5b -0.00020.0001 -O. 0021O. 0007 -O. OO9b0.0010 -0.0002O.0000 -0.00140.0000 -O .0054-O . 00 1 2 -O .000 1O.0000 -0.0008O.0000 -O.0030-O.0004 -O.000 1O.0000 -O.0004O. 0001 -O.0011O. OOOb -O. 0001O. 0000 -O.0005O.0003 -O.OOlb0.0015 -O.000 1O.0000 -O. 0006O.0003 -0.0023O.00 17 -O .000 1O.0000 0.0007o . 0OO? -0.0029O. 00 1S -0.0001O.0000 -0.00070.0001 -0.00320.0008 -O .000 1O.0000 -O. 0006O. 0000 -0.0028O. 0003 -O .000 1O .0000 -O.0005O.0000 -0.0024O.0003 -O.000 1O.0000 -O .00050.000 1 -O. 0022O. 0005 -O. 0001O. 0000 -O. 00050.0002 -0.0021O.0006 -O .000 1O . 0000 -O. OOOS0.0002 -0.0022O. 0007 -O. 000 1O.0000 -O.00060.0001 -O. 002S0.0006 -O .000 1-O . 000 1 -~.0006O.0000 -O.00260.0002

n: O,OC~ cy 6 y t:) 0.0 * \ O-0.05 - L~ 0.1 21 . -0.1 o.o 02 Figure 20 2.5 2 ~o- ~r 6 15 O 10 CY \ 05 QO ~ 0.0 02 Figure 22 A/R = 0.50 ZO/R = 2.00 0.4 ~ o.o- ~ ~ z o.o o' Figure 24 ~4 19 1' A/R = 050 ZO/R = 2.00 12t4 . . n`L 0.1 CY ~ 0.05 Y ~ ~ O ~ 0.0 L~ -0.05 0.0 _ Figure 26 1F 1f ~-~ ~ zo 6 1.5- y O 10- ~ 05 QO- I ~ 16 ~20 0.0 02 Figure 23 14 ~ A/R = 050 ZO/R = 2.00 _ oO ~ ~ C ~ ~ C1 ~ ~ E 0 4 0.8 12 16 2~0 HA~'t. 3 F~ ~) 1 F* (~) I A/~ = O70 7n/R = 70n v.e- tt0.05 * ~0.0 o O - .05 -0.1 - 0.0 02 0.4 Figure 2 0.6 0 ~ 10 12 t4 t6 ~0 PULSATION W*SQRT(R/G) A/~ = n7n 70/~ = 700 ~ C . ., .. ~ 0.4 02 Y _' 0.0 L~ -02 0.0 02 Figure 25 ~o mr~ 1 013 10 ~ 14 PULSATION W*SQRT(R/G) . . 0.4 C A/R = 0.70 ZO/R = ~co o~ ~- i 011 ~* 1 ~ o.c~ 1 Y 1 oo 1 ~ 1 \-o~o5 1 -o.1 1 0.0 0.4 Figure 27 t6 ~20 A/R = 0.70 ZO/R = 2.00 -~1 1 1 0~o ~ ~ tO [1 C] m 1m C1 1 1 1 1 1 0.6 0.8 10 12 t4 16 ~0 PULSATION W*SQRT(R/G) ~1 i I I I 1 ~1 1 1 1 I c~ (5 c 1 D ~C1 [!1 C 1 c~ 1 G [~| 1 . o ~t2 16 Zo PULSATK N W*SQRT(R/G) 77

0.02 ~_ CY 0.01- 6 * y * 0.0 z LL -0.01, 1.0 C.8 .-, O.B 0.4 0.2-1 0.0 ns 0.4 (f - ) o3. 02 C) 1i LU 8 0.1 _ C . _ . _ o.o 02 0.4 0.6 ~o ~ t4 t6 A' R = 0.~0 m 2~0/p,R c,-i.00 c . =~ 0.0 0.2 0.4 O.B 0.8 1.0 t Figure 28 o.o ~ ~ o.o 02 Figure 30 1.4 A/R = 1 c Q ~ ,.. 050 ZO/R = 2.00 ___ ~ OW ~O [ ~ ~ _ (!) ,t, Q Q Q ~ Q ~ ~ ~ ~ 1 0.4 0.6 o ~to 12 1.O 1.8 2.0 O.oB CY * 0.04- 6 y CY o.o -0.04 1.0 0.8 O.B =0.4 0.2 0.0 05- 0.4 L1J 03 ~ 0.2- C) 0.1 0.0 A/R = 0.70 ZO/R = ZOQ o:~ J~ooo m~ o.o o~ Q4 0.6 o~ to t2 t4 t6 tB 21 0 PULSATION W*SQRT(R/G) ~ j~ ~ j 0.0 0.2 0.4 0.6 0.6 1.0 1.2 1.4 Pulsation w*sqrt(R/g) Figure 29 WAVE ENERGY DISPERSION A/R = 0.70 ZO/R = 200 1.6 1.6 2.0 _ = o ~ 2 0 o 0 o. 2 0.4 0;6 o~ to 12 t4 PULSATION W*SQRT(R/G) Figure 31 DAMPING COEFFICIENT 78 LU A ~ t6 t8 zo

