SECOND-ORDER CREST STATISTICS OF REALISTIC SEA STATES

SECOND-ORDER CREST STATISTICS OF REALISTIC SEA STATES MARIOS CHRISTOU Shell Internatonal Exploraton and Producton, 2288 GS Rjswjk, The Netherlands. E-mal: maros.chrstou@shell.com PETER TROMANS Ocean Wave Engneerng Ltd., 5 Pne Walk, Lss, Unted Kngdom. E-mal: pt7185@hotmal.com LUC VANDERSCHUREN Engneerng Systems, Avenue Molre, 5/5, 13 Wavre, Belgum. E-mal: Luc.vanderschuren@skynet.be KEVIN EWANS Shell Internatonal Exploraton and Producton, 2288 GS Rjswjk, The Netherlands. E-mal: kevn.ewans@shell.com 1. INTRODUCTION Current engneerng desgn employs second-order rregular wave theory to predct extreme crest elevatons (Forrstall, 2). However, there are several feld measurements of waves wth crest elevatons sgnfcantly larger than those predcted at second-order; the New Year Wave at the Draupner East platform beng one such example (Haver and Anderson, 2). Consequently, several authors have sought explanatons for the occurrence of these so-called freak, rogue or abnormal waves. These explanatons nclude crossng-seas (Donelan and Magnusson, 25) and hgher-order undrectonal effects (Baldock et al., 1996; Gbson and Swan, 27). Consderng the second-order dstrbuton (Forrstall, 2), there are two dstnct shortcomngs. Frst, the crest statstcs are based on second-order, tme-doman smulatons that employ the JONSWAP spectrum and therefore are only strctly applcable for unmodal JONSWAP spectra. Evdence of ths s gven by Btner- Gregersen and Hagen (23) who have found dscrepances between Forrstall s dstrbuton n bmodal sea states. Second, t neglects nonlnear effects that occur beyond second-order. These hgher-order effects have been demonstrated by (Gbson and Swan, 27) who found local and rapd energy shfts occurrng n the vcnty of an extreme event on account of thrd-order resonant nteractons. These energy shfts are dependent on both the frequency bandwdth and the drectonal spreadng of the underlyng spectrum. The frst shortcomng can be overcome by employng the Spectral Response Surface (SRS) method (Tromans and Vanderschuren, 24), whch has no restrcton on spectral shape and has been appled to combned wndsea and swell sea states by Tromans et al. (27). The second shortcomng has been resolved by Gbson et al. (27) who coupled the SRS method wth the fully nonlnear Fourer-based model of Bateman et al. (21). However, so far ths has only been acheved for undrectonal seas. Therefore, as realstc sea states are drectonal, t s necessary to revert to the SRS wth second-order wave theory and buld on the work of Tromans et al. (27). Ths paper wll add to the lterature by employng the SRS method to nvestgate varous b-modal sea-states. These sea states wll combne varous fetch-lmted wnd-seas descrbed by the JONSWAP spectrum. Some 1

