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N.,E' Ie an was Ch - e -f o.f -th. e :eurea LI of Coasta I) c. Dat f~ iI s i t io)n , i) e bor a h E ~ F i. ac: k D i rec CA o -the D i vi sion of Beac~hes a-rd Shores, and: Dr, I.EC[ton J. G i ss;encanner *the xecCu-tivye D)ir,_-C tcr of t1he F i.or ica De)ear tne-nt of Nla-turaLI R es uce D e bor ah E. ., F l. a C k< D j r e c: t o r I)Iv I s C on- o)f F!ea c: hes aF n d 9 hor)ies 0 C t o b e r 1 9":3 Property Of CSC Library U. S. DEPARTMENT OF COMMERCE NOAA COASTAL SERVICES CENTER 2234 SOUTH HOBSON AVENUE CHARLESTON, SC 29405-2413 C-M CONTENTS Page ABSTRACT................................ 1 INTRODUCTION ..1........................... DISCUSSION.............................. 9 Effect of Shore-Breaker Height ..................10 Effect of Shore-Breaker Height and Bed Slope ...........14 Effect of Shore-Breaker Height, Bed Slope, and Equivalent Shore-Breaking Wave Steepness Parameter..............16 A NOTE ON THE SHORE-BREAKER TYPE....................18 CLOSURE ................................22 REFERENCES...............................23 ON THE DETERMINATION OF WHEN WAVES BREAK IN SHALLOW WATER 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 Prediction of when shore-propagating waves become unstable and break is generally considered to be a function of three factors: 1. wave height, 2. bed slope, and 3. wave steepness. Various relationships have been proposed for the prediction of shore-breaking occurrence. These relation- ships have either been founded on relatively small data bases or have been designed to predict for specialized conditions. In this work a significantly large sample of wave information, including both field and laboratory data, is used to evaluate popularly us~d predictive relation- ships. Graphical and statistical results support the originally proposed relationship of McCowan (1894), where db =1.28 Hb5 as the best prediction of shore-breaking occurrence. INTRODUCTION Not only is the determination of the point at which shore-propagating waves become unstable and break a basic task in many coastal engineering problems, but breaking also identifies the point at which a major change in wave behavior occurs. It is fundamental that the "... ultimate limitation of any wave theory based on potential wave theory is given by the condition at which the wave break s" (Madsen, 1976, p. 79). In terms of the destructive potential of waves, studies have shown (Miller~et al. 1974a, * ~1974b; Miller, 1976) that breaking and broken waves result in greater impact pressures than the more symmetrical, less deformed waves in relatively deeper water. Dune and bluff erosion accompanying shore-incident storm and hurricane impact is a function not only of the storm surge but also of runup and setup produced by final shore-breaking activity. It becomes clear, therefore, that certain conditions at and following breaking necessary for successful coastal engineering design solutions require specialized types of predictive procedures. One of these is the prediction of the point at * ~~which shore-breaking occurs. Results from available research indicate that the water depth, bed slope and wave steepness constitute the major variables influencing wave stability. Considering these variables, ocean waves are generally thought to become unstable and break as follows. Deep water conditions represent one extreme where the water depth and bed-slope do not influence waves passing above, and the waves become unstable and break because they become critically steep. Cri tical steepness occurs under forced wave conditions (Mooers, 1976; Balsillie et al. 1976) wherein significantly high wind stresses produce instability and breaking. Such waves commonly appear as spilling type breakers, often called white caps. The other extreme occurs in nearshore shallow water depths. The water depth is the most critical factor influencing wave stability. The bed slope and wave steepness, although they have been considered to be influential, are apparently of secondary importance. Breaking-wave types, while they may include