*10-3 Submerged Ellitsoid Fr=0.40 *10-9 -0.2 co ~0-o.4 * ~ -o.~- ~- -0.8 -1.0 1- Npan=168 |~. Npan= 601 1 1 1 1 1 1 1 0 10 20 30 40 50 60 70 t*sqrt(g/L) Figure 32 -0.2 CO to-0.4 O ~ -0.8 -1.0 80 ° -9 Submerged Elli o d Fr=0.50 ,,, -9 0 B/L ~ 0.20 PZojL - 0.16 10 -0.2 - ~0.4 o -0.6 -0.8 -1.0 Submerged Ellipsoid Fr=0.45 B/L = 0.20 Zo/I~ = 0.16 . 1 | Npan-168 | | Npan= 60 1 1 1 1 0 20 30 40 50 60 70 80 t*sqrt(g/L) Figure 33 Submerged Ellipsoid Fr=0.80 B/L = 0.20 OWL = 0.16 . -0.2- | Npan=168 | | Npan~ 60 CO _ =-0.4 o -0.~- ~- -0.8 -1.0 1 1 1 1 1 1 1 -1.U- 1 1 0 10 20 30 40 50 60 70 80 0 10 20 t*sqrt(g/L) Figure 34 79 30 40 50 60 70 80 t*sqrt(g/L) Npan= 601 Figure 35

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APPENI)IX 1 INTEGRAL REPRESENTATION OF THE VELOCITY POTENTIAL 1. Green function Let us first introduce the threedimensional Green function for a point source of impulsive strength: G(M9 P,t) = {i(~) . Go(M9P) + H(t) . F(M, P,t) with the notations: Go(M9P) = --+- (A1.2) Or Or' r- F(M,P,t) = --J (gk)~ sin[(gk)~nt] Jo(kR) ek(Z+Z )dk 2~ o M= (x9y,z) P = (x',y',z ) r= [(x-x') +(y_yt)2+(z z,)2]ln r' = [(X-xt)2+(y-y.)2+(z+z.)2l1/2 R = [(x-x' )1/2+(y y, )I/2] JJ S(T) Q(M) 8(t-~) · O(M9 1) = (A1. 10) 41` a a [~P'~) a (p) (M9P,t -I) - ~(p)(~>(P,7) G(M,P,t -I)] dSp where Q(M) is the solid angle under which D is seen from M, when M is in D or S. (1.10) is then integrated with respect to from 0 to t, giving, due to the fundamental property of o: Q(M) (A1. 11) O(M,t) = 4~ J dame [~)(P't) Pa C;(M,P,t-~) ~ Pa O(p,:) G(M,P,t-~)] dSp O S(T) ~ being bounded at infinity (finite energy), and accounting for (A1.8), the integral over SO in the right hand side of last equation vanishes. Next step is to transform the integral over Sf: Isf = J damp [~(P,~)-G(M,P,t -I) - a GAP I) G (M9P,t-~ )] dSp O Sf(T) and where o(t) and H(t) are respectively Dirac's delta function and Heaviside's step function. surface condition, so that: G is the potential induced at point P by a point source of strength 6(t) located at point M, and can be shown to have the following properties: AG(M,P,t) = (or) . 6(t) (A1.4) (A1.5) 2 G(M,P,t) AVG(M9P,t) O fops Sl(z=O) et2 Liz. G(M9P,O ) = G`(M9P,O ) = 0 (A1.6) G(M,P,t) = G(P,M,t) (A1. 7) G = 0(r~2); VG = 0(r~3) when r ~ +oo (A1.8) 2. Green's identity Applying Green's identity in the fluid domain at time to O(P,:) and to G(M(t),P(~),t-~) yields: J: ~ POP t) AG(M(t).P(~). to-) - A~P.~) G(M(t).P(~).t-~)l dV~ = S(T) v u [~(P9~ ) a ups (M9P9t -I) - ~ (p)O (Pa) G (M9P9t -I)] d Sp with S(T) = Sb(~) + Sf(~) + SO Left side of equation (A1.9) is a Stieltjes integral which can be reduced accounting for (A1.4) and (1), to give: (A1. 12) On Sf, ~ and G both satisfy the linearized free ~n(M; Pig) = - g 'I'=(M,P,~) 1 (A1. 13) Gn(M,P,t-~)=- g GTT(M9P9t-l) [At = _ _ J ~ IJ [A GTT G Tl] P = _ g I [A GT T T P (A1. 14) then, introducing the total derivative in ~ of the integral over Sf: (A1.9) 1C J J [<DGT- GOT] dS = J ~ [AGE GOT]I Sf(T) Sf(T) + i; [BIG - GO ] (n ~ dl) . Vc T I Cb(T ) (A1. 15) where Cb is the curve defined by the intersection of the instantaneous body surface Sb and the free surface z=0 and Vc is the velocity of a point on Cb, we may write for Isf: 81

