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El THE CRNf.TRMTI NO Ti--lE WAVF. 1-11 I CY'- DRN U ~~~~~~~~~~~~James H-4 .I~a iAs I L I. i e 0 Ana i~y I ~/R.c~par c~i-iSec:t ion E [:1~~~~~ ~~ ii f a a s .1Cos a I. 1)a Aa A (::,:I u i s i t i on I- Divi si or of ~e :i sand Shcores o F t -~* u aDep)a r-Ife of r Narturlv . Resoutr-c:es U o EEl 1 AND TF-lCRIES COASTAL ZOwNE' ri ~~~~~~~INFORMATION CENTER r El B a c : I- es- an ri dh oII r a.. R eas o u ~- c: o? -n -te r~~~~~l I.~~~~~~~ or i S c ia a. t a Un i va i i i-v o an~~~~~~~~~~~d o. ci:: E. r ivarIo-n- I men-t ~ ~a I. Feu l.atio-I on Na 1: 1 on a UC: t: 1 :7 rl ( (j y I:j ym. :A f ~ nI 0, grvla1 a1 fCom af -i c a.. cs- Cone cia ra a mn 1- Ac: e MaIi a-1 am n cit -r ) + Ocean~~ic Ort::siflIerc:t 11 F, I. c: V. i ci a OF f,. i C: 01, C o aJ,, -t a. 11 a n- a g (am en J. rit1I o r i di a :o) q. a r- man jf o.1 F En1: -- v i r- onmo(an t a 1. IRa u l.a ~-FI tic:-n r U GB 1:7 1. orY cl.ia D ep a r- - if men I oF N a- -I u r a I. R e sour) I.Y c: as U. 459 E . F56b E Th i s 401 p r es e n *t s v-,n u ifter i c: a 1. c",o It It i o n f - t he p -c- i, c t on 0 ae he(: i g h t- b e h a v o r i u r i n t he o. . i -t o r a Li,. wave(-- p ro(jc.ess o sho-re-brcak i< ng ., It i s basic support method o iLogy reqLui red j-n -the deve i.p(I-ent of a muLtIpiesoebekinjwave Itr a n Sf or fna t i on-- mto d e L c de scr i b.)edc i n s ub se -:1 i ie n t woark1 T he tmo r 1k dve r i b ed h e re i n c::v on st i t u t es p a rt i a I. f ut If i L Lmfte n t o f con trYactuLia l o b 1. igat i o ns w it h -the F e der a l C oas--ta 1. Z on e ain aa g ecm- en-- t P rc qr a m l (oastlal. Zone Manf-agafement Act of 9"72., as ame-nuady) thrmv~jh -the F cr i-da COffice of Coasta I ManagemIent gub Jec i-to~ pr ovis i ons of :otratCil-37 enit it .ed "En-rigne er i n oru,:~ct E-nh anc emene-t Pr ografr' U~nd er prov is-i ons of DNR con-tract C,"1037 , -t his work1 was- rev iewed bry -the BeacheS arid S'hores~ R:esour ce Center --, rInStH-k.t ute of .3c ienc~e a-rd F'ulAi ic: Affa irs, For i cla State -niv e TS iu. tdTe ioc ufnent hias b.een adopt ca as a ?.ea c: h(---S and Sho-res Tic h-ni caLI and Des cign Memor a ncdum i n a ccordci-e inc - wit h prov on- of Ch a plt c1 6B--K333, F I.or id a Ad min is t ra tivye Code. At thIne t*i me of submissio-n for contract utwaLI comp Li anc-e James Fl., :Eais i LI.ie was -the contra c:t manager and Ad i-jn i str ator of the AntaLys i sResearch S'ect ion, HalI. N.. Bean was ChI--ief of -the B~urea u of CoataI Dat-a PfcLu is i-i on, Deborah .F iac 1-< Director of- -the Divisiont of B~eaches- and Shores, arnd Dr. E iton, J. 'Gissencianner t'he Exec utivye Di rec tor of -the Fi.or i da iDepar tmentl of N~at uralI. Resources. Deborah E . F La c k, D irecc:tor D ivis io-n of B~eaches anid 9ho-r-._, November, j 98 ft~p~tYOfe CSc irZ U.- S. DEPARTMENT OF COMMERCE NOAA COASTAL SERVICES CENTER 2234 SOUTH HOBSON AVENUE CHARLESTON , SC 29405-2413 CONTENTS ABSTRACT ..1............................. INTRODUCTION ..1.......................... ANALYTICAL APPROACH..........................2 TERMINAL BOUNDARY CONDITIONS..................... 