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Front Cover 1 Front Cover 2 Title Page Page i Page ii Letter of transmittal Page iii Page iv Foreword Page v Page vi Table of Contents Page vii Page viii Page ix Page x Shorebreaking wave height transformation Page 1 Page 2 Page 3 Page 4 Page 5 Page 6 Page 7 Page 8 Page 9 Page 10 Page 11 Page 12 Page 13 Page 14 Page 15 Page 16 Page 17 Page 18 Page 19 Page 20 Page 21 Page 22 Page 23 Page 24 Page 25 Page 26 Page 27 Page 28 Page 29 Page 30 Wavelength and wave celerity during shorebreaking Page 31 Page 32 Page 33 Page 34 Page 35 Page 36 Page 37 Page 38 Page 39 Page 40 Back Matter Page 41 Page 42 Back Cover Page 43 Page 44 

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State of Florida Department of Environmental Protection David B. Struhs, Secretary Division of Resource Assessment and Management Edwin J. Conklin, Director Florida Geological Survey Walter Schmidt, State Geologist and Chief Special Publication No. 41 ShoreBreaking Wave Height Transformation by James H. Balsillie and Wavelength and Wave Celerity During ShoreBreaking by James H. Balsillie Florida Geological Survey Tallahassee, Florida 2000 'V NO C Printed for the Florida Geological Survey Tallahassee, Florida 2000 ISSN 00850640 LETTER OF TRANSMITTAL Florida Geological Survey Tallahassee Govemor Jeb Bush Florida Department of Environmental Protection Tallahassee, Florida 323047700 Dear Govemor Bush: The Florida Geological Survey, Division of Resource Assessment and Management, Department of Environmental Protection is publishing two papers, in this, Special Publication No. 41: "Shorebreaking wave height transformation" and "Wavelength and wave celerity during shorebreaking". The first paper identifies a numerical methodology for transforming wave height during the shorebreaking process. The second paper is a companion paper, which provides a numerical methodology for transforming the wavelength and wave celerity during the shorebreaking process. Both papers provide new technology based on measured data (field and laboratory) and, therefore, do not constitute theoretical approaches currently employed by coastal practitioners. Practical uses for the results presented in these works include the determination of dynamic impact pressures necessary in the design of coastal structures, increased precision in identifying breaking wave parameters leading to increased precision in predicting sediment transport, sediment budgets, beach and coast erosion during storm/hurricane impacts, to name but a few. Respectfully yours, Walter Schmidt, Ph.D., P.G. State Geologist and Chief Florida Geological Survey FOREWORD Whether rocks of our planet (or others for that matter) are igneous, metamorphic, or sedimentary, the final form of the rock deposit is dependent upon forces which led to its formation. The former we refer to as response element (i.e., the final form of the deposit), the latter as force elements (i.e., forces which led to deposition and induration). Force elements include wind, hydraulic forces, gravity, pressure, temperature, chemical reactions. This work is concerned with sedimentary deposits. The majority of geologists have been involved in studying and describing insitu unconsolidated or lithified sedimentary deposits. Unless fossils are present, they will have little idea of the conditions leading to deposition. Moreover, fossils may not indicate specific conditions of transport and deposition. While relatively small in number, there are, however, geologists who have adopted an expanded Earth Science perspective, and have sought to study first currently occurring forces (e.g., water waves, water currents, wind, gravity, etc.) and then describe the resulting sedimentary deposits (e.g., based on their granulometry and/or bedding characteristics). This does, after all, constitute an underlying and basic Geological or Earth Science concept proposed in 1785 by James Hutton, termed the Principle of Uniformitarianism. It states: "The present is the key to the past." It is from the latter school of geology that this work surfaces. It has always been an errand of keen professional interest to the author for two reasons: 1) it allows for the study of natural environmental processes using the robust scientific method that, in combination, 2) provides results that can not only be used to interpret ancient sedimentary deposits and rocks, but provides information that can have significant value for current and future environmental concerns. After all, if Hutton's principle is true, then the corollary must also be true that ... the present is the key to the future. The work presented in this study is concerned with numerically quantifying wave characteristics. Alaska, California, Florida, and Texas have the longest oceanfronting shorelines in the United States. Florida has approximately 1,253 miles of shoreline that front directly upon the Atlantic Ocean, Gulf of Mexico, and Straits of Florida. Moreover, annual average wave energy levels range from near zero for Florida's Big Bend Gulf Coast, low to moderate for the remainder of the Gulf Coast, to high along Florida's Atlantic shores. Only Alaska experiences such wave energy variability, not only because of its large waves but because of zeroenergy occurring along icewedged shores during a significant portion of the year. Alaska does not, however, have a coastal population of any significant proportions. California has a significant coastal population and large Pacific Ocean waves, but not the wave height variability. Coastal Florida and Texas also experience tropical storm and hurricane wave impacts that far exceed annual average wave energy levels. Texas does not, however, have Florida's wave energy variability. Florida, then, does not only have a significantly long shoreline, but it has a large wave energy variability and a significant level of coastal development. Such wave energy variability requires those of us in Florida to be sensitively precise in wave characteristics and wave energy assessment and application. We must remind ourselves that it was and is marine forces (i.e., primarily waves) that, for the most part, formed and form Florida's surficial sediment configurations. James H. Balsillie February 2000 CONTENTS SHOREBREAKING WAVE HEIGHT TRANSFORMATION Page . 1 A BSTRA CT .................................................. . . . . . . . . . . . . . . . . . . . . . . 1 CLASSICAL WAVE THEORIES ........................... WAVE HEIGHT GROWTH DURING SHOREBREAKING .......... Terminal Boundary Condition ....................... Initial Boundary Condition ......................... Transformation of H/Hi ........................... WAVE CREST ELEVATION ABOVE THE DESIGN WATER LEVEL . . Terminal Boundary Condition ....................... Initial Boundary Condition ......................... Prediction of H'/H Transformation ................... CONCLUSIONS ..................................... N O T E . . . . . . . . . . . . .. . .. .. . . . . . . REFERENCES ....................................... NOTATION ........................................ . . . . . . 2 . . . . . . 4 . . . . . . 7 . . . . . . 9 . . . . .. . 10 . . . . . . 1 1 . . . . . . 15 . . . . . . 16 . . . . . . 19 . . . . . . 2 1 ........... 23 ........... 23 ........... 28 .2o , , ... .. .28 LIST OF TABLES Table 1. Field and laboratory data used in analyses. .......................... 5 Table 2. Laboratory data of Putnam (1945) for evaluation of wave height above the D W L. .. .. .. ............................. ......... ....... 17 LIST OF FIGURES Figure 1. Pertinent nearshore wave parameters; those at A illustrate conditions at the initiation of alpha wave peaking, those at B represent conditions at shorebreaking for plunging waves. ................................ 2 INTRODUCTION Figure 2. Wave transformation from deep water to shorebreaking, where the alpha wave peaking process is denoted by b. .............................. 4 Figure 3. Relationship between the water depth at shorebreaking, db, and the shorebreaking wave height, Hb .................. .................. 7 Figure 4. Relationship for prediction of the shorebreaking wave height from the initial equivalent wave steepness parameter. ........................... 8 Figure 5. Relationship between the equivalent wave steepness parameter evaluated at initiation of alpha wave peaking and at shorebreaking. ................. 9 Figure 6. Evaluation of Munk's (1949) parameter for determining the initiation of wave peaking in the shorebreaking process). ........................ 10 Figure 7. Relationship for the prediction of the relative depth of water at which alpha wave peaking is initiated. ................................... 10 Figure 8. Alpha wave peaking predictions using equation (7) plotted as the solid curves. Vertical dashed lines represent d/H = 1.28 = db/Hb. Measured data (solid circles) are for bed slope data of 0.072 and 0.054 from Putnam (1945), and bed slope data of 0.0292 of Buhr Hansen and Svendsen (1979). .... 12 Figure 9. Dependence of design pier deck soffit elevation on wave crest elevation above the design water level (DWL) for two commonly applied theoretical approaches. ................................................ 14 Figure 10. Relationship between the wave height at shorebreaking, Hb, and the trough depth just preceding the crest, db; field data are from Balsillie and Carter (1980) and Balsillie (1980), laboratory data are from Iverson (1952). ... 15 Figure 11. Relationship between the water depth at shorebreaking, db, and the total water depth at shorebreaking, de; data sources as for Figure 10. .......... 15 Figure 12. Relationship between the total depth at shorebreaking, dc, and the wave trough at shorebreaking, dt; data sources as for Figure 10. ............ 16 Figure 13. Relative depth equations for initiation of alpha wave peaking at (di/Hi), and at initiation of wave height increase above the DWL at (d/H,)'. ........... 17 Figure 14. Wave steepness parameter at (d1/Hi)' relative to the wave steepness param eter at (d/H ). ........................................... 18 Figure 15. Relationship for prediction of the incident value of H'/H. ............. 18 Figure 16. Transformation of H'/H using equation (21) plotted as solid curves. Vertical dashed lines represent d/H = 1.28 = db/Hb. Measured data (solid circles) are from Putnam (1945). .................................. 20 Figure 17. Predicted H' from equation (21) versus prototype Beach Erosion Board wave tank data reported by Bretschneider (1960). ...................... 21 Figure 18. Predicted H' from equation (21) versus Lake Okeechobee hurricane wave data reported by Bretschneider (1960). ............................. 21 Figure 19. Ratio of wave crest elevation above the design water level to wave height from equation (21). ...................................... 22 Figure 20. Region of applicability for various wave theories and for the Alpha Wave Peaking methodology of this work. ................................ 