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The Dispersion of Larg - Amplitude Gravity Waves in Deep Water W. Webster (University of California, Berkeley, USA) D.-Y. Kim (Wageningen, The Netherlands) Abstract: The Green-Naghdi (GN) theory of fluid sheets is used to analyze large amplitude, deep water waves in the time domain. Level III theory is used to simulate a train of steep regular waves and a random wave record corresponding to steep seas measured during hurricane Cam- ille. An analysis of the simulated random wave record shows that the linear dispersion assumed for referring a random wave train from one point in space to another does not result in conservative estimates of two important quantities used in de- sign: the crest elevation and particle velocity un- der the crest. 1. Introduction The focus of this paper is on the behavior of large amplitude water waves in deep water, with a particular emphasis on the implications of this behavior for the engineering analysis of the mo- tions of and loads acting on ships and offshore platforms. For the problems of greatest concern here the waves are of a length scale comparable to the horizontal dimensions of the ship or plat- form. For such waves it is common (and reason- able) to neglect both surface tension and viscosity, and we shall do so here. During the last few decades significant advances have taken place in the understanding of these water waves in both deep and shallow water, and there is a very large literature on this research. We will not attempt an exhaustive review this research here since our interest is a fairly narrow one. Much of the research into deep water waves and their applications to design can be di- vided into two principal and almost mutually ex- clusive thrusts: the description of the kinematics and the stability of regular, two-dimensional waves of large amplitude (up to and including breaking), and the description and measurement of random wave systems using analyses which rely on superposition and linearity. This split has its counterpart in the design ounce. A typical problem in design is to deter- mine the adequacy of a structure under consid- eration to withstand the forces imposed by the largest waves it will encounter in its lifetime (the so-called survival" problem). This problem can be thought of as composed of two parts: a descrip- tion of the wave situations (in the absence of the structure) which would lead to the survival con- ditions, and estimation of loads and motions which result from the interaction of these waves with the structure. The focus here is on the first part, the description of the wave system, although it is recognized that the second part is probably the more difficult of the two. This problem is at once nonlinear and random, since the waves which lead to the sur- vival conditions are likely to be breaking, or nearly breaking, local storm waves. This design problem causes a dilemma for the engineer, since he often must choose between an analysis of his structure based on the impingement of a sin- gle, regular, large-amplitude wave (the design wave approach) or an analysis based on the impingement of a random wave system of super- posed linear wave components (the spectral ap- proach). We note that the use of the spectral ap- proach for the estimation of the motions in more moderate seas where linear superposition is probably not a bad assumption (the so-called "operational" problem) has become almost uni- versally accepted, since the use of linear, ran- dom-wave analysis does have several advantages. Its use brings with it the powerful theoretical bases of time series analysis and stochastic pro- cess theory. These provide a rational framework for the estimation of the reliability and operability of the structure. Further, since the wave compo- nents in the spectral decomposition are linear, one can treat with almost equal ease both the fre Department of Naval Architecture & Offshore Engineering, University of California, Berkeley, CA 94707 Currently: Department of Naval Architecture, Seoul National University, Kwanak-Gu, Seoul, Korea 397

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quency domain and time domain problems. Both approaches to the more severe sur- vival problem have advantages and disadvan- tages. The loads used in the design wave ap- proach reflect the sharpening of the crests and the flattening of the troughs due to nonlinear ef- fects, and these effects, in particular, often have significant consequences on the wave loads on offshore platforms and on the shipping of green water on the deck of surface ships. The use of a design wave does yield a deterministic load sys- tem which is relatively easy to incorporate into a design analysis. For this purpose, it is common to use fifth or higher-order Stokes wave approxi- mations or, more recently, the results of stream function expansions (Dean, 1974, Chaplin, 1980~. The methods used to determine these nonlinear waves make use of approaches which can neither be extended to three dimensions nor be general- ized to arbitrary time-domain calculations in which a representation of steep, random wave systems can be made. Kinematic descriptions of regular deep water waves (assuming one can ig- nore viscosity and surface tension) are known to great accuracy (Schwartz, 1974; Fenton, 1988~. It is not difficult to formulate a second-order or higher-order perturbation approximations for non-linear waves in the time domain, but they have been little exploited, if at all, in the design process. The random and three-dimensional char- acter (short-crestedness) of a measured real storm wave system is captured by the usual spec- tral analysis approach. Time series analysis al- lows identification of the spectral composition of the wave surface elevation at the point of mea- surement, and allows identification of some of the directional character of the seaway if many such points of measurement are made close by concurrently. When the spectral representation of the water surface is known, the prediction of the pressures and velocities at and under the free surface at the reference location is usually made by associating the Fourier components of the wave surface with linear (Airy) wave compo- nents. This superposition is only valid if the orig- inal wave system is of a height and character which is consistent with linearization of the free surface boundary condition. Such an assump- tion becomes ever more questionable as the waves become steeper and approach breaking. There are a number of approaches whereby the inter- pretation of the spectral decomposition is modi- fied to improve the prediction of the pressure and velocity fields corresponding to the free surface description. We shall discuss one of these due to Wheeler (1969) in a subsequent section of this pa- per. The prediction of the pressures and veloci- ties at locations remote from the reference loca- tion requires, in addition, an estimate of the dis- persion of the waves. If one supposes the super- position of Airy waves, then each component travels at a different speed which is uniquely re- lated to its own frequency. Thus, the phasing of these components at the remote location is differ- ent from that at the reference location. However, it is known from the study of nonlinear regular waves that steeper waves of the same length travel faster than their less steep counterparts. One can therefore anticipate that there will be a nonlinear interaction between the component waves which will affect their wave speed. For in- stance, consider the case where one analyzes the motions of a large ship in head seas and pre- scribes the wave time history at one point on the ship, say amidships. In order to perform this calculation, it is necessary to predict the wave environment over the whole length of the ship at each instant in time. Since the length of typical large ships is in the order of 400' to 1000' or more, small differences in the estimated dispersion of shorter waves may cause significant discrepan- cies between the wave time history at the bow and at the stern. Further, since the discrepancy at the bow is of the opposite sense from that at the stern (relative to a reference point amidships), these discrepancies may become especially im- portant for pitch or yaw motions which reflect the difference in forces bow and stern. Although linear ship motions analysis can be considered state-of-the-art, nonlinear motions analysis is not. In particular, much of the thrust in recent years in nonlinear ship motions has re- volved about the slow drift problem where second- order forces and waves are taken into account. These endeavors are extremely complex and the prospect of accomplishing in the near future an analysis correct to, say, the third order is not bright. What is troublesome with this state of af- fairs is the fact that the second-order wave prob- lem predicts the same wave celerity as the linear problem and has many of the dispersion charac- teristics of Airy theory. The third-order solution is the lowest order perturbation theory which pre- dicts an increase in celerity of regular waves with steepness similar to that observed in nature and interactions between waves which lead to "phase-locking". In conclusion, it is fair to say that neither design approach to the survival loading (design wave or spectral decomposition) is wholly satis- factory. It is the purpose of this paper to explore the substantial gap which exists between these two design approaches by presenting a different model for the behavior of large-amplitude deep water waves in the time domain. It is of particu 398