spilling breakers, also may include other generally recognized types such as plunging, surging and collapsing breakers. Between the two extremes, the stability of waves is apparently dependent an all three parameters, each of which may play a significant role. Breaker type is commonly of the spilling type. Where breaking occurs in nearshore shallow water depths, resulting in littoral zone activity, the author has adopted the terminology, shore- 2 breaking waves. In deeper water, wave instability is simply termed as breaking. The work presented herein is concerned with conditions at the shore- breaking position, and includes newly acquired field shore-breaker data (Balstllie and Carter, 1980) in addition to the re-evaluation of existing field and laboratory data. The goal of the work is to identify and evaluate criteria useful for least equivocal coastal engineering design solutions which require determination of conditions that induce shore-breaking. PREVIOUS WORK In deep water Michell (1893) found that the maximum limiting wave steepness above which breaking occurs may be given by: L0o7 max where H and L are the deep water wave height and length, respectively, or where from small amplitude wave theory L. = g T2/(2ff), by: (') -14 (2) gTmax 1 Using the data of the Beach Erosion Board (1941) and the fifth order Stokes-Levi-Civita solution (Levi-Civita, 1924), for forced wave conditions given by equations (1) and (2), then: (~) = 0.64 (3) max 3 as discussed by Balsillie (in manuscript) where H' is the height of the deep 0 water wave crest lying above the still water level (SWL). The maximum steepness for progressive waves in any depth of water is given by Miche (1944) as: (H)~~~~~~~~~~~~~~~~~~~~~~4 (L)m - tanhx (4) McCowan (1891) found that at the shore-breaking position the stability of the wave profile is primarily dependent on the water depth to wave height ratio, according to: db db =1.28 (5) Hb subsequently supported Munk (1949), where the subscript 'b' refers to conditions of the water depth and wave height at the shore-breaking position. Equation (5) was developed using solitary wave theory where the entire wave lies above the SWL. However, recent work (Weishar, 1976; Weishar and Byrne, 1978; Hansen, 1976; Balsillie, in manuscript) suggests that: - 0.84 (6) Hb where H' is that portion of the shore-breaker crest lying above the SWL (see definition sketch of Figure 1). Investigation subsequent to development of Equation (5) (Iverson, 1952a, 1952b; Galvin, 1968, 1969; Collins and Wier, 1969; Weggel and Maxwell, 1970; 4 Figure 1. Definition sketch of wave parameters at the shore-breaking position (plunging shore-breaker). Weggel, 1972a, 1972b; and Mallard, 1978) indicate, at the shore-breaking position, that in addition to the water depth, there is a residual dependence on the bottom slope. Galvin (1969) found that: db = 0.92 (7) Hb tan ab > 0.07 where ab is the bottom slope leading to shore-breaking, and: Hb 1.4 - 6.85 tan ab (8) tan ab <.0.07 which are both referenced to the mean water level (MWL) rather than to the SWL used in this work. Galvin (1969) suggests'that for tan ab on the order of from 0.05 to 0.1 the SWL is higher than the MWL (Figure 1) by a factor of 0.04 Hb , and where tan ab is about 0.2, by a factor of 0.08 Hb. Collins and Wier (1969) suggest that: db -1 Hb = (0.72 + 5.6 tan ab) (9) Hbn ab (9) and Mallard (1978) concludes: db 0.997 (10) Hb 0.73 + 2.87 (tan b)10) Equations (7) through (10) are plotted in Figure 2. For equations (8), (9) and (10), values of db/Hb are close for tan ab less than about 0.01 and may 6 db Galvin (1969) '"- - Collins and Wier (1969) N N .-.-.