7 INITIAL BOUNDARY CONDITIONS......................12 TRANSFORMATION OF H/Hi ..........15 CLOSURE................................16 REFERENCES ..............................32 THE TRANSFORMATION OF THE WAVE HEIGHT DURING SHORE-BREAKING: THE ALPHA WAVE PEAKING PROCESS by James H. Balsillie Analysis/Research Section, Bureau of Coastal Data Acquisition, Division of Beaches and Shores, Florida Department of Natural Resources, 3900 Commonwealth Blvd., Tallahassee, FL 32303. ABSTRACT As waves begin to shore-break, the wave crest often tends to rapidly increase in height, reaching a maximum at the shore-breaking position. This phenomenon, termed alpha wave peaking, is primarily dependent on the wave steepness and may be predicted according to: Hb~~~ H t Hi = 1.0 - 0.4 In tanh 100 where Hi is the incident wave height, T is the wave period and Hb is the shore-breaking wave height. Transformation of H/Hi, where H is the local wave height, is given by: - H d ~~~~~0.7 Hi -Hi b2tnh~ in which db is the water depth at shore-breaking, d is the local water depth, and solutions for t, and (2 are developed in the text. I NTRODUCTI ON Generally, as waves near the shoreline, the height of the waves first tend to decrease and then to increase rapidly just prior to shore-breaking. This increase in the wave height beginning just prior to and reachitng a maximum at shore-breaking can, even over gentle bed slopes, be "... remarkably sudden ..." (Munk, 1949) and accompanied by progressive distortion and asymmetry of the wave in profile view, has been termed alpha wave peaking by Balsillie (1980). Alpha wave peaking has been observed as characteristic activity in shore-breaking wave mechanics from many studies (Scripps Institute of Oceanography, 1944a, 1944b; Putnam, 1945; Munk, 1949; Iverson, 1952; Stoker, 1957; Kinsman,!1965; Byrne, 1969; Clifton, Hunter and Phillips, 1971; Komar, 1976; Nakamura, Shiriashi and Sasaki, 1966; Van Dorn, 1966; Buhr Hansen and Svendsen, 1979; Balsillie, 1980; etc.). Two mechanisms occur during shore-breaking: 1. the transformation of H/Hi and 2. the transformation of H'/H, where H is the local wave height, H' is the amount of the wave lying above the design water level (DWL), and Hi is the wave height at the initiation of alpha wave peaking. Pertinent wave height parameters are illustrated in Figure 1. The first of the above mechanisms defines the subject of this paper, the second is the subject of a companion paper (Balsillie, in manuscript). ANALYTICAL APPROACH Of presently available theories, Cnoidal wave theory seems to have gained popularity for predicting the transformation of shoaling waves in shallow water. Svendsen and Buhr Hansen (1976) discuss the applicability of Cnoidal theory using developments of Skovgaard et al. (1974), for the deformation of waves up to shore-breaking (the theory is applicable where d/L0 < 0.10 or d/L < 0.13, seaward of which they recommend the use of Airy wave theory). Svendsen and Buhr Hansen state "... even though cnoidal theory seems to predict the wave height variation reasonably well, no information can be deduced from that theory (or any other theory) about where breaking occurs." Cnoidal wave theory is not, however, simple nor expedient to apply. First, it requires tabulated coefficients for realization of solutions. Second, and more critically, it requires that the 2 A B d-19- Hb OWL ______fl/___ _<____________ dt THti I_ dti tan t Figure 1. Pertinent nearshore wave height parameters; parameters at A illustrate conditions at the initiation of alpha wave peaking, those at B represent conditions at the shore-breaking position. local wave length is known