24 WAVELENGTH AND WAVE CELERITY DURING SHOREBREAKING Page A BSTRA CT .......................................................31 INTRO DUCTION ................................................... 31 DISCUSSION AND RESULTS ........................................... 31 Terminal Boundary Conditions .................................... 33 Initial Boundary Conditions ...................................... 34 Wave Celerity and Wavelength Transformation During Alpha Wave Peaking .... 36 CONCLUSIONS ....................................................37 REFERENCES ................ ......................................37 NOTATION ..................................................... 39 TABLE Table of statistics relating measured and predicted wavelengths at the shore breaking position. ............................................ 34 FIGURES Figure 1. Illustration of wave transformation from deep water to shorebreakinig, where the alpha wave peaking process is given the notation a ... ........ 32 Figure 2. Evaluation of relationships for prediction of the wavelength at the shorebreaking position. ................................. ....... 35 Figure 3. Comparison of predicted and measured wave speeds at the point of initiation of alpha wave peaking; predicted data are from equation (4), measured data from Hansen and Svendsen (1979). ................... .. 36 Figure 4. Illustration of wave speed attenuation during shorebreaking using equation (15) for various initial wave steepness values.. ................. 37 SHOREBREAKING WAVE HEIGHT TRANSFORMATION by James H. Balsillie, P. G. No. 167 ABSTRACT As waves begin to shorebreak, the wave crest rapidly increases in height, reaching a maximum at the shorebreaking position. This phenomenon, termed alpha wave peaking, is primarily dependent on the wave steepness and may be predicted according to: S1.0 0.4 In nh (00 , where H, is the mean incident wave height, T is the wave period and Hb is the mean shorebreaking wave height. For waves considered in this work, Hb ranged from 1.04 to 2.39 times as high as H,. The relative depth of water, di /Hi defining the point at which alpha wave peaking begins (i.e., the initiation of the shorebreaking process) is given by: l = in ta nh 65 ,J H, H, 2 g; 72 Transformation of H/H, during alpha peaking, where H is the local mean wave height, is given by: HH, H , in which db is the water depth at shorebreaking, d is the local water depth, and solutions for 0, and P2 are developed in the text. Many coastal engineering design solutions requiring consideration of wave activity can be accomplished only if the crest elevation of the design wave(s) is known relative to a design water level (DWL). From analysis of field and laboratory data, it is determined that at the shorebreaker position approximately 84% of the wave crest lies above the DWL. The amount of the wave that lies above the DWL during shorebreaking may range from about 0.5 to 0.84. Transformation of H'/H, where H is the local wave height and H' is the amount of H lying above the DWL during alpha wave peaking, may be predicted by: HI Wb 40 tanh 43! j0 H H H Hb where Hb' is the amount of Hb lying above the DWL and the solutions for P3 and 04 are developed in the text. INTRODUCTION wave height, beginning just prior to, and reaching a maximum at shorebreaking can, Generally, as waves approach the even over gentle bed slopes, be shoreline, the height of the waves first tend "...remarkably sudden..." (Munk, 1949) and to decrease and then to increase rapidly just accompanied by progressive distortion and prior to shorebreaking. This increase in the asymmetry of the wave in profile view, has been termed alpha wave peaking by Balsillie (1980). Wave peaking has been observed as characteristic activity in shorebreaking wave mechanics (Scripps Institute of Oceanography, 1944a, 1944b; Putnam, 1945; Munk, 1949; Iverson, 1952; Stoker, 1957; Kinsman, 1965; Byrne, 1969; Clifton and others, 1971; Komar, 1976; Nakamura and others, 1966; Van Dorn, 1978; Hansen and Svendsen, 1979; Balsillie, 1980, 1983b, 1983c; etc.). Two mechanisms occur during shore breaking: 1. the transformation of H/H,, 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 for plunging shorebreakers, where at the shorebreaking position, the front face of the wave crest becomes vertical (to eventually curl, form an air pocket, and fall into the wave trough fronting the crest). Where spilling shorebreakers occur, the shorebreaking position is reached when the top of the wave crest becomes unstable and aerated and turbulent water slips down and across the front face of the crest. Other shorebreaker types such as surging and collapsing waves have also been identified and defined (e.g., Galvin, 1968; Balsillie, 1985). The first of the above mechanisms defines the first subject of this paper, the latter is addressed in the second section. Wavelength and wave celerity behavior during shorebreaking are the subjects of separate works (Balsillie 1984, 1999a). CLASSICAL WA VE THEORIES It is to be noted that existing wave theories have relevance in offshore waters where wave distortion and boundary effects are less constraining compared to conditions in nearshore waters. Application of quantifying methodology for the nearshore zone, however, has been sorely lacking, ostensibly because of the distorted nature of waves and confining boundary conditions. It would, therefore, appear appropriate to recapitulate some of the more conspicuous shortcomings of "classical wave theories" relevant to the problem at hand. From an historical perspective, one should have at least, some appreciation, of early theoretical work of such researchers as Gerstner (1802), LeviCevita (1924), Struik (1926), Gaillard (1935), and Mason (1941). For a more detailed account of early theorists see the bibliography of Mason (1941). Figure 1. Pertinent nearshore wave height parameters; those at A illustrate conditions at the initiation of alpha wave peaking, those at B represent conditions at shorebreaking for plunging waves. Subsequent to this early work, wave theories have surfaced which currently are in general use. Following are descriptions and discussion of such theoritical approaches. Linear (Airy) Wave Theory suffers from the basic problem that the wave is divided evenly above and below the still water level (SWL). This condition clearly does not hold for waves in shallow and transitional water depths or, for that matter, waves undergoing breaking in any depth of water. Also, the calculating algorithum for wavelength in any depth of water is, at best, an envelope fit giving but approximating results (Bretschneider, 1960; Balsillie, 1984a, 1999b). Stokes' Wave Theory attempts to more nearly depict the distortion of the wave profile about the SWL, with the degree of distortion increasing for higher order theory. However, its applicability does not extend into shallow water nor the shallower part of transitional water depths. Dean (1974; fig. 7) demonstrates the same region of validity for Stokes' 5th Order Theory which includes Stokes' 3rd and 4th order region of Figure 1. Cnoidal Wave Theory has been popularly used for predicting shallow and transitional water wave conditions. However, depending upon the investigator, there is variability in defining the region of validity for the theory. Laitone (1963) notes its validity for d/L < 0.125. Svendsen and Hansen (1976) discuss the applicability of Cnoidal theory using developments of Skovgarrd and others (1974) for the deformation of waves up to shore breaking (the theory is applicable where d/Lo < 0.10 or d/L < 0.13 (see NOTATION section at the end of the paper for definitions), seaward of which they recommend the use of Airy wave theory). Svendsen and 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." Similarly, Skovgaard and others (1974) note that where (d/g T2) > 0.014 (i.e., d/L > 0.13) "... Cnoidal theory is meaningless". The Shore Protection Manual (U. S. Army, 1984), on the other hand, appears more restrictive suggesting the use of the Ursell or Stokes parameter such that the theory is valid where U = L2 H/d3 > 26. In addition, as the wavelength becomes long approaching infinity, Cnoidal Wave Theory reduces to Solitary Wave Theory; and as H/d becomes small, the wave profile approaches the sinusoidal profile predicted by Linear (Airy) Theory. Several other problems are entrenched in the theory. First, it is not an easy nor expedient theory in application. Second, it requires that the wavelength is known when, in fact, this wave parameter is not quantitatively known for use in actual design applications. Third, the wavelength is first calculated from Linear (Airy) Theory in deep water (i.e., Lo) then modified to the local water depth of concern (i.e., L), which if only because of the approach, suggests an approximation at best. Softary Wave Theory is a special case of Cnoidal Theory in which the wavelength is infinitely long and the entire wave crest lies above the SWL, is clearly not applicable to natural cases of shorebreaking where shoaling waves have periodicity. More recently developed Stream Function Theory (Chappelear, 1961; Dean, 1965a, 1965b, 1967; Monkmeyer, 1970) constitutes an attempt to mitigate the difficulty of expanding to higher orders the Stokes'type approach, rendering the "... approach computationally simpler than Chappelear's technique, ..., to have wave theories that could be developed on the computer to any order" (Dean and Dalrymple, 1984, p. 305). Stream Function Theory, which has "... some of the same limitations ..." of higher order Stokes' solutions (U. S. Army, 1984, p. 259) has the capability to proceed to the breaking limit in transitional water depths, but not in shallow water. WA VE HEIGHT GROWTH DURING SHOREBREAKING It is, in part, the intent of this work to provide a practicable solution to the shortcomings of the previously discussed wave theories. Mathematical descriptions developed in the following sections require that only the initial wave height, Hi, and wave period, T, are known. In an earlier study, Balsillie (1980) used Ho or Hm as indicators of H,, where Ho is the 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 Ho and Hm and the resulting value of H, are approximately equivalent. Generally, this occurs for waves with higher wave steepness values. However, due to 1.2 C Conltont Depth  1.1 1.0pWot Mo 0t 10 9 011Dirlo OT DIfeelloa Tr4 H _ H i  refraction and frictional effects, etc., where the generated wave steepness is small, H, can become significantly less than Ho or Hm. The importance of the latter phenomenon is illustrated by an example from the laboratory data of Putnam (1945) in Figure 2. In the earlier work (Balsillie, 1980), it was reported that the alpha wave peaking parameter, Hb/H,, is dependent on the equivalent wave steepness parameter, H,/(g T2), and the bed slope, m The influence of these parameters is included in ensuing analyses. In addition, the continuous transformation of H/H, during shorebreaking 