- Mallard (1978) , o I I I,, i I I I , ! I I I I, ,,, 0.001 0.01 .1 1.0 tan cab Figure 2. Comparison of results of predicted db/Hb using the relationships of Galvin (1969), Collins and Wier (1969), and Mallard (1978) given by equations (7) through (10) in text. 7 satisfactorily predict db/Hb where tan ab < 0.1. However, where tan ab is greater than about 0.1 (i.e., ab > 6o), the use of equations (7), (9) and (10) is not recommended. An additional parameter which may significantly influence shore-breaking was investigated by Weggel (1972a, 1972b). The parameter, Hb/(g T2), termed the equivalent breaker steepness parameter, whose derivation is given by Battjes (1974, p. 469) can be introduced into Weggel's empirical result, to yield: r H ~~-1 Hbmax =[c3 - c gT2 (11) Hb max Hg- T where -19 tan ab \ C1 = c2 9 1.0 - e (12) 2 2 in which c2 4.462 m2/s = 1.36 ft2/sec in unit-consistent terms, and: / 1 / ~-19.5 tan ab\ c3 1.56 1.0 + e195 tan b (13) As noted, Weggel's relationship is concerned with predicting a maximum design shore-breaker height. Even so, based on physical reasoning he has incorporated limiting constraints for extreme values of the bed slope and equivalent shore-breaker steepness parameter. He suggests that where the bed slope approaches infinity (i.e., a vertical wall), the minimum value of db/Hb will be one-half the theoretical value (based on the sum of the incident and perfectly reflected wave components) wherein c3 approaches 1.56 8 and the value of (db/Hb)min approaches 0.64 when the equivalent shore-breaker steepness parameter approaches zero. It is interesting to note that for a vertical slope, the maximum value of Hb/(g T2) will be 0.0356. For deep water conditions where according to the Michell (1893) condition (Ho/Lo)max :1/7, then Hb/(g T2) max 1/(14fr) = 0.0227 which is 36% less than Weggel's value. As the bed slope approaches a value of zero (i.e., a flat bed), Weggel assumes that the effect of the slope should diminish and the theoretical value of McCowan shall be more nearly valid,' hence the value of cI approaches zero while c3 approaches 0.78 and the value of (db/Hb)max becomes 1.28. DISCUSSION In this section several commonly recognized relationships for predicting where shore-breaking will occur are discussed and evaluated, progressing from- simplest to most complex. A major problem encountered when dealing with this subject is that one deals with quite small differences between input variables and parameters when, in fact, the errors and variability encountered when measuring hydraulic conditions in the littoral zone are often comparable. The measurement of wave heights and water depths at shore-breaking, whether in the laboratory or field, is invariably difficult, if only because one is dealing with a moving wave form. Laboratory measurements involve highly sophisticated types of sensors for measuring surface elevations. However, no sensors are capable of determining the point at which a wave shore-breaks .that is, for spilling, plunging, surging, collapsing, etc.. ..... shore-breaker types, which is, ultimately, dependent on visual recognition. Hence, it appears realistic to accept that errors creep into laboratory results. In addition, shore-breaking seldom occurs in precisely the same water depth because of complicating factors such as 9 wave reflection and wave interference, and where the bed is mobile by changes in the subaqueous morphology caused by sediment transport processes. Field measurements encounter similar problems. In the'field, however, more than one wave train is usually present which can introduce additional wave interference problems. Because of the variability of breaker depth and the magnitude of impact forces associated with shore-breaking waves in the field, the use of laboratory-equivalent sophisticated sensor equipment is not possible for the higher waves. Hence, less sophisticated measurement techniques can potentially allow additional error to be associated with the results. However, by dealing with the higher field waves, relative to the smaller laboratory waves, field error may in cases be offset o& minimized. From the above discussion it becomes evident, given state-of-the-art measurement techniques, that error will invariably be associated with shore-breaking data. While it is recognized that scientific progress requires the development of measurement techniques which reduce the amount of error, existing data already evaluated plus that developed in the interim periodically deserves re-evaluation. It becomes not only important that as much