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Onshore/offshore transport mechanisms Walsh, Bruce William 1989

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ONSHORE/OFFSHORE TRANSPORT MECHANISMS by BRUCE WILLIAM WALSH B.A.Sc, The University of B r i t i s h Columbia, 1986 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE i n THE FACULTY OF GRADUATE STUDIES Department of C i v i l Engineering We accept t h i s t h e s i s as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA A p r i l 1989 (c) Bruce William Walsh, 1989 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department The University of British Columbia Vancouver, Canada DE-6 (2/88) ABSTRACT The onshore/offshore transport mechanisms are investigated. Careful and d e t a i l e d observations of the surf zone are made to f i n d any differences or s i m i l a r i t i e s between the r e s u l t i n g flows caused by d i f f e r e n t breaking types. Emphasis i s placed on the s p i l l i n g and plunging breakers. Even though the physical breaking properties are s i g n i f i c a n t l y d i f f e r e n t , the time-averaged properties of the surf zone for each type of breaking wave are s i m i l a r . Using t h i s as a basis, a model of the beach i s developed using a control volume that i s bounded by the beach face, the time-averaged water l e v e l , and a v e r t i c a l plane at the breaking point. The momentum acting on the control volume i n the onshore and offshore horizontal d i r e c t i o n s i s balanced. The model shows that the onshore/.offshore sediment transport i s p r i m a r i l y dependent upon the magnitude of the wave setup shoreward of the breaking point, and the permeability of the beach. Increasing the permeability causes a reduction i n the offshore net shear stress acting along the beach face which r e s u l t s i n an increasing slope. Using t h i s simple model, the difference between a gravel and sand beach can be explained, the gravel beach being steeper. The model i s used to calculated the offshore net shear stress for a plane impermeable beach i n the laboratory. The c a l c u l a t i o n gives the r i g h t order of magnitude (<10 N/m2), but proves to be s e n s i t i v e to small inaccuracies i n the measurement of the setup. i i TABLE OF CONTENTS Page Abstract i i Table of Contents i i i L i s t of Tables v i L i s t of Figures v i i Acknowledgement x i CHAPTER 1 INTRODUCTION 1 CHAPTER 2 BREAKING WAVE TYPES 4 2 . 1 Introduction 4 2 . 2 Physical C l a s s i f i c a t i o n of Breaking Waves .... 5 2 . 3 Breaking Indices: H Q / L Q , Beach Slope, and Breaking Types .... 3 2 . 3 . 1 Comparison of the Breaking Indices .... 1 5 2 . 3 . 2 Variation of the Breaking Indices 1 9 2 . 3 . 2 . 1 Experimental Procedures and Calculations .... 1 9 2 . 3 . 2 . 2 Wave Ef f e c t s 2 3 2 . 3 . 2 . 3 Breaking D e f i n i t i o n s 2 7 2 . 3 . 2 . 4 Personal Judgement 2 7 2 . 3 . 3 Breaking Indices on Natural Beaches ... 2 8 2 . 4 Experimental Design and Procedure 3 2 2 . 5 Experimental Results .' 34 2 . 6 Discussion 3 6 i i i Page CHAPTER 3 WAVE HEIGHT AND DEPTH AT BREAKING 38 3.1 Introduction 38 3.2 Definition of the Breaking Height and Depth .. 39 3.2.1 The Height-to-Depth Ratio 42 3.3 Influence of the Wave Steepness 47 3.4 Influence of the Depth on Breaking 57 3.5 Influence of the Beach Slope on Breaking 60 3.6 Discussion 65 CHAPTER 4 FLOW DYNAMICS WITHIN THE SURF ZONE 68 4.1 Introduction 68 4.2 Transition to Rotational Flow 70 4.3 Established Turbulent Flow 75 4.4 Mass Transport 76 4.5 Setup i n the Surf Zone 88 4.5.1 Experimental Design and Procedure 92 4.5.2 Experimental Results 94 4.6 Discussion 99 CHAPTER 5 DEVELOPMENT OF THE BEACH FACE CONTROL VOLUME 102 5.1 Introduction 102 5.2 Formulation of the Model 103 5.2.1 Assumption of an Impermeable Beach .... 105 5.2.2 Inclusion of the E f f e c t of Permeability 115 5.3 Different Beaches Subject to the Same Wave Attack 117 5.4 Same Beaches Subject to Varying Wave Attack .. 127 iv Page 5.5 Oblique Wave Attack 131 5.6 Discussion 132 CHAPTER 6 TESTING OF THE BEACH FACE CONTROL VOLUME MODEL .... 136 6.1 Introduction 13 6 6.2 Model for an Impermeable Beach 137 6.3 Results 139 6.4 S e n s i t i v i t y of the Model 147 6.5 Discussion 152 CHAPTER 7 SUMMARY AND DISCUSSIONS 155 BIBLIOGRAPHY 163 APPENDICES A - Shoaling Characteristics of an O s c i l l a t o r y Wave .... 