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 studies of Balsillie (1980, 1983b), new laboratory data have become available (Table 1) to I I I * 0.054 Mb T2 0.0086 Wove Figure 2. Wave transformation from deep water to shorebreaking, alpha wave peaking process is denoted by ab. Corrected H NH where the Table 1. Field and laboratory data used in analyses. T H( d. d,H Investigator m T (s) E 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 0.072 1.15 1.29 3.13 2.27 170.0 131.0 0.072 1.22 1.29 2.69 2.67 208.0 147.0 0.072 1.50 1.66 4.77 3.17 391.0 236.8 0.072 1.54 1.58 4.45 2.92 428.0 267.0 0.072 1.97 1.86 5.01 3.11 916.5 462.8 0.054 0.86 1.08 2.22 1.66 87.4 78.6 0.054 0.965 1.11 2.50 1.93 108.9 103.8 0.054 1.34 1.48 3.96 2.96 312.0 213.7 0.054 1.50 1.58 4.39 3.16 425.7 278.7 0.054 1.97 1.84 5.94 3.47 1039.1 557.9 Step" 1.05 1.16   126.2 108.5 Step" 1.19 1.19  142.6 119.4 Setp" 1.35 1.31  237.4 181.5 Step" 1.50 1.60   375.0 235.0 Step" 1.98 1.86   997.8 441.0 Hansen and 0.0292 0.833 1.32 3.64 2.55 207.0 156.3 Svendsen (1979) 0.0292 1.00 1.13 2.57 1.66 104.2 91.8 0.0292 1.00 1.21 2.95 1.89 153.1 126.4 0.0292 1.00 1.35 3.92 2.66 258.4 191.8 0.0292 1.25 1.30 2.50 1.51 162.9 124.3 0.0292 1.25 1.37 3.15 2.04 229.2 167.6 0.0292 1.25 1.46 3.95 2.42 395.3 270.1 0.0292 1.67 1.46 3.57 2.29 283.2 194.8 0.0292 1.67 1.42 3.57 2.15 302.5 209.8 0.0292 1.67 1.47 3.84 2.27 340.3 233.5 0.0292 1.67 1.48 4.13 2.41 389.1 261.3 0.0292 1.67 1.66 5.05 3.09 675.7 408.2 0.0292 2.00 1.69 4.47 2.66 608.6 359.3 0.0292 2.00 1.95 5.50 3.01 1048.2 537.8 0.0292 2.50 1.84 5.22 2.63 874.9 479.7 0.0292 2.50 2.20 5.95 2.75 1531.4 704.1 0.0292 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) 0.025 1.55 1.25  173.8 138.8 0.050 1.038 1.21  162.4 134.5 0.050 1.55 1.20   162.9 135.5 0.050 1.55 1.28 2.43 1.56 216.0 168.2 0.050 1.55 1.27 2.68 1.22 210.2 165.8 Table 1. Field and laboratory data used in analyses (cont.). T Hb d, d, ( ) Investigator m d H, H, db Singamsetti and 0.100 1.035 1.11   160.3 144.8 Wind (1980) 0.100 1.555 1.25   173.8 139.3 0.200 1.038 1.35   157.6 116.7 Wang and others 0.067 1.00 1.19   122.5 103.2 (1982)"' 0.067 1.33 1.20   148.2 123.8 0.067 1.34 1.28   195.5 153.0 0.067 1.47 1.22   178.0 146.0 0.067 1.56 1.49   283.9 190.8 0.067 1.65 1.21   196.2 161.7 0.067 1.89 1.32   280.1 212.2 0.100 1.22 1.17   145.9 124.7 0.100 1.50 1.34   191.7 143.2 0.100 1.58 1.13   222.4 197.3 0.100 1.61 1.33  267.4 201.6 0.100 1.80 1.55   396.9 256.1 0.100 1.89 1.52   350.1 230.3 Nakagawa (1983)"" 0.364 1.40 1.27  179.5 141.2 Mizuguchi (1986)"' 0.050 1.22 1.39   331.5 239.1 Hattori (1986)"' 0.050 0.85 1.23   160.9 131.1 0.050 1.00 1.38   316.1 227.9 0.050 1.40 1.81   738.8 408.7 0.050 1.00 1.30   208.5 160.7 0.050 0.80 1.05   96.5 92.2 0.050 0.85 1.20   138.8 116.1 0.050 1.00 1.42 3.16  257.9 181.5 0.050 1.40 1.96   768.3 392.0 0.050 0.80 1.06  96.3 90.9 0.050 0.84 1.02  123.5 121.3 0.050 0.99 1.25   240.1 192.1 0.050 1.40 1.70 5.04  711.4 417.6 Watanabe and 0.050 1.19 1.40 2.91 1.6 252.3 180.2 Dibajnia (1988)'" 0.050 1.18 1.28 2.50 1.6 213.2 166.4 0.050 0.94 1.17 2.50 1.6 135.3 115.5 Takikawa and others 0.050 2.08 1.36 2.62 2.08 229.2 197.2 (1997)"'____ __ ' Based on 400 consecutive wave measurements; step had a slope of 0.444, poststep slope was 0.009; New data. represent a wide range in bed slope conditions. Terminal Boundary Condition The terminal boundary of alpha wave peaking is defined as the shorebreaking point. Galvin (1968) provides a comprehensive description of the various types of shorebreaking waves. Of the principal types, however, spilling and plunging shorebreakers constitute those more commonly applied in design considerations. The shorebreaking point of a plunging breaker is defined to occur when the front face of the wave crest becomes vertical (Figure 1); the shorebreaking point of a spilling breaker occurs when the top of Laborat x Field Di 10 o Field Di SAField Di Field Di *Field Da Field Dg WAVES CAN BREAK IN THIS REGION DUE TO CRITICALLY HIGH WIND STRESSES 1 0.01 L 0.01 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, 1983a, Balsillie, 1999b, Balsillie and Tanner, 1999), illustrated in Figure 3, and given by: H 1.28 d, where Hb is the shorebreaking wave height, and db is the water depth at shorebreaking. 0.1 1 10 Hb (m) Figure 3. Relationship between the water depth at shorebreaking, db, and the shorebreaking wave height, Hb (after Balsillie, 1999b). Enhancement in precision of db/Hb prediction has been attempted by incorporating the bed slope and wave steepness (Weggel, 1972a, 1972b; Mallard, 1978). Balsillie (1983a, 1999b) found, however, that equation (1) as yet constitutes the most reliable predicting equation, and that equation (1) applies equally well to both spilling and plunging shorebreakers. The second parameter, Hb/Hj, 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, 1980), both wave steepness and bed slope were indicated to affect alpha wave peaking. However, based on new data, and subsequent analyses and testing, the following relationship can be recommended: illustrated in Figure 4. Additional attention was given to the bed slope and no refinement was found to improve equation (2). In fact, equation (2) is evaluated for a wide range of bed slope conditions, wherein any scatter might easily be attributed to the difficulty in identifying where shorebreaking occurs (Balsillie, 1999b). Other work suggests that the bed slope is probably more instrumental in influencing the type of shorebreaker that will be produced (Balsillie, 1984b, 1984c, 1985, 1999b). Due to scale differences between axes of Figure 4, wave steepness data from Table 1 are plotted in Figure 5 where now the axes are comparable. Dividing both sides of equation (2) by g T2 yields: S2_ T2 1.0 gr gT2  0.4 tanh(l00 H  0.4 n tanh 100 1rann 4annn 1U IUU H IUUU Iv UU Figure 4. Relationship for prediction of the shorebreaking wave height from the initial equivalent wave steepness parameter. Hb = 1.0 H, /Hb o0.067  _T2) a 0.072 gT A0.100 X 0.200 O.6388 step A Equation (3) 100 100 1 1000 /H g T2 Figure 5. Relationship between the equivalent wave steepness parameter evaluated at initiation of alpha wave peaking and at shorebreaking. which is superimposed upon the data of Figure 5 to show excellent agreement. Initial Boundary Condition 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 shorebreaking process. He applied the Rayleigh assumption (Eagleson and Dean, 1966) given by: d Equation (5) is plotted in Figure 6, from which the nonrepresentative nature of the equation is apparent. Additional analysis indicates that if we solve for d,/H, rather than d,/db of equation (5) and consider the incident wave steepness (bed slope produced no consistent results), the following relationship, plotted in Figure 7, provides good agreement: c, E, = c, E, d, d, (4) H H HI Hb, { In tanh (65gH) \ [ g^}\ 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 wave peaking and at shorebreaking, respectively. Using the Rayleigh assumption and Solitary wave theory, Munk (1949) suggests that: 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. 3 0 N Of 12 x db l ^4 .^Bed Slope n 0.0292 A 0.050 x 0.0556 Equation (5) o 0.072 0 1 2 3 4 Hb Hi Figure 6. Evaluation of Munk's (1949) parameter for determining the initiation of wave peaking in the shorebreaking process. Transformation of H/H, 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 shorebreaking waves, and in sediment transport predictions. Data tabulated by Putnam (1945)and Hansen and Svendsen (1979) are used to determine the nature of the transformation. The general equation is given by: HHb 02 tanh [) (7) H Jd d ) H, H, H HJ where 0, is a coefficient which determines where the transformation of H/Hi begins, given by: e 4nnn innnn IU IUU I ,W1 Figure 7. Relationship for the prediction of the relative depth of water at which alpha wave peaking is initiated. in which (d,/H,) is given by equation (6), e is the Naperian constant, and (2 determines the local peaked height during the shore breaking process given by: Hb 0I2 1.0 (9) H1 where Hb/H, is given by equation (2). Equation (7) is evaluated 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 shorebreak on the step slope over which measurements were widely spaced). Data of Wang and others (1982), Nakagawa (1983), and Mizuguchi (1986) provide only data for H,, H. and T. The more recent data of Hattori (1986) are not plotted here; it is to be noted, however, that equation (7) resulted in essentially precise representation of his data. 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 4, 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 shorebreak (i.e., since this must be visually determined). 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). WAVE CREST ELEVATION ABOVE THE DESIGN WA TER LEVEL While wave crest height is useful in practical applications, there is also a need to know the wave crest elevation above some reference plane. For instance, where a storm tide is used in assessing coastal engineering design solutions, wave crest/trough elevations relative to the known storm tide still water level (i.e., DWL) are needed. For example, suppose that the task is assigned to design the elevation of a fishing pier deck where the shorebreaking wave height at the structure from other calculations is estimated to be 4.5 m (approximately 15 feet). If, from theoretical calculations, the sine wave assumption is used (Figure 9) then onehalf, or 2.25 m, of the wave will lie above the design water level (SWL). If, however, the Solitary wave assumption is used, the entire 4.5 m wave lies above the DWL. This results in a large design uncertainty of 2.25 m (7.4 feet). While the sine wave assumption may actually be too low to insure a safe deck elevation, the Solitary wave assumption may be in excess, particularly in view of the high costs associated with construction and maintenance in the littoral zone. The above example, though it states the basic problem, is an oversimplification. It is well known that in addition to the design wave crest elevation above the DWL, other considerations, in particular the expected horizontal and vertical design wave impact loads, should be applied. The latter is possible only if the nature of wave transformation during shorebreaking is known. An estimation of the amount of the local wave crest that lies above the DWL, H', can be attempted using various wave theories. However, the applicability of classical theories, although they have 2.0 ' I I I I I 1 I 1 I I I 2.0 Ism 0.29 ! = 0.0292 m = 0.0292 : Hi = 0.064 m Hi = 0.0388 m T = 1.0 s T = 1.25 s 1.5 I 1.0 I _ 2.0 m = 0.0292 m = 0.0292 Hi = 0.0668 m I Hi = 0.0961 m T = 1.25 I T = 1.67 s 1.5 1.0 20 m 0.0292 m = 0.0292 Hi = 0.0941 m Hi = 0.09 m I T 1.25 s T = 1.67 s 1.0 8  2.0 m = 0.0292 m =0.0292 Hi = 0.0801 m Hi = 0.0403 m ST = 1.67 a I\ T = 1.67 s 1.5 1.0 2.0 2 I m = 0.0292 m = 0.0292 SHi 0.07 m Hi = 0.0644 m 1. T 1.67 s T = 2.0 s 1.0 I 2.0 m = 0.0292 \ m = 0.0292 ,Hi = 0.0374 m Hi = 0.04 m T = 2.0 s T = 2.5 s 1.5 I .0 I  1.0  2.0  m = 0.0292 I m = 0.0292 : *Hi = 0.07 m Hi= 0.0428 m .