NEW YORK UNIVERSITY College of Engineering RESEARCH DIVISION University Heights, New York 53, N. Y. Department of Meteorology and Oceanography MODELS OF RANDOM SEAS BASED ON THE LAGRANGIAN EQUATIONS OF MOTION by Willard J. Pierson, Jr. ‘ Technical Report Prepared for the Office of Naval Research ee Lil under contract PF Nonr-285(03) (Gb : @ ThSEhOO TOEO O UMA AN IOHM/18lN) NEW YORK UNIVERSITY COLLEGE OF ENGINEERING RESEARCH DIVISION Department of Meteorology and Oceanography Models of Random Seas Based on the Lagrangian Equations of Motion by Willard J. Pierson, Jr. Technical Report Prepared for the Office of Naval Research under contract Nonr -285(03) April 1961 Page 10, Page 20, Page 21, Page a2, Models of Random Seas Based on the Lagrangian Equations of Motion eqn (25) eqn (38) eqn (43) eqn (47) eqn (48) eqn (53) line l ERRATA for Zoe read Zo tt second integral, read exp( mea/2) for k read K E(2°) = -34, 4, for ¥, read ye E(2!)> = -2b,° The skewness is not zero, and vedd wo hraget sane ~ gopkboh te oiedas i ATARAR ir. hast 1s oe (25) npe {OF oyek is\"s- jqxo bast lntgetth Baoowe (8° i) ape ,0S at 4 bpee # xok = LER) ope AS oget B, ee (5)2 (te) aps ¥ beg Sexo! (68) ape | | Basie Fiteya 2) nga ve aes : aolt: hae jorss ton at apnewede oft i i ‘ ay Aol, r i oa Esa rs me oe i Ser ay eee ao ee ee ee niet lla “> a eh tal > ene 4, cca Introduction The present state of theoretical knowledge of wind generated gravity waves, when studied as a random process in nature, is based ona short crested Gaussian sea surface obtained by linearizing the Eulerian equations of motion (Pierson [1952], [1955]; Longuet-Higgins [1957]), on a second order extension of the Eulerian equations for long crested waves in which the underlying linear waves are Gaussian (Tick [1959]), and on various results involving wave spectra that do not specify the probability structure of the waves, some of which involve nonlinear properties of the waves (Phillips [1958], [1961]). The short crested Gaussian model has proved fairly useful in explaining the angular spreading and dispersion of swell, the marked variation in height from wave to wave in a storm sea, wave refraction, bottom pressure fluctuations caused by waves passing overhead, and the motions of ships in waves (St. Denis and Pierson [1953], Lewis [1955], Cartwright and Rydill [1956]). These applications are dependent on either the gross features of the waves or on natural processes that appear to linearize the system even further. For example, the wave profiles are ina sense averaged over the length and beam of a vessel at sea, and the nonlinear effects are reduced. The higher harmonics ina nonlinear profile and the high fre- quency linear components are also attenuated with depth in such a way that the application of Gaussian noise theory to the zero crossings of bottom pressure fluctuations leads to useful results (Ehrenfeld, Good- man, et al [1958]). The spectrum of the time process at a fixed point has been most frequently studied. Since the actual wave is recorded, more or less, the full nonlinear motion is recorded, and the spectrum of this nonlinear motion is estimated. The accuracy of computations in a linear theory based on spectra estimated from a nonlinear record is open to question. In particular, higher moments of the spectra are quite open to question. Theoretical developments in the study of the short crested Gaussian sea surface require numerous higher moments of the linear part # ALOney dir - y ai rca eal pai in tate ae rss, Ral i - | PD eB ” int a 40% “ ny a. anys aden a’, ray weikue: 5 bse va role i pada ity thins eet ) oat mauler le « yon eel desis bls ‘pc borkeaa bidet gy 17 eh! eam taal p2eer . pani \ ee Od rOrey tes i ~~ bultane gabe a. 4 jiekpa patie poe CROP pT ste ' mT) a. rig) hat a yi . pe bic OSU AE? ah ipeu ar aes 4 wea att A abi ah} dioastey inh eat Eis eo anh ya wane #5; ah rae ane ee ‘in grrr feng fii a Leet anh Ti ells et oad runaeeve we Pape oc Sula ie Shrie save wht te PU ass . . ae teach) .obat) peween ; | dayt St ha ot 7) y ee lot to ADSM s levees " funda eh de i _ Lithia el ews vs qsge lS Uae yethertiqw edigis ad? galalelqee Ae AGITSATIOT Gv . eer ret. 7) OCH OF BVI. 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(OBO LaDy ic oy Hine SwuaB a 49 Shale | bsibirr i raditp ont rT wit) bua’ Ao? bo rurtioeige any Dale elem wey 3 Heine vie: ary iisre ‘pia: wilt Pros Seat) so eri ei ew Ay seni a oa) bewonties at agoHm NOPeaUy Ws ASI ey Pomtery - Aas On inf? Bowe HL Pee HEIDI: aie hooad = anita ot Tag wane we aipoeg toed! be. sina: me ‘ine i tee ed al j ss y PAE: Ate DY) aa tesraseon WR) HD rey okt apa eons , a | aia vu ste =2i= of the process. If computed spectra are used to estimate these moments, many of the higher moments are obscured by high frequency white noise in the estimate or made questionable by the high frequency nonlinear part of the spectrum so that the results predicted from them are doubtful. If the various theoretical forms for the spectra that have been proposed are used, the fourth or fifth moment of the frequency spectrum becomes questionable and the meaning of the second or third moment in the wave number spectrum is obscure. These problems are further complicated by the fact that sea surface slopes depend on the very high frequency capillary waves. With capillary waves, breakers, whitecaps and turbulence in the upper layers of the water, any results on curvature, as they depend on the higher moments of the spectrurn, are highly doubtful for a storm sea. The actual sea surface has gross features that are not in accord with the short crested Gaussian model. No artist, for example, depicts wind waves as irregular sinusoidal waves. Wave profiles along a line as a function of distance show sharp crests and long shallow troughs that do not occur in the model. This point will be discussed further in a later part of this paper. Purpose of paper The purpose of this paper is to derive three new random sea - way models that appear to have some properties of actual seas not illus - trated by previous models, to derive a few of these properties and discuss other possible properties, and to show how these models appear to be more realistic by comparing them with selected observations. These new models explain some of the difficulties that arise with the higher spectral moments. They may also be capable of giving some information about whitecaps, breaking waves, and sharp crested waves. Since all mathe - matical models of nature fail in one way or the other when the conditions and assumptions in the derivation are not met, these models will provide a second choice when observations of waves are to be compared with theo- retical results. Some questions are raised that can only be answered by additional theoretical work and by precise measurements of waves. 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Whee ees du BGITRSOTI SCION oval « Se a tial alee over acre Wavoash has bert? s eters coal to wet Hover cet of allay aye ore id tihng . v ? ae of thwce ais aes awell wuhe OF Dae, SatrRO sg | akdys mare] eesti \ A : i won veel) Aijorlavsosdo DeJSATSR Gl) fan) garters ya aihetlegy aha, en tsar of aa ead oleic OS24 4 Delt Hal sth} BS wid b> Gian A Lajday .% efsbort ioe walgenrianid ones potylp tn Aadiqges &- oale phar ye wt ee iwi (ih Seine) oN e Bates > Gy roa bee anv ( guidgorl ph > ie wedi: 3 ob pot wee ser: ew mew a ligl eau 3 htt Pell nlobsier Lao tine,’ : x shires ; tLiw ato ‘ie aged: fond ic fe Sts Ro avin Sih mi anes Ney ees Yeas bre - oond af big Banh BRITON? Gi G3: STA BD Be Ts REGEN S vada hal Peres bee bia > o8 a: a la 317 Ae cee TH 8 hrsine se finer aahaee oe, ip Ores Anco fealsenha ‘Comtibba ioe vel baie wad se! Me th Gs neds beatae +9 avohenigy arvind. ailvact teclieg 1350 167 io ih t da ant vii bi ) " & + eee aad adobe ie oud eh ne livencta (abate asdf gore properties of those models described above in order to design more definitive experiments concerning the nature of wind generated waves. The Lagrangian equations Let a, B, and 5 be the x, y, and z coordinates of a particle of fluid. In the Lagrangian system of equations, a solution to the equations consists of finding the positions x, y, and z of all of the particles in the fluid as a function of time and the initial positions of the particles, a, B, and 6. The Lagrangian equations, according to Lamb [1932] are given by equations (1) where subscripts denote partial differentiation. mati ales igs 8 CIES oe EU = © (1) nee Sieg (Arg 8 IA lf) = =n0) XeXs * YeeVs + (Ay, + B%5 + Pg /P = The equation of continuity is most conveniently expressed by (2) da aty.2) | — 0 dt | 8(a, 8, 6) Such solutions need not be irrotational, but, if a function, F(a, 8, 5,t) , can be found such that (@)i dh = (x, x, +r Yt May 2 z,)da + Coan Ve Yat Z, aay eee (x, X¢ PMs V5 +r (z, z~)dé is a perfect differential, there is no vorticity. A zero order solution to these equations is given by x =a yaar (4) = Po=p, 7 ge 8 in which all fluid particles are at rest in hydrostatic equilibrium under the force of gravity. We expand, following the concepts described by Stoker [1957]; about the zero order solution in terms of a small parameter,é, as in (5) in which Fe is a constant. Here we think of e as equalto ak, ab i we wl LN eae | ” f fine Le ad a viele sie) , . | ; ee ee o payee rs ogee my ae TAR } i, Binge salynatgact ad ; alatavac bo Ub = ee oe ey ee ‘ ii Ps e tend! ee G AL file GS, Ge ye