168 B - Observations of P a r t i c l e Motions Under Breaking Waves 171 C - Wave Setup/Setdown Data 184 D - Wave Reflection Data 207 E - Laboratory Study of Breakers (Iversen 1952) 209 v LIST OF TABLES Table T i t l e Page 2.1 Transition points 22 3.1 Maximum height-to-depth r a t i o s for s o l i t a r y waves (after Galvin 1972) 43 3.2 Beach slope e f f e c t on breaker height for o s c i l l a t o r y waves 61 3.3 Beach slope e f f e c t on breaker height for a 1:15 slope 61 6.1 The calculated shear stress using the data c o l l e c t e d for each wave condition 140 6.2 S e n s i t i v i t y of the impermeable beach model 149 v i LIST OF FIGURES Figure T i t l e Page 2 . 1 Breaking wave p r o f i l e s 6 2 . 2 Breaker types for three beach slopes (Galvin 1 9 6 8 ) . 1 0 2 . 3 Breaker type as a function of the offshore parameter (Galvin 1 9 6 8 ) 1 1 2 . 4 Breaker type as a function of the onshore parameter (Galvin 1 9 6 8 ) 1 1 2 . 5 Variation of the breaker type with the beach slope and H Q / T 2 (Iversen 1 9 5 3 ) 1 2 2 . 6 Variation of the breaker height with the deepwater wave steepness and the beach slope (Iversen 1 9 5 3 ) 13 2 . 7 Variation of the breaker type with the deepwater wave steepness and the beach slope 14 2 . 8 Regions for which surging, plunging, and s p i l l i n g breakers occur (Weggel 1 9 7 2 ) 1 6 2 . 9 Non-dimensional depth at breaking versus breaker steepness (SPM 1 9 8 4 ) 1 7 2 . 1 0 Breaker height index versus deepwater wave steepness 1 7 2 . 1 1 Breaker height index versus deepwater wave steepness for d i f f e r e n t slopes 1 8 2 . 1 2 Spill-Plunge t r a n s i t i o n 2 0 2 . 1 3 Plunge-Surge t r a n s i t i o n 2 1 2 . 1 4 The breaker types included i n Galvin's ( 1 9 6 8 ) study 2 5 2 . 1 5 E f f e c t of backrush on the breaker type 2 6 2 . 1 6 Transition zones 2 9 2 . 1 7 Comparison of the breaker indices for natural beaches 3 0 2 . 1 8 Overall flume setup (not to scale) 3 3 v i i Figure T i t l e Page 2.19 Experimental results for H Q / L Q and the breaker type on the 1:15 slope 35 3.1 Variables of the wave at the point of breaking .... 41 3.2 Breaker height index versus deepwater wave steepness 45 3.3 Non-dimensional depth at breaking versus breaker steepness (SPM 1984) 45 3.4 Dependence of H^/d^ on deepwater wave steepness and beach slope 4 6 3.5 Plot of data (Iversen 1952) for 1:10 slope . .. 48 3.6 Plot of data (Iversen 1952) for 1:20 slope 49 3.7 Plot of data (Iversen 1952) for 1:30 slope 50 3.8 Plot of data (Iversen 1952) for 1:50 slope 51 3.9 Comparison of experimental r e s u l t s with Miche's formula 53 3.10 Miche's formula and the shoaling equation for selected values of H Q / L Q 55 3.11 Miche's l i m i t 56 3.12 Breaking response of a s o l i t a r y wave 59 3.13 H b/H Q versus H 0/L Q 62 3.14 d b/H Q versus H 0/L Q 64 5.15 Trends of important r a t i o s for o s c i l l a t o r y waves and s o l i t a r y waves 64a 3.16 Relationships between the beach slope, the deepwater wave steepness, and the breaking conditions 65a 4.1 Surf zone regions 69 4.2 Vortex chain produced by a symmetric s p i l l i n g breaker 72 4.3 Asymmetrical plunging breaker and the splash-plunge cycle (Longuet-Higgins 1953 .... 74 4.4 Turbulent bore 77 v i i i Figure T i t l e Page 4.5 Typical mass transport v e l o c i t y p r o f i l e s 79 4.6 Effe c t of increasing beach roughness on the mass transport v e l o c i t y 80 4.7 V e r t i c a l d i s t r i b u t i o n of the d r i f t v e l o c i t y at the breaking point 82 4.8 Transport mechanism offshore of the n u l l point .... 84 4.9 Transport mechanism onshore of the n u l l point 86 5.10 a) Ci r c u l a t i o n c e l l s on either side of the n u l l point, b) Mass transport v e l o c i t y p r o f i l e s i n the surf zone 87 4.11 Def i n i t i o n sketch of the wave setup .. 90 4.12 Setup curves of S w/H b versus H^/gT2 93 4.13 Setup p r o f i l e s for experimental run 2 95 4.14 Calculated t o t a l setup versus measured t o t a l setup for an experimental 1:15 slope 97 4.15 a) Total setup versus deepwater steepness b) Relative t o t a l setup versus deepwater steepness 98 4.16 Components of the t o t a l setup for experimental run 7, run 10, run 11 100 5.1 Beach face control volume and the forces acting upon i t 104 5.2 Simplified beach face control volume 106 5.3 Forces acting upon a p a r t i c l e laying on the beach . 108 5.4 A speculative explanation for an offshore net shear stress 110 5.5 The offshore control volume used to calculate Mb' . 112 5.6 I n f i l t r a t i o n and e x f i l t r a t i o n on a beach face 119 5.7 Undamped solution for the same wave attacking d i f f e r e n t beaches (equation 5.22) . 