2 CHRISTOU, TROMANS, VANDERSCHUREN, AND EWANS authors have observed the occurrence of set-up below feld measurements of rogue waves (Toffol et al., 27; Taylor et al., 26). Toffol et al. (26) undertook tme-doman smulatons based on the second-order model of Sharma and Dean (1981). These smulatons modelled two dentcal JONSWAP spectra wth dfferent mean drectons of propagaton. When the separaton between the mean drecton of propagatons of the two seas was approxmately 9, a set-up was predcted by the second-order model. One of the ams of the present paper s to nvestgate whether a set-up s predcted by second-order theory when fetch-lmted wnd-seas nteract wth swells as often occurs n the real ocean. Ths paper contnues by brefly summarsng the SRS method. Followng ths, the smulatons undertaken and results obtaned from the SRS wll be dscussed. Fnally, the last secton wll draw the conclusons and present the wder mplcatons of the work. 2. SPECTRAL SURFACE RESPONSE METHOD The applcaton of the spectral response surface method to crest statstcs s descrbed elsewhere (Tromans and Vanderschuren, 24). In ths secton, t suffces to summarse t brefly and to dscuss the enhancements made to enable the descrpton of the wnd-sea and swell sea state and the solaton of the sum- and dfferenceterms. A random lnear sea can be represented as the sum of many frequency components, where each component obeys a Normal dstrbuton. Usng the second-order wave nteractons of Sharma and Dean (1981), the nonlnear surface elevaton can be expressed as a functon of the lnear components and ther Hlbert transforms. The resultant expresson allows us to use a Frst Order Relablty Method (FORM) type of analyss, treatng a constant value of ocean surface elevaton as the lmt state. Ths s a wdely applcable approach, requrng only that the varable chosen to defne the lmt state can be expressed n terms of underlyng lnear wave components. The underlyng lnear ocean surface elevaton, η (1), s a sum of N components and can be wrtten n the usual way as N N η (1) = η = a cos (k x ω t + φ ), (1) where, for each component, a s the random ampltude, ω s frequency, φ s the random phase angle, and t s tme. The values of a are chosen such that the frst-order surface satsfes, on average, a drectonal b-modal spectrum. At each step n frequency the spectrum was dscretsed over drecton n equal energy packets. Most of the waves n a random sea are small and obey a lnear model. However, the larger waves that determne the ar-gap problem exhbt hgher, sharper crests assocated wth non-lnearty. We can account for ths by the second-order correcton to the ocean surface gven by Sharma and Dean (1981) η (2) = η (2)+ + η (2), (2) η (2)+ = 1 N N a a j K + j 4 cos (ψ + ψ j ), (3) η (2) = 1 4 N j N a a j K j cos (ψ ψ j ), (4) j where ψ = k x ω t + φ, η (2)+ and η (2) are the second-order sum- and dfference-terms wth ther correspondng kernels K + and K respectvely. The contrbutons of η (2)+ and η (2) to the total water surface elevaton are consdered n ths paper. The frequency components can be transformed nto standardsed (unt-varance, zero mean) varables by dvdng each by ts standard devaton, σ, x = η σ and x = η σ, (5)

SECOND-ORDER CREST STATISTICS OF REALISTIC SEA STATES 3 where the tlde ndcates a Hlbert transform. The Hlbert transform of a varable may be thought of as the varable wth all ts Fourer components advanced by π/2. The jont densty functon of the standardsed varables, {x, x }, s then unt-varance Normal wth no correlaton between the varables. Usng the expressons above, we can wrte the total (frst- plus second-order) surface elevaton as N N N η = σ x + {F j σ σ j x x j + H j σ σ j x x j }, (6) j Settng η to be constant n equaton (6) defnes a hyper-surface (of constant ocean surface elevaton) n the space of {x, x }. We recall that {x, x } have a jont Normal dstrbuton wth zero cross-correlaton. Thus, surfaces of constant probablty densty are concentrc hyper-spheres n the space of the standardsed varables. The probablty densty s hghest at the orgn and falls monotoncally as a functon of dstance from the orgn. Under these crcumstances, the hyper-surface defned by a constant value of η can be treated as a lmt state n a FORM type of analyss. The pont on the lmt state hyper-surface where the dstance to the orgn s shortest and, therefore, the probablty densty greatest s the desgn pont To a good approxmaton ths s the pont where a maxmum n η s most lkely. The dstance, β, from the orgn to the desgn pont provdes an estmate of the probablty, Q, of a maxmum above the value η and, hence, an estmate of the statstcs of crest elevaton ( ) β 2 Q = exp. (7) 2 The ncluson of b-modal sea s acheved by wrtng the combned sea state frequency-drecton spectrum, G(f, θ), as the sum of the swell component and the wnd-sea component G(f, θ) = G swell (f, θ) + G sea (f, θ) (8) In turn, the swell spectrum s wrtten as the product of the frequency varance densty spectrum, S swell (f, θ), and the frequency-dependent drectonal dstrbuton, D(f, θ) G swell (f, θ) = S swell (f)d swell (f, θ), (9) wth a smlar expresson for the wnd-sea component. A log-normal functon s used to descrbe the swell varance densty spectrum, and a JONSWAP spectrum s used to descrbe the wnd-sea varance densty spectrum. The drectonal dstrbutons of Ewans (1998) and Ewans (21) are used to descrbe the drectonal dstrbutons of the wnd-sea and swell respectvely. 3. SIMULATIONS Several smulatons were undertaken to nvestgate the effects of dfferent envronmental condtons wth combned wnd-sea and swell. The wnd-seas were generated from JONSWAP spectra (Hasselmann et al., 1973) wth fetch relatonshps developed by Carter (1982) for the Phllps parameter, α, and peak frequency, f p. In terms of the fetch, x, and constant wnd speed, u, these are gven by α =.662X.2 (1) f p = 2.84X.3 (11) where X = gx/u 2 and f p = f p u/g wth g beng the gravtatonal acceleraton. Based on these equatons, fve fetch-lmted JONSWAP spectra were generated wth dfferent fetch lengths but a constant wnd speed of u = 2m/s. The propertes of all fve fetch-lmted JONSWAP spectra (labelled A to E) are presented n Table 1 and plotted n Fgure 1. All fve wnd-sea spectra were combned wth a lognormal swell of H s = 3m, standard devaton of σ =.15Hz and a varety of peak perods from 5s below to 1s above the peak perod of the wnd-sea. Ths range of swell peak perods was chosen to nvestgate a varety of relatve peak perods and also mnmse the dfference between the peak perods of the wnd-sea and swell. Concernng the latter, the second-order soluton of Sharma and Dean (1981) s known to produce spurous