Ade. era fa aK Ahan So jor fc i Ait Ly ‘gat i’ tle vs € ain Sy i ane twee tr! yuiba i) Stele Oy fe) BAe es mo te atic tiga iy) Ort See oi Hetaliaers hub tsinyvan « ’ }) ¢j aik 1 U J \ n , 4 ; a qt\s - : ; ¥ i a% ‘le mohioan? a vee or > i c = ) j pox * ae hia Oo: ‘bale ab sigeamdne ssade (ft) nsatiaupe gain ie ase areesi 7 ic. ri ao I nit Te in att Ata 7 9 yy : 7” rm Os aviareuks iO. 10 notayge nebdde fae frat wp TE od male (3A a . Yolen a a! Seed yt ys GF afl ial De mi noltuloes v abi a¥o3" A . m - m y 7 - be 08g o mils rat i prod hive, Orie eb yi ey aay i =the ai ditzad isi; a ila diac he dit i, oe crs 7 tes ‘yea a) rie i redo uy Oatisane aire 2ao% avis griwet| rei os a. / f rl he MOREL te yao Bi i eis tenes Bac ae iuke Wnt aM vn tl weds ae . ‘Aw: 6 leope ae 7 hy Aovield ‘ag, ah nieteao - ua iat iE fey. y i i but the first order terms must have the dimensions of a length, and so a = ¢/k is used in the various solutions that are obtained. The parameter € never appears explicitly. See, for example, Longuet-Higgins [1953], and Pierson and Fife [1961] for different ways that e can be interpreted. (5) (6) (7) ne 72 x=atex, te Xo (4 Yah cheery iia Yo 2 z=6+tez +e Z5 2 PiSPR ae PO) cD iece Po) 2 eS Elance amcce F, The equations in ¢ become | (>) cee teeta Pig ® = | oO Viena Sie Sip! © = ane, ig yale w ar + 2) = (0) lat’ Yipt bt dF) = x), da + ¥ 148° + Zz), 06 The equations in ¢ become Fe EZ et Poe O aeee er a me tp que St tt ola Yatt + 8228p * Pog/P = —*i1te*18 ~ Yite Yip ~ Zitt71p Zou, +t B26 + Pog! P = - X1e¢*15 7 ite 16 ~ Z1tt 216 Palm Zeb nec on iGealian sslamlors Uipic low tlauls Gk ma | *ot = ZeOila yap lalate aes + + PeSeana Uieeia” Sie ieleney Dine sss) 5 + (25, + Z1,Y16 * YieYis © 1 215)% are ni sieei6 ae PLA eat . a be 4 TseecSprves « lari + ‘Mer .@ Tae ryt OEE ® 0% mona Loven * : ie a ands a awe Pew wohhrh iy (Oey et hat mt elt Bala ed, ‘ ; it af é PLT TAAA, i aS MOG AES er Ee i a 2s oo optim Jeet dn ied hudey vel, Hn hme chy f Lat RTE Aa 4 iy, ir Dawe at at \> +e 1 A ; un ge? 2 Oe@ v ° 7 > f aaa att uF a | o ' al i i\ . * — At \ a) . 7 i= 4 Th f ; CD ae ; t \. . ; val a aan i | v4 Oi n™ iw ¢ y' s ‘ t T ' i = / 4? } ai* r ‘ ar fy ha ’ r ' ; ‘ c - i Agi Y pa i! Th, : ; ' aet ‘y r at mt my ; AS ‘ ) Py j 4 ihe is Equations (6) are linear. A solution of them in turn determines the right hand side of (7). Since the left hand side of (7) is linear, in principle at least, a solution can be found. Also in principle (though not in practice), it is possible to proceed to as high an order as desired by this procedure. The Gerstner wave A solution to equations (6), applicable to waves in deep water, when added to the zero order solution, yields equation (8). ak ; i Tice sq 5S (el, ap Ex a Coane sin(k,a + kB - wt) ak : ! 7 aie 3 Vea aiee ys) am ste © sin(k,a + kB wt) k6 (8) z=6+ez,=6tae cos(k,a - k,f - ot) P=p +P, =P,- gpd aw k4 =—e FY Sis sin(k,a + k,B - wt) In order to impose the condition that p(a, B, 6) = P, at © 0); ake is required that 2 Ww — (9) wes The continuity equation imposes the condition that 2 2 The first order solution has no vorticity to first order as FY has (10) Kk +k been found. An alternate form for the solution given by (8) is (ll). 2 2 x =a-acos@e” 78 sin( S (a cos 6+ 6 sin 6) - at) 26) we (11) y=Bp-a sin Oe” 8 Sin te cos 6 + & sin 6) - at) 2 2 z=&+tae’ 5/8 cos( —— (a cos 8+ B sin®) -at) : | | | - Le: j i ede x in anes ey! went Gi un act ee ly olay Bynes, fie uel ans, mites « énotlbupeh a sloponciti me) Siamsieh eh (7) Ay her dinar, fist rity wotihee weit LAs obik ‘bevaut, day 7 Ea atTIAYG Al dood dggrcut)) Hg ORAS ai vali heat iret, ods en agitulos &. tmibed i, wtabaoo ty whine vel tel a oe wa rete: ine Wid wee il basaerc, ew aldiesog af) a: . | ; ¥ gouw 74tta 108) za a) non om awe ty ial AP Gphw SI shen 63) Cyakie® mS voit ulde Bee ss — ae - ao nda) nelinvod whitahy DR ae tibet axbe, ot of bobba cee) Aig i) A, N boty += mi *p=¥ i 7 a , ‘2 ne ii «'? - . Me aime bo] or ci ” qj = ,23 +9 = % {ha a = 2h 5 fy Fe cat de € ' Ray - ; pa? * oun = 4 s - se oe “oe Ng ote a oo ; qf 40:28:98 Je (8.6 iq tant api Ebinos ad) sagdeur on webs 3 al 5 5 u . . | tert Peslegas m _ . oI al d $= | fun? coltibeays arts g9 ebgir! soleus wiivalaaas 9 aT y s Pe) s j ws so ous 7 ¥ 1 1 Dai 7 t PTT, {2 de xebze texdt of YIP DIDTO Q BSH MORI) loa rebitg ‘text wat - . . seco aaa po OEP et fAbgd swig hoettiion ai! aigl moat sraniaste ah i i i , . A ne i t we oe Die aoe of peste \ale ? > 68 Hon am we on r ¥ §. iat ~ (8 ade a + he abo x) =a wkte cos 6, Yo = Pag sin® , and Zo =0, equations (1), (2), and (3) are still satisfied to second order but third order complications arise. These two terms are the equivalent of the s econd order current found in irrotational waves in the Eulerian system of equations as described in Lamb. Extension of these results to finite depth would probably follow the results of Biesel [1952] who achieved a remarkable representation for periodic breaking waves by means of the Lagrangian equations (see also Pierson [1955]).* A randomized short crested model The expressions for X),y),and zy in either (8) or (11) are solutions of equations (6). These equations are linear. Therefore, a sum, over dif- ferent parameter values, of solutions of the form of XY and zy will also be a solution. If it is assumed that each particle has a displacement from its rest position, ao: Bo 6 _, that is described by a stationary Gaussian fo) vector process with specified coherency relationships, a solution to equations (1) that is correct to first order is given by equations (13). co oT 2 5) 2 x=a- f fe® ° cos@ sin(=—(a cos 6+ B sind(- at + €(w,0)) J25(w, 6) d6@ dw fo) = Ti eu Oo 6) ee (13) y=6- 4 fe 8 sind Sia ae cos0+B sinO) - wt + €(w,6)) V2S(w, 8) dO dw re) -T T 25 / 2 z=6+ if (ee 8 cos Cs (acos0+fsin@) -wt + e(w,6)) V2S(w, 6) dé dw Oo -T *The remarks concerning the Gerstner wave are in error as these results six years later show. ey Re soley fy eaugaih ‘ Hg 2 Wie ini bet hs age) ini ‘goedT : pa Bi Pi gh: 5 ie a ora i “Bins Wi abe ee shia emt oh) Ne dpaietsd: wyld ol hav ionteea: ‘wrk wpe rel hpon t Pa Dv ewe ot: \pspdivw Bed ew fokteaeib ot wvittees us ‘eh ta wile Tee ae nt feed bew'ir bales ye ae onde: Aled alhopaeg aly Lael it reliyow THES bntasa esas €b, fut wikd vais qvciite wd as5 4 we ¥¢ qo ly: osu aang an 6s ai sbeyetess es tabs baosem r " Sai re. 8s CAS. Ae i i ie sane Aha Py Bien Sis i =, ae ehh ¥ Aaah nynegnue aol Hid anon wadtonA Atle. Bis {t). do A ae) ols nupSs Ou ss Gas. Bris.” OAS. fia” Be Py ‘owl baw) seine ane ‘Hesllgrtoan tsbte bility JU. Poe | handoe og beiteitem a Aneel! Batons 2 be ued dew tird cab bueses sd? to Mentastups wit ore aan . inat a) hodiroeah wi ano Leip 0 rads ven of Fear nt aie nM ‘acl wotiat vidude te blinww dvaob stad OF buney peods te cotnastadt | | #1) Heiseineamteeod sided: sot a ‘bore. ioe odw flee i } isaalal te edlows outa ses) siplinuys cabguasyget aha io gcdore Yet BovaW, ‘pakkawsd ‘pbetieang i *; (ec ana ; Loire botser> tone bos : re one hase meromraien Line ray (enna low ais “i1) “er (G\ nerivie AY Ue | Binks uf" hae ro) snolenorqae ont, ‘TB sev0l aoe é sWrolot8dT hen @ys ecoflerps sesnt ahs) soot aaspe Me onla lw pe BRe Pi eo orto edt lp anoles te .woylay: reternaisa aren ek: a non jpamocalgalh: * wad ploiisaq apne tua bemupes eit nu snoltutog ®: anbedusD yrstoiiaie & ‘yet bs divseud ©) tactd ‘ > te, an aalnog 249% att ‘endltaupe Oo) noltulee e ,eqidenqiialeg yoasssdop ReAighes dliw a6oD07H volo ‘ it * nm ‘ 7 : y 4O}) atoteupe vd e ovin wi TATE reiki oi ieate> at dantg ay a : ra cant oy taiky 9% G,aes\ ({6 woke + % inant! | sas ohecr\n cola a ; aL y rr ie . . ge ce ee ee 7 a. hen Boles 640 a)8 * Pao a {-n “vos A : 4 : i oe re u\a” r ‘< \ ee Thee 8 AC ecole the = (pase 4 FAedn 9) << sos a P| 4 Nich 2 ne vi i i 7 i v Bi : ‘ 7 ‘ : Sear ier rnsemreors aaa oeh eilire’ Sey bad Be TOTEN pt 48 eo Bw rerdeted ea yethecodite ‘etienay! ae a ar, : et avons trial bing asec ou The notation here is that used by Pierson (1952, 1955) except S(w, 6) is the resolution of the variance of the particle motions into fre - quency and direction. For some fixed particle, say, one at the surface given by a = 0, 6 =0, 6 =0, the motions x = x(t), y = y(t), and z = z(t) form a stationary vector Gaussian process such that the spectrum of x(t) 1S JS(w, @)(cos6)“do , the spectrum of y(t) is [S(w, )(sine)“ae , the spectrum of z(t) is [S(w, 6) de , the cospectrum of x(t) y(t) is [S(w, 0) cos® sin@ dé , the quadrature spectrum of x(t) z(t) is [S(w, 8) cos dO, the quadrature spectrum of y(t) z(t) is [S(w, 6) sin® de , and all other cross spectra are zero. Other notations are often useful. For example, a notation similar to that of Longuet-Higgins [1957] would yield (14) as the complete equi- valent of (13). xX=a-Za EE IS Seal Charles (omy tear) many Kan m n mn mn Ss k6 (14) y=B-2 2s Si sin(k a + kp - Oy ea tf coe) ké Ze OM eae! Or Comix. Garis wow. ea al 1) mn m n mn mn In (14), added conditions are that 2 2 2 (ne) Lay Sa = Kinn and that “aati (tke 1 Aaet) sso yd ae ie 4 sik biatdet sod | saat hut) Beerebs Yeni at cater ost wit te a | aD cnaBtowes eal bi (6 erst SHE TR AO THR Ais teay eae eure aa% . ialiiore vib he tie o's ban lee # Oar 8 oe keeobincse aie hee ee le, D di: bes mej tle Wald, aad dave snore sete oual): my pemraiate a % gn fonns)(t ore | . _ tay Xs, cet ) . WS Soi Ke ‘ae : us (2) tq Lot 9 ga an (s ae, ra fix le To" mate ; ; ys nie Beod (a ole, oe = oye (ihe Ber ser 135096 eat bas Bh Dano (8 aye) : froih fija ay 46 cou eq e eure . | Bb Bain, (0 wel | ; Tee wie erioege aWeien sotteril telirnin molsian = Ane tot tulsa uabio sh acoiTernn rah2Q at re > bros aiolgmie > ‘od! ae (4) Bisty blo: 1e trees) ier. Void | ‘ oe ; tet): ta 8 . oe ntl tg 7 6 , wi, Abad tes ag mo ' a : le j ‘on . 