123 5.8 Damped solution for the same wave attacking d i f f e r e n t beaches (equation 5.23) with selected data from Dalrymple and Thompson (1976) 126 ix Figure T i t l e Page 5.9 Undamped solution for d i f f e r e n t waves attacking the same beach (equation 5.28) 129 5.10 Damped solution for d i f f e r e n t waves attacking the same beach (equation 5.29) with selected data from Dalrymple and Thompson (1976) 130 6.1 Shear stress required for a moveable bed based upon Shields entrainment function 14 3 6.2 Calculated shear stress versus H/gT2, depth = 47.5 cm 144 6.3 Calculated shear stress versus H/gT2, depth = 45.0 cm 145 6.4 Calculated shear stress versus H/gT2 for changing wave conditions 148 x ACKNOWLEDGEMENTS The author i s very grateful for the f i n a n c i a l support, ouragement, and guidance given by his supervisor, Professor C. Quick. x i C H A P T E R 1 : I N T R O D U C T I O N I t i s well documented that beaches can undergo quite large changes when exposed to severe storm attack. I t i s known that large quantities of material can be moved offshore, causing a reduction i n beach slope and permitting wave attack to cut at the backshore region, often with very destructive r e s u l t s . On the other hand, during periods of calm weather conditions, gentler waves can move sediment shoreward so that the beach repairs e i t h e r p a r t i a l l y or completely, depending upon the i n t e r v a l u n t i l the next storm. Damage during the storm can be considerable and i t i s economically desirable to be able to p r e d i c t the nature and extent of damage. The motion of waves i n deep and shallow water i s well known and well modelled. However, as the waves shoal and break upon a beach, the c h a r a c t e r i s t i c s and action of the breaking waves are not very well understood and are d i f f i c u l t to model. Incident waves generally a r r i v e obliquely to the shoreline and can be resolved into longshore and onshore/offshore components. The 1 main focus of research has been on the wave-induced longshore currents and the longshore transport. L i t t l e has been done on the onshore/offshore components and many questions regarding these a r i s e . Among the more pressing ones are: 1 . Why are there d i f f e r e n t types of breaking waves and what are the c h a r a c t e r i s t i c s associated with breaking? 2. Does the breaker type influence the on-offshore currents and transport mechanisms i n the surf zone? 3. What are the relationships between the breaking waves, beach slope, and sediments for onshore/offshore transport to occur? 4. How does the onshore/offshore transport change f o r changing wave and beach conditions? This t h e s i s w i l l t r y to answer some of these questions, remembering that the nearshore processes are complex and interconnected, and that many are s t i l l not properly understood. Presented i n t h i s thesis i s a breakdown of the events occurring i n the surf zone with respect to only the onshore/offshore d i r e c t i o n . Chapter two describes the physical c h a r a c t e r i s t i c s of each breaker type and the re l a t i o n s h i p between the breaker type, deepwater wave steepness, and beach slope. Experimental measurements are compared with those found by other researchers. Chapter three defines the wave height and depth at breaking, and discusses the influence of wave steepness, breaking depth, and the beach slope on the breaking process. 2 Chapter four describes the f l u i d motions produced within the surf zone with p a r t i c u l a r emphasis on the transformation of i r r o t a t i o n a l wave motion into r o t a t i o n a l and ultimately turbulent flows. Both the mass transport v e l o c i t i e s and the water l e v e l changes, which are r e s u l t s of these flows, are discussed. Experimental observations of f l u i d and p a r t i c l e motions, as well as ca r e f u l measurements of water l e v e l changes, are used. Chapter f i v e shows the development of an on-offshore transport model based upon a control volume of the surf zone which uses the findings of the previous chapters as a basis. In chapter s i x , the model i s tested using the experimental r e s u l t s found f o r water l e v e l changes mentioned i n chapter four. Lastly, conclusions drawn from a l l the r e s u l t s are given i n chapter seven. 3 C H A P T E R 2 : B R E A K I N G W A V E T Y P E S 2 . 1 I N T R O D U C T I O N The most v i s i b l e aspect of a breaking wave i s i t s shape as i t breaks. As with any complex phenomena, an important f i r s t step i s to categorize the events that are observed. For breaking waves, the shapes have been c l a s s i f i e d into four breaker types, each type dominating for c e r t a i n values of deepwater steepness and beach slopes. Although the d i s t i n c t breaker types are r e l a t i v e l y easy to d i s t i n g u i s h , the t r a n s i t i o n s that separate the breaker types are d i f f i c u l t to determine. Since the flow dynamics vary for d i f f e r e n t breaker types i t i s important to understand the conditions that are associated with each breaker type. This chapter describes the physical c h a r a c t e r i s t i c s of each breaker type and the r e l a t i o n s h i p between breaker type, deepwater steepness, and beach slope. 