\& T = 2.5 s \ T = 3.33 s 1.5  l n_ I i i i i I I I P I I 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 10 d/H Figure 8. Alpha wave peaking predictions using equation (7) plotted as the solid curves. Vertical dashed lines represent d/H = 1.28 = db/Hb. Measured data (solid circles) are for bed slope data of 0.072 and 0.054 from Putnam (1945), and bed slope data of 0.0292 from Hansen and Svendsen (1979). 1 I I I I I I I I I I I Sm = 0.072 I m = 0.072 Hi = 0.093 m Hi = 0.0701 m T = 0.865 I T= 1.22 a II I m = 0.072 m = 0.072 Hi = 0.0762 m Hi = 0.0564 m T= 1.15 T = 1.5 . I ____,,_______*_*___ __ I 0*" e m = 0.072 m = 0.054 I Hi = 0.0543 m Hi = 0.0829 m T = 1.54 a T = 0.86 s '\ I I* *,*o *o* _ *'! m = 0.072 1 m = 0.054 Hi = 0.0415 m Hi = 0.0838 m T = 1.97 T = 0.965 Sm = 0.054 m = 0.054 SH = 0.0564 m Hi = 0.0366 m T = 1.34 a T = 1.97 s ) I m = 0.054 I m = 0.0292 H = 0.0518 m Hi 0.0329 m T = 1.5 T = 0.833 a Sm m = 0.0292 H = 0.0942 m Hi = 0.0379 m ST' 1.05 T=1.0a 5  ; .________k,,1 0 1 2 38 4 3 6 7 Figure 8. (cont.) 1 2 3 4 5 6 7 8 9 0 12345d/H d/H n a I 1.( Pier Deck Solitary Wave Assumption Sine Wave 2 m Vertical 1 m Scale O 0m1 Figure 9. Dependence of design pier deck soffit elevation on wave crest elevation above the design water level (DWL) for two commonly applied theoretical approaches. demonstrated relevance for predicting "deeper water" wave conditions, are not specifically designed to predict wave behavior during the shorebreaking process, particularly since shorebreaking waves are not symmetrical in profile view. Rather, the crests become progressively asymmetrical and distorted (e.g., Adeyemo, 1968, 1970; Weggel, 1968). Despite the underlying importance of the issue considered, surprisingly little work has been produced which addresses the phenomenon. In fact, of the work done, the paper of Bretschneider (1960), that by its singularity, becomes a classical accomplishment. Bretschneider's nomograph (Bretschneider, 1960; U. S. Army, 1984, p. 7107) provides measures of H' relative to the design water level (i.e., DWL which in this paper represents the SWL). However, for the entire range of conditions represented by the nomograph no mathematical description has been developed. This author (Balsillie, 1983c) attempted to formulate a mathematical representation using a modification of second order Stokes wave theory. The results, however, were less than satisfactory. A problem associated with the data used by Bretschneider is that his wave measurements were single point source samples. It was determined that, in addition, it would be valuable to know how the value of H'/H behaves as the wave progresses across a shoaling bathymetry to shore breaking. Hence, the author (JHB) conducted a reanalysis. While the amount of continuous wave height transformation data is not large, the data of Putnam (1945) for two bed slopes in conjunction with Bretschneider's (1960), Iverson's (1952), Weishar's (1976), Hansen's (1976) and Balsillie and Carter's (1980, 1984a, 1984b) and Balsillie's (1980) point source data provide sufficient information on which to conduct an investigation. As before, it becomes necessary to identify initial and terminal boundary conditions of the process. Terminal Boundary Condition The first boundary condition occurs where the value of H'/H is evaluated at the shorebreaking position, that is, the value of Hb'/Hb. Using the laboratory data of Iverson (1952) and field data of Balsillie and Carter (1980) and Balsillie (1980) the relationship between the shorebreaking wave height and the wave trough depth, d, (i.e., the vertical distance from the wave trough located just shoreward of the breaking wave crest to the bottom, referenced to the DWL), may be found. These data are plotted in Figure 10, and indicate that dt/Hb = 1.092 for 88 wave samples. Weishar (1976) reports that dt/Hb = 1.124 for 116 field measurements. A weighted average from the two groups of data yields: d, 1.11 Hb Iverson's (1952) laboratory data and field data of Balsillie and Carter (1980) and Balsillie (1980) also suggest that the depth at breaking can be related to the total depth, dc (i.e., the vertical distance from the top of the wave crest at shorebreaking to the bottom), according to: d 0.590 d. (10) (11) as illustrated in Figure 11. Hb(m) Figure 10. Relationship between the wave height at shorebreaking, Hb, and the trough depth just preceding the crest, d,; field data are from Balsillie and Carter (1980) and Balsillie (1980), laboratory data are from Iverson (1952). With the same data source used to develop equation (11), dt and dc can be related according to: dt 0.529 d, (12) which is illustrated in Figure 12. de (m) Figure 11. Relationship between the water depth at shorebreaking, db, and the total water depth at shorebreaking, d.; data sources as for Figure 10. Combination of equations (1) and (10) through (12), where Hb' = dc db, yields the average result that: 0.84 H, which is referenced to the Hansen (1976) reports that: (0.85),, = (0.82)swL Hb (13) DWL. (14) where the values are referenced to the mean water level (MWL) and still water level (SWL) as indicated by the subscripts (definitions of MWL and SWL are those given by Galvin, 1969). Regardless of the slight discrepancy between the DWL coefficients of equations (13) and (14), Hansen's result has been presented to show that Hb'/Hb is in the mideighty percent range and not some significantly large or small value. It is appropriate, also, to address the effect of shorebreaker type. The field data of Balsillie and Carter (1980) and Balsillie (1980) represent plunging and spilling type shorebreakers. However, no correlation was found between Hb'/Hb and the shore breaker type. Weishar (1976) reports the results of a field study using ground photography including the shorebreaker type. Using the established relationship of db/Hb = 1.28, Weishar's values of Hb/db may be transformed to yield: Hib C bP and: H,,P r, (0.88),M = (0.85)s, = (0.86)m = (0.82)s, where the subscripts PI and Sp refer to plunging and spilling shorebreaker types. Weishar (1976) is careful to note, however, (ml* Field Data .. '" D Lab Data (n =48) So0.5 1.0 1. dc (m) Figure 12. Relationship between the total depth at shorebreaking, dc, and the wave trough depth at shorebreaking, dt data sources as for Figure 10. that considerable variation occurred in the data and that the difference between mean values of the last two equations may not be statistically significant. It is concluded, therefore, that while there may be dependence of the value of Hb'/Hb on the shorebreaker type, sufficient data are not yet available nor work on other methods (e.g., Iwagaki and others, 1974) accomplished to justify such a commitment. Hence, equation (13) shall prevail as least equivocal guidance. Initial Boundary Condition The other boundary condition occurs at the beginning of the shorebreaking process. It has been suggested (see Balsillie, 1983c, p. 7) that the maximum value of H'/H in deep water is about 0.64. This maximum value, then, represents forced wave conditions (i.e., the waves are subject to the wind forces from which they were generated, and maximum wave steepness is maintained). A value of less than 0.64 represents free or coasting waves (i.e., the waves are no longer subject to original generating winds, but have left the generation area and have or are undergoing dispersion). (See Mooers, 1976; Balsillie and others, 1976) Upon reaching transitional water depths, the bottom slope begins to introduce an additional effect on the value of H'/H. Therefore, H'/H may have a value greater than 0.5 when the wave reaches the point of initiation of the shorebreaking process. When a wave reaching the initiation point is forced, one may expect the progressive increase in the value H'/H to be minimal, provided that bed slope conditions do not change significantly during the shorebreaking process. It was found, however, that for free or coasting waves H'/H does not begin to significantly increase in value until shore breaking begins, which has been determined to occur when the critical alpha wave peaking depth is encountered. Near the point of initiation of shorebreaking, the value H'/H shall be given the notation (Hi'/Hi)' which requires quantification. It (Balsillie, increase was also found 1983b) that the in the wave height Figure alpha wave I above the DWL begins somewhat earlier (i.e., further offshore) in the shorepropagating wave transformation history than does the initiation of the alpha wave peaking process. In terms of design approach, this complication should not be of undue concern, since even though the height of the wave above the DWL might be slightly increasing, the total wave height typically is still decreasing...until the initiation of alpha wave peaking after which H/Hi and H'/H both increase to reach a maximum value at the shore breaking position. In addition, the initial values of both H/Hi and H'/H are significantly small. Even though vertical changes might be slight in the design application sense, it would be prudent to account for this apparent discrepancy. Using the data of Putnam (1945) illustrated in Figure 13 and listed in Table 2, the relative water depth (di/Hi)' indicating where the increase in wave crest height above the SWL appears to begin may be approximated by: (Y = IH/ SIn tanh _201 2 Lj (17) T 13. Relative depth equations for initiation of wave peaking at (di/Hi), and at initiation of height increase above the DWL at (di/H,)'. Table 2. Laboratory data of Putnam (1945) for evaluation of wave height above the DWL. m T (H,'H)' (d,/H,)' (H/g T)' 0.072 0.865 0.575 3.5 71.4 1.15 0.550 4.0 158.7 1.22 0.525 5.0 200.0 1.50 0.525 6.0 384.6 1.54 0.500 6.5 434.8 1.97 0.525 7.0 909.1 0.054 0.86 0.575 3.5 75.2 0.965 0.550 4.0 96.2 1.34 0.550 5.0 303.0 1.50 0.520 6.0 400.0 1.97 0.525 7.0 1000.0 NOTE: H,' is evaluated at the point where the height above the DWL begins to rise, not at the initiation of Alpha Wave Peaking ... use equation (20) for transformation. Also plotted in Figure 13 is the relationship for di/Hi from equation (6) for comparative purposes. The wave steepness parameter at the two points are slightly different in value. Using the data of Putnam (1945), the transformation of wave steepness may be approximated (Figure 14) by: f 1462 \ 1.055 ( H, = 1.462 1. [g gr1Tg (18) in which (Hi /g T2)' is the wave steep ness parameter where crest height increase above the DWL is initiated, and (Hi/g T2) is the wave steepness parameter at the beginning of alpha wave peaking. The relative incident wave height, (Hi'/Hi)' conforming to relative depth conditions, illustrated in Figure 15, may be given by: ( 0.5 + 1.25 g (19) where (Hi2/(g di T2))' is evaluated at (di/H,)'. Analytically, it becomes of value to be able to predict the magnitude of H'/H at the point of initiation of alpha wave peaking (i.e., at d,/H, rather than at (d, /Hi)'). Using the notation (H'/Hi)o, numerical analysis results in a closely