when, in fact, the wave length is seldom known short of additional theoretical calculations for an approximation. It is the intent of this work to provide a practicable solution to the problem. Mathematical descriptions developed in following sections assume that the initial wave height, Hi, and wave period, T, are known. In an earlier study, Balsillie (1980) used H0or Hm as indicators of Hi where H.is the specified deep water wave height, and Hm is the wave height measured in the constant depth portion of a laboratory wave channel. In many laboratory investigations it has been found that initial wave characteristics are in the range where H.and Hm and the resulting value of Hi are approximately equivalent. Generally, this occurs for waves with higher wave steepness values. However, due to refraction and frictional effects, etc., where the generated wave steepness is small, Hi can become significantly less than H or H.' The importance of the latter phenonmenon is illustrated by an example from the laboratory data of Putnam (1945) in Figure 2. Therefore, in this study only data which describe the continuous transformation of waves across a known bathymetry, from which Hi can be determined, are considered. In the earlier work of Balsillie (1980), it was reported that the alpha wave peaking parameter, H b/Hi, is dependent on the equivalent wave steepness parameter, H.i(g T2), and the bed slope, tan a. The influence of these 1~~~~~~~~~~~b parameters are included in ensuing analyses. In addition, the continuous transformation of H /Hi during shore-breaking is investigated. First, however, determination of where alpha wave peaking is initiated and terminated require identification. Where possible, both field data and laboratory data are considered. It is to be noted, however, that laboratory information by far constitutes the bulk of available data. However, since the study of Balsillie, new laboratory data have become available (see the table). In addition to the study of Putnam (1945), results are reported by 4 1.2 ---Constant Depth tan OL = 0.054 IB _----Deep Water - b -=0.00963 1.1 - :0.010H =0.00887'-~ �/ Directlon of Wove co00 Trave I - 1.2 1.0-� - I . O _-- _ \ I o Hm-0 . 0.9- _ a Corrected ap 0 o "_'0 0 ' -I HI 0.7- 1 - T=O . 965 s. b . 28; I -0.9 d/H Figure 2. Illustration of wave transformation from deep water to shore-breaking, where the alpha wave peaking process is given the notation ap. p ,~~~~~~~~~~~~~~~~~~~~~ Table of data used in analyses. tan ab T Hb i di b (s) H. H1 d T2 g T FIELD DATA Wood (1970, 1971) 0.0556 3.49 1.58 4.23 2.00 451.2 286.2 LABORATORY DATA Putnam (1945) 0.072 0.865 1.04 2.34 1.76 78.8 74.4 ." 1.15 1.29 3.13 2.27 162.0 131.0 1.22 1.29 2.69 2.67 189.4 147.0 1.50 1.66 4.77 3.17 367.5 236.8 1.54 1.58 4.45 2.92 381.1 267.0 1.97 1.86 5.01 3.11 745.7 462.8 0.054 0.86 1.08 2.22 1.66 85.3 78.6 0.965 1.11 2.50 1.93 112.7 103.8 1.34 1.48 3.96 2.96 279.3 213.7 1.50 1.58 4.39 3.16 386.8 278.7 1.97 1.84 5.94 3.47 927.6 557.9 step 1.05 1.16 ---- ---- I26.2 108.5 1.09 1.19 ---- ---- 142.6 119.4 1.35 1.31 --- ---- 237.4 181.5 1.50 1.60 ---- ---- 375.0 235.0 1.98 1.86 ---- ---- 997.8 441.0 Buhr Hansen and 0.0292 0.833 1.32 3.64 2.55 207.0 156.3 Svendsen (1979) 1.00 1.13 2.57 1.66 104.2 91.8 1.00 1.21 2.95 1.89 153.1 126.4 1.00 1.35 3.92 2.66 258.4 191.8 1.25 1.30 2.50 '1.51 162.9 124.3 1.25 1.37 3.15 2.04 229.2 167.6 1.25 1.46 3.95 2.42 395.3 270.1 1.67 1.46 3.57 2.29 283.2 194.8 1.67 1.42 3.57 2.15 302.5 209.8 1.67 1.47 3.84 2.27 340.3 233.5 1.67 1.48 4.13 2.41 389.1 261.3 1.67 1.66 5.05 3.09 675.7 408.2 2.00 1.69 4.47 2.66 608.6 359.3 2.00 1.95 5.50 3.01 1048.2 537.8 2.50 1.84 5.22 2.63 874.9 479.7 2.50 2.20 5.95 2.75 1531.4 704.1 3.33 2.39 6.67 3.08 2544.5 1069.2 Singamsetti and 0.025 1.28 1.27 ---- ---- 170.8 134.9 Wind (1980) " 1.55 1.25 ---- ---- 173.8 138.8 0.050 1.038 1.21 ---- ---- 162.4 134.5 1.55 1.20 -- --. 