data as possible is available but that the range in data is as large as possible. General characteristics of the field and laboratory data used in this study are listed in Table 11. To date no study known to the author has included such a large data base with such a large range in values. Effect of Shore-Breaker Height The most straightforward and first known successful relationship predicting where shore-breaking will occur was suggested by Mc~owan (1894), given by equation (5). The relationship is evaluated in Figure 3 using 418 simultaneous measurements of d b and Hb (i.e., 167 field and 251 laboratory data pai-rs). Equation (5) is superimposed upon the data of Figure 3 and 10 Table 1. General characteristics of the field and laboratory data used in analyses. i~ Al. arVI (1i904) 63 0.0200-0.01ii54 () .61A2. 96 2 0. /2,-.456 .3 10. 91: 0.00201 0.01103 Scriptiv (1945) Spec. tleas!32 0 . I 7I A 4 ~9 .....1. . . .... .... Sc~ripts- (1945S) Leica Type .1 .:5 0O A1:5 9 i A" 23 4/` - 5 4-1 'i ' 2 A13.7 O0,001030006I 2 5cr I p P (394"5) I e it:a Typ-e XI1 1 0. 049 2 ('0 II I '8 A ,uO 3.. '2 '7 0 -13.0 0.001007. 0 0057,'2 Fta I(,sil I e - anvd tar t er (1A980) 26 0.005 .-o ).'5 O 0')5 0 5 41A 0.i -I0.6 A I 38-- 8.57 0. 00040O.00560O P~utniah, Hunik and Trayltar (194'?> NatunralI Sand Beach 7 .6 6 0 08 8 0, A 43 O I i 2 0 Q, 9 A1 0 A. .32 O.045 1.) . 0A15 49 *~~~~~~~~~~~~~' A 044 O .0 b4a 01,~G 0 0 O0 0 .0 5; 1 9 1; 2.22 0.000950O. 001A3 9 * S 4 0.241 0 0.., 0 10? 0 0*3' C) 1 >11 0 72- 1.22 0.00459--b..0i673~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(8 ~i5 0i2 0.O0 0.( Smooth i Met al anrd Cloric ret e 4 ( 0. A 00 0.0419 O 0 O '3 0 6''0. 0980 O ;99-- I . 91 O0.001O 280 O0760 *~~~ ~~~~~~ ~~~~ O 0.139 O0,O061 --0) .4015 0.0/.0A131 O. (3- 1.35 0.(00342-0,(.01259'. * S ~~ ~~~~~~~5 02 60 O0.06S1 01 04 OQQ7 -.1A89 0.1 S A1.27 0.0O0O4 1 1--O.A6513 A /A - i -fic:h Pea Gra-vel. 4 0.09(1 0.0 3-7-0.079 O.0513. 0. AII 0 .95, 1 .99? 0. 00O09 50O.00893 � N ~ ~~ ~ ~~~~~~~~ A ~ .4:3 O0.061- 0).1A0I 0. 0 -9 0 .1 4~ 1. 0(3 2. 32 0.00(1)O127 O0.003:38B4 Hunik (1949), 1cr ke Icy ExpiA 5 ((.39 O (B(7( A100 O0.A118-0 1 45- A . 05-- 1 . 98 0.00226 0 00926 5 0.054 0 .06(3-0.092 O0.063-O .1 AI 0i 0B6- A ~97 0.001 79-0).(126.9 6 O 0.07 2 0.08C-2-O0.099 0. 071A-0 1 26 O0.90- 1.97 ). 0026-Ab0 12"'47 Hnkial (1949), Ei.E:.Ii. ExPsA I 0. 030 0031 --. 054 0, 043-0.08EJ1 0.75- 1 .03 0. 0032"?-Ob. O0.59 9 15 0049 O 4 3- 13 '3 0 . O4 3 -0 1 87 0.~73--- 1 .0(8 0.00-446-0.0OA137 1 3 .159 O .034--b121A 0. 044-0.170O 0, 74-,- A .09 0.~00376-0. 01039? tvei-son (1952? 13 0. 020 0 .050.O1 21 A.0650.1 56 (0.90-- 2. 65 0.001 42- O. 00939 115 0.03:3 0. 053-0--. 1 26 O.070--b 1 55 A - 25- 265 0. 00080--0 .00990 A19 O 0.O05 0 0. 043b A128 0 04'9?- 0. A165 0.7 '?4- 2.4 b.001(- A95(0. 0iA081A A165 0. A 00 O O 4 9-b A322 0.( 04 3-bO. A1:37 0.8- 2.50 0.001A19--b.01092 Mar. ir san ad C.vacikc (1953)3 :3 0.020 006--b .0(3 l4 0,70-A3. 1I 0. Th" 2.62 0O03O~ 009 39 3 0 A100 O00O7.3 0 A113 O407 70A129 A1.0 --2.5 0.b11A9--0.01092 Boawen , Tveian avid Siomoans (196(3)0 It 0 ( l.08 0.04 --0.)127 O .0O42 -0.O097' 0.82- 2. 37 O0.002 A4-(0.0O() I9 S Komar anld Eimmoinns (1968). A l 00 0.136 O0O 7 4-O. 66 0.09 .'?.212 A .1 4-- 2.~37 0.~()0 A9,3-O-0.009158 1 4 0 . 07( O O030- 0.1VW 0.0410--Owl 64 0.81A -- 2.37 O00O01 " '3 -0 A.01036 AIA 0.036 ). 042-0.1A47 0. 043A0.16 2 0.81-- �.:s71 0.001 1'14--0.009EP? 9 0.1015 0.035i-0.A"170 0. 03e)-I0.17O 0.81-- 2. 37 0 . 000 913-0.01123 We.ggel. and Ha xwe (1 (1 97() )3 O .057 0.089--b. I 62 0.0 070. A169 A .26-- 2. 05 0.00216--b. E 00(30 Van Darn (j9733)3 4 0.022- 0.13 -0.16ej 0.~1 13 -0.208 A . 65 - 4.8E 0.00062 0.00622 4 0.040 0.1A9A0A162 0).1 --A A1 A1? .65-- 4 .13 0 . O0 05 3--O.0O0O6 )7 4 O 0.0(3:3 0,. A08--D 15'6 O.093-0. 154 A 6 4.8 0.00040.-I0.00585 Iiuhr [haisc-n- anid svenlds~en (19,79? AO3 16 0.0292 0.~ 0 4 3--bO. 1 29 0.047--b. 1 49 0.3:- 3,33 0. 