fitted value according to: S= 0.515 + 12 Ha s g 1 T' Figure 14. Wave steepness parameter at (di/Hi)' relative to the wave steepness parameter at (d/Hi). '" See text 0.5 / Equation (19) (/) 8 0.054 ^ f H .72 o HA \h 00.0<72 o~ ~ _B . ..5  , .. . , .j I , , , I 10 IoX 103 104 g d1T2/ Figure 15. Relationship for prediction of the incident value of H'/H. (20) It is interesting to note that where limiting conditions are imposed, equation (19) provides consistent results. Suppose that a profile occurs where a wave, initially in deep water, suddenly encounters an abrupt slope change and it must shore break. Suppose, also, that the wave in deep water is fully forced so that (Ho'/H) = (Hi'/Hi)' = 0.64. For such conditions, from equation (19), (Hi /g (di T2))'1 = 56.4. From another viewpoint, where Hb /db = 1/1.28 = 0.78 and (Hi /g T2)' = 1/(14 n), then (Hi /(g di T2))' = 63.8, and (Hi'/Hi)' = 0.648. Both solutions are close and represented by the asterisk in Figure 15. Also, where the maximum possible value of (H'/H) in equation (20) is 0.84, representing maximum forced wave conditions in shallow water, the value of (Hi /(g T2)) becomes 0.0271. Where Hb/Lb = 0.79 (Hb /(g T2))0.5 given by Balsillie (1984a), then by substitution (H/L)max for shallow water waves becomes 0.13 or 1/7.7. This approximation is consistent with the Michell (1893) criterion of 1/7. Prediction of H'H Transformation Incorporation of the preceding boundary conditions leads to the following development describing the H'/H transformation during the shorebreaking process. The general equation is given by: H H 4 tanh '(  where db/Hb = 1.28, and: 3 = ee (dIHi)' (dblH,) (21) (22) in which e is the Naperian constant, (db/Hb) is 1.28, and: H H/ H,,r (23) where from equation (16) Hb'/Hb = 0.84, (Hi'/Hi)' is given by equation (19), and (d,/H,)' is given by equation (17). Equation (21) is plotted in Figure 16 as the solid curves. It is to be noted from these plots that equation (21) appears to successfully represent initial values of H'/H (i.e., incident waves with larger wave steepness values tend to have larger initial H'/H values). There is, however, discrepancy between measured and predicted values of H'/H as the shore breaking position is closely approached. Putnam's (1945) laboratory data tend to consistently underestimate the value of H'/H very near and at the shorebreaking position. One must recall, however, that the behavior of the curve predicted by equation (21) is determined by the terminal boundary condition that Hb'/Hb = 0.84, which is based on prototype wave data and results from other investigations. Hence, laboratory conditions or measurement techniques may account for the apparently low values of Putnam's data near shorebreaking. While the terminal boundary condition must surely be refined by future research efforts, equation (21) would appear to provide a satisfactory and, certainly, a useful method for predicting H'/H during the shorebreaking process. Equation (21) is also tested using the prototype laboratory wave (Figure 17) and field hurricane waves (Figure 18) reported by Bretschneider (1960). Figure 17 indicates very good agreement. Figure 18, however, indicates that equation (21) underestimates H'/H. One must keep in mind that the measured data represent the maximum wave height occurring during oneminute recording periods, not the average wave height. Therefore, one would expect equation (21) to underestimate the measured data, and that the plotted line represents an expected upper limit as also found by Bretschneider (1960). Based on the success of equation (21), the value of H'/H as a function of H/(g T2) and d/(g T2) is given by Figure 19 for transitional and shallow water depths. Figure 19, then, provides an alternative to the nomographic approach originally proposed by Bretschneider (1960). 0.9 H m =0.072 m = 0.054 Hi = 0.04184 m Hi = 0.0949 m 0.7 T= 1.97 s T = 0.965 a 0.5 I 0.9 I m = 0.054 m = 0.054 Hi = 0.05807 m H = 0.03803 m 0.7 T = 1.34 a T = 1.97 s 0.5 1 0.9 .I I m I 0.054 5 10 15 20 25 Hj = 0.05513 m 0.7 T = 1.5 s I .. , 0.5 I * * 0 5 10 15 20 25 d/H Figure 16. Transformation of H'/H using equation (21) plotted as solid curves. Vertical dashed lines represent d/H = 1.28 = db/Hb. Measured data (solid circles) are from Putnam (1945). Hprod Hpred S* (m) * 0 1 2 H"mMe (m) Figure 17. Predicted H' from equation (21) versus prototype Beach Erosion Board wave tank data reported by Bretschneider (1960). CONCLUSIONS Classical wave theory, among certain coastal practitioners, has been viewed as the "last word" in the prognostication of water wave behavior. However, when applied to the progressive distortion of waves as they approach and reach shorebreaking, classical wave theory falls short of providing realistic solutions. A major shortcoming of wave theories is that they require specification of variables not known or which can only be approximated, in particular the wavelength. Shortcomings of theories have been presented in the second section of this work. The results of this work, however, provide a methodology for wave height transformation during the wave breaking process up to the point of shorebreaking, requiring only the specification of initial wave height and period; bed slope is apparently not a factor affecting the transformation (other research suggests that bed slope is more nearly instrumental in influencing the type of shorebreaker that will be produced (e.g., Balsillie, 1999b)). H .a (m) 1  0 Figure 18. (21) versus wave data (1960). H;m.a (m) Predicted H' from equation Lake Okeechobee hurricane reported by Bretschneider While wave generation equations (e.g., for storms and hurricanes) and classical theory can be used to generate and propagate waves to the initial or incipient shorebreaking depth (i.e., d/H,)', the importance of the numerical methodology developed here is that it requires no a priori historical knowledge of wave height behavior seaward of d/H, or (d/H,)'. Equations can be simply evaluated using a handheld scientific calculator; other equations may be as simply evaluated, while wave height transformation equations detailing the shorebreaking process may be more easily evaluated using a programmable calculator or computer. For the data used in this study, wave peaking transformation is initiated in water depths of from about 1.5 to 7 times the incident wave height. Two mechanisms of wave breaking transformation have been numerically quantified. The first deals with change in total height of the wave, and the second with the change in crest height above the still water level. The first H' 0.7 0.6 0o Shallow and Tranmltlonal Water 4 Deep I I I 11111 I I I I1iII I I I gpI l Water 0.0001 0.001 0.01 0.1 d gT2 Figure 19. Ratio of wave crest elevation above the design water level to wave height from equation (21). mechanism is based on a larger set of data from a variety of sources and representing many bed slope conditions. For the data used in this study, the total wave height at shorebreaking ranged from 1 to about 2.5 times the height of the incident wave. The second mechanism is based on a much smaller data set, but the reader should note that it is essentially a "finetuned" version of methodology presently existing. The amount of the wave crest lying above the still water level ranged from 0.5 to 0.84, with a constant value of 0.84 at the shorebreaking position. Some of the major numerical attributes of the introduced prediction methodology which address shortcomings of classical wave theories include: 1. The methodology does not require that the wavelength is known to calculate the basic wave parameters through the shorebreaking process. 2. The wavelength and wave celerity, however, can be calculated using empirically derived expressions addressed in a companion work (Balsillie, 1999a). 3. The point of initiation of shorebreaking (i.e., incipient shorebreaking given by di /Hi) can be determined from Hi /g T2. Therefore, one needs only to know of the incident wave height and period to determined the water depth at which shorebreaking begins. 4. The final shorebreaking wave height can be numerically determined from the incident wave height and wave period. 5. Continuous quantitative peaking of the wave height can be numerically determined through the shorebreaking process from incipient shore breaking to final shore breaking. 6. The amount of the wave crest lying above the still water level can be numerically determined continuously through the shorebreaking process. 7. A shorebreaker classification can be redefined as a numerical continuum. 8. The methodology requires no apriori knowledge of the wave behavior seaward of di /Hi rather based on incident wave height and period, it is dependent upon water depths. Figure 20 is a modified version of Figure 27 from the Shore Protection Manual (U. S. Army, 1984), originally proposed by LeMehaute (1969). It has become a "standard" among coastal engineers for identifying regions of validity for wave theories. It is modified as follows. First, the terminology validity is invalid, and is replaced by "regions of applicability"; this is so because theories by definition have no validity since they are unproven. Second, the "breaking" region in the upper lefthand part of the figure is renamed "broken" since, by definition, waves have broken when d/H = 1.28 or H/d = 0.78 is reached. Third, the "nonbreaking" region has been moved to the right and, fourth, a "shorebreaking region has been identified. In addition to wave theories, the region of applicability of results from this work are plotted on Figure (20) as the stipled region. The line labelled "incipient shore breaking" identifies the beginning of the shorebreaking process given by equation (6). It is to be noted that as waves become longer (i.e., smaller values of H/g T2) wave peaking becomes greater (i.e., Hb /Hi becomes greater in value). Very steep waves (i.e., large values of H/g T2) peak very little or not at all. The delineation of "shallow water" and "transitional water" depths at a value of d/L = 0.04 would appear to be inappropriate. The results of this study suggest that the delineation should be dropped and that shallow water should begin at the line segment BC of Figure 20 (which is given by equation (6)). NOTE It is important to note that variables in this paper represent average wave heights and periods, etc. Moment variable statistics relating to significant or maximum wave heights, for instance, may be found in other works (e.g., Balsillie and Carter, 1984a, 1984b; U. S. Army, 1974). ACKNOWLEDGEMENTS Florida Geological Survey staff reviews were conducted by Paulette Bond, Kenneth Campbell, Thomas M. Scott, Joel Duncan, Edward Lane, and Steven Spencer, whose editorial comments are gratefully acknowledged. The review comments of William F. Tanner of Florida State University are gratefully acknowledged. Nicholas C. Kraus d =0.04 L d '2 0.00155 gT d  = 0.13 L d  = 0.014 9T2 d  0.5 L d g 0.0792 gT2 .Shallow Water Intermediate Water Deep Water  I 0.01 d g T2 0.05 0.1 0.4 Figure 20. Regions of applicability for various wave theories and for the Alpha Wave Peaking methodology of this work. of the U. S. Army, Coastal Engineering Research Center (now the Coastal Engineering Laboratory) also reviewed the manuscript. REFERENCES Adeyemo, M. D., 1968, Effect of beach slope and shoaling on wave asymmetry: Proceedings of the 11 th Conference on Coastal Engineering, p. 145172. 