162.9 135.5 1.55 1.28 2.43 1.56 216.0 168.2 1.55 1.27 2.68 1.22 210.2 165.8 0.100 1.035 1.11 ---- ---- 160.3 144.8 1" 1.555 1.25 ---- ---- 173.8 139.3 0.200 1.038 1.35 ---- ---- 157.6 116.7 Based on 400 consecutive wave measurements; step had a slope of 0.444, post-step slope was 0.009. Buhr Hansen and Svendsen (1979) and Singamsetti and Wind (1980) to represent a wide range in bed slope conditions. TERMINAL BOUNDARY CONDITIONS The terminal boundary of alpha wave peaking is defined as the shore-breaking point. Galvin (1968) provides a comprehensive description of the various types of shore-breaking waves. Of the principal types, however, spilling and plunging shore-breakers constitute those more commonly applied in design considerations. The shore-breaking point of a plunging breaker is defined to occur when the front face of the wave crest becomes vertical (Figure 1); the shore-breaking point of a spilling breaker occurs when the top of the wave crest becomes unstable and water and foam slides or spills down the front face of the crest. Two parameters identifying termination of alpha wave peaking are db/Hb and Hb/Hi. The first parameter may be straightforwardly given by the McCowan criterion (McCowan, 1894; Munk, 1949; Balsillie, in manuscript), illustrated in Figure 3, and given by: db = 1.28 Hb where Hb is the shore-breaking wave height, and db is the water depth at shore-breaking. Enhancement in precision of db/Hb prediction has been attempted by incorporating the bed slope and wave steepness. Balsillie (in manuscript) found, however, that equation (1) as yet constitutes the most reliable equation with accuracy limits to the 95% C.I. of � 0. 029 Hb. Additional analysis indicates (Balsillie, in manuscript) that equation (1) applies equally well to both spilling and plunging shore-breakers. 7 10 � LABORATORY DATA FIELD DATA .+% ~ + + Gaillard (1904) + a Scripps (1945), Leica Type I + 7 Scripps (1945), Leica Type II + x Scripps (1945), Spec. Meas. o Balsillie and Carter (1980) 1.0 - + x x (m) db o 0 db .28Hb 0.01 0.01 0.1 1.0 Hb (m) Figure 3. Relationship between the water depth at shore-breaking, db, and the shore-breaking wave height, Hb (from Balsillie, in manuscript). The second parameter, Hb/Hi, describing the relative height attained as a result of alpha wave peaking, is more difficult to quantify. It is, however, considered to be a terminal boundary parameter since Hi is understood to be specified as input. In the previous work published by the author (Balsillie, in manuscript), both wave steepness and bed slope were indicated to affect alpha wave peaking. However, based on new data, and subsequent and considerable dimensional analyses and testing, the following relationship can be recommended: H. - 1.0 - 0.4 In tanh 100 (2) illustrated in Figure 4. Additional attention was given to the bed slope; no refinement was found to improve equation (2). In