4, a : Jonaet Vea ant 8 4 %, ,, _> r hs : a a o. | it )n03 Sg iro + hoe - turks et Rao ttbinie boleh) adel) a ‘e ; e fun iq. ret = 2 Ww (16) mil 2 /je oe g Vim n The free surface Those particles on the free surface are described by the condition 6 = 0 and (13) becomes co oT 2 x= x(aspst)imiai— 4} fsin(=-(a cos0+B sin@)-wtt+e)cos@ V2S(w, 0) dO dw O -7 oo 6 2 (i?) ya= yilas8st)= 16 ' — fh Jsin(—a cos0+ 8 sin@) -wt + €) sind V2S(w, 0) d@ dw Oo -T co oT 2 Z= 2 (asBet)— el J cos(— (acos@ + Bsin6) - wt + e) rf 2S(w, 6) dO dw OT, In principle, and perhaps admitting triple values for the inverses over limited regions in the x, y,t space, x = x(a,8,t) and y = y(a, 8, t) imply inverses of the form a=a(x,y,t) and B so that the free surface can be found from (18) B(x, y, t) (19) PF AGM (oe S418), [Rey e)) = FACS S18) o The surface so defined is certainly not the equivalent of the short crested Gaussian sea surface. The short crested Gaussian sea surface is equivalent to this representation when the amplitude of the particle motions becomes so small that a = x and B = y are satisfactory approxi- mations to the inverses given by (18). Otherwise, further study suggests that z = 2(x, y,t) has many features of actual waves in that the crests can be quite sharply pointed and in that the higher nonlinear harmonics that occur in the higher order derivations in the Eulerian system are already present in this model. It is believed that the problems that arise in the adequate probabi- listic description of the surface defined by (19) in terms of (17) and (18) will be very difficult to solye and that considerable effort will be required. All of the problems that arise in the study of Gaussian noise and in the study of the short crested Gaussian sea surface have their analogue in this pe ‘ ’ ©, : : i . a , manmnmemt alam ; Y hh, % et ae’ i \" rrr, hele 1) e) ? an” ry a | \ De 4 ‘we tyua sorts AT iden 4 iaegn olive Rd) a tok whe sosisue asi! adt coa5e aa omsa't | | ee Goad’ ei) ee 0 CEE, tc eo a. ae) ee a. hOB 1 HS Wao y+ 1 -(Onte Gt Hage A) bmanY of 9 (2.8.0) | | - . . : a? ea BONG WES Gale (s + kis (Waia® *Ove2 ole wie (> §- (toe Te . | sa mm 2 —— j : § ; ec we &h (0 w)eS¥ (> - ha (One® © Yeone): 5 ec 1 l-,4ee8 beupnvn! aHy Tal anutay etqit1 qutt safes & sascieeg Bs .siqhoniag al (t,Q,0)¢ 2 y Daw (1,3 oe] =) ome t. 9 edt al mrolges basen ; neat mtd Ags weet oval ¥! ap hon (ayy anise * ff 1 gy eyes ij (. ft Binds av oa? seine each wht sai fy vy wiles. £97 ¥ me 49.N 2)ejs> 8 4 teste adic taelavives.odtijon yiniatse> 4) benlie® o¢ scadusit adT wa sttiss eee mutea an) baryveato Prone oT eos aoe cane ‘heise alisiriga adil Wo obutiiqais act amiw Eoifaineee lade elif of Taal avingng ixprdqa vritcsiaitar 216 ¥ @ Bne w= & Jat) Ltqire se wecsenad. enol! a tea? efengace yiuidale sitrod /6eiwpeceiie)) et nevi iy paetavnd ‘ade at anet Sel ruscad ooze oct $64) ch aoe ow deblas do, wet uTeMt Ykie wad. (Sy, ee 299% tad Agindavied 72 ania ahdgid dt yorlt nb Sod Setnkoa qiqvaite Vie dete eeceleye ve cohen with at AAO % ‘ob HAYS riihg kit outa _ : | - Aebout oh ap sinipobe ser wi.otiys tant eiryltorg a3 sand? bovoitoal a) at), Fit D eo ora yias r badges os a 1), ers 9) Lydd einer a 121 “ed barwthnls ‘sat sit wild ts ici hl ll aired han a Aaah ed Bw Palle pian abs ianec tan} ile ayioe ot HuaGiee'yeav od Uilew onthe Abie selon HaiMdend to ybete et at oben Luni ombidory fst te HA: side ch Sugelend. clot syed. suntque ope, oe oni a audi) bareatn rowte watt bs vbuite ( model, Second order effects in the short crested model For reasons to be discussed later, it may be necessary to proceed to second order in this model and determine the effects of equations (7) on the wavy surface especially if breaking waves are to be studied. The long crested linear model If S(w, 8) becomes concentrated at the angle 9 = 0 and degenerates to a function of w only, equations (13) become a 2 72 xi= x(a5.0, t)) = a finer 8.8 sin(~—a - wt + €) N2S(w) dw fo) co 62 2 Z =eziq,O,1t)i= sont fe° §/g cos(=—a - wt + €) V2S(w) dw fo) An alternate notation is given by k_6 x=a-Za e ™ sin(k a-wtte_) n n n n (21) k 6 z=6-Za e " cos(k a-wtte ) n n n n For 6 equal to zero, the inverse of the first equation is (22) oh et (oll (54,1) This implies a free surface given by (23) Zaz (ascetic) az (cent) Second order effects in the long crested model The long crested model is more amenable to an investigation of second order effects because it is simpler. We consider the problem of two waves, and then generalize to the randomized process. For two waves the linear solution is given by nm Mig iin sy Pion! aoatle | ib wet cilia Laren: RE a - a. at * nner oo seer sit we we ita eorinw Staind ‘i “eitaea» saul? ire _ 4 ee * : : : - ; . vn ‘tabgip ie pil aes esuneneges bie > W atxne at's a bare etioe wren! epray a td (0 bie | . arnuned (24). aan ss! ‘alr i ee mos) 7 ie $ 2, 2 oe vimentin ay Bw es a 8 a Webeon eet -_ DD estes eee eine Mel Wy anht 8 alee re) ‘qe eRe. 5 a ee Pee “ws (aha ote ye ~# ayer 8 wL ide (2,2 aie 78 ud way is, ai coitadon | 4 = % 4 b 1 = OA o #f- 07% a: ‘ Te —_ , "at oa : — a (atiw-o fines. # aot-tee 7 rT] it ft oe hoe et ‘ at-nelisap® 27H. ott lo seni) Od pO LHe ot laupea 8% eos 2 (Smee (9 (4 ,e)D HH . Sobor be jtasn5 Bol moe ni ws atie 4 tobi! ba he mathe fantnl Ak oF Gide snie stom: Bt Lobait beiaaia gio! ai att ic “i reatdour <9 x abinnos 3W waltzes be et: ee sivaced gieity vance re Reva Owe vod ihasgosg he ¢imobast ont oF ‘peiles on ands bee nae % NOs k,6 k,6 x) 77a ,e sin(ka - wa + «) -a,e sin(k,a - wot + €5) k, 6 k,6 (24) Zz) =a,e cos(kja-w t+e))+a,e cos(k,a - wt + €,) Pp, = 0 ao) k,6 AW k,6 Fy - K e sin(k,a - wt + €)+ iS e sin(k,a- watt t,) It is irrotational to first order. For simplicity, we assume that w5 > w)- The second order equations to be solved are obtained by substitut - ing (24) into (7) where y is set equal to $8 only and all partials with respect to B vanish. These equations are given by oeeuec2q ibaqe! Pls 0 3 2 2ks 2k, 6 224 * 8225 + Pog/p = gla, ke ~ +a, ky e (k, +k,)6 + 2a,ajk)k,e cos(k, - k))a-(w,-w)t te, -€,)) (k,+ k,)6 [25] [x5 +25 -2a)a,e cos((k, -k,)a-(w,-w,)t+ €5-&) }, = 0 (AG PANS t6) 2k,6 = Zz eal 1 2 2: SE anlot aoe er ety (k) +k, )6 + aja,(wk, + w>k)) e cos((k, - k))a- (o, - w,)t + €& -€) ) Jda aa (k,+k )6 Z (w, - @1)(w5%)) e sin((k, - k,)a- (w, - w,)t + €- €,) ]aé + [lzoe7 In (25), we have four equations inp, z and x. The first three de- termine a solution, subject to appropriate boundary conditions, for p, z, and x. The last is then used to see if the solution is irrotational and to make the solution irrotational if possible. Note that x = x, (6)t can be added to (23), if desired, as part of the total solution to the equations. The boundary conditions are that Dai O at 6 = 0 and that X> and Z5 approach zero as 6 approaches - © If the right hand side of the second equation is represented by g[F(5) + G(a, 6,t)] and the third equation is written as x, + Zo 6 -G(a, 6, t) =0, 2a ‘egal Wah at Bal Wien ty baron tidia. © aneeneae te : a ik ” *,, a Ri am, ' t Lae Pi 7 a 5 ve ; 5 cam fe p > mae om 7 ¢ es ae ee f) Rm, o ee a hg * 4p - 2 ctl +. niga . oye ¥ 5 Bale of eh ae ; * per: 44 dba - ss ar tae - oa 1g at reread Kena tort a. oe ae iting: 8 bose ty t * ail aide ah wT ala oy ‘ehaal > ae 7a De yt 1 aewee ye ow vytiodancis 20% nadie UnaeEe ul ae ‘tw igiedye ped Porat f adet ‘aed dice of eT Sie aueh Seb sg Dao> oi wgtT Pal ay) ehabIr ae Wi Ane vlan.) 2 leups nd ex Wd seed C7), odiak ea) a 7 va wewly ota endincope oaae® NSS Go ae, ae kt ie vail ar £ ee = “ s 3" 4 a" 3e is tae ag" a rs.. oe) : uy i i H ; Ye } a (1.2% oF vf es « i nts ato _ tet joey 7 Bet Ah. 7, : 7 he / oat a ye ph pw, af ye es ite _ i ; guad”, 6 a eye ae oh 7 oe an | LT ar 7 Bu 7 f i« q” ' A a o gil) ” aT od ‘ ‘ a iy sOO | af 1 Signe a g lee a a a ye ae ACY | i : 5" wtf pt > ger oe oA Viasat ° _ = mt (ge A o gare. Jt oh t - | F ' yi - 1 are substituted into the last equation, it is seen that the vorticity can 2 2k,6 2 2k,6 be made zero by adding ay k),e tta, k, Ws € t to x,. The second order terms therefore become aja, w pte, (kj +k))6 a) Se : ———): sin(kk, -k,)a - (w, -w))t + €5-€)) Ww, - W 2 1 aa, (k,-k,)6 + : (o, + wo, e sin((k, -k,)a - (wy - w,)t + €,-€ ) 2k,6 2k, 6 2 i 2 2 ta) wikje tta, wok, e t aera (k,+k,)6 mene a2 2 2 i, 2 z (27) Zo = (o, + W1%> + Ws Je cos((k, -k,)a - (5 w,)t + €5 - &) aja, (k-k))6 - w,(w, + 0) e cog(k, - k))a- (w, -w,)t + E> - €1) 5 2,6 a oe 2k,6 = Weyl 1 (472 2 1 Dy SED IG SN) ale =) (k) +k, )6 - 2p aa, 050, € cod(k, - k))a- (w, - 0,)t + €5-€)) (k -k,)6 + 2p A] a,W,0) € cod(k, -k,)a-(w.) )t +P €,-€)) F, = a) a5, e(k2-ky) sin((k, - k)a- (w5 -w,)t + €,-€ )) (k,+k)) +a,a (a, -w je sid(k, - k,)a-(w,-@)tt€,-€) Mi af toa tetas ‘anehinenps, ai ahd: de yt ‘ont gritajacen au Aeor> it vidieweg ay va dart), itvas Ee + ail} rinse a? a: oipe buids sed aot ne Hynes bail amend fata me , aes a gas® anes = oat . aT A .a)ee vO! nmi) rherds numacrsgaete ttn? arte sabhien att | Sree elt’ platy Salida : vil) Bara 19 a x) ah sonia atnb mil? no tkadpe babi” . ef fot nohiutos, ~ ’ ia 6bna yraiiibaos seit east nell ylatzan Jon a9ah eo aovewort babbs aedw tod) bri/ot ad oodt