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tar interest to use this model to investigate the dispersion of a random wave system from one lo- cation to another so that some insight into the omissions of current linear and second-order theories can be obtained. The foundation for this development is the Green-Naghdi theory of fluid sheets (hereafter referred to simply as GN theory) and, in particular, the extension of this theory to deep water waves (Green & Naghdi, 1986 & 1987~. This study could also have been performed using other nonlinear formulations, but GN theory was chosen since it is particularly efficient computa- tionally. Following the introduction of the GN gov- erning equations for level III theory below, the remainder of the paper will consist of two parts: (a) validation of the theory using known results of steep regular waves, and (b) use of the time-do- main solution of these equations to simulate a steep, random seaway. 2. GN level III theory of deep water waves GN theory is a model for three-dimen- sional fluid flow which, since it involves one fewer independent space variable than three-di- mensional space, is called a fluid sheet model. The basis of this model is rather different from traditional models derived from potential theory using perturbation methods or from the special- ized methods often introduced to compute with high accuracy the characteristics of regular, two- dimensional water waves. When viscosity and surface tension are ignored and the fluid flow is assumed to be irrotational, the field equation (Laplaces's equation) is linear. The only nonlin- earities are found in the boundary conditions on the free surface. The treatment of the field equa- tion and nonlinear boundary conditions by per- turbation methods and GN theory are the an- tithesis of one another. In the perturbation method, the field equa- tion is retained exactly and the boundary condi- tions are approximated; in GN theory the field equation is approximated and the full boundary conditions are retained. However, it is not our purpose here to give a detailed discussion of the consequences of these different approaches. The reader is referred to Green & Naghdi (1986, 1987) for a precise exposition of GN approach to water waves, and to Webster & Shields, (1990) for an overview and commentary on the method. Since perturbation parameters or scales are not used in its development, the limits of ap- plicability of GN theory are implicit and must be determined by physical or numerical experi- ment. For the problem of steep water waves we choose GN level III theory, as defined in Webster & Shields (1990~. Although this theory is com plex, this level theory was necessary for the treatment of even a narrow-banded spectrum. We introduce a coordinate system Oxyz, with the Oz axis oriented vertically up and the Oxy plane horizontal and corresponding to the undisturbed free surface. In the GN theory used here, the vertical dependence (i.e., the depen- dence on z) of the kinematics of the fluid flow is restricted. That is, we introduce a set of func- tions (n(Z) which will serve as a basis for the ver- tical dependence. These functions play the same role that "shape functions" play in finite element analysis. We assume that the fluid velocity, v~x,y,z;t) = (u,v,w) can be approximated with three of these basis functions (for level III). Thus, 3 v~x,y,z;t) = I, vn~x~y;t) kn(Z) n=1 ~1 where vn = (un,vn,wn) are vector coefficients as- sociated with the function An. Following Green and Naghdi (1986), we select basis functions given by kn~z) = zonk) eaz n = 1 2 3 (2 where a is a constant, the choice of which will discussed below. The exponential factor, \~ = eaz was selected since it has the same form as the z dependence found in the Airy wave solution. The other terms in the basis can be regarded as sys- tematic variations of the Airy wave velocity pat- tern. The kinematic assumption (I) is inserted into the equations for conservation of mass, con- servation of momentum (Euler's equations), and the kinematic boundary condition on the free sur- face, z = ,B(x,y;t). It is possible to satisfy all of these equations identically except for conserva- tion of momentum, which is satisfied only ap- proximately. Euler's equations are multiplied by \1' \2, \3 and integrated with respect to z. The result is a set of three vector equations which re- flect conservation of momentum in a weighted average sense. These together with exact state- meets of conservation of mass and the kinematic boundary conditions are the evolution equations for this model of the flow. The final evolution equations can be expressed in rather compact general form (equations 3.4, 3.8 & 3.11, respec- tively, in Webster & Shields (1990~) but these equa- tions will not be repeated here. The determina- tion of the evolution equations in terms of deriva- tives of the primary variables requires a prodi- gious amount of algebraic manipulation. This manipulation is, however, not difficult if one uses any of the new symbolic processors now avail- able. (A program called Mathematically was used for this manipulation). 399