4 2.2 P H Y S I C A L C L A S S I F I C A T I O N S O F B R E A K I N G W A V E S The breaking wave can be generally c l a s s i f i e d into one of four groups: s p i l l i n g ; plunging; c o l l a p s i n g ; or, surging (Galvin 1968, 1972). The physical appearance of each type i s shown i n figure 2.1 and the description of each type as given by Galvin are as follows: 1) S p i l l i n g - White turbulent water appears at the crest preceded by a small j e t of water. The turbulence " s p i l l s " down the face of the wave. The wave shape as a whole remains symmetric. 2) Plunging - The front of the wave face reaches v e r t i c a l and the crest forms a j e t which plunges ahead of the wave. 3) Collapsing - The front of the wave face reaches v e r t i c a l and the lower portion of the wave acts as a truncated plunging breaker. 4) Surging - The wave stays r e l a t i v e l y smooth as i t moves up the beach, except f o r minor turbulence at the wave-shoreline interface. Turbulence i s generated by bottom boundary shear. The d e s c r i p t i v e terms are used to d i s t i n g u i s h the breaker at the point of i n i t i a l motion. However, three problems a r i s e by the use of t h i s c l a s s i f i c a t i o n system When the breaking p r o f i l e i s an intermediate of any two of the above p r o f i l e s i t i s d i f f i c u l t to determine how the breaker can best be described. This i s solved by using a breaker type index which predicts the breaker type based on the value of H 0/L 0 and the beach slope. One must remember that a continuous sequence e x i s t s between s p i l l i n g , plunging, co l l a p s i n g , and surging breakers and t r a n s i t i o n types are possible. Breakers can f a l l outside of t h i s continuous sequence. An example of t h i s i s a surging wave that i s forced to act as a plunging breaker due to the action of the backrush. For 5 -b) P lung ing Figure 2.1 Breaking wave prof i les 6 example, a breaker index can predict a surging breaker, but a "forced" plunging type breaker a c t u a l l y occurs. Obviously, i n t h i s case the permeability and slope of the beach play important r o l e s . The f i n a l problem i s one of scale. The i n i t i a l breaking motions of both the plunging and s p i l l i n g breakers are the same except the scale for which they occur (Galvin 1968, 1972; M i l l e r 1976; Basco 1985). Plunging breakers are generally categorized by a large-scale, v i s i b l e c u r l i n g of the wave crest around an inner a i r core. The f a l l i n g j e t h i t s the oncoming trough suddenly transforming the i r r o t a t i o n a l motion to r o t a t i o n a l motion over a large percentage of the water column. An i d e n t i f i a b l e time and distance are required f o r the wave to c u r l over before reaching the j e t impact point or plunge point. For the s p i l l i n g breaker, on the other hand, the i n i t i a l , low-level impact of the j e t generally occurs quickly and i s not v i s i b l e to the eye. The small scale j e t r a p i d l y creates turbulence at the crest which s l i d e s down the face of the wave. V o r t i c i t y and turbulence are created and the flow gradually changes from i r r o t a t i o n a l to r o t a t i o n a l motion over an increasing percentage of the water column. Both s p i l l i n g and plunging breakers are i n i t i a t e d by a j e t h i t t i n g the forward face of the wave. The i n i t i a l behaviour of each type i s i d e n t i c a l , except f o r the difference i n scale, but the r e s u l t i n g motions are very d i f f e r e n t . Care must be taken to i d e n t i f y the breaker based on i t s t o t a l motion. 7 Aside from these problems, the c l a s s i f i c a t i o n of breakers given by Galvin i s widely used as a q u a l i t a t i v e description, but does not lend i t s e l f to producing quantitative values. 2 . 