00013-0.0OA092 liepoe!crted by Munk (1949)i 2repar ted by Gau~hKoaear anid Ma-tti'( 3973) ; (it Ii (: A fi xetl- bds. I 10 o LABORATORY DATA FIELD DATA + Gaillard (1904) a Scripps (1945), Leica Type I + v Scripps (1945), Leica Type II +. x Scripps (1945), Spec. Meas. A+,/ o Balsillie and Carter (1980) + 1.0 - b 0 (m) (mn) 4-db = 1.28 Hb 0.01- 0.01 0.1 1.0 10 Hb (m) Figure 3. The McCowan equation superimposed on the data of Table 1. Data include 167 fielgure 3. The McCowan equation superimposed on the 251 laboratory points. 12 appears to successfully represent the central trend of the data. It is to be noted that the relative magnitude of scatter about the line appears to be approximately equivalent for both laboratory and field data. Statistical methods provide better assurance of the goodness of fit. Commonly used regression techniques employ predictive regression which upon regressing x on y minimizes only the sum of the squares of the horizontal distances from the points to the fitted line. However, enhanced assessment of the goodness of fit can be determined using functional regression procedures discussed by Ricker (1973). He suggests, based on the work of Teissier (1948), that the central tendency of the data might be more adequately determined by finding the line which minimizes the sum of the products of both the vertical and horizontal distances of each point from the line. The slope of this line forced through the origin (i.e., x = 0, y = 0) is given by: y m : 2 (14) Plus and minus limits, s', of the central line fitted by equation (14) to the 95% confidence interval limit, are given by Ricker (1973), according to: 2~~~~~ s:_+ t1/2(n- 2) n m2 (15) where n is the sample size, r is the Pierson product-moment correlation coefficient, and t /,(n - 2) is the Student's t value for n - 2 degrees of freedom. However, Equation (14) as it applies to the data of Figure 3 is influenced more by the larger data values. For instance, for the field data, m =1.426 and for the laboratory data, m = 1.168, but for the data combined m =1.425. Hence, even though there are more laboratory data than field data, the laboratory data influences the overall slope by only 0.07%. This condition introduces a significant problem since it is the smaller magnitude laboratory data which probably represent the more precise information due to the more sophisticated measurement techniques used. For this reason, functional regression techniques are applied separately to the laboratory and field data and the fitted slope, m, determined as a weighted average. Resulting statistics are listed in Table 2, where for the data of Figure 3 the weighted average fitted slope is 1.271. In fact, if the listed values of m for the laboratory and field data and the sample size of the field data are held constant, then only 3 more laboratory measurements would be re- quired to result in the Mc~owan coefficient of 1.28 for the laboratory and field data combi'ned (i.e., weighted average value). Plus and minus limits of the McCowan equation (i.e., where m = 1.28) are, from equation (15), 1.251 and 1.309. While it could bel viewed that the statistical results listed in Table 2 for equation (5) are fortuitous, two considerations should be noted: 1. both the laboratory and field data samples are significantly large, and 2. recognizing that statistical methods cannot always provide definitive answer~s to specific numerical problems, the graphical approach can be used to allow the reader to observe and render his or her own .judgements. Effect of Shore-Breaker Height and Bed Slope Of the previously introduced equations which consider the bed slope in addition to the shore-breaker height, the equation of Mallard (1978). ... i.e., equation (10). ...has been selected for evaluation. Reasons for 14 Table 2. Statistical results -- goodness of fit of functional regressions. I. . IField Da'. : ..rbbt c) V. Y 1)a ta At t l) at a Ii VI II i .r 5, D I I V V I , Y t1::CCOAN REA-AT I.INlI-11P F i gue 3) Iiidepel)(ewit Fit 1 67 1 .426 091i25 0090 251 1 i~ C .16 E8776e 0). 070 4113 i .27i I? .6 36 0037 M c C .) waI)~ E -ual aio1n V1 67 i .28 0.Y1 215 007 0 251 1 .2"'I 0 . 0./ 0 .0515 418: `1 .20 0. 636 0 . 029 MALl ARD RELATIONSIIIP (Figure 4) Inodependent F it 1 313 i . 1 75 0.403 0., 108 251 I i05: 0.84 13 .7 2 38IS9 i .098* - 09 447 0.039 Cor r er, t ioan 138 i . 