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The final set of four evolution equations for unsteady two-dimensional flow is presented in Appendix A. The components of the vertical components of the vn (w~,w2,w3) have been elim- inated, as have the so-called "integrated pres- sures" pi, P2 and pa. The pressure on the free surface, 13, is taken to be zero. The remaining four variables are the free surface elevation, p~x,t), and the three horizontal components of the vn: ul~x,t), u2(x,t) and u3(x,t). These evolution equations will be used for all of the nonlinear computations in this paper. 3. Large-amplitude waves of permanent form In this paper we use the GN level III the- ory for time domain calculations of the dispersion of random waves. The authors know of no high accuracy calculations of these wave systems to compare with. Since this theory is a new one, it seems prudent to first compare the characteris- tics of large amplitude regular waves predicted by this theory with known very accurate results for these waves. To determine waves of permanent form, the transformation 8/at ~ uO 3/0x is applied to the evolution equations in Appendix A (these equa- tions are Galilean invariant). The equations now only depend on the primary variables ,B, us, u2 and U3 and their derivatives with respect to x. The problem of determining a wave of permanent form of a given length and elevation at the crest can be posed as a two-point boundary value prob- lem over a domain equal to the half-length of the wave, with a symmetry condition imposed at both ends of the domain and an elevation condition at the crest imposed at one end of the domain. In addition, global conditions stipulating that the flow is irrotational on the average and the mean water depth is zero need to be applied. A proce- dure based on Thomas' method described by Ertekin (1984) and generalized by Shields (1986) was used to find these solutions. For the compar- isons below, the x domain was discretized into 200 equally-spaced intervals (201 nodes) for one wave length. Central difference formulas were used throughout. Several characteristics of these waves are obvious candidates for comparison. These in- clude: wave celerity, wave profile and velocity profile. The parameters which are of importance here are wave steepness, wave length and the constant, a, which appears in the basis functions (2~. As mentioned in the previous section, this constant governs the exponential decay of the ve- locities in depth. A value of a equal to the wave number, k = 2~/\, where ~ is the wave length, produces the same decay with z as predicted by Airy wave theory and this choice yields the best comparison with finite regular waves. We intro- duce the notion of "bandwidth", the ability of GN theory to predict waves of wave numbers which are different from a. We anticipate that there will be a range of wave numbers kit ~ a ~ k2 for which the theory will produce satisfactory re- sults. Accurate, high-order stream function re- sults computed by Sobey (1989) are used for the comparisons below. a. Wave celerity. The celerity, or phase velocity of the wave, is the speed, uO, of the coordinate system neces- sary to yield a time invariant wave form. Figure 1 shows the ratio of the celerity of infinitesimal waves predicted by various levels of GN theory to the celerity of Airy waves. It is seen that the bandwidth of GN level I for a relative celerity er- ror of, say, 2% is very narrow, that of GN level II is broader and that of GN level III is broader still. Since the focus of this paper is a random wave train, it seemed appropriate to choose the theory with the broadest bandwidth and therefore GN level III was selected primarily on this basis. 1 1 >` 1 = ~ o.s ~ 0.8 ._ a) a) 0.7 c' is 0.6 - /. . ~ levell ;,! . level 11 0 level 111 3 Figure 1. The ratio of celerity of infinitesimal waves predicted by various levels of GN theory to that predicted by Airy wave theory. It is well known that the celerity of a regu- lar wave depends on its steepness. Figure 2 shows the results of GN level III theory for waves of various steepness for the special situation where k _ a. The error between the GN results (the line) and the stream function results (the black squares) is much smaller than 1% and cannot be detected on this figure. For values of k different from a, Figure 3 shows the error in celerity as a function of steepness. It is seen that for values of 2.25 > a/k > 0.5, the celerity error is within 1% for all values of steepness less than 0.12 (breaking waves correspond to a steepness of 400

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about 0.14). That is, the celerity error is less than height). For wave of small wave height the eleva 1% for waves of one-half of the length of the wave lion varies almost sinusoidally in x. As the wave for which k - a to waves which are well over steepens, the crest becomes sharper and the twice the length of the wave for which k _ a). For trough flattens. Figure 5 shows the profiles for waves of lower steepness the bandwidth Is some- waves of steepness 0.106 predicted for various what larger, but the bandwidth is, of course, al- values of a/k. It is seen that even for this very ways smaller than that for infinitesimal waves. steep wave, the variation in profiles is very little for the range of am between 0.5 and 2.0. 1.10 ' ' ! ! ! 0 1.08 ~ So fey ~Zz 0 4~ ~ 106 F'''''''''''.'''''''''''''.''''''''''''''-.''''''''''''''.''''''''~'''''''-''''''''''''] ~ 0.2 ~: ~ ~ / ~ ~ ~O C 0 0.04 0.08 0.12 steepness, h/\ ~04 ) 0;785 1.57 2.36 3 14 Figure 2. The variation of celerity with steepness 2~ x/\ for GN level III theory for am = l.O (curve) and Figure 4. Wave profiles predicted by GN level III for numerically accurate results (black squares). theory for regular waves of various crest heights for a/lr = 1. j ~ 2.25 c 0.5 B ~ W_~ Hi= ~L D IS / 0 0.02 0.04 0.06 0.08 0.1 0.12 -0.30 0.785 1.571 2.356 3.142 steepness, h/\ 2~ x/\ Figure 3. Error in prediction of the celerity of Figure 5. Profile of waves of steepness 0.106 pre regular waves by GN level III theory as a dieted by GN level III theory for various values of function of steepness for elk ~ 1. a/k. b. Wave profile. c. Particle velocity Figure 4 shows the wave profiles computed From the point of view of design of many for waves of various elevations at the crest for the offshore platforms, the horizontal particle veloc case k- a. These profile shapes deviate less than ity under the crest of the wave is probably the one line width from high accuracy profiles (the most important. It is this characteristic of the deviation is much less than 1% of the wave flow which causes the most significant loads on 40~