3 B R E A K I N G I N D I C E S : H Q / L Q , B E A C H S L O P E , B R E A K I N G T Y P E S There i s agreement that breaker types form a spectrum of shapes going from s p i l l i n g to plunging to c o l l a p s i n g to surging, and v i c e versa. The cycle i s complex because the beach slope depends upon the breaker type and the breaker type depends upon the beach slope. General observations of beaches can be made. Steep waves on low angle beaches tend to form s p i l l i n g breakers, whereas less steep waves on higher angle beach slopes tend to form plunging or surging breakers. Results of Iversen (1953), Patrick and Wiegel (1955), Galvin (1968), and personal observations confirm these statements. Thornton et a l . (1976) measured the kinematics of various types of breaking waves and f i n d that the manner i n which waves break depends very much on the c h a r a c t e r i s t i c s of the deepwater steepness and the near-shore bottom slope. Breaking waves were o r i g i n a l l y c l a s s i f i e d by Galvin (1968) who found that two dimensionless parameters could be used to determine the type of breaking wave r e s u l t i n g from offshore or inshore wave c h a r a c t e r i s t i c s . The breaker types f o r waves on impermeable beaches with slope m, wave period T, deepwater wavelength L Q, and eith e r deepwater or breaker height, H Q or H^, can be sorted by two dimensionless combinations of these 8 v a r i a b l e s . The offshore parameter i s , H 0 / ( L 0 m 2 ) (2,1) and, the inshore parameter i s , H b / (gmT 2 ) (2,2) Both parameters are empirically determined. I n i t i a l l y , H 0 / L 0 and ( H b / g T ) 1 / / 2 are plotted for each slope (figure 2 . 2 ) . The r e g u l a r i t i e s i n figure 2 . 2 suggests that the slope can be included i n the c l a s s i f y i n g parameters (figures 2 . 3 and 2 . 4 ) . For the offshore and inshore parameters, respectively, the surge-plunge t r a n s i t i o n s are 0 . 0 9 and 0 . 0 0 3 and the pl u n g e - s p i l l t r a n s i t i o n s are 4 . 8 and 0 . 0 6 8 . As either of these parameters increase, the breaker type goes from surging to co l l a p s i n g to plunging to s p i l l i n g . Iversen ( 1 9 5 3 ) provides a graph of the v a r i a t i o n of the breaker c h a r a c t e r i s t i c s with beach slope, deepwater wave height, and period (figure 2 . 5 ) . The parameter H Q / T 2 i s equivalent to the deepwater wave steepness, H 0 / L Q / because LQ=gT2/27T, and i s plott e d i n figure 2 . 6 . Iversen gives only the ranges of H Q / T 2 over which each of the three breaker types are observed; breaker types f o r i n d i v i d u a l waves are not ava i l a b l e . In addition, for two of the three beach slopes investigated, the plunge-surge t r a n s i t i o n may not have been reached; therefore, the lower l i m i t of H 0 / L 0 f o r which plunging breakers are observed are used as the t r a n s i t i o n points. Patrick and Wiegel ( 1 9 5 5 ) provide a graph of the v a r i a t i o n of breaker c h a r a c t e r i s t i c s with deep water wave steepness and offshore beach steepness (figure 2 . 7 ) . The laboratory data upon 9 A Spilling O Plunging • Collapsing • Surging ® Plunging affected by reflection 0.05 L °-2°L •—g-oo c—9-am 9 Jo °-,0' D—rJ-O-*—#CDO—L-c—®0-O-OO——OO OcL o «n .c u o <= I 1 1 1111 l lJ 1 1 ' I < I I I I I l I l l t i i I I0"4 I0"5 I0"2 10* Computed Deep-woter Steepness, H 0 / L 0 0.20I—c&Jbo—' to • o.ioL_a3#U>j^-o_o-c oL-0.05 I O—OO-Q o b &Cr-o to 0.02 0.04 0.06 0.08 0.10 0.12 Breoker Sleepness, ( H b / g T 2 ) l / 2 F i gure 2 , 2 Breaker "types f o r 3 beach s l o p e s ( G a l v i n 1968) 10 A Spilling O Plunging • Collapsing • Surging <g) Plunging affected by reflection Surging-Collapsing Plunging Spilling 10 -A 10 -3 I0' 2 10"' H0/(L0mz) 10° 10' 10' F i g u r e 2 ,3 Breaker type as a f u n c t i o n of the o f f s h o r e parameter ( G a l v i n 1968) Surging-Collapsing Plunging Spilling 10" 10" 10' 10' F i gure 2 ,4 Hb/{gmn Breaker type as a f u n c t i o n of the onshore parameter ( G a l v i n 1968) i i • I-10 p p L U N G I N G S P I L L I N G S U R G I N G 5 P I L L I N G gin q < Btach Slop, Btach S/opt - 1:20 S P I • /.SO P L U 001 0 02 0.04 OX* 0.1 0 2 0 4 0.6 Q 8 D E E P W A T E R W A V E , H0/T* gure 2 , 5 Var i at i on of the b r e a k e r "type w i "th beach s I ope an d H o / T 2 ( I versen- 1953) 12 Beach sio 38 BeaoT Slo >e B e a c i Slo >e 1:10 Su 2 0 5') Flu ige Dlungu Plunge Sail 3D O.0O1 0.01 0.1 Deepwater steepness, Ho/Lo F i gure 2 ,6 Var i a t i on of the b r e a k e r type w i th the deepwater wave s t e e p n e s s and the beach s l o p e ( I ve rsen 1953) 13 u, i " 0 a 1/20 z o < U 1/20 1/50 1/10 1/10 1/20 1/50 a aoi 0.00B 0.01 o.o< 0.04 M. , DCEP WATER WAVE HEIGHT L . , OCEP WATER WAVE L£N€TM 1 1 "*" •A • I JTTT II 1 1 i J Spilling Plunging Surging OJM 0.0* ( a f t e r I v e r s e n 1953) F i gure 2 ,7 V a r i a t i o n o f the b r e a k e r type wi th the deepwater wave s t e e p n e s s and the beach s l o p e ( P a t r i c k and Wiege I 1955) 14 which Patrick and Wiegel base t h e i r graph has not been published. For two of the three beach slopes investigated, the plunging to surging t r a n s i t i o n may not have been reached. The lower l i m i t of H 0 / L Q for which the plunging breaker are observed are used as the t r a n s i t i o n point. Weggel ( 1 9 7 2 ) plots the r e s u l t s of Patrick and Wiegel i n terms of H Q / L 0 , beach slopes, and breaker type. The regions of t r a n s i t i o n are shown i n figure 2 . 