APPENI)IX 2 FREQUENCY DOMAIN GREEN FUNCTIONS FOR 1~ BODY-NONLINEAR PROBLEM + Isf = (A1. 16) 1 J-1 d1 o J do f [(Pa), 7) F~(M(t),P(~),t-~) - FO ] (np~dl p).Vc(P) O Cb~) [~(P(~),~)G`(~P, A) - G(M,P,t-~)OT(P,~)] dSp The first term is zero, due to the initial conditions for G and A, and gathering terms, we finally obtain the following integral representation of the velocity potential in the fluid domain D(t), in terms of a mixed distribution of sources and normal dipoles on Sb. (Go terms in the integral over Sb have been extracted accounting for the integral property of 6(t), and eliminated from the line integral, for Go is identically zero at the free surface) Q(1~ ~ O(M,t) = (A1. 17) Ji [~(P,t)~ Go(M,P)~Go(M,P)~(P,t)]dSp Sb(t) + tdz(t + g J do f [~P(~) to F'(M(t),P(~),t-~) - FO ] (np~dl p.V~P) O Cb(~) a ~ [~(P,7) ~(p)F(M P. t-`c) (Pa,) F(M, Pity)] dSp When M is on Sb, Q(M) = 2~ (if the normal is continuous in M) and (A1.17) gives the integral equation to be solved for ~ on Sb, considering amen to be known (forced motions). After solution, (A1.17) may be used to compute everywhere else in the fluid domain D, where Q(M) = 4~. When the body is fully submerged or when Cb is time invariant, the line integral in (A1. 17) simply vanishes. At last, note that the solution may also be represented by a distribution of sources only. The derivation of the corresponding integral representation follows comparable steps and, for brevity, will not be given. We are interested here in the steady-state periodic free-surface potential generated by a submerged point source following an arbitrary periodic motion of frequency lo, with a source strength of frequency pal, p 2 0, i.e. solution of the following problem: AGip(M(~),p,[) = arm)) mS pet (i=l) sin pot (i=2) (A2. 1) 32C2,p + g amp = 0 for z. = 0 (A2.2) VGip ~ O for r > ~(A2.3) + ra liadon condition (A2.4) for M(t) = [x(t),y(t),z(t)] prescribed modon of frequency P = [x ,y',z' ] fixed point r(t) = [(x(t)-x ) + (y(t)-y') + (z(t)- z')2]~/2 (A2.5) An exhaustive study of the solutions of this problem may be found in Ferrant (1988), where these Green functions were initially developed for the solution of the body-nonlinear problem in the frequency domain. Only the results liable to be useful for the present paper will be repeated in this appendix. 