deuce mnoltiaups euoone gored ud? 92 eottplion © x =o? sn RL Bs eBlely oun tulow, gut? 2, fees Bae «x troy ena taupe eoicly Iault edd of scoitujor gabiry so nuit . AAS, Yilvifioy 209 fade neew Bi tt aolisupa: (aah at aint Byiir? unde, oe @ Fe 4 4 - ‘, “5 7 , fig »x av 7 a ee ar y ) , = ta ; ts sind s ¥d ore Abs ttt wed i + ~ i*a@ ve ' 4 ul amooed otoletedl? astiat ieahio bonuses adT = ae aya} P we . ms sail to (js o> Fi w) ae ab miola os a ne ee { s ) ow te : ; : fer Me ected Bf -t- ) adh ta Lf t pe ag) { f)} uM : ake | Falk SAT ada ws Ra wale ed ae Se as : i as : “4 }~ Js gi eie gs +o ° ('e a i BL WF a) [y ‘¢ lig ® . u Sib= SH 900 * ui) al, 4-a)leadn |” ae ee cee) pee oe perigh Fale = gil) (3 sAyiecs © OU 68, *, gta : i a? ge Spe . Al x= eat} ch ph - 64°30 Air Jin} we Be jf - 2 booty : Ain Ub pub t at Pe bao TM Mgr ish Niger See «MA | tans le Bem BR. fins (b= 7 8) regeeig’y HER Bh teeing ge ogc : oa | a(t we} ee { ane + Ais = lo)" (a+ eA lkos, ie os Miclt'g SA ass © \ i ah, * * nelle oo LSB MA oe (por 59 9 A bes oes sal (er shee ae qe “sr gh a f gh a ioe iy My v4 3". + Tete ie) - ol ata ciljade {3 nail, > eh ie a i : ih r ) : : ; vn p ; { a rgaey ay At spa yd) st e ; cavhig: yi ets pede f Th) uit igs The full solution is obtained by combining equations (5), (24), and (27). This solution satisfies the equations to second order. It is irrotational to second order. If more terms are added to the linear solution subject to the con- dition that w) < W5 < Ww, Kseed wy the terms in the linear solution inter - act in a predictable way to generate appropriate second order terms. The randomized second order solution for x(a,6,t) and z(a,6,t) are given by equations (28). k.§ x(alont)="ae—= aa. e 7 sin(k.a-w.t + e.) i i i i a.a, eae (k.+k.)6 = =z z Talc e J 1 abet (gs Sho ae SG) jie jrit g a a.a, (k.-k.)6 + EEA. tw.)w,e J * gin((k.-k,)a-(w.-w,)t+e,-€.) jpii 8s jo Jot Joi Joi 2k.6 +Zavu.k.e *t (28) k,6 ZA(,O5 6) =) Oates e cos(k.a-w.t +.«€.) i i i i aia. 9 2 spe + # nares (w; PEEP APs Je eos ( (ess i) aio aa ec een) aa. (k,-k,)6 = pa Oe —J w (w.+w.)e J cos((k.-k.)a-(w.-w.)t+ €.-€.) j>i i Sd a) eee J ekesl Ay et Jou The solution given by equation (28) has features that are quite different from solutions obtained by the assumption of irrotationality in the Eulerian system. The trigonometric second order terms involve only the difference between two frequencies. One of the second order terms dies out rapidly with depth. The other dies out slowly with depth. The term that is linear in t is the average drift of the particle, and as a group of higher waves passes, the effect of the second order terms is to increase the drift. Correspondingly, as low waves pass, the drift is decreased. The positions of those particles that are on the free surface are obtained by setting 6 = 0, and the result is equation (29). tot) (8) fire ery yrtmicternia: we Linithnide UL asi thn ott ci tl. ‘Telau Dngnne wh enwtinnpe ual Awitittdn jemtaiybie. adt 1*$} Ira an 7 . aha heres ne, ot toncidesoneh mntry 90) ere suntan podiolow «s raat Liat) cad ahi ¥ "e deeds diryistn, qv avhnabid: eae! aris eh wiht sed wale oe v oa olka “ee se Tail fete CEES OT (ebte bab o4. src iano Tete adavoinny of yaw whites atba've & AP ia wre (y.4 oe Bere [) oi Teg? He re, yah 70 breveqa nb beable ot 2 ah masteaup wid th {0 ! ¥ deen dalh ea at 1h 21s. 5 .o)m, * 1 t L< / : ; a> 2) 7 er | ave i , ‘ | ‘ ‘ 5 » | Pe i, wb = af abe ai ee ; i Lo — 4 ~ 23 i H i i ; i Le bons bd dp 2 A i a | i Lt se me, ; . [feck Uw be ats we i|le ¢ mite 4 hh § a a t j I : . A ee | \ ho pre} i ads i § 2 . ' a | © 5A at » ‘ - (aS : a4 1.6 °F Pow 1 Abe & os dof), 6 ale s | " oA, 5 ‘ = a- a OO ia oT ia S)) foo : wt ? eh ol ny ee 4 ~ 9 7 r ( j t ‘ A. #. boast : éiug , AVA {3d 3 +i : ph) emt = Lear mn ies bag, Wiens 7 a ? a a | I \ ' 4h per viv S76 104) eure nea) Maug “yo novig fomuloe mT fil ehiianouled ee a ts | PULP Tey i ‘gikoved Dadi aio at iiaioe reed) loeretiih: t : ‘ u » f Vind Sviduar eis Tab VACA R SPTT Si TOA h 47 al \ iy UE ORR Y fe RB rolod, $4} . eiiat vebata lnwrosnae edt th adh gain yoes bo nwt esney With qT Nagab ptiw yiwola jeg sai vous + ee (1g 1h kth bier. hha: eothy Guta & we mak Nolixaq sili t rai ayrtewe ‘sutt fi y ob taomil we tat) erie), S28 eat zee! ey e? 4¢nted yshto bes YISR sith iW? neh E) Ts Laadeng aoyuwr a oitgid lo eeedeiosh a? Tab watt anne vevawowol ea ‘vigethwoquesto Ab ails N14 aetive Sov? ord no-o'rw badd wala ttn ties enor iw eres oA'T, 40 S$) witlenpe ‘ae: Sta w's nit bes 0's apaiies xe benteido Silo x(a,t)=a- Da. sin(k.a- w.t + €.) i i i i a.a. w.(w.+w.) r a) : 2 f, a : 2 ea 5 378, sin((k; k,)a (o; ote. €)+ Za, w kit (29) z(a,t)= Za. cos(k.a-w.t + €.) i i i i aa. 5 tes pe eaicy: cos((k,-k.)a-(w.-w.)t + €.-€.) oil d i jin rd i} re ia The parametric equations, x = x(a,t) and z = z(a,t), imply an inverse for x = x(a,t) such that a = a(x,t), and so (30) Ze ez (oxsub) eit) is the equation for the free surface correct to second order as obtained by this solution of the Lagrangian equations of motion. Comparison of the Lagrangian and Eulerian models Results obtained in the Lagrangian system of equations are difficult to compare with results obtained in the Eulerian system of equations. In the Eulerian system the fluid velocity at a fixed point below the surface will be Gaussian in the linear Gaussian model. The velocity at a fixed point in the linear Gaussian model in the Lagrangian system has not been found but it involves finding w= Gee BX, FARE ON Ore5 F45 1)n 15)) 5 (31) ai—salcazet)ee and OF = 10l(ecsezeat) from equations (21), even in the long crested case, and this velocity is not Gaussian unless the waves are so low that a = x and 6 =z are suf- ficiently accurate inverses of x = x(a,6,t) and z = z(a,6,t). However, a higher order model in the Eulerian system of equations may well yield velocities comparable to those given by a model to a lower order inthe Lagrangian equations. The first and second order models in the Lagrangian equations appear to have advantages over the comparable first and second order models in the Eulerian equations, as will be shown later. Since the wy a, r to ian > he: a! : ot viet some, n thei peas a 7 yee ee . . Psi tw. ahah i i : ¢ ‘ +4, oa Hee Hh paenat eee nh thin “Mala rr es vip + aennainen id 0h —_ - : y ac Fu ~* + Me 1 joa ph pina, Tt hace = be be Ato ae ylomi it phe = baa aie > x sanoltaupe atyromaeg oAT ; ae) | on bas (3 x)p = tet due Ws abt ne yok sovovnt ae (732, alos | yi ben tanto és tsb boosea bi 3anavos pal Yaar sexi ‘ait gor woltuaps ods. a“ ta otion oe mufeeneany & anipanayat offs Io abby too si . elabom semi isu’ bow autgzorgint ai neti to ngelitea Ste, wuiioliaupe to nisi8y4 NBiyosr nat edt a bectarda axiiina sl _ Yo mainys natistud sft ot boalide stiveet site steymos oF 1 200% woled into bexil = 3= yitoolew biel tert eet asizvelu 5 oral angtta \ilzoler siT dJabor naleareD 74 sail ols at Keieaysd au Atti es al'niiy eae trsieye nilynexpal edz nl taba! apelin 1 Heal ods st talog: pax ate . gribrid es vlowat Ht tnd bac saad bi ae a(t. =a) nea extol - ap - ¥. . , bes, i eae. PTE (6 e}de | \ oi i f 17 { ‘ > th z at Yiialev palit bas. ,ebe3 bajests mrrcll wild ni neve Ais } ‘sale cama -hasate a 23 baa ‘= 9 2ed2 welos orn bore edi ew dion WelsscRD ton (4, én), oe ban ace ype = %, bo usaravnd cee Uanokaih ‘waollsaps to ho rae a re adr at Labor tabs io todyin & revewoH - f awe! 4 o? leh: mi & yo asvry esau oy aldexeqenn> asi tioolow. Blaby Lai: vai | a. \ . -BAOLSUPS, nalgies yal wi att aobte | sooitsups & Higaaryad ods ni sivbom Tabyo. faozse bins tent ad a 4eh7o. beroaas bas Vasil: Idexaqeno's od 7 780 ‘aeysieavbs, oved 7s nasaae pds. ania - is¥at te eda od iliw as \esoltaupe asivelug nt ai ‘eloboar i Vv im os ilvsbes resulting free surface is non-Gaussian it may be just as difficult to obtain results on the probability structure of the first and second order Lagrang - ian models as it would be to obtain results on a much higher order Eulerian model. If the problems prove intractable, numerical computation is a possibility that should yield some results. The real problem is thus to compare the six available random models with nature and to devise relevant experiments to see which model comes the closest to agreeing with that which is observed. Any linear non- Gaussian model that can be devised along with its higher order extensions should also be allowed to enter the competition. We suspect that such models will not do as well. Recapitulation So far in this paper, three new random sea surfaces have been obtain- ed. They are the short crested sea surface given by (17), (18), and (19) based on a linear superposition of the particle motions; the long crested sea surface given by (21), (22), and (23) based on a linear superposition of the particle motions; and the long crested sea surface given by (29) and (30) based on the second order correction to the long crested model. Each model is irrotational to the order to which it is carried out. The notation chosen to represent the second order long crested model is not as useful as the notation used by Tick [1959] in his study of the second order long crested model in the Eulerian system. These results, however, show the existence of such a model. Properties of the models The multivariate probability structures of these models have not yet been found. Only a few of the properties of the long crested linear model have been found. A possible realization of z = z(x) has been constructed, and the probability density of z(x) for a fixed x has been found. a i i hehe ror ke 4G my ae Pear ot, 7 sere i ‘inie'ae wie tw Me ‘Sari anid . mpliaees, r apes | wah ‘ha hnanhbl Bony eae wets to arate feito, gost Hes ROT, © 7 ela Wj Tab4g tor yes Rete e ee ania ns rivers, of wel arene tt we mide, sd) ee WF! 