0.04 0.01 0.05 H g T2 0.001 0.005 0.001 0.0005 0.0005 0.005 0.001 1970, Velocity fields in the wave breaker zone: Proceedings of the 12th Coastal Engineering Conference, p. 435460. Balsillie, J. H., 1980, The peaking of waves accompanying shorebreaking: in Tanner, W. F., ed., Proceedings of a Symposium on Shorelines Past and Present, Tallahassee, FL, Department of Geology, Florida State University, p. 183247. 1983a, On the determination of when waves break in shallow water: Florida Department of Natural Resources, Beaches and Shores Technical and Design Memorandum No. 833, 25 p. 1983b, The transformation of the wave height during shore breaking: the alpha wave peaking process: Florida Department of Natural Resources, Beaches and Shores Technical and Design Memorandum No. 834, 33 p. 1983c, Wave crest elevation above the design water level during shorebreaking: Florida Department of Natural Resources, Beaches and Shores Technical and Design Memorandum No. 835, 41 p. 1984a, Wave length and wave celerity during shorebreaking: Florida Department of Natural Resources, Beaches and Shores Technical and Design Memorandum No. 841, 17 p. 1984b, Attenuation of wave characteristics following shore breaking on longshore sand bars: Florida Department of Natural Resources, Beaches and Shores Technical and Design Memorandum No. 843, 62 p. 1984c, A multiple shore breaking wave transformation computer model: Florida Department of Natural Resources, Beaches and Shores Technical and Design Memorandum No. 844, 81 p. 1985, Redefinition of shore breaker classification as a numerical continuum and a design shore breaker: Journal of Coastal Research, v. 1, no. 3, p. 247254. 1999a, Wavelength and wave celerity during shorebreaking: Florida Geological Survey, Special Publication No. 41, p. 3140. 1999b, On the breaking of nearshore waves: Florida Geological Survey, Special Publication No. 45, p. 1155. Balsillie, J. H., and Carter, R. W. G., 1980, On the runup resulting from shore breaking wave activity: in Tanner, W. F., ed., Proceedings of a Symposium on Shorelines Past and Present, Tallahassee, FL, Department of Geology, Florida State University, p. 269341. 1984a, Observed wave data: the shorebreaker height: Florida Department of Natural Resources, Beaches and Shores Technical and Design Memorandum No. 842, 70 p. 1984b, The visual estimation of shorebreaking wave heights: Coastal Engineering, v. 8, p. 367 385. Balsillie, J. H., Campbell, K., Coleman, C., Entsminger, L., Glassen, R., Hajishafie, N., Huang, D., Tunsoy, A. F., and Tanner, W. F., 1976, Wave parameter gradients along the wave ray: Marine Geology, v. 22, p. M17 M21. Balsillie, J. H., and Tanner, W. F., 1999, Stepwise regression in the earth sciences: a coastal processes example: Environmental Geosciences, v. 6, p. 99105. Bretschneider, C. L., 1960, Selection of design wave for offshore structures: Transactions, American Society of Civil Engineers, v. 125, pt. 1, paper no. 3026, p. 388416. Byrne, R. J., 1969, Field occurrences of induced multiple gravity waves: Journal of Geophysical Research, v. 74, p. 25902596. Clifton, H. E., Hunter, R. E., and Phillips, R. L., 1971, Depositional structures and processes in the nonbarred high energy nearshore: Journal of Sedimentary Petrology, v. 41, p. 651 670. Dean, R. G., 1974, Breaking wave criteria: a study employing a numerical wave theory: Proceedings of the 14th Conference on Coastal Engineering, chap. 8, p. 108123. Eagleson, P. S., and Dean, R. G., 1966, Small amplitude wave theory: in Ippen, A. T., ed., Estuary and Coastline Hydrodynamics, New York, McGrawHill, Inc., p. 192. Gaillard, D. D., 1904, Wave action in relation to engineering structures: U. S. Army Corps of Engineers Professional Paper No. 32, p. 110 123. Galvin, C. J., Jr. 1968, Breaker type classification on three laboratory beaches: Journal of Geophysical Research, v. 73, no. 12, p. 3651 2659. 1969, Breaker travel and choice of design wave height: Journal of the Waterways and Harbors Division, American Society of Civil Engineers, v. 95, no. WW2, p. 175200. Gerstner, F. V., 1802, Theory of waves, etc. (Theorie der wallen): Ahb. Kgl. Bohm. Ges. Wiss. Prag. (also: 1809, Gilberts Annalen de Physk, v. 32. p.412). Hansen, U. A., 1976, Wave setup and design water level: Journal of the Waterway, Port, and Ocean Division, American Society of Civil Engineers, no. WW2, p. 227240. Hansen, J. B., and Svendsen, I. A., 1979, Regular waves in shoaling water: Series Paper No. 21, Institute of Hydrodynamics and Hydraulic Engineering, Technical University of Denmark, 233 p. Hattori, M., 1986, Experimental study on the validity range of various wave theories: Proceedings of the 20th Coastal Engineering Conference, chap. 18, p. 232246. Iverson, H. W., 1952, Laboratory study of breakers: in Gravity Waves, National Bureau of Standards Circular 52, U. S. Government Printing Office, Washington, D. C., p. 932. Iwagaki, Y., Sakai, T., Tsukioka, K., and Sawai, N., 1974, Relationship between vertical distribution of water particle velocity and type of breakers on beaches: Coastal Engineering in Japan, v. 17, p. 5158. Kinsman, B., 1965, Wind Waves, New York, Dover Pulications, 676 p. Komar, P. D., 1976, Beach Processes and Sedimentation, Englewood Cliffs, New Jersey, PrenticeHall, 429 p. Laitone, E. V., 1963, Higher approximation to nonlinear water waves and the limiting heights of Cnoidal, Solitary and Stokes waves: Beach Erosion Board Technical Memorandum 133, 106 p. LeMehaute, B., 1969, An introduction to hydrodynamics and water waves: Water Wave Theories, v. 2, TR ERL 118POL32, U. S. Department of Commerce, ESSA, Washington, D. C. LeviCivita, T., 1924, Uber die transportgeschwidigkeit einer stationaren wellenbewegung: Vortage aus dem Gebiete der Hydro und Aaerodynamik, Berlin. Mallard, W. W., 1978, Investigation of the effect of beach slope on the breaking height to depth ratio: M. S. Thesis, Department of Civil Engineering, Newark, New Jersey, University of Delaware, 168 p. Mason, M. A., 1941, A study of progressive oscillatory waves in water: U. S. Army, Beach Erosion Board Technical Report No. 1, 39 p. McCowan, J., 1894, On the highest wave of permanent type: Philosophical Magazine Edinburgh, v. 32, p. 351 358. Michell, J. H., 1893, On the highest waves in water: Philosophical Magazine Edinburgh, v. 36, p. 430435. Mizuguchi, M., 1986, Experimental study on kinematics and dynamics of wave breaking: Proceedings of the 20th Coastal Engineering Conference, chap. 45, p. 589603. Mooers, C. N. K., 1976, Winddriven currents on the continental margin: in Stanley, D. J., and Swift, D. J. P., eds., Marine Sediment Transport and Environmental Management, New York, John Wiley and Sons, p. 2952. 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. Nakagawa, T., 1983, On characteristics of the waterparticle velocity in a plunging breaker: Journal of Fluid Mechanics, v. 126, p. 251268. Nakamura, M., Shiriashi, H., and Sasaki, Y., 1966, Wave decaying due to breaking: Proceedings of the 10th Conference on Coastal Engineering, chap. 16, p. 234253. Putnam, J. A., 1945, Preliminary report of model studies on the transition of waves in shallow water: University of California at Berkeley, College of Engineering, Contract nos. 16290, HE116106 (declassified U. S. Army document from the Coastal Engineering Research Center), 35 p. Scripps Institute of Oceanography, 1944a, Waves in shallow water: Scripps Institute of Oceanography, La Jolla, CA, S.I.O. Report No. 1, 28 p. 1944b, Effect of bottom slope on breaker characteristics as observed along the Scripps Institution pier: Scripps Institute of Oceanography, La Jolla, CA, S.I.O. Report No. 2, 14 p. Singamsetti, S. R., and Wind, H. G., 1980, Breaking waves: Report M1371, Toegepast Onderzoek Waterstaat, Delft Hydraulics Laboratory, 66 p. Skovgaard, O., Svendsen, I. A., Jonsson, I. G., and BrinkKjaer, 0., 1974, Sinusoidal and Cnoidal gravity waves, formulae and tables: Institution of Hydrodynamics and Hydraulic Engineering, Technical University of Denmark, 8 p. Stoker, J. J., 1957, Water Waves, New York, Interscience Publishers, Inc., 567 p. Stokes, G. C., 1880, On the theory of oscillatory waves: Mathematical and Physical Papers, Cambridge, Cambridge University Press, v. 1, p. 314326. Struik, D. J., 1926, Rigorous determination of periodic irrotational waves in a canal of finite depth: Math. Ann., v. 45, p. 595634. Svendsen, I. A., and Hansen, J. B., 1976, Deformation up to breaking of periodic waves on a beach: Proceedings of the 15th Coastal Engineering Conference, chap. 27, p. 477496. Takikawa, K., Yamada, F., and Matsumoto, K., 1997, Internal characteristics and numerical analysis of plunging breaker on a slope: Coastal Engineering, no. 31, p. 143161. U. S. Army, 1984, Shore Protection Manual, Coastal Engineering Research Center, 2 vols, 1262 p. Van Dorn, W. G., 1978, Breaking invariants in shoaling waves: Journal of Geophysical Research, v. 83, no. C6, p. 29812988. Wang, H., Sunamura, T., and Hwang, P. A., 1982, Drift velocity at the wave breaking point: Coastal Engineering, v. 6, p. 121150. Watanabe, A., and Dibajnia, M., 1988, A numerical model of wave deformation in surf zone: Proceedings of the 21st International Coastal Engineering Conference, v. 1, chap. 41, p. 578 587. Weggel, J. R., 1968, The impact pressure of breaking water waves: Unpublished Ph. D. Dissertation, University of Illinois. ,_ 1972a, Maximum breaker height for design: Proceedings of the 13th Annual Coastal Engineering Conference, p. 419432. 