1. General case - Arbitrary periodic motion The preceding problem may be solved using different methods (Fourier transform, time asymptotic limit of unsteady solution,...), the details of which being given in Ferrant (1988). Although involving some tedious algebra, the results may be expressed in a very concise form, involving the frequency domain Green function for a fixed point source. In the general case of an arbitrary periodic motion, G1p and G2p are given by: ~ . Glp( M(t), P. t) = 2~'1 costar [ G`30(0, M(~),P) 00 + 2Re2, G~,o(lt0,M(~l~),P) e ( ) ] do 2n r-- G2p( M4t),P, t) = ~J S~Pt [ G - (0,M(~),P) 00 + 2 Re A, Go,o(lc~, M(~/cd)7P) e ( ) ] do (A2.6) (A2.7) where G`~o(~7M,P) is the complex Green function for the frequency domain diffraction-radiation problem. G`,,o(O,M7P) is the zero frequency limit of this function, i.e. for infinite depth: G`~,o(O,M?P) = - Cur ~ Or' (A2.8) 82

Nota: Although initially derived for the infinite depth case, expressions (A2.6), (A2.7) can be easily shown to hold for finite depth too, provided the finite depth diffraction- radiation Green function is substituted to Gino. 2. Sine heave motion 2.1 Basic formulations Explicit forms of the body-nonlinear frequency domain Green functions may be derived by first imposing the analytic form of the source path, and then introducing appropriate formulations of Gino (Near-Field, Haskind, ...). In the case of a time-harmonic heaving motion, we have: M(t)=(xO.yO. zO+acoscx) (A2.9) Then, substituting for example the Near-Field formulation of Glaxo in (A2.6), (A2.7), we get, after some transformations: cos pot r 1 1 ~ G~p[M(t).P, t] = ~ 4~c Lr(~) + r'(t) (A2. 10) I, 12 kn ~ c" pa c" near ~ ~ em ME [a (/)] don do . cos nix + ~ e n ° [Ip+n(akn)+Ip n(akn) ] . [Ho(knR) cOs not - Jo(knR) sin not] and: sin pot ~ 1 1 G2p[M(t),P,t] ~~ 47c r(t) r (t) (A2.11) -I ~-3 J sin p~·sinn~ Re IT e n E~[~(il~l))] do d~.sinnC0t +7en ° [Ip-n(akn)~Ip+n(akn)] [Ho(knR) sinn()t+Jo(knR)cosnot] where Ho and JO are respectively the Struve and Bessel functions of order 0, Ip is the modified Bessel function of order p, E1 is the exponential-integral function, and: kn = (no) 2/g R=[(xo-x) +(YO-Y)] (n(t) = kn [z(t) + z' + j R cos 0] These expressions are mostly appropriate for the computation of Gip when R is small. For larger radial distances, formulations based on the modified Haskind form of Gino are to be preferred; see details in Ferrant (1988). 2.2 Far-field behaviour Accounting for the asymptotic behaviour of the special functions Ho, JO and E1, the following far-field expressions for Gip are easily derived, for a source motion being given by (A2.9): G,p(M(~).P.~) = (A2.12) n~27TR)I e [Ip+n(akn)+9n(akl,)].sin(k,,R )7/4) +o(R~) for R woo 00 . ~/2 G2p(M(~).P.~) = (A2.13) t2~R] e [Ip n(akn)-Ip+n(akn) ] cos(knR-ncot-~/4) + o(R~~) for R - - oo As can be deduced from these expressions, the far- field is a superposition of regular circular waves of velocities Cn = gnu, 1 < n < oo., the amplitudes of the harmonics being explicitly given by the coefficients in front of the sine functions, involving modified Bessel functions of arguments akn. 83