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(Ul) yal vis eo6tre gee cerca ve sore 6d! 3%e vad? eal batonto ey tof oi? saectom sloliteg BAP ke ire i miue.teomll & ee te ro eoqisqua teecil © no Kapad fe apes Ass) sid peey 2 wt 8 ai bain (04) vel Ay iy San £38 98 o> a ve ad) box bag Ktoett elim, eet dud feborm Baaevs pool adi of goalie 1” > seb1 bao +p i: os eG Soand (ty. ngli pious aT ..J0G,b0i1795 6 4 al isl i Of msiiu site oF Leaciteto' $i af Tepe) | whee 2a Jon ai seborn b siibie wrol aBlac bi a eatld cat Cay oF wrote gol zsbs6 basses 14 tes vhote able ine ah) asa ya bwhw mabaton od? Lo OH} woods , xan Wood). etvans aead'l reat va ia ed 2 lx” { mitt? ai Anbioit: bulasad) , . babum Ay Fare Te * eyed hte . °* | © § - See to neltvequaat | pwaedl stabory Sbs tO OR TION TAN WIN idadorg byte é hata Oey) | : belsevs. guint alt hi apitretoms Sas ty wal Byinny “Ubawal ead joy TOM. \ hs (hls © & be coviastless oldladagiA bawot Niod aval tok gS) er ahaa acc «= baksi VO} tele Ye etinirab writ Seton ai rit shows: Te ieaets sbiceh ry aire = Iisys Construction of z = z(x) for the linear model With 6 and t equal to zero in (21), the result is 5 Sh ap oe SS Cho pyc Epbal(e Ghar G)) 1 n n n (32) N TT = ar Z1 Zar cos(k_a ce) The quantities xy and z, forma vector Gaussian process.* The spectrum 1 of x] is S(k), the spectrum of z, is S(k), the co-spectrum is zero, and the quadrature spectrum is “aes The coherency is one. Tick [1961] has developed procedures for the construction of vector Gaussian processes with required spectra, cross spectra, and coherencies. A vector process with a reasonable spectrum and with the proper cross spectra was available, and it was a simple matter to prepare graphs of x(a) and z(a) as a function of a. The results are shown in figure 1, where x, (a), z(a), anda + x, (a) are all graphed as a function of a. For any value of a inthis figure, a pair of values for x and z result that can be plotted as a point in the x, z plane as shown immediately below. The points in the x, z plane then trace out the indicated curve that represents z = z(x). Only two crests are shown intrgure: ll: Figure 2 shows the result of preparing much longer graphs of x(a) and z(a). Eight crests are shown in the figure. In many ways, this artificial record appears to be much more realistic than one that might be obtained with a linear Gaussian model. These results show that, although the spectrum of the orbital motion could be band limited (i.e. , identically zero outside of a certain wave num- ber range), the spectrum of z = z(x) could be very rich in higher harmonics of the orbital motion spectrum. With one or more simple discontinuities in slope, as would be achieved at the sharp crests, the asymptotic form pre- dicted by Phillips [1958] would be found as applied to the long crested case. However, it would occur at wave numbers much higher than are presently capable of being resolved in spectral analyses. One of the difficulties with the Eulerian models is that they do not indicate within themselves unrealistic wave forms. A spectrum that would yield an impossible sea condition does not appear to be different from one that would yield a realistic sea condition. The linear Lagrangian model is *In fact, Zz is the Hilbert transform of x): talkers i Bon ke madi id (su a ae! + So ab ap et it Ciara fates ae - i oN hen : ; A A 1 } a SY iio: ait) CLA ators ot Lasigr> * hk As 3 rad a ei N Lies potas: od a Y oe . r a0 ee i, ,| . : . fs te ROD ae ee : - ie Fae a a ont a ae a6 eis heed ww wey (oe'csv' it rii! 3 Bax ;* ao Poitncmnayy 7 i L ‘ bas 089s 8) pow tsaegm av aal-. eet ye bo sree POU é on) (ble ab 4 i , , ' i { ' ' ini aw at (ora 79da% alt Mie = vag ested: sho hile. Puls lay Uli oo shee 5 iit 4 " werkt e070 ha eS a Lie 17 eset. Melaivotedos bac .SYdoeqw Seong ett) aye b jtliipg Mi w, «28 RS ars? ReGth Sso0719: a T dtiv be res 7981 6 ‘aldas wegen B bi wees 2049 < talon iy eto eh e1sondg Bl Sisian alicia 7 mar YY Bas pidalrare enw Pe . a te N2nUt # 68 tp)! bow niin thas ibs oth) px acetie fo wtogh «i Pe von e1lo557, edt A) bag ait ‘nie 6 "Shey Whe O38: 4x Hethorul € es bedaerg tha wee, Bi el ivloy o se bettoly od mes tad) 4 vag 2 tos 2 70) nouhae, w 3072 est sly x Sis ni) ere aye cr yolod #fooaibormwitineode wade a1 4 WiaaT ert Yl (xia 5 ey ora fail? 2Ysis Quetn ord te bovaw ee wet hiya v Be ‘| BERD | betest> simil edt of) Dellags ag bavsl'* dbluow (¢ARF} ponies wo £ Xx inaheye sts Goqli + sare ASU. Steomvr Hyveaw os LIS sina 3k - Poe souvileda lari: mS HE Ravinwsy qeted ‘Te. ald yaads tas ded alabor: sgitaln A silt alta sails eas 9 eit im, ott t a x" bleow fads nated 7998, des Bey yh aes Helias ane 4 tomers only inh tir stata HO MONS. FEST wT ht ae ; a in acres: Ihr Beek sect & ae alsin expires: £06 bisiy @ i diclokors an) rn 3 haere ee SQ ERAT > apsloirotinag its bibary bine’ sonal j ag og he moolah Aioae “ait Aa?) & toh eT See Z(x) Fig. 1. GRAPHS OF Z(a), X,(a), AND X(a) TO SHOW HOW Z=Z(X) IS OBTAINED hiya SHONOYL MOTIWHS GNY SLS3YD dYVHS HLIM S31IdO0Y¥d 3AVM ONV (x)Z=Z NIVLGO OL (%)X ONV (9)Z JO NOILMIOS DIdL3NVeVd ‘7 oinsty 1 y ' 1 i} ‘ v4 fi i ; ; i P ea mi Ne : : i t i he aera q ts py sli ‘ i ‘ r : 1 the Ni “\ 7 t i a ; i ), o } i i = i \ ’ t ' m i 5 ) oat See | Ty , : { 1 u : t , whe ' ye! i ‘ iy , i itn fi : ca of i an : i in Vi 7 JOY, b i < it 1 ir i ; ; At 1 , i ' \ l : Py i % Ta ; i s i 1a ‘ } j i= f =~ i i i in i i i tees i ae! ‘ H i A i : i i ! : i i uy bt pain , 1k i ; Kot ‘ } Y i i (es “ { ioe ; i f f iy hy ah I i i ; fe i Wh i i oe Z(a) <2? ]\\ OC a Figure 3. Parametric solution of z(a),x(a) to obtain free surface z = z(x) with loops. mo ‘ « ——. LAGFUCS FI ‘ y ee oe — i 7 r oc TN - ae nt eel (iorneiss -19- already capable of differentiating between completely unrealistic spectra and more realistic ones. If the space scale of x) and z, is changed for the same vector realization used to construct figures 1 and 2, the result is figure 3. Here a is triple valued for a certain range of x, and when z(x) is plotted the surface forms a large loop. This may be indicative of a very unstable state in the waves that could not be attained in nature be- cause breaking waves would form each time such a loop would be initiated in the space-time representation and the kinetic energy of the wave motion would be dissipated by turbulence in the whitecaps to an extent that the particle motion spectra would be modified to a more realistic condition. Conditions for a loop in z = z(x) A loop occurs in z = z(x) if z is triple valued as a function of x. This in turn occurs if x(a) is not monotonically increasing. Thus if m <0 where a 2s — - - (33) Wear l Zalk cos(k a w itt c)) a loop will form. The probability that (34) Dawe KmCoOs (kia@i-iwert teem) eal nen n n n must therefore be investigated. The three moments given by (35) a = f S(k) dk (eo) (36) #, = fS(k) k dk (o) and (37) vo = T S(k) mora {e) need to be considered in this connection, and from (34) it is evident that b, must exist and be bounded. The probability that m will be less than zero is given by bi " t- vam a: Sa Ramee Rte etaienionny re euied prs ue aiding ad ‘ Vikawate pes sath cs im tebe = +8 ennsiatad piame “arid b+ peat esuigh hy ee oO) foes wopepattiven yorn ey Tipo way thy - ieee Pew se bolsgnas pigsty « yo! heuts jy, Sa avixhalbel ad yer weet : | ‘Patairst st bibow goal s Nog erry dose rien! biyaw « RRgW gail este 1d : oituagd att hus poriasceReeyen ail Se ce Abs 4t ssceldze! yd’ baasnivels a how! NTT) y siJellank Ota @ ed baetiGem ef blonw Arteade om Lasse al ad peng, 7 eoiserd. avo an} lo yg terns Ai yesth, Tre tow male aqaaeiuiw adi { ote vhliswl, » ie devlay elytis €) i tw) YW audT { @t'F U's te ie 7 Ba i 4 ft (y+ + = alia sf ny # I Sy “oe Td) & ' ME * (eis " ae {{tje i i th” vere saehevs of 45 (hE) Geos? ba vis as is 2! orton inedt scab ad iw anieest ial vilysivasonnm tun SG Sra03 wo RAL Oe ST OND bn nireae 48 ¥ onyit at ee ak 3 aeqal sytel:s ereyal ooel ioe omar, tenrtgentry wh tml ge net winter ni bere wit a wt You bla. That eavour wit) ah atake oly dost yore & th gool x 19) eal ihaees b] s mi W300 gqoch sz a 4i toyn Lh ataauc 107 ab eT bo he | adwe it} > oer x) : ; Pam OY (te) S co wwe ihe: ait « Sout? ait aasong, stT rm ot = re wy. 4 a’ (wey , ; a \balaghbaynl wd beatae ted) ada an) ive i 1H? at Ra a (ey bam i} With at Dp zap! 1h 24 0) DOGH Sshnued od bere tehew Panai evi ae : ; : a hd inal yiNipedoig eat S20" (38) plit< 0) =—————— Ne dt'=)——— { e dt . Equation (38) may be interpreted to be the probability that some particle, a.) will be found at the top of a loop between the points where z = z(x) is vertical for some particular time of observation. Sucha particle is surely involved in a breaking wave. Other particles that do not satisfy this condition also may be involved in a breaker. Since the spectrum S(k) has not as yet been measured, it is necessary to investigate possible relationships between S(k) and spectra of z(x,0) and z(0,t) before estimating the probability in (38). Marginal distribution of z=z(x) The graphical construction used to illustrate the generation of the random function z = z(x) can