1972b, Maximum breaker height: Journal of the Waterways, Harbors and Coastal Engineering Division, American Society of Civil Engineers, No. WW4, Proc. Paper 9384, p. 529548. Weishar, L. L., 1976, An examination of shoaling wave parameters: M. S. Thesis, College of William and Mary, Williamsburg, VA, 86 p. Wood, W. L., Jr., 1970, Transformations of breaking wave characteristics over a submarine bar: Technical Report No. 4, Department of Natural Science, East Lansing, MI, Michigan State University, 116 p. 1971, breaking wave submarine bar: Department of state University Transformation of parameters over a Ph.D. Dissertation, Geology, Michigan ,East Lansing. NOTATION The following symbols are used in this paper: c = local wave speed. cb = wave speed at shorebreaking. ci = wave speed at initiation of wave peaking. d = local water depth measured from the DWL. db = water depth at shorebreaking. dc = water depth from the wave crest to the bed at shorebreaking. di = water depth at initiation of wave peaking. dt = water depth from the trough fronting the wave crest to the bed at shore breaking. dti = water depth from the trough fronting the wave crest to the bed at initiation of wave peaking. DWL = design water level (in this paper it is the SWL). e = Naperian constant. Eb = total wave energy at shorebreaking. E, = total wave energy at initiation of wave peaking. g = acceleration of gravity. H = mean local wave height. Hb = mean wave height at shorebreaking. Hb' = mean amount of wave crest at shorebreaking lying above the DWL. Hi = mean wave height at the initiation of wave peaking. Hi' = mean amount of wave crest lying above the DWL at the initiation of wave peaking. Ho = mean deep water wave height. Hm = mean wave height measured in the constant depth portion of a laboratory wave channel. Hti = water depth from wave crest to bed at the initiation of wave peaking. L = local wavelength. Lo = deep water wavelength. m = bed slope. MWL = mean water level. PL = subscript denoting the plungingtype shorebreaking wave. SP = subscript denoting the spillingtype shorebreaking wave. SWL = still water level. T = wave period. ( = relating coefficients developed in the text. WAVELENGTH AND WAVE CELERITY DURING SHOREBREAKING by James H. Balsillie, P. G. No. 167 ABSTRACT Prediction of wave phase speed and, hence, wavelength at shorebreaking has remained a controversial issue. Based on available field data (n = 47) and laboratory data (n = 40 to 71), a family of relationships is derived for predicting wavelength at shorebreaking. Assuming approximate linear wave speed attenuation, a method is derived for prediction of wave speed during the shorebreaking process. INTRODUCTION Wave height, H, wavelength, L, wave period, T, water depth, d, and bed slope tan a, constitute the fundamental hydraulic variables forming the basis for derivation of composite parameters (e.g., wave steepness, H/L; or equivalent wave steepness parameter, H/(g T2); surf similarity parameter, tana/(H/g T2)1/2; etc.) utilized in most coastal engineering design applications. It becomes not only desirable to be able to provide for determination of such parameters over a wide variety of conditions in order to accurately describe a natural process, but to be able to provide the simplest and most straightforward procedures possible. As the number of basic variables becomes large, the solution of any problem invariably becomes proportionally more complex. It becomes desirable, therefore, to provide methods for predicting as many of the variables as is feasible. One such variable is the wavelength. As will become evident, determination of the local values of H and d as waves shorepropagate over a shoaling bed is complex, since following specification of their values, which may exhibit a wide range, H relative to d experiences additional and significant progressive transformations as shoaling continues. However, the wave period, once initially specified, is considered to be conserved (i.e., remains invariant) across the shoaling bathymetry until breaking occurs, and a simplifying condition emerges. The wavelength, however, experiences attenuation during shore breaking, thereby introducing additional complexity. The wavelength not only appears in many shoaling design wave equations, but usually just when one has little insight as to its local value short of tedious theoretical calculations for obtaining an estimation. Most importantly wavelength is related to the wave speed and wave energy. It becomes important, therefore, to provide methods) for prediction of the wavelength and wave speed. In this paper such prediction is investigated during the shorebreaking process. DISCUSSION AND RESULTS As shorepropagating waves approach the shoreline across shoaling I I I I I I I I a Constant Depth s Deep Water  S0.0110 0~ 0 tan ab = 0.054 Hi 0.0086 OTZ Direction of Wave Travel Data from Putnam (1945) T = 0.065 s Hb gT2 * d= = 1.28 H Hb Figure 1. Illustration of wave transformation from deep water to shorebreaking, where the alpha wave peaking process (segment A B) is given the notation ap. bathymetry, the wave height tends to initially decrease due to a number of factors such as friction, etc., and then begins to increase rapidly in height just before shore breaking occurs (Scripps Institute of Oceanography, 1944a, 1944b; Putnam, 1945; Munk, 1949; Iverson, 1952; Stoker, 1957; Kinsman, 1965; Balsillie, 1983a, 1983b, 1983c, 1999a). The transformation is illustrated in Figure 1 (notation is defined at the end of the paper). It is the increase in wave height which accompanies the shore breaking process (note: waves may break in relatively deeper water due only to critically high wind stresses which cause waves to become critically steep (i.e., forced waves); shorebreaking occurs primarily because water depths become critically shallow). Shorebreaking wave mechanics are described by the alpha wave peaking concept (Balsillie, 1980, 1983b, 1983c, in 1999a) and denoted by the symbol a in Figure 1. Alpha wave peaking, then, describes the zone of interest investigation of the wave celerity wavelength. The speed with which a group of waves, comprising a wave train, travels is not always equivalent to the speed of individual waves within the group. The individual wave speed, termed the phase speed, is given by: L C 7 and group wave speed, c9, by: L C9 = n r In deep water (i.e., d/L > 0.5) n = 0.5; in intermediate water depths (i.e., 0.04 < d/L < 0.5) n increases in value to become, finally, n =1 in shallow water (i.e., d/L < 1.0 0.9 I 0.8  0.7 I 10 Adjusted H H1 9 8 7 6 5 4 3 2 d/H for and 0.04) and c = Cg. According to small amplitude (Airy) wave theory, the phase speed and wavelength in any depth of water may be given by: =L = g tanh 2d T 2w L which is evaluated in the following section. In this work the alpha wave peaking process is assumed to occur in shallow water where n = 1. In order to determine the transformation of c and L during shore breaking, one first needs to have knowledge of the governing boundary conditions. Terminal Boundary Conditions Equation (3) is has been applied to predict conditions at shorebreaking. Shore breaking defines the terminal boundary condition for the phenomena considered in this work. A more appropriate application from small amplitude (Airy) wave theory is given by: Lb Cb = i (4) or where Solitary wave theory is applied, by: Lb g(d + H;) (5) where Hb' is that portion of the wave height at shorebreaking lying above the design water level (DWL). [Note: the entire wave crest of a solitary wave lies above the still water level and Hb' = Hb.] About equation (5), Smith (1976) states: "Although this equation is widely used in the literature on wave theories and is generally accepted, few discussions have been presented which establishes its validity." The same appears to be true of equation (4), while a general misunderstanding about equation (3) seems to have been proliferated in the literature. Van Dorn (1978) found that at shore breaking, the wave speed was always greater than the small amplitude speed of equation (4), and smaller than the solitary wave celerity of equation (5). He reports that: C, = 2H which was found to agree roughly with that predicted for limiting Stokes waves in deep water. Available field and laboratory data (see the Table) are used to evaluate the above equations. The wave celerity is analyzed in terms of the wavelength rather than the wave speed since the length yields a much wider range of values (from 0.5 to 100 m, or 2 orders of magnitude). The data are plotted in Figure 2. Figure 2 illustrates that equation (3) does not predict Lb and, hence cb, with the precision of the other fitted relationships. It is to be noted, however, that equation (3) was developed from theoretical considerations to represent an upper limit envelope curve (see equation 4 for a suggested correction factor and fig. 8, both from Bretschnieder, 1960). In addition, because equation (3) is an algorithm it is awkward to apply and not generally recommended for use in design work, at least not in the breaker zone. Based on empirical evidence (field and laboratory data of the Table), equations that more successfully predict expected values in shallow water are closely given by: L, b=T d, 5 L, = T 2 g Hi 2^ Table of statistics relating measured and predicted wavelengths at the shore breaking position. Investigator L" = m TV L, = m Tr L, m TgH and n Category m r m r m r FIELD DATA Gaillard (1904) 26 0.9462 0.9608 0.7514 0.9587 1.241 0.9404 Balsillie and Carter (1980) 21 1.226 0.9692 0.9101 0.9695 1.359 0.9689 Field Results 47 0.9832 0.9737 0.7737 0.9757 1.259 0.9720 LABORATORY DATA Galvin and Eagleson (1965) 24 1.342* 0.3603* 0.8843* 0.3842* 1.178 0.3658 Eagleson (1965) 7 1.084* 0.8399* 0.8131* 0.8850* 1.233 0.7985 Van Dom (1976, 1978) 12 1.264 0.9636 0.8895 0.9889 1.254 0.9933 Hansen and Svendsen (1979) 28 1.113 0.9947 0.8036 0.9962 1.162 0.9969 Laboratory Results 4071 1.205 0.9858 0.8561 0.9947 1.211 0.9933 ALL DATA Total Results 87118 0.9972 0.9801 0.7794 0.9210 1.254 0.9836 Weighted Results 87118 1.111  0.829  1.251  Adjusted Weighted Results** 87118 1.1176 . 0.8374  1.2644  NOTES: Unless otherwise indicated all m and r are from regression analyses. These results represent db referenced to MWL and are not used in determination of m, all others used in the analysis are referenced to SWL. ** Adjusted values were determined so that resulting equations yield intraconsistent results. and we now have a family of design relationships for the prediction of Lb and cb. and: L = T 2 ( + H 9 Initial Boundary Condtions Where db = 1.28 Hb (McCowan, 1894; Munk, 1949; Balsillie, 1983a, 1985, 1999a, 1999b) and Hb' = 0.84 Hb (Balsillie, 1983c, 1985), the previous three relationships can be modified to yield two additional equations, as: L, v H (10) 5 and L, T g (d, + H) (11) S4 With the exception of the results of Hansen and Svendsen (1979), there is little if any, data available which will allow for determination of the wave speed at the point of initiation of alpha wave peaking (i.e., at cb). Based on other alpha wave peaking investigations (Balsillie, 1980, 1983b, 1983c, 1999a), it may be reasonable to assume that ci can be related to cb. However, the problem is encountered that the difference between ci and cb is slight, at least compared to natural variability in the data and possible measurement errors. 