fact, equation '(2) is evaluated for a wide range of bed slope conditions; scatter might easily be attributed to the difficulty in identifying where shore-breaking occurs. Subsequent work by the author suggests that the bed slope is probably more instrumental in influencing the type of shore-breaker that will be produced. Due to scale differences between axes of Figure 4, wave steepness data from the table are plotted in Figure 5, where now the axes are comparable. Dividing both sides of equation (2) by g T2 yields: g T g -~ 1.0 -0.4 In tanh 100 g H1T g T2 g T2 go n which is superimposed upon the data to show good agreement.- 9 2.5 tan ab - T 0.025 _ O 0.0292 2.0 - � 0.050 _A 0.054 Hb - 0.0556 Hi_ o 0.072 -- 0.10 0 � 1.5 - V 0.20 j . 1- ~+ step o V O HL = 1.0- 0.4 In[tanh -00 HgT Figure 4. Relationship for prediction of the shore-breaking wave height from the initial equivalent wave steepness parameter. tan ab 3I 0.025 103- 00.0292C - O 0.050 A 0.054 * 0.0556 ye1 o 0.072 * 0.10 0 H -1b v 0.20 g- T2).+ step / V equation (3) 2 _ a*'. 102 I 103 Figure 5. Relationship between the equivalent wave steepness parameter evaluated at initiation of alpha wave peaking and at shore-breaking. INITIAL BOUNDARY CONDITIONS Various investigators (e.g., Stokes, 1880; Galvin, 1969; Dean, 1974) have conducted studies to delineate constraints of breaking. It was Munk (1949), however, who considered in some detail wave peaking in the shore-breaking process. He applied the Rayleigh assumption (Eagleson and Dean, 1966) given by: ci Ei = cb Eb (4) where c is the phase speed (shallow water condition only, where wave period is conserved and no energy is lost) and E is the total wave energy, and the subscripts i and b refer to conditions at initiation of alpha wave peaking and at shore-breaking, respectively. Using Solitary wave theory, Munk (1949) suggests that: - = - (5) db :H~i Equation (5) is plotted in Figure 6, from which the non-representative nature of the equation is apparent. However, based on what is known about shore-breaking wave activity, it is possible to develop a representative mathematical relationship. It has been demonstrated (Balsillie, in manuscript) that the water depth is the most influencial factor causing shore-breaking. Therefore, the solution may be dependent on db/Hb for which there is a solution (Figure 3), and from which it follows that we wish to solve for di/Hi rather than di/db of equation (5), where di/Hi must remain larger than db/Hb. Of the factors remaining, there may be dependency on the incident wave steepness and/or bed slope. Since required input parameters include the incident wave height and period, the incident wave steepness parameter is considered first. Data from the table are plotted in Figure 7, and the following equation is suggested: 12 4 i ~3 M �o o o - d, O d 2- 4 / db /l~i'/ 1- 3/4 0 ' I . I 0 1 2 3 4 Hb Hi Figure 6. Evaluation of Munk's (1949) parameter for determining the initiation of wave peaking in the shore-breaking process (symbols as for Figure 5). 