be used to obtain the marginal distribution of z= 2(x). The joint density of z(a,0) and & m(a,0) is bivariate normal. The variance of z(a) is ee the variance of m(x) is bo, and the covariance is oT as defined in (34), (35) and (36). If f(z,m) is the bivariate normal distribution of z and m, we note that (39) F(z, m) = f(z, m)m nearly has the required properties of a probability density function since m is dimensionless and E(m) = 1. Also since x and a have the same dimensions, a given number of equally spaced points in a small interval of a given length over a produce m times this number of points with this same spacing in an interval of a different size on the x axis. The function F(z,m) is not a probability density function strictly because it is negative for m< 0, but this corresponds physically to im- possible configurations of z = z(x), and points on z = z(x) generated when m <0 will be shown to have extremely low probabilities. Actually even more points on z = z(x) are probably destroyed by the breaking pro- cess, and z = z(x) is modified in form long before m becomes zero for some sea conditions. However, as an approximation, we can truncate F(z,m) at m = 0 and renormalize. Thus el) VS ethes & »eioterady, oeoe fortz Yodo rg at ad OF hateny ‘ghia ot gars wes aaa al es “9 ainhw witthoy arti awh ee (Th ot OY ap tactics tal atte oF ol Ttag & Hou | duaiter wo he nari alohlaed Stow vat, twat Vistias Jon ob dity aslsitven cede! rew gothded & af Godden ries + . Voxan oe a nk hee foam) gat yinass oaks adi hatborinsy: & ef 4 Bersivsde »: pq aay #h tow Geel {ihz tea) 19mg wil sonke : antdeqe ban! TAYE woawied saidenclialey eldigany arkyitwnveii od teen i {BF} ai ilidado 14 alt yidsornires, sroled (yO )m Siva (0 awa . | thaws te jo abiwdtssuth Gui? Vx noite oct eit ata vréolli ;. ; lanisan @seisevid ef (i) (alert = ae Bae {(O,oie Wo ystaneh taloj atl of) ‘sonoitevad of? bow ,. &i (4) Ao spmhis ay air, aw wi lode To = 208 bee ; a ot) brs (ee) ifGE) nl Henttod » aw Ale aw oft O@n6 4 We nojtvdi«iieib lantiam wieray id edd al (on mY ie ‘eotite mulicas? yledab ytilidaddsa sy to wail wuotG, basins ett wnt elas Sizae will oved oe Done «) annie all 7 jre)ct mie sbolanienamih at ore Satal ile 6 ai wi uPmaq © VilsupYy to Toda 12 neva sy a. » WCE shit fale, Woatog 3c Tears wild Karri =i Souba% q GoTeNE Atynee navi ay ) abig on) Gia MEG fret ab gn ‘Asyxeial fie mi grioags ar plidiaie ngbsaut yinenah gilidenorg a iow wi Rit motto wt ot} i ae m ANZA a vildalnyda Bpgny 68470 - OL Sy CF Dat ol ayvitepen ey vi outage aedw HWieTeity {<)o >a ac B¥ntud Had ia. =e to anghtertugitans aldt | WEI A nel Mdidart ‘wot, ybarree td he WES of inurona: od iltw 0 Mp a arg, golatiseyd eifiyd Doyissesh vidadorcd aia har a MO wintoG e1cn ewe “36% ey ee S oie 300 al oreted, youl wa8%% y amt bite gaia bale inin-c @ Soa one” : | ry | . seoitihaes aoe amom ry het yt) wisdae 1? a3 “a AGhiart: Te sagas is aa ey owed | . | ‘mud’ ea tlaar toned baw 0 2 aes co oO co 2 -1/2 (40) (Src. m)am lz = =| fort Vere +NU, e | eK Dees Van -1/NG, Then (41) E(z) = x ST \e F (2, m)am |az =< atl fet 2] aa AAEM | 20 2 W 2 Y -1/2y 2 ome) -E°/2 1 2 (42) E(z‘) = Ve dé + (+ wo e | ae ee Ua ae -1/Nb, “b, yo -1/2y 3 -£7/2 1 2 (43) (2°) = == [3¥, i. ere | Van k “1G, aie the probability density function is given by me serie bY P-L 2/4 Pb [oh V2 5 (44) pla) ao 7) Nh Be pea noucinet atte to “O24 NO wi ton Wy Hy If b, is small compared to one, (45) E(z) = - (46) E(z°) = ¥, (47) E(z°) = 34,b, d 2 i (2) = cae og 48 p = @ - ) Z ip ve oS Zz Let (49) PAN ee v4 oe by Then (2th, )°/2, tr 2 (50) (Eh) = eee are Nan \ M5 Ge re) GRE: er "ee eh eb a se He er? te | mm wi HY deal (dvr, 2) All i J ey wa t . “} ; | df g let us assume that iK/ie yee Se (57) S'"(£) =| q 0 otherwise From equations (35), (36), (37), (55), (56), and (57), one can obtain the result that (58) ue -¥,*(Tep) 2 Wales ai (59) == Gos)! : — He | (res) sf g n- 3 fo) 1 - (£,/f oa SPD) as. u and b<é ("sit (ta) “8 a) on (sa) 5 ef ale (€2} b 5 Bs (a) ef sean bo aeioitena ~ii¢ Stee beee €} 244cwoda oie . i ur wt, ye : hd bas , he Py lo botenrtiin® AL @eraiteyv “it qe O1608 oh (ale Yo suscora ‘itt ($2) nvet4 L4 PGO@ 601! 4 yd heoambiae of tad? suasiisey seit wi alt T ow « oF = * yaw b:aee svaew 43g) & 22 : — : a! wh fia)" « awa (=e . Ab ia}e | [e2} rr ¥ th (1)"'R = felgqriaxs 40)) sonje bad . aes a ib + edt )2e wh “a idj2 (82) tut exvoese ae fol '>4 > ,! "S\M ea (3)"S (Ta) se lwteidG 0 SAD each’. (T2) Daw (O02) (22) (78) 18e) (82) wooiiaupe oo18 jens 3fsae7 edi alegdo Varict ) toh ne a r “ys bp f S i e a we - at Ges Vea ae =| V( aor it (Wey ent« 4 Lema (60) y. = Len — | feu | 2 ye 2 Re Ge: ° ee fe (135 a] where ae [Eve Gan. a ¥ For one fairly severe storm at sea We was about equal to 116 £t? and f, was equal to 0.046. The results above permit an investigation of the sensitivity of the particle motion moments to the values of n and i : Tables 1 through 5 give the values of vp Uo 2? 1/Nb, , bP /b 5 for various values of n and for f,, = 0.46, 0.92, 4.6 and ©. The values of ie correspond to periods of 2.16, 1.08, .216 and zero seconds. Table l. Values of _ a various values of n and i for * = 116 £2 anil @ = Ose BOSS ee - ne 0.46 0.92 4.6 co f 5 5) 116.26 116.26 116.26 116.26 Bait IM6n28 WlGs2'8) | LN6.28° lle.28 5s5 IMNGeS. 0) LLGesi0O 6330" 1162310 Dae! NINA WIG SYA M4 eA WING S72 Ball LUG BH UNG NNO BS WilSsse) 459 16239 6.39 116539) 116.39 4.7 116.42 116.43 116.44 116.44 4.5 116.47 116.49 116.50 116.50 area - thigh * fi : i. 7 Ss Oh1 09 laupa iieds 266 a i oe. Ses foitsyMeovdd @s 4! “ aeeds atleast ofT, A600 ap Leupe saw yh a) 4 ¥ rar 7 ‘ > ry ae i o . ) 00 Te itOls wtevan ia ans VoD c , (Epa 7 ' : oe aT ' ou! « 1c) esneimiort BOltoat alah ag ml? To #7lv ithe ee —_ ; TS ’ : | patie baa , gw jf i Y to pavley adi avig Hotere 1 palitst ‘@ } Mok co .6 i es 8 ot ae a %, ' ky cel + ai * ‘ ie Oe ys SHY 2.8 (2) 2a8 1) 3 ai gy Biri TAY - 6al, gv ar \ ny , ? . 6 al > (SRwasee Geof fhecen 215, oHt0, 0. ¢)1\8 Jo adhoivad of foocweTiog ~} to seglay iaieliiee , ee : x P = : idat Pi - bas o to aguliry ww ! BY TOR” hs! i) Bouhav | | a 1 Rae oy : i . 4 ou P ty i Pig 7 i 7 : aR alt batt OG Ath Ld it a , t ee | ‘ tad < ' i ao oe di OF 8 | roAbl: “SO ) bE tl 2 5! aS 7 bit née hl PF af Pati 4¢ Alt * ia ‘ if | P j ahi La of fie 7 e) 4 ’ ’ f fs : othe Table 2. Values of by ae various values of n and fis for ee Ge ain £, = 0.046 sec”. n tu 0.046 0.92 4.6 20 5.9 0.51 0.51 0.51 0.51 5.7 0.53 0.53 0.53 0.53 5.5 0.55 0.55 0.55 0.55 5.3 0.57 0.57 0.57 0.57 5.1 0.59 0.59 0.59 0.59 4.9 0.62 0.62 0.62 0.62 4.7 0.65 0.66 0.66 0.66 4.5 0.69 0.70 0.71 0.71 Table 3 Values of b, for Reus values of n and f for » * = ay ft” and f, = 0.046 sec’ f nee 0.46 0.92 4.6 oo 5.9 0034 .0036 .0038 0039 5.7 0038 .0042 .0046 0048 5.5 0044 .0049 .0057 0064 5.3 0051 .0060 .0076 0102 5.1 0060 .0075 .0105 0290 4.9 0072 .0097 .0162 2 4.7 OOSiza aa0li2i 90261 eo os tidal : tina a to ied abl avoitey vo! Vie tp geulaV 7 . . ¢ PS ~) po. © ub e (6606.0 « bit } ie + 18 4 1. . tat ae ul ' ij ane E ar: he uo ; o ask $0.0 dbo rf _ —— ~~ a TS — ~ A — cae lh ST ia eee iinet cra a ny ry" “be a £00; BEOU e700, 209, ty Be06. .d)00.0 Se00.. Bz00. ¥2 200 fPO0, O850, bPFOD: Ot S0i4. Avoo., OA0d. 0200, ee ot co, ela, arog . nant 1.6 o Soro.” Food. -Sto0. oe Pe - ‘ 1456 : PSO, TBAOG ee ; 7 SbbD., TO1D, FOIG, b . =26 5 Table 4. Values of LNG, for various values of n and i for ,* = 116 RO Ge ON ee n f 5.9 eZ eG) elGeee elowl 5.7 16.2 15.5 14.8 14.5 5.5 IB - WA Te 5.3 WE | Ay eG Ee) 5.1 1Ze9 ee eS On88 5.9 4.9 Wi NOK AS 0 4.7 10.7 8.9 6.2 0 4.5 Oe ae BY 0 Table 5. Values of bP Mb for various values of n and i for ¥,* = 116 ft” and f, = 0.046 Bae, Be n 0.46 0.92 4.6 ae 5.9 0.67 0.63 0.60 0.59 D6 0.63 0.58 0.53 0.51 5.5 0.59 0.52 0.45 0.40 G8) 0.54 0.46 0.37 0.27 Sell 0.50 0.40 0.29 0.10 4.9 0.46 0.34 0.21 4.7 0.42 0.29 0.14 0 4.5 0.38 0.25 0.10 als oe re > Ts ; i iy Ab tp a ind nal A ‘ 5% 4,¥ ® ; j A> | 4? | S401 rx Ios set 5.8% mete es? ms e ve < P.Si Lod ef 2 gu! fil efi bet : ’ aa Gaet Yi { ..¢g i i) 7? Vi ti t 4 A eteT Seniey eaunisa: 1 WV" 34 batten 4 H yaa OF) .0 4) ine “mh bil =” WW to + Soe ff iC : 4 ‘J A? 3 i. boo 4) o-~-+ goa - bal —— + CF ——— — ve 9 re G 1’) Hf) es ae | *2.0 nA £44) Tic i . 