09 0. 8403 0. 099 251 I .098 .841i3. 0 . 065 389 1 .0()98 0.9447 0 .0)35 WCE.RELATIONSHIP (Figure 5) Cia ~~~~~~~~~~~Independent Fit 1 26 1 .254 ). 871i3 0.1I09 251 1. 159 0. 9258 0.055.317 i I19* 0.9552 0038 or re c tin j)V 126 1 .191 0, 87 i3 ). 09?2 251 1 .191i .9 2 581 0.046 377 I .191 09 "552 0033 *Weigh ted aver age upon wh ich asseisseent o:)f I he u Mr.:Co:way eqjua tion (a .s-umni ng 1 .271 and. I .283 are es-senit ialt y eviuiva Lent ) and the cc r ec-ted: eq.ua tions, are based, selection are: 1. the wave heights and water depths are referenced to the SWL, and 2. Mallard's analysis used a significantly large sample (i.e. n = 213) including seven laboratory studies and one field study. Application of equation (14) to the available data indicates that Mallard's equation underestimates the central tendency of the data by 10% (i.e., from Table 2, 1001[l-(s,/s')] = 100[1-(0.035/0.039)] = 10.2%, where Sc is associated with the corrected equation and so is associated with the original equation). The corrected equation is given by: db 0.997 l = 1.098 0.73 - 2.87 (tan cb) (16) Hb illustrated in Figure 4. Plus and minus limits of the coefficient correcting the Mallard equation (m = 1.098), from equation (15), are 1.063 and 1.132 (Table 2). Visual inspection of Figure 4 illustrates that scatter associated with equation (16) is somewhat greater than for the McCowan equation illustrated in Figure 3. Statistics listed in Table 2 support the visual comparison, wherein the correlation coefficient associated with the McCowan equation has a larger value and the degree of relative scatter aboutithe fitted regression lines, given by equation (15), is 17% less than that associated with Mallard's corrected equation. Effect of Shore-Breaker Height, Bed Slope, and Equivalent Shore-Breaking Wave Steepness Parameter While (Weggel, 1972a, 1972b) introduced physical constraints in order to yield more reasonable results for extreme conditions (equations (11) through 10 I I ' ' o LABORATORY DATA FIELD DATA + Gaillard (1904) * Scripps (1945), Leica Type I , Scripps (1945), Leica Type II - o Balsillie and Carter (1980) 'L~~ + 1.0 - + -00' ++ C~~~~~~~~~~~-. ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~-o O . I6d ~~~~ /o /~~~~~~~~~ o0 1 - I * 0.01 0.1 1.0 10 Predicted db (m) Figure 4. Measured water depth at shore-breaking versus the depth predicted from the corrected Mallard relationship given by equation (16). Data include 138 field and 251 laboratory points. 17 (13)), he was primarily concerned with predicting the maximum shore-breaker height. In this work, however, the intent is to determine the water depth at shore-breaking for a shore-breaker more closely representing average height conditions. Application of equation (14) to the data yields a corrected equation given by db _ = 1.191 [c3 - cl J (17) illustrated in Figure 5, where c1 and C3 are given by equations (12) and (13) and c2 remains as specified earlier. Plus and minus limits of the corrected equation, from equation (15), are 1.158 and 1.224. Equation (17) results in better predictive precision than equation (11) for average conditions by 13% (i.e., from Table 2 lOO[l-(s'/s;)] = 100l[1-(0.033/0.038)] = 13.2%). Visual comparison of Figures 3 and 5 illustrate that there is less scatter for the McCowan equation than for equation (17). The correlation coefficient for the McCowan equation is higher than for equation (17), and the relative scatter about the fitted line is 12% less for the McCowan equation. Similar comparisons also show that Weggel's modified equation results in somewhat better predictive precision than the corrected Mallard equation A NOTE ON THE SHORE-BREAKER TYPE Weishar and Byrne (1979) report a statistically. significant difference in the average value of db/Hb for plunging and non-plunging shore-breaker types (note that Weishar (1976) originally defined the non-plunging waves 18 o LABORATORY DATA +++- FIELD DATA + +.t.. + Gaillard (1904) V2 a Scripps (1945), Leica Type I '': '" v Scripps (1945), Leica Type II r- o Balsillie and Carter (1980) 1.0- - 2 - ., oo ' 0.1 -- S.