13 8 11111 d I .r I 1 tan ab () = 1.28- In [tanh (5 )/ 6 - � 0.0292 T V 0.050 o A 0.054 1 3 ( 0.0556 i 4 - 0.072M D0 7 32 <__ - ()mn 1.28 1 I I I ' I I I I I 1 1 l 2 _l 103 102 3 .1(y1 Figure 7. Relationship for the prediction of the relative depth at which alpha wave peaking is initiated. dbL < i (6) Hi - Hb -2 I n tanh (65 (6) in which it is assumed that db/Hb = 1.28. The equation represents a significant range of bed slopes (i.e., 0.0292 to 0.072) for data from a variety of sources. TRANSFORMATION OF H/Hi In addition to specification of the boundary conditions, it is desirable to be able to predict the continuous behavior of alpha wave peaking. Such behavior, for example, may be important in determining horizontal and vertical impact loading potential of shore-breaking waves, and in sediment transport prediction. Data tabulated by Putnam (1945) and Buhr Hansen and Svendsen (1979) are used to determine the nature of the transformation. The general equation is given by: H Hb ah / db~ 0.7 Hi Hi 2 tanh l H (7) where DI is a coefficient which determines where the transformation of H/Hi begins, given by: e 2.7183 : = : (8) (di/Hi) - (db/Hb) (di/H)' - (db/Hb) in which (di/Hi)' is given by equation (6), e is the Naperian constant, and t2 determines the local peaked height during shore-breaking given by: 15 Hb H - 1.0 (9) in which Hb/Hi is given by equation (2). Equation (7) is evaluated (dashed curves) in Figure 8 for various bed conditions. Data from Singamsetti and Wind (1980) are not plotted because the authors did not tabulate the transformation information. Only four data points are available for the field data of Wood (1970, 1971) and are not plotted. Data from Putnam (1945) for the step slope could be plotted, but would require considerable license in estimation to determine the value of di (since the waves began to shore-break on the step slope over which measurements were widely spaced). In many of the plots of Figure 8, the laboratory data suggest that db/Hb is closer to unity than to a value of 1.28. From Figure 3, however, it is evident that laboratory data "tend" toward a lower value. This may be symptomatic of difficulties in determining precisely when small laboratory waves shore-break (i.e., since this must be visually observed and cannot be measured). The terminal boundary condition of db/Hb = 1.28 is, therefore, maintained. Scatter of data relative to equation (7) is noted in some of the plots. Overall, however, the. shape of the transformation appears to be well represented by equation (7). CLOSURE The inapplicability of Airy wave theory to represent shallow water waves is well known. Cnoidal wave theory has been recommended (Svendsen and Brink-Kjaer, 1972; Skovgaard et al. 1974; Svendsen and Buhr Hansen, 1976) where d/Lo < 0.10 or d/L < 0.13. However, Cnoidal theory is not easy to apply. First, it requires the use of tabulated ellipitical function. Second, and more problematic, it requires that the local wave length is known when, 16 2.0 Hi 0.00134 - 1.8 gT2 T = 1.97 s. - 1.6 / _/ _ / - 1.4 / _X/ - 1.2 / _ . . / HI .........._'-..........._-------------- 1. o a 4 3 2 1.6 g-T ', - 1.4 d/H tan = 1.540.072; . shore-breaking occurs at where d/H = 1.28. _ 7 ~/ ___ ___ ___ _ ___ _ _ _ __________ __ _ _ - - - - ---- - - ,-! 1.0 d/ H tan ab = 0.072; shore-breaking occurs at A where d/H = 1.28. 17~~~~~~~~~~~~~~~~~~~~~~~~~ - - ~~~0.00272 -1.6 g T2 / -1.4 /1.2 0.~~~~~~~~~~~~~~~~~~~~~~~'-1002 142 T ~~~~~~~~~~~~1.422 gT2 T 1.1�- 1. 0~~~0 8 ~~7 65432 1 d/14 Figure 8a. (cont.) 1 8 j , | i J 1.4 I = 0.01269 gT2 - 1.2 T = 0.865 s. H I I I , I _ 8 7 6 5 4 3 2 1 d/H Figure 8a. (cont.) 19 2.0 -1.9 I' Hi = 0.00108 / - 1.7 gT / l T= 2.0s / - 1.6 / �. / t- 1.5 - 1.4 / - 1.3 I / -1.1 .............. - .....