4 CHRISTOU, TROMANS, VANDERSCHUREN, AND EWANS H s /λ p [-] Spectrum Fetch [km] H s [m] T p Deep Shallow A 2 5.9 9.19.39.42 B 4 7.21 11.31.36.44 C 6 8.83 12.78.35.46 D 8 1.19 13.93.34.48 E 1 11.4 14.89.33.49 TABLE 1. Parameters for the fetch-lmted wnd-sea JONSWAP spectra wth a constant wnd speed of 2m/s (see Fgure 1) results when two components are wdely separated. Ths problem s generally not of great concern f the energy contaned by the two components s small, such as the nteracton between the low- and hgh-frequency tals of the spectrum. However, when the two peaks of the b-modal spectrum themselves are wdely separated, ths causes sgnfcant spurous values that produce physcally unrealstc results. As such, the present smulatons were chosen so as to avod ths problem. Consderng all JONSWAP spectra, the peak perods of swell vary from a mnmum of 4.19s to a maxmum of 19.89s. Whlst the smaller peak perods n ths range are unlkely for swell seas, they are not mpossble and could be generated by short duraton wnds that occur some dstance from the pont of nterest. Every combnaton of wnd-sea and swell was frst constraned to be run as a undrectonal smulaton. The drectonal spreadng of Ewans (1998) and Ewans (21) for the wnd-sea and swell respectvely was then consdered. For the drectonal cases, the angle between the mean drecton of propagaton of the wnd-sea and swell was also vared. Furthermore, the whole group of smulatons was undertaken n deep and shallow water wth depths of d = 2m and 3m respectvely. 4 35 3 2km 4km 6km 8km 1km E(f) [m 2 /Hz] 25 2 15 1 5.5.1.15.2.25.3 f [Hz] FIGURE 1. JONSWAP spectra for seas generated from dfferent fetch lengths.