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fixed (jacketed) platforms. Once again, the hori- zontal velocity finder the crest of even very steep waves is predicted with much less than a 1% er- ror if k _ a. For k ~ a deviations occur. Figure 6 shows the variation with z of the horizontal veloc- ity under the crest for a wave of steepness- 0.106. The velocity is non-dimensionalized with the Airy celerity. Curves for various values of a/k are shown, where the value for a/k = 1 is coincident with numerically accurate results. The water surface at the crest of this wave is at a non-di- mensional value of fizz/\ = 0.403, and the undis- turbed water level corresponds to a value of z = 0. Also shown on this figure is the horizontal parti- cle velocity prediction from Airy wave theory. 0.5 ~ ~ ~ 1 ~ ~ ~ 1 ~ ~ ~ ~ ~ ~ ~ -'I a''''''''''''''''''' ''''''.? ''''''''''''''''''''''~ a/k - - -0.49 - -0.64 1 .0 - - - - 1 . 5 6 ----- 1.96 - ----Airy 0 0.2 0.4 0.6 0.8 horizontal particle velocity / ~ Figure 6. Variation of horizontal particle velocity under the crest of a wave of steepness 0.106 as a function of non-dimensional altitude. It is seen that Airy wave theory uniformly over-predicts the horizontal particle velocity un- der the crest for all z. In general, the prediction of the GN level III theory is good, except very near the crest. If 0.64 ~ a/k < 1.56 the error is ev- erywhere less than 5% of the numerically exact result. For a/k = 1.96 the error is 25% at the wa- ter surface but becomes less than 5% for values of 2~z/l < 0.25; for elk = 0.49 the error is 14% at the water surface but does not drop to less than 5% unless fizz/\ < 0. Separate investigations were also made at different values of steepness and at different loca- tions along the wave. At the crest at a steepness of 12%, the error for 0.64 < a/k < 1.56 increased slightly to 6%; that for a/k = 0.49 and 1.96 in- creased to 20% and 35%, respectively. Beneath the trough much smaller particle velocity errors were observed for all values of a/k discussed here and, thus, it appears that the crest is the most critical location. d. Summary of regular wave comparisons. The bandwidth for particle velocity error is much narrower than for either wave celerity er- ror or for wave profile error. Let us denote the wave length for which a/k = 1 by DO. The above comparisons indicate that as long as 2/3 TO < ~ < 3/2 TO, we can anticipate errors of less than 1% for celerity or wave profile and less than 6% for horizontal particle velocities for a steepness up to 12%. These limitations imply that the GN level III theory may be a good model for a steep, nar- row-banded seaway. 4. Time domain results Two different wave situations were investi- gated using the time domain version of the Green-Naghdi level III equations in Appendix A. The evolution equations are second-order in time and third order in space. At each instant the time derivatives (on the left-hand side of each equation in Appendix A) can be found as a solu- tion to a two-point boundary value problem. Since this is an initial-value problem starting from an initially quiescent condition, the global conditions for irrotationality or for mean water level used for determining waves of permanent form need not be applied here. The difference formulation for the two- point boundary value problem is the same Thomas' algorithm used for the waves of perma- nent form. Integration in time is performed us- ing a modified Euler method. Both integrations in space and time are second-order accurate and variations in both time and space steps were made to assure that convergence was adequate (less than 1% error). One of the particular advantages of GN theory in general is that it yields differential equations in the horizontal coordinates. Since the computational effort required to solve the two- point boundary value problem grows linearly with the computational domain, the overall time integration retains this property. In this sense, the GN theory allows one to compute larger spa- tial domains than, say, boundary element meth- ods were the effort typically grows with at least the square of the size of the number of nodes. The left-hand boundary for both problems below was considered to be a "wave-maker" where values of ~B(xw,t), ul~xw~t)' u2(Xw~t) and u3(xw,t) were prescribed (xw is the x location of the wavemaker). In general, the values of the three u's are not known a priori for the nonlinear wave system. We used the values obtained from a 402