8 . Iversen ( 1 9 5 2 , 1 9 5 3 ) , Galvin ( 1 9 6 8 ) , Goda ( 1 9 7 0 ) and Weggel ( 1 9 7 2 ) e s t a b l i s h that H B / H Q and d^E^ depend on the beach slope and the incident wave steepness. The U.S. Army Corps of Engineers presents the re s u l t s of Weggel and Goda as two graphs (figures 2 . 9 and 2 . 1 0 ) . Weggel's r e s u l t s are based upon those of Patrick and Wiegel. Figure 2 . 1 0 presents Goda's empirically derived r e l a t i o n s h i p between the breaker height index, H ^ / H Q , and deepwater wave steepness, H Q/gT , for various slopes. Patrick et a l . present ranges of H 0 / L Q for several beach slopes for which each type of breaker can be expected to occur. This information i s also presented i n figure 2 . 1 0 . A s i m i l a r r e s u l t i s reached by Iversen ( 1 9 5 2 ) and, again, the t r a n s i t i o n points given by Patrick et a l . are plotted i n figure 2 . 1 1 . 2.3.1 Comparisons of the Breaking Indices Iversen ( 1 9 5 3 ) , Patrick et a l . ( 1 9 5 5 ) , and Galvin ( 1 9 6 8 ) provide the only independent data on the r e l a t i o n s h i p between beach slope, deepwater wave steepness, and breaker types. I f i t i s possible to predict the breaker type given the slope and 1 5 007 0 06 0.03 0.04 0.03 0.02 0.01 H i = 0.01 0 v ion H - Spilling I t 0.30 m i n - piuo O l ( l - « ! f 1 Region I - Surging 1 1 0 0.01 0.02 0 03 0 04 0 0 3 0 06 0.07 0 08 0 09 0.10 Beach Slope, m F i g u r e 2 .8 Reg ions f o r which s u r g i n g , p l u n g i n g , and s p i l l i n g b r e a k e r s o c c u r (Weggel 1972) 16 0 0.002 0.004 0.006 0.008 0.010 0.012 0.014 0.016 0.018 0.020 Figure'2,10 Breaker height index v e r s u s deepwater wave s teepness (SPM 1984) 00004 00006 0001 0002 0 004 0 006 0 01 002 0 03 HJ, (olUr Goda, 1970) •J1 F i g u r e 2 ,9 Non-d imens ion aI depth a t b r e a k i n g v e r s u s b r e a k e r s t e e p n e s s (SPM 1984) 17 Figure 2,11 Breaker height index v e r s u s deepwater wave s t e e p n e s s f o r d i f f e r e n t slopes 18 H 0/L 0, then t h i s information could be use to qu a n t i t a t i v e l y understand the breaking process. Figures 2.12 and 2.13 are the plo t s of the values of the breaker indices presented i n table 2.1. Comparison of the s p i l l i n g to plunging t r a n s i t i o n show agreement between Patrick et a l . and Iversen, with Galvin's r e s u l t s p l o t t i n g well below. In fact, f o r values of low beach slope, the difference i s of an order of magnitude. For the plunging to surging t r a n s i t i o n there i s l i t t l e agreement between the researchers, with Galvin' s r e s u l t s d i f f e r i n g by two orders of magnitude. 2.3.2 Variations of the Breaking Indices Differences of up to two orders of magnitude between the re s u l t s of the three researchers suggests that s i g n i f i c a n t factors are causing the v a r i a t i o n s . These factors may ar i s e from differences i n the procedures, c a l c u l a t i o n s , and d e f i n i t i o n s used by the researches. These and others w i l l now be discussed. 2.3.2.1 Experimental Procedures and Calculations Galvin uses a 71 foot wave flume and a piston-type wave generator. Iversen uses a 54 foot wave flume and a flap-type wave generator. Patrick et a l . do not provide any information of t h e i r equipment. To c a l c u l a t e the deepwater wave height, H Q, Iversen and Patrick et a l . use the wave height measured i n the constant depth section of the wave flume and back ca l c u l a t e H Q using l i n e a r wave theory. Galvin uses the wave height predicted t h e o r e t i c a l l y for a given displacement of the piston (Biessel 19 Deepwater wave s teepness, H o / L o 0 . 0 7 0 - i 0 . 0 6 0 -0 . 0 4 0 -0 . 0 3 0 -0 . 0 2 0 -0 . 0 1 0 -o . o o o -\ \ 1 1 1 1 1 1 — 0 . 0 0 0 0 . 0 2 0 0 . 0 4 0 0 . 0 8 0 0 . 0 8 0 0 . 1 0 0 0 .120 0 .140 Slope ( m ) Figure 2,12 Spill-Plunge transit ion 20 0 . 0 1 0 Deepwater wave steepness, Ho /Lo 0 . 0 0 8 Plunge 0 . 0 0 6 0 . 0 0 4 0 . 0 0 2 0 . 0 0 0 I v e r s e n ( i 9 5 3 ) -}— P a t r i c k et a l . ( 1955 ) -?K- G a l v i n ( 1 9 6 8 ) Surge 0 . 0 0 0 0 . 0 2 0 0 . 0 4 0 0 . 0 6 0 0 . 