26.0 $2.5 S c.4 SC 70.0 1 ou ee. Ec ‘ ha be 4] G2.0 Lae i iA 0 ? ray | t ' i; . ih n cb Vie os b Qbt . Weise a 6. L2G Discussion of tables The assumption that the range that has been assumed for the values of fi and n for the spectral forms for S(k) must first be questioned. The range of ie is probably more than adequate. However, the relation- ship between particle motion spectra and free surface spectra is not known. Perhaps n can be even smaller than 4.5 and still yield spectra for the free surface that behave over a given range like 1/fP where p is greater than 5 as wave observations seem to suggest. If these assumed values are reasonable, it would follow that a loop ina realization, z = z(x), would be a very rare event. The small- est value (except the zero) in Table 5 is 4.7 and the associated proba- bility is less than lo, If loops were identified with breaking waves, the results would imply that waves shorter than 0.2 feet and n's less than 5 produce such breakers and this does not seem at all reasonable. Stated another way, results based on spectra like K/f£ depend very strongly on the high frequency tail of the spectrum where the waves are less than 3 inches long and on the fact that the exponent is exactly 5 and not 5.1. The original conditions correspond to waves with a significant height of 42 feet and with representative lengths of 1500 to 2000 feet. Breakers in sucha sea certainly occur and surely have dimensions exceeding 3 inches. Thus although such loops identify un- realistic spectra, the limitations on the spectral form appear to be due to effects that occur before such loops even have a chance to attempt to form. The tables also suggest that equation (49) will be approxi - mately correct as to general shape, and that the higher moments are both difficult to detect theoretically and to measure in practice. Equations (45), (46), and (52) are satisfactory approximations, but the approximate higher moments and skewness and kurtosis values will be in error by large percentages. The distribution of z as given by equation (44) when graphed for two conditions, one for which Le = 0.46 and n = 5.9, the other for which ie = 4.6 and n = 4.5 can hardly be dis - tinguished visually from the normal curve. “ahs ‘. Weng, x01 bomunas wo ait sang ognet orld Paty ary ieiiniok ont? thw, yoreew! ioe sargand 02 nointange : baat ‘Rat = tal rie sop)nit mee taht earraped séhau Sones. er pokba res Biasinatorst rout wie we ya 2 SetOiaupe & ixeloD, ali yaivine ial beaiaida haays Soadg eat T td mevly & noltan naottenaert aot 4ouka sant oS - Hee) eae hrasetinas 10n 80a? san Toho fuloome w? ey nw weinnktgcdt fenoeiortt 2iboiteag shb10 Sndoee oAhT eB b Lid? 1 saviwe aalzolds 3 “bad bends saany Yecal! xii} 2) as gril aver) ad o3 Laer sivew #93 Lamoktato? th athaireg Grebe brid? ideaxte Ata aid” oe Bebba: eligi . sontézem 444 4 aniée Sz wl ded testy high » ever Syl ins 404 bveqh senig 447 10 Sad yino et nolioar zs bone Genny ait pedal rly vel boolaliyns we. Qeo.¥ nivea atte th worl Ss tates vara eTae soll Ciao K -28 that there probably is a residual vorticity field at third order in the La- grangian solution that may account for the difference. The particle velocity at the crest for the Gerstner wave is dx 2 66 = = (66) u(0, 0) Gia 0t=0 aw ta kw, and when the phase speed is set equal to the particle velocity, the limiting form (67) ak = 1 is still obtained. With ak = 1, the second order phase speed and the particle velocities at the crest are exactly double that of the linear theory, but it is known that the phase speed due to finite height cannot exceed about 1.2 times the linear phase speed. These results are all indicative of the need for different wave models and for measurements of higher order effects in waves. Construction of z = z(x) for the second order equations From recent results of Tick [1961], it may be possible by start - ing with equations (28) to construct the vector process given below by an appropriate sequence of operations on the linear realizations constructed above. x(a, 0, 0) z(a, 0, 0) x(a, T, 0) z(a, T, 0) x(a, 0, 61) z(a, 0, 61) (Gh, Sep 61) z(a, 7, 5,) x(a, 0, 0) x,(a, 0,6) ) i! sHivee Wend! ar s Je5I9,07, al dseeoo ed Var (foe gp galt Yo ofl Ce aew MAGed Pret w yd wolsd “s7!) &8o507¢ 7 oe Jar syeco of (287 ay | iigep ler i ba Svirtinaes saohsedlilens i nil do edolUinleye lo eo hse mielaae R ' iP : TH) to vg on Ys Fi adie c i q ‘ if a4 4 i i i= Bit DU 6)... ' { \ : =20— The construction may even turn out to be considerably simpler than that employed by Tick [1961] because only low frequencies are in- volved. In the above process, T is to be a fraction of a second after G =O), 5) is to be a foot or so below the free surface. The first two pairs will yield an estimate of the instantaneous speed of acrest. If x, (a, 0,0) exceeds this value, then the crest could be considered to be a breaking crest. The other terms should yield some information on the size of the region of unstable motion. A range of possible spectral forms can be used to construct this vector process and some idea of the fraction of breaking waves in the total can be gained. However, from the results given above on the linear random model and on the periodic second order irrotational model, the results obtained from such a construction will probably not be quantitatively cor- rect as they may yield waves that are too high physically by a factor of two, even for irrotational waves to second order. Breakers in a random sea Whitecaps and breakers in deep water are an integral part of the problem of wave generation. Many photographs of waves in deep water show such breaking waves. The equations governing the wave motion fail locally and a breaker is produced. The water in the breaking part of a wave is certainly governed by physical laws that are completely different from those that govern the wave motion. A breaker is probably produced when the water particle speed exceeds the speed of the crest, but other effects such as the wind actually blowing off the crest of a wave before this re- quirement is met may also play an important part. Moreover, the turbulence produced by a breaking wave, governed by eddy viscosity laws, will tend to damp out the higher frequency linear components in the spectrum of the wind generated sea. The absence of the local chop in the wake of a ship illustrates this effect. Whenever a breaker disorganizes the fluid motions, there is a mass of water moving forward at the crest of the waves. This effect may contribute to both the drift current at the surface and to the growth of the lower frequency waves. A 1) dhe eo De ersgee tone’ My ei Taig ervonte eveds ads i s i htt D axl GY ¢ , 4 “a @ >» domted bp bal lew eaten ool ee @aeiea tion 19 1 OF ee uM dado Ga MoT) Bains ohh la fat 4 ay" ; iwier ty +... i ‘| 7 » Ibo ray agli ar jue dabiunl ? ‘ ; ; pal lung tee o) Lenheeta Yr pao w Tite s witha se eviw hie} ceymdcik By DP \ S604 ae nln eete Shih 7 r Ss SinATe & 14n, 4070 2 oe me re yile Vo Onn ey vip eee 1 (Aus BO! | (28 T4089 snd? iy asi th 1 ower OT) VilLAvIr Ghiw eel te Cee ‘ b 968 i i 5 AS | i yl bo Ral Vane ve f iyi ; Mi? ¥ Pa! Gd AS 7 shan iit qh hay pee! ieeted 2 Gove i atw ¥) an fel iho ho 4 pace. @ heat eal yc i) ste dbtiaiyn. yee Bay ot ra BA Si Sowa, 8d) ta = ie Also there is a wind drift current in a local sea that is nearly in the same direction as the traveling waves. This wind drift current pro- duces a field of vorticity in the same sense as the direction of propagation of the waves and may well cause breaking to occur for considerably lower wave heights than even irrotational theory would predict. The work of Longuet-Higgins [1953], [1960] has shaken the con- cepts of irrotational motion for gravity waves to their very foundation . His results predict that, even in the absence of wind drift currents and turbulent effects due to breakers, the mass transport velocity in deep water may be considerably stronger than irrotational theory predicts if the waves have been running long enough. Moreover, as Longuet-Higgins [1953] has pointed out, Duvriel- Jacotin has shown that any mass transport velocity can be used as the starting point and a wave motion can be superimposed thereon. It is also known from classical periodic wave theory that a sharp angle at the crest of 120° is another way to define the limiting height of a periodic wave of the form z = z(x). It may be possible if the other con- cepts discussed above fail, due to the fact that the results are only to second order, to study constructions of z = z(x) as a random process to see how many waves approach this limiting form. With all of these preliminary remarks it would seem that equation (29), with perhaps still an arbitrary second order vorticity field added, can provide some insight on the growth of a wave spectrum. Con- sider two orbital motion spectra with the same variance, one with con- tributions with relatively higher frequencies than the other. The mass transport at the surface will be stronger for the spectrum with the richer high frequency content but at the same time the speeds of the crests will be less. The interaction of these effects could conceivably cause the higher frequency waves in the spectrum as they are overtaken by and ride up on the crests of the longer lower frequency waves (Longuet -Higgins and Stewart [1960]) to break and dissipate by turbulent action. At the same time the low frequency components can continue to grow and even be caused to grow by the breaking of the shorter waves. Considerations such as this may explain various wave spectra where forms like Kyoue Sow ifasore and Kye have been proposed and where the exponent “) x Pii« a 7 , iy rh we HAS or sy i ft Ve iT ify ' ‘ § ' Mae ash rit vii Neh): a Te meet tA hina ; . aged L yet Re ee ed j pathy to) Rae By “= wh ty i ’ _ wine ais AR a Dr oy ity Lee ee ‘) 8 i ee 1 qilgied oe i é es Ava a) Mhawe sahy a: : bey cir iy alos § Ee oteoa- ; ; theta “Hens ai: Ee ene . wh tro gitiy Seating eaay : ny ' | hSOEACLS & , he th olew r j 1 wel oVidi SOV ES | Tix . ’ ware hA i ta te woe nai ii hat i wr & tite nia toi Sete if F 1. 