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linear solution of the GN level III equations to re- late these quantities to p. The local disturbance caused by the not quite correct values of the three u's appeared to die out quickly (as it does with a real wave-maker in a wave tank). In order to avoid any initial disturbances, a cosine-squared ramp was provided at the start of the wave- maker. The ramp was applied only to the first full cycle of the wave maker. A simple Sommerfeld boundary condition was imposed on the right-hand boundary for both examples and this condition was based on the as- sumption that all waves have a celerity equal to: that of infinitesimal waves of length lO. In prac- tice, little reflection was observed, but in both ex- amples the right-hand boundary was taken far enough away to minimize any possible adverse consequences from reflections. a. The generation of regular waves. In order to assure that the time domain in- tegration scheme and the wave-maker were per- forming correctly, a set of regular waves was generated. Although the internal calculations were all performed non-dimensionally, the re- sults are reported dimensionally, corresponding to typical ocean wave scales. The computational domain consisted of 1300 space steps of 12.28' and the time steps were 0.2 sec. The waves had a wave length of 809' and final height of 70' (corresponding to steepness of 8.65%~. Figure 7 shows a snapshot of the wave elevation profile 160 seconds after the start-up of the wave-maker. Since the wave packet had not progressed past 6000', the remainder of the computational do- main is not shown. This figure shows that the eldest two waves (the rightmost two waves) are somewhat distorted, and that the train of notice- ably steep, but regular waves follows. Figure 8 shows the wave elevation time history as seen by an observer 644' from the wave maker. This observer sees almost twice as many waves as seen in the surface elevation view be- cause the group velocity is much less than the celerity (a manifestation of dispersion). A very short time after the time of figure 7, the leading wave of this packet became quite steep and reached a breaking condition. Local snapshots of this process at 2 second intervals are shown in Figure 9. The solution algorithm breaks down when the wave reaches breaking conditions and the computation can not continue. It was not clear, at first, whether the breaking wave at the front of the group was real or simply an artifact of either the start-up of the wave maker or of the GN theory. Longuet- Higgins (1974) demonstrated, both theoretically and experimentally, that the leading wave in a packet of generated waves will steepen. Further, his experiments showed that the leading wave can break if the generated waves were steep enough to begin with. Unfortunately, his linear a) a, -20 ct ~ 40 60 - r - a ._ ~ 20 Cl) 0-0 a) ~ -20 ct -40 A, P ~.i ~', ~ 0 1000 2000 3000 4000 5000 6000 distance from wave maker Oft. Figure 7. Wave surface elevation at time = 160 seconds after initiation of wavemaker (regular waves of length = 809' and height = 70') ~.V V V ~V V V, V V V ~ 1 120 160 Figure 8. Wave elevation time history at a point 644' away from wavemaker. 403 time (see)

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100. c o 40 0' 20 a) ~0 V -20 Figure 9. Details of breaking wave at leading edge of wave packet. analysis was unable to predict such an occur- rence. In order to investigate this further, we at- tempted to duplicate the simulated wave situation in the Ship Model Towing Tank at U.C. Berkeley. To our surprise, we found that we were unable to make a train waves at this steepness without the wave in the front of the group breaking after about 12-14 wave maker cycles (that is with 6 or 7 waves in the tank). The breaking occurred even when the wave maker was turned on very smoothly over three wave cycles. b. Steep random waves. In order to investigate the dispersion of random waves, a numerical experiment was conducted. An existing measurement of steep waves, recorded during Hurricane Camille in the Gulf of Mexico, August 16-17, 1969 was used as a foundation for this experiment. The particu- lar record covered 512 seconds in real time with two measurements per second. Although this record was taken in a water depth of 325 ft (which corresponds to a depth at which shallow water ef- fects are just beginning to be perceived), it was felt that this sample was a good representative of the survival conditions one might encounter. The waves were simulated in exactly same fashion as the regular waves in a. above were. However, it was desired to generate a wave sys- tem like that from Hurricane Camille at a given reference point removed some distance from the wave-maker. The coordinate system was chosen so that this reference point was x = 0. A finite, untruncated Fourier transform of the record was determined, 2 nmax p(t) = A, an sin At + bn cos Ant, (3 n=0 where An In =- at nmaX _ . ~.......... .................................... ~; , 3500 4000 4500 5000 5500 6000 distance from wave maker (ft.) fit is the time interval between data points, nma~ is the number of data points in the record. Using linear dispersion (Airy theory), the record (3) was referred to a new location xw, as suming that the waves are two-dimensional and progressing in the positive x direction, yielding 2 nmax p(t) = A, an sin (t) + bn cos (t), (4 n=0 where the phase (t) = can (it (cog w) For the numerical experiment described below, xw was taken to be -644' (i.e. 644' up weather from the point x = 0'). This new record was used to drive the wave maker. Three wave probes were "mounted" in the computational do main, at x = 0', x = 400' and x = 800'. The dis tance between the probe at the reference point at x = 0' and that at x = 400' is comparable to the length of a typical offshore platform; the distance between the reference point at x = 0' and that at x = 800' is comparable to the length of a typical large ship. The constant a was selected to corre spond to waves for which TO = 809' (i.e. a = 2~/809) and the computation was run for the same 512 seconds of the original record with temporal steps of 0.2 sec. and 1300 spatial steps of 12.28'. Figure 10 shows a comparison of the wave elevation measured at the probe at x = 0 and the original Hurricane Camille record. In general the two traces compare very well except near t = 130 and t = 450. The computed wave elevation history is smoother than the original record pre sumably because the bandwidth of the GN level III theory is limited. We do note however, that most of the waves do lie within the wave length range of 500' to 1200' corresponding to the range 404