SECOND-ORDER CREST STATISTICS OF REALISTIC SEA STATES 5 4. DISCUSSION OF RESULTS Consderng the deep water (d = 2m) runs, Fgures 2 and 3 present the undrectonal and drectonal smulatons respectvely. For all JONSWAP spectra these fgures present the total water surface elevaton (η/h s ), the second-order sum-terms (η (2)+ /H s ) and dfference-terms (η (2) /H s ) normalsed by the sgnfcant wave heght. A probablty of exceedance of.1 was chosen, representng a three-hour sea state. These contour plots demonstrate the effect of varyng the relatve peak perod (T p,swell T p,sea ) and relatve mean drecton of propagaton ( θ = ) between the wnd-sea and swell. Focusng on the deep-water, undrectonal smulatons presented n Fgure 2, t s clear that for all cases, as the relatve peak perod decreases the ratos η/h s, η (2)+ /H s and η (2) /H s ncrease. Ths can be explaned by the fact that the steepness of the swell ncreases as T p, and thus λ p, decrease. As a result, the second-order correcton factors are larger. Movng down the rows of sub-plots n Fgure 2 the ratos η/h s, η (2)+ /H s and η (2) /H s decrease as the fetch, and thus the peak perod, of the wnd-sea ncreases. Once agan, ths can be explaned as the shorter fetches produce smaller peak perods, whch result n a steeper wave feld (see column H s /λ p n Table 1), and therefore, these ratos are larger. Furthermore, a pont to notce s that all the dfference terms are negatve, predctng a set-down beneath the extreme wave group as expected. Movng onto the deep-water, drectonal smulatons presented n Fgure 3, for all cases, as the relatve peak perod decreases the rato of η/h s ncreases, once agan due to the ncrease swell steepness. Consderng the relatve mean drecton of propagaton, the largest value of η/h s occurs very close to the co-lnear condton, and the mnmum appears at approxmately θ = 11. As n the undrectonal cases, the steepest wnd-seas produce the largest η/h s ratos. Consderng the sum- and dfference-terms, ther magntude ncreases and decreases respectvely as the relatve mean drecton of propagatve decreases. However, as the sum-terms are approxmately an order of magntude greater than the dfference-terms, the former domnate to produce the largest η/h s ratos when θ s smallest. Moreover, unlke n the deep-water, undrectonal smulatons, there are condtons n whch postve dfference-terms are predcted, yeldng a set-up beneath the extreme wave group. Although these set-up values are small n magntude, they are more apparent n the steeper wnd-sea states when θ > 9. Fgures 4 and 5 present the same contour plots as Fgures 2 and 3, however, for the shallow water (d = 3m) smulatons. The shallow-water, undrectonal smulatons llustrated by Fgure 4 predct the same behavour for the rato η/h s as dscussed for the deep-water equvalent. The sum- and dfference-terms, however, dffer from ther deep-water counterparts. In ths case the locaton of the mnmum value for each wnd-sea spectrum shfts. Ths s due to shallow water effects becomng more sgnfcant as the peak perod of both the wnd-sea and the swell ncrease, and thus the relatve water depth decreases. Consderng the shallow-water, drectonal smulatons presented n Fgure 5, the mnmum value occurs at θ 12 and the maxmum at θ 1. These results are consstent wth Tromans et al. (27) and the latter agrees wth the fndngs of Forrstall (2), who plotted the second-order nteracton kernel for shallow water and llustrated that ts greatest value occurs just off the co-lnear condton. In the shallow water cases, set-ups are also predcted for the combned condtons wth the steepest wnd-sea states. However, once agan, the magntude of the sum-terms s sgnfcantly greater than the dfference-terms, and thus the former domnate. 5. CONCLUSIONS Ths paper has concerned the extreme crest elevaton occurrng wthn a varety of combned undrectonal and drectonal sea-states n deep and shallow water depths. The nonlnearty of the wave feld was consdered up to a second-order approxmaton usng the soluton of Sharma and Dean (1981) and the probablty of exceedance of a gven crest elevaton was calculated usng the response surface method of Tromans and Vanderschuren (24). Furthermore, the role of second-order sum- and dfference-terms n generatng these extremes was examned. The followng conclusons can be drawn from the present study.