0 8 0 Slope ( m ) 0.100 0 .120 0 .140 Figure 2.13 Plunge-Surge transit ion a) Patr ick and Weggel (1955) S L O P E SURGING PLUNGING S P I L L I N G 0.020 0,033 0,050 0.100 Ho/Lo<0.0039 <0.0079 <0.0095 0.0039<Ho/Lo<0.020 0.0079< <0.035 0.0095< <0.060 Ho/Lo>0.020 >0.035 >0,060 b) Galvin (1968) S L D P E SURGING PLUNGING S P I L L I N G 0,020 0.033 0.050 0,100 Ho/Lo<0.000036 <0.000100 <0.000225 <0.000900 0.000036<Ho/Lo<0.00192 0.000100< <0.00520 0.000225< <0.01200 0.00090CK 0.04800 Ho/Lo>0,00192 >0.00520 >0,01200 >0.04800 c) Iversen (1953) S L D P E SURGING PLUNGING S P I L L I N G 0.020 0.033 0.050 0.100 Ho/Lo<0.00195 <0.00390 <0.00878 0.00195<Ho/Lo<0.023 0,0039< <0.0375 0.00878< <0.0605 Ho /Lo>0 .0233 >0.0375 >0.0605 Table 2,1 Transition points 22 1951) and back calculates H Q using l i n e a r wave theory. This method e f f e c t i v e l y eliminates the problem of at t a i n i n g a s p a t i a l l y uniform wave height i n the wave flume, but Galvin (1964) finds that the t h e o r e t i c a l height i s greater than the measured wave height i n the flume. Therefore, for a given breaker type on a given slope with i d e n t i c a l conditions i n the wave flume, Galvin's r e s u l t s should p l o t above Iversen's and Patrick's, regardless of the wave generator. However, figures 2.12 and 2.13 show t h i s not to be the case. Iversen's and Patrick's r e s u l t s are consistently above Galvin's which indicates that other factors are responsible f o r the difference i n the r e s u l t s . 2.3.2.2 Wave Ef f e c t s Wave e f f e c t s that i n t e r f e r e with the development of t y p i c a l breaking waves are secondary waves, wave r e f l e c t i o n s , and backrush. Secondary waves are caused by the breakdown of a large wave into a primary wave and a number of smaller secondary waves. Wave r e f l e c t i o n s and backrush are influenced by beach slope and beach permeability with both increasing for increasing slope and decreasing permeability. (The importance of beach slope and beach permeability w i l l be described i n l a t e r chapters.) Galvin uses an impermeable concrete beach and slopes steeper than those commonly found i n nature. Therefore, r e f l e c t i o n s and backrush may a f f e c t the breaking wave, but w i l l not a f f e c t Galvin's c a l c u l a t i o n of H Q/L 0 since he determines t h i s from the displacement of the wave generator. To account 23 for v a r i a t i o n s i n the breaker shape Galvin l a b e l s breakers according to a l i s t of possible types (f igure ,,2.14) and requires that the data s a t i s f y the following r e s t r i c t i o n s : 1) A dominant breaker type e x i s t s when at le a s t 70% of the waves i n a steady-state sequence have the same breaker type. 2) The breaking of the primary wave i s not hindered by a secondary wave and i s s a t i s f i e d when the breaker i s not given a code number of 7 or 8, according to figure 2.14. These precautions reduce the influence of secondary waves and wave r e f l e c t i o n s . None of these precautions, though, takes backrush into account. Backrush i s capable of forcing a coll a p s i n g or surging breaker to act as a plunging breaker. If forced plunging breakers are recorded as a plunging breaker then the plunge-surge t r a n s i t i o n w i l l move down (figure 2.15) since plunging w i l l occur over a greater range of H 0/L Q. Iversen's cal c u l a t i o n s of H 0/L Q are affected by secondary waves and r e f l e c t i o n s since the value of H Q i s dependent upon wave heights measured within the wave flume. The si g n i f i c a n c e of t h i s has already been mentioned. Iversen, though, takes backrush into account so that the plunge-surge t r a n s i t i o n given by Iversen should be higher than the t r a n s i t i o n given by Galvin, which i s the case. Beach permeability i s an important factor c o n t r o l l i n g backrush. I f a beach i s permeable the backrush w i l l be reduced since the uprush can permeate into the beach. I f the beach i s impermeable, the backrush w i l l be strong since a l l of the uprush must return as backrush. Coda Type of Breaking Description 1 Spilling 2 Well-developed plunging 3 Plunging 4 Collapsing 5 Surging 6 Plunging altered by reflected wave 7 Plunging altered by secondary wave 8 Surging altered by secondary wave 0 Secondary wave washed out Bubbles and turbulent watai spill down front face of wart. Tbe upper 2aTc of the front fact may become vertical befort breaking. Crest curls over a large air pocket. Smooth splash-up usu-ally follows. Crest curls less and air pocket •mailer than in 2. Breaking occurs over lower half of wave. Minimal air pocket and usually no spLajh-up. Bub-bles and foam present. Wave slides up beach witif little or no bubble production! Water surface remains almost plane except where ripples majt be produced on the beach fats during runback. Small waves reflected from the preceding wave peak up tbt breaking crest. Breaking other-wise