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60 - - o g a) a) 3 -2C 4G 20 4r - ~Camille time history ''''''~''''-'''''-IT'----'-------------'--''''''-------~--'''''---''----------'''--''''------------------'----- ------------------------------------- -:- ill - v GN level 111 simulation ~ , , . . 0100 200 300 400 time (see) 500 Figure 10. Comparison of GN level III simulation at x = 0 with recorded Camille time history 2/3 NO ~ ~ < 3/2 TO for which the GN level III the- ory yields uniformly excellent results. The useful part of the wave elevation records at the three probes cover a somewhat smaller time interval than the original Camille record because of the time it takes for waves to progress from the wavemaker to the probes. In fir ~ 600 .~ - 400 Cal Q o, 200 o probe at origin O probe at400' probe at 800' Camille Al ~ ~ ~ i . .~, O0.2 0.4 0.6 0.8 1 1.2 frequency transect Figure 11. Spectrum of the original Camille record and that measured at the three wave probes. Finite Fourier transforms of these records were also made and spectra formed. Figure 11 shows the spectrum of the original record, as well as the spectra of the time histories recorded at the three probes. These spectra have been smoothed using a seven-point moving average. 60 40 20 0 -20 -40 All four spectra are very nearly the same except the original Camille spectrum has a peak at a wave frequency of about 0.45 ra/sec which does not occur in the simulated record. The three spectra from the wave probes can be considered identical. Several additional simulations were per- formed using the same input record but with in- put to the wavemaker multiplied by a factor. The simulation with a factor of 1.2 (i.e., the input was 20% larger) produced waves which were almost breaking. Larger factors produced waves which did break and in these cases it was not possible to complete the simulation. The spectra of the time histories of the three wave probes were all about equal for the 20% larger simulation and all were almost ex- actly 44% larger than the corresponding spectra for the original simulation, as one would antici- pate. The actual wave profiles, although quite similar in form, were measurably more "peaked" near the highest waves. In the discussion below we will use both the simulation using the orig~- nal wavemaker input (labelled 100% Camille in- put) and that resulting from the 20% larger input (labelled 120% Camille input). The time histories of the wave elevation at x = 0' are now parts of a consistent description in time and space of nonlinear wave systems, and these descriptions afford an opportunity for assessing of the effects of nonlinear dispersion. Let us suppose that the time history recorded at the numerical wave probe at x = 0 is a realistic 405

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40 O 20 ._ ' O a) -20 -40 Figure 12. Comparison of waves at x = 400' predicted by GN level III and by linear dispersion from x = 0' (100% Camille input). record of a possible realization of a storm wave system (its closeness to the measured Hurricane Camille record lends credibility to this supposi- tion). This time history will be taken as a refer- ence time history. The time histories of wave ele- vation at the probes at x = 400' and 800' recorded in the simulations are part of a nonlinear wave system, but can also be estimated from the refer- ence time history at x = 0' using finite Fourier transforms and linear dispersion (as was done in (3) and (4) above). Such a process is spectrum- preserving and therefore these estimated time histories at the other two probes will have exactly the same spectrum as the simulated time history at x = 0'. However, the spectra of the simulated time histories at the x = 0', 400' and 800' are all sensibly the same (see Figure 10~. Thus, the two sets of time histories: the nonlinear GN level III simulation, and that derived by linear dispersion will have essentially the same spectra at each probe. In other words, each represents a differ- ent realization of the same spectrum. The comparison between the time histories at the two alternate probe locations is essentially a comparison between linear dispersion and non- linear dispersion. Figure 12. shows a compari- son of the time histories at x = 400' for the origi- nal wavemaker input (100% Camille input). It is clear that the character of both wave systems is more similar than the comparison between the simulation and the original Camille record, but upon close examination one finds the trace from nonlinear dispersion shows sharper peaks and flatter troughs than that from the linear disper- s~on. Ll ' I ' I ' ' ' ' ! ' ' ' ' ! ' ' ' ' ! ' ' ' ' ! ' ' ' ' ! ' ' ' ' I ' ' ' ' GN level 111 simulation .- II11,,,,1,,,,1,,,,,,,,,,,,,,1,,,,1,... 100 150 200 250 300 350 400 450 500 time (see) 40 20 O . 0 ' a) -20 ~ 3 -40 The relation between linear and nonlinear dispersion can be perhaps more clearly seen in Figure 13. The top three graphs in this figure show the results of matching the elevations of the individual crests and troughs from the record produced by the nonlinear dispersion simulation and that from the linear dispersion. The values resulting from each are plotted along a different axis. For the probe at x = 0' there is a perfect cor- relation between the crest and trough elevations derived from the time history and those derived from the finite Fourier transform, since the transform was determined using the whole wave time history at this point and no terms were thrown away. The two time histories at the probes at x = 400' and 800' were not identical and, in some cases, were not geometrically similar. Thus, identifying the corresponding crests and troughs was sometimes ambiguous and scatter occurred. For these two probes, the points show a signifi- cant deviation from the 45 line which would indicate perfect correlation. It is seen that, in comparison with the nonlinear dispersion simu- lation, the linear dispersion results show smaller crest heights and larger trough depths. This dis- crepancy is worse for the probe at x = 800' than for that at x = 400'. What is important in design of many ships and platforms is the combination of the maxi- mum wave elevation at the crest and the horizon- tal particle velocity at the crest. For the superpo- sition of waves given by (3), Airy theory predicts 406

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o - ~ - : - ~ ~ l l ~ ~ ~ ~ ~ ~ l Hi: ~ ~ to Juntas JalaaU.M - U!SJadS!P J~aU!1 5- uolsJads!p J~9U!1 of:- off- M~- ; I I > ~ ~ ~ O ,_ A o ~ g N ~0) S uo!s~ads!p~eau!l 6u!qo~a'~s~alaa4M-Uo!s~ads!p~eau!l ;z x Kit o ~ 9 r ~, ~, ~0 T ~--.- --- ----- ~--------~0 Cock 01 O O O O ~ O O O O O O 6ulq~a3~S Jala6UM uo!~!sodwo~ap ~a!'nd 91!U!d 407