6 CHRISTOU, TROMANS, VANDERSCHUREN, AND EWANS (1) In both deep and shallow water, the steepest wnd-seas (produced by the shortest fetch lengths) nteractng wth the steepest swells resulted n the largest value for the crest elevaton to sgnfcant wave heght rato. (2) For drectonal smulatons, the largest crest elevaton to sgnfcant wave heght rato occurred when the wnd-sea and swell were propagatng at a angle of approxmately θ = 5 and 1 to each other n deep and shallow water respectvely. (3) Lkewse, the smallest rato occurred at approxmately θ = 11 and 12 n deep and shallow water respectvely. (4) The dfference-terms from the undrectonal smulatons n both deep and shallow water depths always produced a set-down beneath the extreme wave group. (5) In contrast, the drectonal smulatons dd predct set-up beneath some extreme wave groups, n partcular when the steepest wnd-seas combned wth the steepest swells propagatng at θ > 9. (6) Generally, the second-order sum-terms were sgnfcantly larger than the dfference-terms (up to an order of magntude greater n the deep-water smulatons). Therefore, they domnated the behavour of the total water surface elevaton. Consequently, even when a set-up was predcted by the dfferenceterms, ths had a neglgble effect on the total crest elevaton. ACKNOWLEDGEMENT The authors gratefully acknowledge the support of Shell Internatonal Exploraton and Producton. REFERENCES T. E. Baldock, C. Swan, and P. H. Taylor. A laboratory study of nonlnear surface waves on water. Phlosophcal Transactons of the Royal Socety of London A, 354:649 676, 1996. W. J. D. Bateman, C. Swan, and P. H. Taylor. On the effcent numercal smulaton of drectonally-spread surface water waves. Journal of Computatonal Physcs, 174:277 35, 21. E. Btner-Gregersen and Osten Hagen. Effects of two-peak spectra on wave crest statstcs. In Proceedngs of the 22nd Internatonal Conference on Offshore Mechancs and Artc Engneerng, OMAE, 23. D.J.T. Carter. Predcton of wave heght and perod for a constant wnd velocty usng the jonswap results. Ocean Engneerng, 9(1):17 33, 1982. ISSN 29-818. M. A. Donelan and A. K. Magnusson. The role of meteorologcal focusng n generatng rogue wave condtons. In Proceedngs of Aha Hulkoa Hawaan Wnter Workshop, Unversty of Hawa at Manoa., 25. K. C. Ewans. Drectonal spreadng n ocean swell. In Proceedngs of the 4th Internatonal Symposum Waves, 21. K. C. Ewans. Observatons of the drectonal spectrum of fetch-lmted waves. Journal of Physcal Oceanography, 28(3):495 512, 1998. G.Z. Forrstall. Wave crest dstrbutons: Observatons and second order theory. Journal of Physcal Oceanography., 3:1931 1943, 2. R. Gbson and C. Swan. The evoluton of large ocean waves: the role of local and rapd spectral changes. Phlosophcal Transactons of the Royal Socety of London A., 463:21 48, 27. do: 1.198/rspa.26. 1729. R. S. Gbson, C. Swan, and P. S. Tromans. Fully nonlnear statstcs of wave crest elevaton calculated usng a Spectral Response Surface method: Applcaton to undrectonal sea states. Journal of Physcal Oceanography, 37:3 15, 27. K. Hasselmann, T. P. Barnet, E. Bouws, H. Carlson, D. E Cartwrght, K. Enke, K. Ewng, J. A. Ewng, H. Genpp, D. E. Hasselmann, P. Kruseman, A. Meerburg, P. Müller, D. J. Olbes, K. Rchter, K. Sell, and H. Walden. Measurments of Wnd-Wave Growth and Swell Decay durng the Jont North Sea Wave Project (JONSWAP). Deutsches Hydrographsches Insttut, 1973.

SECOND-ORDER CREST STATISTICS OF REALISTIC SEA STATES 7 S. Haver and O. J. Anderson. Freak waves rare realzatons of a typcal populaton or typcal realzatons of a rare populaton? In Proceedngs of 1th ISOPE Conference., 2. J. N. Sharma and R. G. Dean. Second-order drectonal seas and assocated wave forces. Socety of Petroleum Engneerng Journal, 4:129 14, 1981. P. H. Taylor, T. A. A. Adcock, A. G. L. Borthwck, D. A. G. Walker, and Y. Yao. The nature of the Draupner gant wave of 1 st January 1995 and the assocated sea-state, and how to estmate drectonal spreadng from an Euleran surface elevaton tme hstory. In Proceedngs of the 9th Internatonal Workshop on Wave Hndastng and Forecastng, 26. A. Toffol, M. Onorato, and J. Monbalu. Wave statstcs n unmodal and bmodal seas from a second-order model. European Journal of Mechancs B - Fluds, 25(5):649 661, 26. A. Toffol, M. Onorato, A. V. Babann, E. Btner-Gregersen, A. R. Osborne, and J. Monbalu. Second-order theory and setup n surface gravty waves: A comparson wth expermental data. Journal of Physcal Oceanography, 37(11):2726 2739, 27. P. Tromans and L. Vanderschuren. A Spectral Response Surface method for calculatng crest elevaton statstcs. Journal of Offshore Mechancs and Arctc Engneerng, 126:51 53, 24. P. Tromans, L. Vanderschuren, and K. C. Ewans. Second order statstcs of hgh waves n wnd sea and swell. In Proceedngs of the 26th Internatonal Conference on Offshore Mechancs and Artc Engneerng, OMAE, 27.