-, - ' '0..01 0.1 1.0 10 Predicted db (m) Figure 5. Measured water depth at shore-breaking versus the depth predicted from the modified Weggel relationship given by equation (17). Data include 126 field and 251 laboratory points. 19 o Spilling ,A Spill- Plunge * Plunging 0.8- d~~~~~~~~~~ 0.6- db (m) 0 0.4- A/ - - o gdb/Hb = 1.28 Af 0.2- - 0 J 0 0 / I I I I I I 0 0.2 0.4 0.6 0.8 Hb (m) Figure 6. Illustration of the lack of dependence of shore-breaker type on the depth of water at shore-breaking. Data from Balsillie and Carter (1980). 21 CLOSURE With the recognition that conditions at shore-breaking are,.,complex and that the difference between d b and Hb is relatively small, then one may expect a certain amount of inherent variability (which can bommonly be significant relative to the small difference between d b and H b) and, hence, scatter in the measured data. For this reason, a significantly large sample of data, characterized by a wide range in values, has been used to reassess commonly used relationships for prediction of shore-breaking occurrence. Relationships evaluated proceed from simple to complex incorporating, progressively, the wave height, bed slope, and wave steepness. The more complex relationships such as that in the form of the corrected Mallard equation which incorporates the wave height and bed slope (valid only where the bed slope is less than about 0.1), and that of the modified Weggel equation which incorporates wave height, bed slope and wave steepness, result in greater scatter, both statistically and graphically, than does the McCowan equation which incorporates the wave height only. This result does not absolutely discount the applicability of the more complex equations (bearing in mind any noted domain limitations). It does., however, on the basis of existing data and its associated variability which may be expected from existing measurement techniques, suggest that until refined measurement methods are found which may be used in both the field and laboratory, the McCowan equation provides the best predictor of db/Hb 22 as spilling shore-breakers). For plungers the average value of db/Hb was 1.15 (n = 70), for non-plungers 1.47 (n = 46), and for the data combined 1.27 (n = 116). Weishar obtained his data from a photographic study in the field. The data base was obtained from three film runs over a one hour period. Assuming that wave conditions may change significantly within a 20-minute period (Balsillie and Carter, in manuscript), then at a minimum the data represent three wave trains. Judging from shore-breaker type frequency plots (Weishar and Byrne, 1979, Figure 6, p. 494) three to four wave trains may have been shore-incident during the experiments. Hence, assuming a maximum of four wave trains, their data may actually represent a maximum of 12 points. In other words, it may have been more reasonable to calculate the average value of db/Hb for each wave train, rather than averaging all the data. The data of Balsillie and Carter (1980) were manually measured by two individuals using a staff at the shore-breaking location (i.e., where the front face of the wave crest was vertical for plunging shore-breakers, and where the top of the wave crest began to break and "foam" for spilling shore-breakers). Crest and trough heights were measured for thirty shore-breakers taking care that the measurements represented a single wave train, from which average values of db and Hb were determined. Results for 26 sets of data, representing 780-individual crest-height measurements, are plotted in Figure 6. Spill-plunge shore-breakers occurredwhere, in the longshore. direction, a combination of shore-breaking characteristics were noted to consistently occur along the wave crests. Such mixed shore-breaker type occurrence per wave crest probably corresponded to local alongshore differences in bed slope, and diffraction effects, etc. It is suggested from Figure 6 that db/Hb applies equally to shore-breakers regardless of the shore-breaker type. 20 REFERENCES Balsillie, J. H., in manuscript, Wave crest elevation above the design water level during shore-breaking. Balsillie, J. H., and Carter, R. W. G., in manuscript, Observed wave data: the shore-breaker height. Balsillie, J. H., et al., 1976, Wave parameter gradients along the wave ray: Marine Geology, v. 22. Balsillie, J. H., and Carter, R. W. G., 1980, On the runup resulting from shore-breaking wave activity: Shorelines Past and Present, Department of Geology, Florida State University, Tallahassee, FL, v. 2, p. 269-341. Bowen, A. 