28. af an Jf ' ent? an ia atta ts a iil @uit : oe hw albu Ta ee me fu jel vurite baeeyvou ly oreo i ar ireed pial it, Tia & 429 ow PRAM wil PT , ‘ x) 9 W ? nn) hw rt & aap ; ie ; { * Fhivesyg We bohbar i fefas a , me, i hyeiack rite ed) de Prog hee uo doSbaicl y aan ed ie ' he aol 2ai ail at hwas wet 4 + cry y JM OTD imnhyid mvoley F ¢ that F j tye shee mT ae | \ ty tty out te! = Hea , uy Ma “ceyoett wil wht era) epee ! fe aks mit ad, are 8 od oe WS ee ' we didland haw abdt weed J m é - 3l- appears to depend on the heights of the waves that were measured. Comparison with observations The purpose of this part of this paper is to present the results of some observations that suggest that the various models proposed above are more nearly in accord with actual waves than models based on the Eulerian equations. Figure 4 is a copy of a photograph given by Roll [1957] for waves a few centimeters high and 8 to 35 cm long. The third photograph from the top shows a remarkable similarity to figure 2. For various wave heights and apparent lengths, either a rounded crest or a sharp crest occurs. Note the marked departure from normality that slopes and curvatures would have if they were evaluated from sucha record. Figures 5, 6, and 7 show some reproductions of seven selected wave profiles obtained by stereo-photogrammetric techniques from photo - graphs taken aboarda ship. The height scale is five times that of the horizontal scale. The profiles show the presence of sharp crests and shallow troughs as these models predict. Note also that the profiles are much richer in high wave numbers than corresponding time histories are rich in high frequencies. These profiles were kindly furnished the author by Dr. Norman F. Jasper of the David Taylor Model Basin. Additional information on these records is given by Brooks and Jasper [1957]. Figure 8 shows some selected time histories obtained in Buz- zards Bay by Harlow G. Farmer [Farmer and Ketchum (in press)]. The original records have been reversed, so that time increases from left to right, and blackened below the original trace. The wave sensing sys- tem consists of a very fine wire (0.015 inch diameter) that appears to follow the water elevation more accurately than previous wave poles or wave wires. A number of the crests are very sharply pointed. There is evidence that a breaker occurred in the top trace where one crest shows a nearly vertical rise at the crest. Since these records are much richer in high frequency content (sharp crests, and vertical rises), spectra computed from them will be richer at high frequencies than spectra pre - viously reported for waves in this general range of heights and periods. The figures and illustrations in Schulejkin [1960] also suggest that the randomized Lagrangian model corresponds well with reality. Ne die ‘he: ‘ieee! scam indd BONew, Per fa silat wal ie bi anh ot vasa . , 7 aontiuranenda suf m6 age hr athe: ats MipamEy'Os OL taqad ebty to rk aA? Ba, seogityg “ntl ; YOUN Donugoatiy utube eopiiay 3A) veal) 1899 ym Yarty sit a7 sete bai ofl) ao head ‘elabawm aad? peviw J Mo Peat HCD OM, benone at Viveod OO By me ere) eke al (T2E0 Wo wd res} De Re | niley ow vata i nny ht i oi ios mmergerals b.idt oT , gyal a 22 ot & One oyid at otomliian way Riliyiod atAw guodsay WC .f @ecryll at ytivatlonin Videmianer swt eek aut’. 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Ty ee YaFMAIS@ Ta Bee pare at POSE @, Jang Co ' ri Ae ae oTA ay. "et 1 ei | t eR4 da Th GIRLS Oe OAR, ire hice (2g > Seis References Biesel, F. [1952]: Study of wave propagation in water of gradually varying depth. In Gravity Waves. Natl. Bur. Standards (U.S.) Circ. 521, pp. 221-234. Brooks, R.L., and N.F. Jasper [1957]: Statistics on wave heights and periods for the North Atlantic Ocean. DATMOBAS Report 1091. Cartwright, D.E., and L. J. Rydill [1956]: The rolling and pitching of a ship at sea. Trans. I.N.A. (London), vol. 98, p. 100. Ehrenfeld, Goodman, et al. [1958]: Theoretical and observed results for the zero and ordinate crossing problems of stationary Gaussian noise with application to pressure records of ocean waves. Tech. Report No. 1, Bureau of Ships, Contract No. Nobs 27018 (1734F), New York University, College of Engineering, Research Division. Farmer, H.G., and D.D. Ketchum [in press]: An instrumentation sys - tem for wave measurements, recording and analysis. Proc. 7th Conference on Coastal Engineering. Lamb, H. [1932]: Hydrodynamics, 6th Ed. Cambridge Univ. Press. Lewis, E.V. [1955]: Ship speeds in irregular seas. Trans. SNAME, vol. 63. Longuet -Higgins, M.S. [1953]: Mass transport in water waves. Phil. Trans. Roy. Soc. of London, Series A, vol. 245, pp. 535-581. Longuet -Higgins, M.S. [1957]: Statistical properties of a random moving surface. Phil. Trans. Roy. Soc. of London, Series A, vol. 250, pp. 157-174. Longuet -Higgins, M.S. [1960]: Mass transport in the boundary layer at a free oscillating surface. J. Fluid Mechanics, 8, 293 -306. Longuet -Higgins, M.S., and R. W. Stewart [1960]: Changes in the form of short gravity waves on long waves and tidal currents. J. Fluid Mech., 8, 565-583. Michell, J.H. [1893]: The highest wave in water. Phil. Mag., 36, 430-437. Phillips, O. M. [1958]: The equilibrium range in the spectra of wind-generated waves. J. Fluid Mechanics, vol. 4, part 4, pp. 426-434. ot _ pe re) in he a ry OG - 7 : - : t U ‘i 1 a : 7 -_ i r - : + 7 ry i ne : i i — ra a if S fe i , i >) el in aie re : v wi Peay vi pie gnisde ry nilis. oO nec st rae : ae Sadyiec raw An ozizeteat®. “iven sy tages’, ti bak ‘val sm ARS Hogan Rp HOM TAG Mmeoa ohne dees welt wot tg = agi | (eet wilvedy b MSteedt » ‘ ve >eva 15 ay pre yi! SWAN We iecegdieramm se% ne a S Peys Ge Rua Suipsouet Mast Saat at OV iw 70), (nat yin WHGs ev tathw al repavicbet enh ECR TT eM seat a Gy -? i a 26) y Mi to 34 “ec art biota? DV es Cio 4 tS Vaew Ri it agleimasieche aerw to vite sae it) ebuabasi2 we a fom a a) bre geilior aft 18261) Airest igh Bak, y ae ov anbont) ” ee et anaet nen 20 aiia’ sersado bag Laviie7oa47 ear of 18 erntook bf adore spuitele iO einsiddta gilesai1s osacthrre baa Oveg AAI. hevewamas ¥d ab 10801) Teawste .W A bow, eM satgaitt- sonyaod Hgbekvdct HA ‘Peokw nt) winlsied 0-0 bas OH: tore : yarteas wat Lada io HO 2 eS) FH? a505% sitio sit - biidiag2,, bet 240 ppdlorseybasbyit F {Seely iH Seto 4 base Via Hi iboats gine ers) Be wh vbw {, mw! sO ville ? tov ,” aehac protien iio 1930 ie sYOR ee ke A ssiitoaorg loptianate (Veet) ‘ety souyaod Mohnint to 1908 yo sankeT 0 re S5ettuw harind Xo TL 49 uit Ht Sioqamett weald ([0dGl}.25M suigp ih increta dso biol t .sasitve goisellioen opal & 3x 7 eavave noo! wo seve VIVETY J1Ous Yo etor H#A~858, 8 Boot Slats 1 7 "Sstpw ui avEw taeiyirt ant HECRI) H Aa — VEh-DER: + act? oh ogiet prulsdiliops aft 6800)" LO sea ringisehM bint! > Jae beierenas. berw’. fe-dSh ioq. 30 Phillips, O. M. [1961]: The dynamics of random finite amplitude gravity waves (preprint). NAS-NRC Conference on Ocean Wave Spectra, May 1-4, 1961. Pierson, W. J. [1952]: A unified mathematical theory for the analysis, propagation and refraction of storm generated ocean surface waves. Parts I and II. Research Division, College of Engineer- ing, New York University. Prepared for the Beach Erosion Board, Department of the Army, and Office of Naval Research, Department of the Navy. Pierson, W.J. [1955]: Wind Generated Gravity Waves. Advances in Geophysics, vol. 2, Academic Press. Pierson, W.J., and P. Fife [1961]: Some nonlinear properties of long crested periodic waves with lengths near 2.44 centimeters. Journal of Geophysical Research, vol. 66menowels Roll, H. U. [1957]: Oberflichen-Wellen des Meeres. Handbuch der Physik, (48), Berlin, Springer Verlag. Schulejkin, W.W. [1960]: Theorie der Meereswellen. Akad. Verlag- Berlin. St. Denis, M., and W. J. Pierson, Jr. [1953]: On the motions of ships in confused seas. Trans. Society of Naval Architects and Marine Engineers, vol. 61, pp. 280-357. Stoker, J.J. [1957]: Water Waves. Interscience Publishers, New York. Tick, L. J. [1959]: A nonlinear random model of gravity waves, I. Jour. Math. and Mech., vol. 8, no. 5. Tick, L. J. [1961]: Non-linear probability models of ocean waves (preprint). NAS NRC Conference on Ocean Wave Spectra, May 1-4, 1961. woITdge Wa Hae ome BonerebidS "aye. BAD Rio Aa tn iy wp tins Fradactidee n€ attach mole: iQ * heh ort Cbreety at O pee (aca Sa ethan odtarrs sisi i A absnent ay ive 08 4 worote to oltvwrtst bas seta g OOS ; iD doce. Th brea Date behead “is oe va thas av hats - gaat eae trie ye Waco Anranye es igeel? i eo Tel irk? D +5 | 1 Ya lees ha PAST ot | ee me . av) ort ie weratveued ne . f { uP 7 fed eS sare bis 7 iva bdisetectws): Oat Ww 34 ‘ae i, Laem — 2 - é } i . al) see ae “4 fi G7 > eulNViLeo te? Being nae Tne pair ' ° 7. tees oy 7 t ( ‘ ee | ach. TTY dys 4 Pat ‘ t a 1 ser bale? % , a Fox er J if hd as 1 ibe = . it 4 ‘ ~~ re a) ; i : - 7 (mh ‘ne ‘ , q = 47) ; Mai ‘ , ; 7 ae 8 j vs eee he bt * l x Aipaewi ' ov" ,: re tJ w t : We fore bY T y iv i “ vi Bié ; 74 rit 5d m - ~a Py eu f say 4 aot _ ae n ‘ iy i ; f Hh. 7am J on = meri . F tere i A, be 4 ‘ ’ Lan ty ue ‘ ; bal He! a faab att ilies A | : Foe eee fe =: 9 oa pe eee ; Ire aces i iF : BOLL Pret 244 ‘ . SEHR ey std von: Walthy te Ge UNH! Rotman itt y : ial 4b 16 BAYH Pep vi tic eget ee eB ho ee i NEU Ra MAAK lhl. CateMMGA PS Re % CCAM TC RUCK Ea iM tS an ad ({ Ihnen ah SS ee St esas ects 5 : g <5 ee. ee as : Beef Z = iS Feast = “tr ik = Wana eee ewes eigen syese *, See LS f peer es: oF = 7 tS: S Py. f sick 5 oy. ssi =f ae aa tet ae ina anus tt ett ¥ i eee | i 7 Se = me Sets vse: F he tt . at ney et: rhs 8 ¥ > dilitadate Haett: wF bh Mpg i’ ot ate ed 2 ~~ - 3 oes gee m . < nets 4 : ~~ a = S o> — poe creer : RL od cee as. hag? t Ve-seerr a ——— S sae = ——- % 5 Sheer ‘. 25 :, 1 Ar Lede Ebest Testy et" Bs iti eure ES Bact Me bot op ore ape x aes a Sat oc eRe 7 xeeipeerst aj & ) de) Hasty Sr i ' H 1 HPs tilt iss ‘ aa uf saul ce i} i te nt i uy oa bik my