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the horizontal particle velocity, u~x,z,t), in the waves to be u~x,z,t) 2 nmax ((I)n ~ Ad, ~ e g (an sin cut + bn cos ont) n=0 Referring again to Figure 6, it is clear that Airy theory yields particle velocities which are much too high. It is typical in many offshore applica- tions to use an approximation called Wheeler stretching" (Wheeler, 19691. In this approach, the exponential decay factor in (5) is modified, yielding 5(x,z,t) 2 nmax fin Z it) Ad, ~ e g (an sin ant + bn cos Ate, n=0 ~6 where z' = z - p~x,t). That is, the value of z used in the exponential decay measures the relative distance below the free surface rather than the absolute distance below the undisturbed free sur- face level. The bottom graphs in Figure 13 are corre- lation plots showing particle velocities both under the crest and under the troughs. It relates the corresponding peaks in the horizontal particle ve- locity at a distance 10' below the free surface at x = 0' computed by using the GN level III model, (1), and that determined using the finite Fourier sum (3) and Wheeler stretching (6~. The sum in (6) is unrealistically dominated by the large number of high-frequency components in the fi- nite Fourier decomposition when the exponential decay factor is unity, as it is when z _ ~ (i.e., z' = 0~. The effect of these high frequency components is unimportant for very small values of z' ~ O and thus a value of z' = -10' was selected. The correla- tion between the prediction of horizontal particle velocity at x = 0' from GN level III theory and Wheeler stretching is very good under the crest (positive velocities) and only slightly less good under the troughs (negative velocities). A signif- icant deviation occurs only for the very highest waves and the deviation remains no more than about 5%. Although we do not show the results here, finite Fourier decompositions and estimates of the particle velocities using Wheeler stretching were performed for the GN level III time histo- ries recorded at the other two wave probes. This information was used to develop correlation dia- grams similar to the bottom graph in Figure 13. These correlations were almost identical in character to the left-hand bottom graph in Figure 13. As a result, we conclude that if the time his- tory of the wave elevation is known at the point of interest, Wheeler stretching is a very good esti- mator of the peak velocities beneath the crest of a wave and a good estimator for the velocities be- neath the trough of the wave. Let us now investigate the situation when the wave elevation history is not known at the point of interest and must be determined by lin- ear dispersion. The middle and right graphs on the bottom of Figures 13 are correlation diagrams resulting from comparing the horizontal particle velocity at the probes at x = 400' and 800'. The GN level III prediction is based on (1) using the val- ues of us, us, Us and ,B determined at these loca- tions by the nonlinear simulation; the spectral method prediction is based on the Fourier de- composition (3), linear dispersion (4) and Wheel- er stretching (6~. It is obvious that the compari- son at the probe at x = 400' is significantly poorer than that at x = 0', and that at x = 800' is poorer still. In particular, GN theory predicted particle velocities under many of the crests in excess of 30 fps, whereas the linear dispersion result did not predict any velocities this large. When viewed as a whole, the graphs in Figure 13 show that the relationship between hor- izontal particle velocity discrepancy and the wave crest and trough discrepancy is nearly constant. That is, when the wave crest and trough predic- tions are good, the horizontal particle velocity predictions are good; when the wave crest and trough predictions are poor, the particle velocities are corresponding poor. It appears therefore that linear dispersion is the weak link in the predic- tion process rather than the Wheeler stretching. Perhaps a more instructive view of the dif- ference between linear dispersion can be gleaned from a comparison of the wave elevation profiles. Figure 14a shows snapshots of the wave elevation computed using linear dispersion for x = -400' to x = 2000' at 2 second intervals from t = 442 to t = 462 seconds. Figure 14b shows the same set of snapshots for the simulated waves using GN level III. Both sets correspond to the 120% Camille input to the wavemaker and by construc- tion, both sets of records have exactly the same time histories at x = 0'. The linear dispersion record shows many small wiggles which are the result of the high frequency terms in the 2100 terms of the finite Fourier sum. In general these effects are localized near x = 0. A cursory glance shows that the two sets of snaphsots are similar, but a significant differ- ence occurs between t = 454 and t = 460. The GN level III simulation predicts a large, nearly breaking wave crest which persists for about 6 408