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Swift, eds.), p. 65-87. 23 McCowan, J., 1894, On the highest wave of permanent type: Philosophical Magazine, Edinburgh, v. 32, p. 351-358. Miche, R., 1944, Mouvements ondulatoirs des mers en profondeur constante on decroissant: Annals des Ponts et Chausses (University of California at Berkeley translation, Wave Research Laboratory, Series 3, Issue 363, 1954). Michell, J. H., 1893, On the highest wave in water: Philosophical Magazine, v. 36, 5th Series, p. 430-437. Miller, R. L., Role of vortices in surf zone prediction: sedimentation and wave forces: [In] Beach and Nearshore Sedimentation (R.A. Davis, Jr., and R.L. Ethington, eds.), Society of Economic Paleontologists and Mineralogists, Special Publication No. 24, P. 92-114. Miller, R. L., et al., 1974a, Field measurements of impact pressures in surf: Proceedings of the 14th Coastal Engineering Conference, Copenhagen, v. 3, chap. 103, p. 1761-1777. Miller, R. L., et al., 1974b, The effect of breaker shape on impact pressures in surf: Fluid Dynamics and Sediment Transport Laboratory, Department of Geophysical Sciences, University of Chicago, Technical Report No. 14. Mooers, C. N. K., 1976, Wind-driven currents on the continetal margin: [In] Marine Sediment Transport and Environmental Management (D.J. Stanley and D.J.P. Swift, eds.), p. 29-52. Morison, J. R., and Crooke, R. C., 1953, The mechanics of deep water, shallow water, and breaking waves: U. S. Army, Beach Erosion Board, Technical Memorandum No. 40. Munk, W. H., 1949, The solitary wave theory and its application to surf problems: Annals of the New York Academy of Sciences, v. 51, p. 376-424. Putnam, J. A., Munk, W. H., and Traylor, M. A., 1949, The prediction of longshore currents: Transactions, American Geophysical Union, v. 30, no.3, p. 337-345. Ricker, W. E., 1973, Linear regression in fishery research: Journal of Fisheries Research Board of Canada, v. 30, no. 3, p. 409-434. Teissier, G., 1948, La relation d'allometrie: sa significance statistique et biologique: Biometrics, v. 4, p. 14-18. U. S. Army, 1941, A study of progressive oscillatory waves in water: U. S. Army, Beach Erosion Board, Technical Memorandum No. 1. Van Dorn, W. G., 1978, Breaking invariants in shoaling water: Journal of Geophysical Research, v. 83, no. C6, p. 2881-2988. 24 Weggel, J. R., Jr., 1972a, Maximum breaker height: Journal of the Waterways, Harbors and Coastal Engineering Division, ASCE',ino. WW4, Proc. Paper 9384 p. 529-548. jWeggel, J. R., Jr., 1972b, Maximum breaker height for design: Proceedings of the 13th Coastal Engineering Conference, chap. 21, p. 419-432. Weggel, J. R., Jr., and Maxwell, W. H. C., 1970, Numerical model for wave pressure distributions: Journal of the Waterways, Harbors and Coastal Engineering Division, ASCE, no. WW3' Proc. Paper 7467, p. 623-642. Weishar, L. L., 1976, An examination of shoaling wave parameters: M. S. Thesis, College of William and Mary, Williamsburg, VA. Weishar, L. L., and Byrne,R-. J.-, 1979, Field study of breaking--wave characteristics: Proceedings of the 16th International Conference on Coastal Engineering, Hamburg, p. 487-506. 25 ~~R~E~ JI~~i ANSON LEXINGTON COLUMIAN AIKEN ~~~~~~~KERSHAW AA~AIKEN~ .d U1 /DEN CHESTERFIELD RICHMI 1~~~~~~~~~~~~~ I -.Z CHE RAW li BRNWEtLL-..C'. RCANN R I, R L HOE( SUMTER~~~~~~~~~ - -I ORANGEBUR'\H0 -;-II ,-.- --- Hi ALLEN- AD �, AT N BE NA BAONO MFORENCE R I S I~~~~~~~~~~~~*~LRNCEA-LMERT ISAN L R R LU/uGETHOUSE HA Mp6 A ( Fi' L. LY I - V E~~~~~~~~ii~~~~FLL EAH SOUH AOLN RJHRY>> ~ SCAE IN MILES GE. 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