0.3 0.5 0.1 0.5 0.1 0.5 0.1 0.5 1.0 5 10 100 Predicted Lb (m) Figure 2. Evaluation of relationships for prediction of the wavelength at the shore breaking position. From left to right figure equations are given by text equations (3), (7), (9), and (8), respectively. Another approach using theoretical reasoning proves useful. The total energy of a wave is the sum total of its kinetic and potential energies. The kinetic energy is that portion of the total energy due to water particle velocities associated with wave motion. Potential energy is that portion of the total energy resulting from the wave fluid mass lying above the still water level (SWL). Based on solitary wave theory, Balsillie (1984) determined expressions for the kinetic, potential and total wave energies at the initiation of alpha wave peaking and at the shorebreaking position. Application of the Rayleigh assumption (Eagleson and Dean, 1966),i.e., ... ci ETi = cb ETb (where ci and ETi are the wave speed and total wave energy, respectively, at the initiation of alpha wave peaking, and cb and ETb are their counterparts at shorebreaking), resulted in a working relationship given by: C/ Cb L/ L = 1.841 .8 L, L H T (12) in which (H'/H)ia is the percent of the wave crest height lying above the SWL at the initiation of alpha wave peaking (Balsillie, 1983c, 1999a), given by: SH' H), (13) H 0.515 + 12 H g Comparison of the results from equation (12) with the data of Hansen and Svendsen (1979) suggests a slight modification to the coefficient of equation (12) by a factor of 0, = 1.073, and: LI S 1.84 1, 1 + 14) Cb Lb Lb T Measured data and predicted results from equation (14) using Hansen and Svendsen's (1979) data are illustrated in Figure 3. Wave Celerity and Wavelength Transformation During Alpha Wave Peaking The data of Hansen and Svendsen (1979) for a slope of 0.0292 and that of Hedges and Kirkg6z (1981) for slopes of 0.2247,0.1404,0.102 and 0.0667 suggest that the transformation of c, the local wave celerity during shorebreaking, is nonlinear but only slightly so. For three bed slopes of 0.022, 0.040, and 0.083, Van Dorn (1978) illustrates that the transformation of c is only slightly nonlinear and that it accelerated at a rate close to 0.5 g tan ab. Unfortunately, Van Dorn did not publish his transformation data, Hedges and Kirg6z did not publish their breaker data, and only the data of Hansen and Svendsen, for a single slope, are available. A power curve is used to more nearly represent the slightly nonlinear attenuation of the wave speed during shorebreaking. Two points were used in the curve fitting analysis: 1. ci /Cb from equation (14) and, 2. c/cb = 1.0 at the shorebreaking position. The following equations provide for computation of wave speed (and, thereby, wavelength) attenuation with a slightly non linear character: Measured ci (m/s) Figure 3. measured initiation predicted measured Svendsen Comparison of predicted and wave speeds at the point of of alpha wave peaking; data are from equation (14), data from Hansen and (1979). where c is the local wave speed, I 0.3 02 = 0.1475 tanh .00074 (H' (16) 0074) and 03 = 0.9987 0.233802 (17) Examples from equation (15) are illustrated in Figure 4. Equation (15) is valid where (di /Hi) (d/H) 2 1.28 where di /Hi is the relative water depth at which (H'/H), occurs given (Balsillie, 1983b, 1999a) by: d, d, H, Ho W {n[tanh(65 ) 2 I (18) The shorebreaking wave height, Hb, used in computation of the wave speed at the shorebreaking position, cb, may be evaluated from incident wave conditions (Balsillie, 1983b, 1999a) according to: L c C, Lb T L Lb, 03 2 (15) Hb = 1.0 0.4n tanh 100 HlO H, [[ g2 ) CONCLUSIONS Three issues concerning the prediction of the wavelength and wave speed during the shore breaking wave process 1.30 I have been addressed. First, a family of relationships based on field and laboratory data has been defined for determination of L and c at the shorebreaking position. These relationships have been used to refine theoretical predictions from small amplitude (Airy) and solitary wave theories. The family of derived relationships provides for alternate data sources to l/cb Third, using the above two equations (19) as boundary conditions, the transformation of c and L during alpha wave peaking may be predicted using equation (15). The results presented herein are important because they provide a methodology for predicting wavelength and celerity requiring that only the incident wave 1 2 3 4 5 6 7 d/H Figure 4. Illustration of wave speed attenuation during shorebreaking using equation (15) for various initial wave steepness values. more closely facilitate the needs of the coastal engineer whose completeness in data may differ from project to project. In addition, the commonly used algorithm given by equation (3) is assessed. Not only is the expression difficult to apply, but in surf zone applications it has often been incorrectly used since it represents an upper limit envelope fit. In view of the developments presented in this work, the continued use of equation (3) in surf zone applications is no longer necessary. Second, based on consideration of Solitary wave theory and the Rayleigh assumption, the wavelength and wave speed at the initiation of shorebreaking (i.e., beginning of alpha wave peaking where the wave crest begins to increase in height) may be predicted using equation (14). height and period are known. As more data become available, refinement or corroboration of the relationship for prediction of the initial or incipient wavelength or celerity is desirable. The same is true for prediction of wavelength and celerity transformation throughout the shorebreaking process, although as noted the slight nonlinearity in attenuation should not pose significant problems. ACKNOWLEDGEMENTS Florida Geological Survey staff reviews were conducted by Jon Arthur, Paulette Bond, Kenneth Campbell, Joel Duncan, Ed Lane, Jacqueline Lloyd, Deborah MeKeel, Frank Rupert, and Thomas Scott. Their editorial comments are greatefully acknowledged. The review comments of William F. Tanner of Florida State University are gratefully acknowledged. Nicholas C. Kraus of the U. S. Army, Coastal Engineering Research Center (now the Coastal Engineering Laboratory) also reviewed the manuscript. REFERENCES Balsillie, J. H., 1980, The peaking of waves accompanying shorebreaking: in Tanner, W. F., ed., Shorelines Past and Present, Department of Geology, Tallahassee, FL, Florida State University, v. 1, p. 183247. 1983a, On the determination of when waves break in shallow water: Florida Department of Natural Resources, Beaches and Shores Technical and Design Memorandum No. 833, 25 p. 1983b, The transformation of the wave height during shore breaking: Florida Department of Natural Resources, Beaches and Shores Technical and Design Memorandum No. 834, 33 p. 1983c, Wave crest elevation above the design water level during shorebreaking: Florida Department of Natural Resources, Beaches and Shores Technical and Design Memorandum No. 835, 41 p. 1984, Wave length and wave celerity during shorebreaking: Florida Department of Natural Resources, Beaches and Shores Technical and Design Memorandum No. 841, 17 p. 1985, Redefinition of shore breaker classification as a numerical continuum and a design shore breaker: Journal of Coastal Research, v. 1, p. 247254. 1999a, Wave height transformation during shore breaking: Florida Geological Survey, Special Publication No. 21, p. 130. 1999b, On the breaking of nearshore waves: Florida Geological Survey, Special Publication No. 45, p. 1155. Balsillie, J. H., and Carter, R. W. G., 1980, On the runup resulting from shore breaking wave activity: in W. F. Tanner, ed., Shorelines Past and Present, Tallahassee, FL, Department of Geology, Florida State University, v. 2, p. 269341. Balsillie, J. H., Campbell, K., Coleman, C., Entsminger, L., Glassen, R., Hajishafie, N., Huang, D., Tunsoy, A. F., and Tanner, W. F., 1976, Wave parameter gradients along the wave ray: Marine Geology, v. 22. p. M17 M21. Bretschneider, C. L., 1960, Selection of ( design wave for offshore structures: Transactions, American Society of Civil Engineers, v. 125, pt. 1, paper no. 3026, p. 388416. Eagleson, P. S., 1965, Theoretical study of longshore currents on a plane beach: Massachusetts Institute of Technology, School of Engineering, Hyrdodynamics Laboratory Report No. 82, 48 p. Eagleson, P. S., and Dean, R. G., 1966, Small amplitude wave theory. in A. T. Ippen, ed., Esturay and Coastline Hydrodynamics, New York, McGrawHill Inc., chap. 1, p. 192. Gaillard, D. D., 1904, Wave action in relation to engineering structures: U. S. Army Corps of Engineers Professional Paper No. 31. p. 110 123. Galvin, C. J., Jr., and Eagleson, P. S., 1965, Experimental study of longshore currents on a plane beach: Coastal Engineering Research Center Technical Memorandum No. 10, 80 p. Hansen, J., and Svendsen, I. A., 1979, Regular waves in shoaling water: Technical University of Denmark, Institute of Hydrodynamics and Hydraulic Engineering, Series Paper No. 21, 233 p. Hedges, T. S., and Kirkg6z, M. S., 1981, Experimental study of the transformation zone of plunging breakers: Coastal Engineering, v. 4, p. 319333. Iverson, H. W., 1952, Laboratory study of breakers: in Gravity Waves, National Bureau of Standards Circular 52, U. S. Government Printing Office, Washington, D. C., p. 932. Kinsman, B., 1965, Wind Waves, Pentice Hall, Inc., Englewood Cliffs, New Jersey, 676 p. McGowan, J., 1894, On the highest wave of permanent type: Philosophical Magazine, Edinburgh, Series No. 5, v. 32, p. 351358. Mooers, C. H. K., 1976, Winddriven currents of the continental margin: in Stanley, D. J., and Swift, D. J. P., eds., Marine Sediment Transport and Environmental Management, New York, John Wiley and Sons, p. 29 52. Munk, W. H., 1949, The solitary wave and its application to surf zone problems: Annals of the New York Academy of Sciences, v. 51, p. 376424. Putnam, J. A., 1945, Preliminary report of model studies on the transition of waves in shallow water: University of California at Berkeley, College of Engineering, Contract Nos. 16290, HE116106 (declassified U. S. Army document from the Coastal Engineering Research Center), 35 p. Scripps Institute of Oceanography, 1944a, Waves in shoaling water: Scripps Institute of Oceanography, La Jolla, CA, S. I. O. Report No. 1, 28 p. 1944b, Effect of bottom slope on breaker characteristics as observed along the Scripps Institution pier: Scripps Institute of Oceanography, La Jolla, CA, S. I. O. Report No. 24, 14 p. Smith, R. M., 1976, Breaking wave criteria on a sloping beach: M. S. Thesis, U. S. Naval Postgraduate School, Monterey, CA, 97 p. Stoker, J. J., 1957, Water Waves, New York, Interscience Publishers, Inc., 567 p. Van Dorn, W. G., 1976, Setup and set down in shoaling breakers: Proceedings of the 15th Conference on Coastal Engineering, chap. 41, p. 738751. 1978, Breaking invariants in shoaling waves: Journal of Geophysical Research, v. 83, p. 29812988. NOTATION Symbols c local wave (phase) speed or celerity cg group wave speed d local water depth measured from the DWL DWL design water level g acceleration of gravity H mean local wave height H' mean local wave height lying above the DWL L local wavelength m coefficient n number of data points comprising a sample r Pearson productmoment correlation coefficient SWL design water level represented by the still water level T wave period tan a bed slope ab alpha wave peaking 0q1 coefficients Subscripts b parametervalue at the shorebreaking position (i.e., at termination of alpha wave peaking). i parameter value at the beginning of shorebreaking (i.e., at the initiation of alpha wave peaking) m parameter value before entering transitional water depths 828 8 "5234 84/29/1 3476 B F 