8 CHRISTOU, TROMANS, VANDERSCHUREN, AND EWANS Second order η / H s 1.28 (a) 1.26 1.24 1.22 1.2 1.18.35 (b).3.25 Sum terms / Hs.2 Dfference terms / Hs.6 (c).8.1.12 1.14 (d) 1.13 1.12 1.11 1.95 (g) 1.9 1.85 1.8 1.74 (j) 1.72 1.7 1.68 1.66 1.64 1.62 (m) 1.6 1.58 1.56 1.54.21 (e).2.19.18.17.18 (h).17.16 (k).165.16.155.16.155 (n).15.5 (f).55.6.65.46 ().48.5.52.44 (l).46.48 (o).43.44.45 FIGURE 2. Undrectonal, deep-water smulatons (d = 2m). (a), (b) and (c) JONSWAP spectra A; (d), (e) and (f) JONSWAP spectra B; (g), (h) and () JONSWAP spectra C; (j), (k) and (l) JONSWAP spectra D; (m), (n) and (o) JONSWAP spectra E. All combned wth a swell of H s = 3m and σ =.15Hz.

SECOND-ORDER CREST STATISTICS OF REALISTIC SEA STATES 9 18 (a) 9 1.241 1.2222 Second order η / H s 1.235 1.1847 18 (b) 9.15 Sum terms / Hs.1 18 9 (c) Dfference terms / Hs.6.4.2.2 18 (d) 9 1.1121 1.158 1.995 18 (e) 9.9.13.9.1.11.12 18 (f) 9.2.1.1.2 18 (g) 9 1.769 1.733 1.697 18 (h) 9.12.115.95.1.15.11 18 () 9.1.5.5.1.15 18 (j) 9 1.566 1.59 1.541 18 (k) 9.11.95.1.15 18 (l) 9.5.5.1.15 18 (m) 9 1.462 1.48 1.444 18 (n) 9.11.95.1.15 18 9 (o).5.1.15 FIGURE 3. Drectonal, deep-water smulatons (d = 2m). (a), (b) and (c) JONSWAP spectra A; (d), (e) and (f) JONSWAP spectra B; (g), (h) and () JONSWAP spectra C; (j), (k) and (l) JONSWAP spectra D; (m), (n) and (o) JONSWAP spectra E. All combned wth a swell of H s = 3m and σ =.15Hz.

1 CHRISTOU, TROMANS, VANDERSCHUREN, AND EWANS Second order η / H s 1.28 (a) 1.26 1.24 1.22 1.2.4 (b).35.3 Sum terms / Hs.12 (c).14.16 Dfference terms / Hs.18 1.15 (d) 1.14 1.13.38 (e).36.34.18 (f).19.2.21 1.12 (g) 1.115 1.11 1.16 1.14 1.18 (j) 1.12 1.14 1.12 1.1 (m) (h).245 ().44.25.255.42.26.265.4 (k).52.5.48.61.6.59.58.57 (n).35 (l).31.315.32.365 (o).37.375 FIGURE 4. Undrectonal, shallow-water smulatons (d = 3m). (a), (b) and (c) JON- SWAP spectra A; (d), (e) and (f) JONSWAP spectra B; (g), (h) and () JONSWAP spectra C; (j), (k) and (l) JONSWAP spectra D; (m), (n) and (o) JONSWAP spectra E. All combned wth a swell of H s = 3m and σ =.15Hz.

SECOND-ORDER CREST STATISTICS OF REALISTIC SEA STATES 11 18 (a) 9 1.253 Second order η / H s 1.2345 1.2186 18 (b) 9 Sum terms / Hs.15.2 18 9 (c) Dfference terms / Hs.5.5 18 (d) 9 1.1814 1.1675 1.1721 1.1767 18 (e) 9.2.22.24.26 18 (f) 9.2.4.6.8 18 (g) 9 1.188 1.188 1.1844 1.1881 1.1918 1.1955 18 (h) 9.26.27.28.29.3.31.32.33 18 () 9.4.5.6.7.8.9.1 18 (j) 9 1.2111 1.2148 1.2185 1.2222 1.2259 18 (k) 9.33.34.35.36.37.38.39.4 18 (l) 9.6.7.8.9.1.11 18 (m) 9 1.2492 1.2529 1.2567 1.264 1.2642 18 (n) 9.4.41.42.43.44.45.46 18 9 (o).9.1.11.12.13 FIGURE 5. Drectonal, shallow-water smulatons (d = 3m). (a), (b) and (c) JONSWAP spectra A; (d), (e) and (f) JONSWAP spectra B; (g), (h) and () JONSWAP spectra C; (j), (k) and (l) JONSWAP spectra D; (m), (n) and (o) JONSWAP spectra E. All combned wth a swell of H s = 3m and σ =.15Hz.