--: --~---- ---- ~--- - t- -- - - - - -- - - ----- ---- ---4--- 1.0 H1 - 1.5 2 0.00259 / -1.4 T =1.50 s / -1.3 / *' -1.2 - ~/ /I- 1.1 -- -- -- -- -- - ------,- - - -- --- - - - ----- ---- --- - -- ---- 1..0 * I ! , . I I I , 8 7 6 5 4 3 2 1 d/H Figure 8b. Alpha wave peaking; data from Putnam (1945) for tan ab = 0.054; shore-breaking occurs at A where d/H = 1.28. 20 I I ~~~~~~~~~~~~~1.6 H 0.00358 11.4 gT I T = 1.34s. - Hi~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ,12 -1. I ---~~~~ - ------ I--- -~- . 0.00887 1 gT T =0.95s. 2 -------���- 1.0 =0.01172 - gTZ T 0.86s. 42 d,/H ~ ~ ~ ~ / *~~~~ I --~~~~~~,,,-- --r I ,* *,** 0 8 7 6 5 4 3 2 d /H Figure 8b. (cont.) 21 - 2.2 Hi 'I = 0.000393 / -2.0 gT /I T = 3.33 s / i- 1.8 / // / / * - 1.4 2 1 / d/H / / / ' - 1 ,4 / 22 ? - 1,2 � * / 8 7 6 5 4 3 2 1 d/H Figure 8c. Alpha wave peaking; data from Buhr Hansen and Svendsen (1979) for tan ab = 0.0292; shore-breaking occurs at z where d/H = 1.28. 22 H 22 = 0.000653 - 2.0 9T2 | T = 2.50 s. I ~/ 'H / / e _/ =-1.4 . / ,,1 / _ * -/_ d/H Figure 8c. (cont.) 23 23 I I ~~~~~2.0 Hi 0.000954 -8 gT T =2.0 s./ -1.6 0/~~~~~. d/H~~ Figure~~~~~ /c -1.2t. 24~ I~~~~~~~~~~~~~~~~~~~~ I H. I: Hi g T2 0.001143 T = 2.5 0s. /� -1.4 F '-~~~~~~~~~~~~~~~~~~~~~~~~1.2 H 0/ ~~~~~I / -1.2 Hi ,,,,i T 1.667 s. 4 8 7 6 4 3 2 1 dim Figure 8c. (cont.) 5 - 0,00164 -1.5 gT T =2.0 s -1A / ~~~H d/ H~ Figur 8c. cont7 020 0.00253 gT I-1.4 T :1.25s./: T~~~~~~~~~~~~~~~~~' 1.07 1.2 ~~~~~dH1 Fi~gure 8c 0cn.005) 27~~~~~I:-. / 1.4 = 0.00294 gT2 T = 1.667 s. / -1.2 Io 1 -1.4 = 0.00335 f1 gT2 / -r H T =1667 s. / -1.2 Hi /. /, aHii- 1.4 -2 = 000353 / T = 1.667 s. / -1.2 * /I Figure 8c. (cont.) 28 28 H1~~~~~~~~~~~~~~~~~~~~~~ A 1.4 =0.00387 * -~~~~ gT 2 T 1.Os / 1.2 H~~~~~~~~~~~~~i -1.4 HI = 0.00436 T U .33s H1 I- 1.0 Hi~~~~~~~~~~~~~~~ 1A -1 0,00483 T =0.833s.' -~~~~~~~~~~~ - - - -1.2 8~ ~~~ ~~~~~~~~~~~~ 7 6 I d /K * ~~~~Figure 8c. (Cont.) 29 -1.4 Hi = 0.00614 gT T = 1.25 s. / i- .--------------------------------- ----------- 1. - - 1.4 .i2 - 0.00653 gT2 i H_ / 1 ...................................... -~---- ....~ r'- 19 - 1.4 -- =0.00960 _i 1.0 s. i - 1.2 I - ---- 1.0 I I I I I I ! 8 7 6 5 4 3 2 1 d/W Figure 8c. (cont.) 30 in fact, it is seldom known. It has been the purpose of this paper, therefore, to provide a practicable solution to these problems, which represents wave height behavior during the shore-breaking process. Two basic processes during shore-breaking have been identified as: 1. the total height of the wave tends to increase reaching a maximum at the shore-breaking point, and 2. the amount of the wave crest lying above the still water level tends to increase during the process. The latter process has been addressed in a companion paper (Balsillie, in manuscript). The former process defines the focus of attention in the present paper. The point at which shore-breaking is initiated (i.e., incipient shore-breaking point) is given by equation (6), the wave height at the shore-breaking point by equation (2), and the wave height transformation by equation (7). The equations are dependent on the equivalent wave steepness parameter, H/(g T2), rather than d/L. Hence, dependence on the wave length is removed. The wave period is a readily available variable and, in addition, i.s conserved at least until shore-breaking occurs. ACKNOWLEDGEMENTS The work of L. J. 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