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seconds and has a maximum elevation of 65' above the undisturbed free surface. This wave was the highest crest obtained by the simulation. The corresponding wave for the linear dispersion case is never more than 50' high and lacks the coherence of the simulated wave. The difference appears to be that the Airy wave components are not "phase-locked" and can not remain together for any length of time. 5. Conclusions A nonlinear fluid sheet model for predicting the dispersion of random wave sys- tems in the time domain was introduced and compared with known results for steep waves of permanent form. It was found that the particu- lar model used here, Green-Naghdi level III, compared extremely well with the results for wave celerity and wave profile for a wide range of steepness and over a fairly broad bandwidth of wave lengths. The comparison with wave parti- cle velocities was good over a fairly narrow, but useful, bandwidth of wave lengths. The GN level III model was used to predict the generation of regular waves and it was found (and confirmed by laboratory experiments that the leading edge of a packet of relatively steep waves always appears to break before very many waves are created. This nonlinear model was also used to model a real steep wave record, that measured during Humcane Camille in 1969. The purpose of this study was to investigate the effects of non- linear dispersion. These results can be summa- rized as follows: linear dispersion leads to under- pred iction of both the wave elevation aru! the wave particle velocities at a point remote from a loca- tion where the wave elevation history is known. This under-prediction may represent a signifi- cant lack of conservatism in the use of the spec- tral method for design to withstand survival con- ditions. The time histories at all probes either recorded from the nonlinear simulation or from linear dispersion from the probe at x = 0' all had spectra which were sensibly the same. That is, all were acceptable realizations of the same spec- trum. Yet those time histories of either wave ele- vation or particle velocity resulting from linear dispersion did not compare well with those which resulted from nonlinear dispersion. Thus, we can further conclude that not all realizations of a spectrum correspond to realistic wave systems, if the waves high enough to lead to significant non- linear effects. Acknowledgement This research was in part sponsored by the Office of Naval Research, United States Navy, under contract N00014-88-K-0002 with the Univer- sity of California, Berkeley. References Chaplin, J. R. (19801. Developments of stream function wave theory. Coastal Engineer- ing, Vol. 3, pp. 179-205. Dean, R. G. (1974), Evaluation and development of water wave theories for engineering ap- plication. U. S. Army Coastal Engineering Research Center, Report SR- 1 (two vol- umes). Ertekin, R. C. (1984~. Soliton generation by mov- ing disturbances in shallow water. Ph.D. Thesis, Univ. of Calif. Berkeley. v + 352 pp. Fenton, J. D. (19881. The numerical solution of steady water wave problems. Comput. Geosci., Vol 14, pp. 357-368. Green, A. E. and Naghdi, P. M. (1986~. A nonlin- ear theory of water waves for finite and in- finite depths. Philos. Trans. Roy. Soc. London Ser. A, Vol. 320, pp. 37-70. Green, A. E. and Naghdi, P. M. (1987~. Further developments in a nonlinear theory of wa- ter waves for finite and infinite depths. Philos. Trans. Roy. Soc. London Ser. A, Vol. 324, pp. 47-72. Longuet-Higgins, M. S. (1974), Breaking waves - in deep or shallow water. 10th Symposium on Naval Hydrodynamics, MIT, pp. 597- 605. Schwartz, L. W. (1974) Computer extension and analytic continuation of Stokes' expansion for gravity waves. J. Fluid Mech. Vol. 62, pp. 553-578. Shields, J. J. (19861. A direct theory for waves ap- proaching a beach. Ph.D. Thesis, Univ. of Calif. Berkeley. v + 137 p. Sobey, R. J. (1989~. Variations on Fourier wave theory. International Journal for Numer- ical Methods in Fluid Mechanics, Vol. 9, pp.1453-1467. Webster, W. C. & Shields, J. J. (19901. Appli- cations of high-level, Green-Naghdi theory to fluid flow problems. IUTAM Sympos- ium on Marine Dynamics, Brunel Univer- sity, London (in press). Wheeler, J. D. (19691. Method for calculating forces produced by irregular waves. Off- shore Technology Conference, Houston, Texas, OTC 1006. 409

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50 o -50 50 o -50 50 o -50 50 o -50 50 o -50 -400 0 400 800 1200 1600 2000 x (it) Figure 14a. Wave profiles predicted by superposition of Airy waves and linear dispersion. 410 442 444 446 t = 448 t =450 t =452 t =454 t =456 t=458 t =460 t =462 /

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so o - 5 0 50 o - 5 0 - 5 0 50 o - 5 0 50 o - 5 0 -400 0 400 800 1200 Figure 14b. Wave profiles predicted by GN Level III theory. 411 1 600 2000 x (ft) 442 444 t = 446 t =448 t = 450 t = 452 t -454 t = 456 t=458 t =460 t =462

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~ ~ | ~ m1 ~ - ~ C N HI K ~ | Nat = C';1 At, Cal =1 cat ~ 1 ~ -, ~C = ~C - ~ ~ 1 X ~ ~- ~ =~ ~ o =. ATE ~ ~ ~ ~, ~ ~, cat ~hi,, , 5 + ~a, Ct ~ ~Cal $ ~+ Cd ~- N + ^ ~CQ _ C:~ ~t ~at CD acts + ~+ Cal, ~ Cad Am, 0: ~Ct , - Hi a' + CD , , o | | = 1 ~o = = = (3o (1,, ~ X ~ =| K ~' ~= Q _ - ~ ~ ~$ N , - `'=L Cq N N ~C~ C+ ~C~ ~C'l+ C: C: + c ~e~; + ~cc, c~ ~a~, ~6 (= O c: ~ C , ~.:, ~ C K `:, t ;| C ~ct C~;l L I + 1 1 ~ 415

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DISCUSSION Krish Thiagarajan University of Michigan, USA The authors have prescribed a certain velocity distribution at the wave-maker position in their numerical wavetank. They also claim that disturbances caused due to this specified distribution are localized and die out quickly with horizontal distance. This claim may not be entirely true. Satisfying the no-flow condition on the wave-maker surface had been a known problem. Existing second order wave generator theories (Ref[1] reviews some of them.) indicate the existence of a second order free wave of frequency twice the fundamental wave frequency generated by the wave maker. This free wave is parasitic in nature as it travels along with the wave of interest, i.e., it does not die out. My own experiments have confirmed this (Reftl]). The above discussion, while pertaining to a physical tank, may also be applicable to a numerical wave tank. Ref.[1]: Thiagarajan, K. "An Experimental Study on Higher Order Waves and Hydrodynamic Loading on Vertical Surface Piercing Cylinders," M.Eng. Thesis, Faculty of Engineering, Memorial University of Newfoundland, St. John's, Canada, 1989 i . 416