unaffected. Primary may ride in on secondary immediately befon it , or secondary immediately behind ride* in on primary is front. First kind difficult to distinguish from 8. Plunging secondary may break just in front of surging primary. Difficult to distinguish from 7. Runback from previous pri-mary carries tbe secondary ws*s offshore, where it may break out of field of view or just disappear. Figure 2.14 Breaker "types included in Galvin's(1968) s tudy 25 © Slope (m) (T) Sp i l l -P lunge transit ion (2) P l u n g e - S u r g e transit ion where forced plungers are recorded as surging breakers (3) P l u n g e - S u r g e transit ion where the effect of the backrush is neglected Figure 2.15 The effect of backrush on the breaker type 26 Surging breakers generally occur closer to the shoreline. Therefore, a surging wave on a permeable beach can be forced to plunge on an impermeable beach. Both Galvin and Iversen use impermeable beaches steeper than those commonly found i n nature. I t i s not su r p r i s i n g then to f i n d the surge region to be r e l a t i v e l y small for small values of H Q/L 0 (see figure 2.13). The same cannot be d e f i n i t e l y said for waves i n the spill- p l u n g e t r a n s i t i o n . Generally, these waves break further offshore than surging breakers and are less susceptible to the e f f e c t of backrush. 2 . 3 .2.3 Breaking D e f i n i t i o n s Galvin defines s p i l l i n g , plunging, co l l a p s i n g , and surging breakers, but finds that the col l a p s i n g breakers group together with the surging breakers and i n the end only shows the t r a n s i t i o n f o r s p i l l i n g , plunging, and surging breakers. Iversen and Patrick et a l . define s p i l l i n g , plunging, and surging breakers. They also grouped the co l l a p s i n g type with the surging breaker. Therefore, the researchers use the same d e f i n i t i o n s for each breaker type which does not help explain the d i f f e r e n t r e s u l t s . 2.3.2.4 Personal Judgement A breaking wave close to a t r a n s i t i o n state w i l l exhibit c h a r a c t e r i s t i c s of two breaking types. In t h i s s i t u a t i o n , the personal judgement of the researchers w i l l have a s i g n i f i c a n t influence on the f i n a l r e s u l t s . At a t r a n s i t i o n , the breaker d e f i n i t i o n s begin to overlap, so for example, what one researcher defines as a plunging breaker, another researcher 27 could define as a c o l l a p s i n g breaker. Even though the d e f i n i t i o n s of the breaking types are the same and d e f i n i t e , the i n t e r p r e t a t i o n of the d e f i n i t i o n are d i f f e r e n t between ind i v i d u a l s and are applied at d i f f e r e n t points i n the breaking process. This may explain the consistent difference found between the breaking t r a n s i t i o n s . Figure 2.16 shows the range of v a r i a t i o n between the r e s u l t s of the researchers. Three regions define d e f i n i t e breaking types and two regions define areas of overlap that, according to the previous discussions, may form t r a n s i t i o n breaking zones. This seems reasonable since some form of a t r a n s i t i o n breaker i s expected. 2.3.3 Breaking Indices on Natural Beaches The breaking indices have been determined by using plane impermeable beaches i n c o n t r o l l e d laboratory environments. Natural beaches, however, have bars and steps, continuously changing composite slopes, and varying degrees of permeability. I t i s not s u r p r i s i n g to f i n d that the breaking indices found for laboratory conditions do not adequately predict conditions i n nature. Weishar and Byrne (1978) use the r e s u l t s of 116 waves filmed at V i r g i n i a Beach, Va., on the A t l a n t i c U.S. coast. They f i n d that the c l a s s i f i c a t i o n of Galvin does not s i g n i f i c a n t l y discriminate between plunging and s p i l l i n g breakers (figure 2.17). Both plunging and s p i l l i n g breakers occur over the whole range, with plunging breakers tending to bunch at the t r a n s i t i o n 28 29 a » n an jfta m~J A n'Sm-B-ftfll. - " n__» 1 ft a , a T> a * E , 10* i o ' t o » ie> Ho/ L Q ma A 6 A«aAAA&< J I . .1^8 a m nlfi8h$nVBAll).A n a n \o- io' to A # * % *J AA 8 ^  .. 6 ^ AAA A A T T ii i v r | r ir — | H b / g m T 2 A A • A I0"» IO* „ t»* •<>' Figure 2.17 Comparison of the b r e a k e r indices f o r natural beaches points given by Galvin. The t r a n s i t i o n s given by Patrick et a l . do not f i t the r e s u l t s either. More data points are needed for the s p i l l i n g breakers to determine i f there i s a greater tendency f o r them to bunch. The poor r e s u l t s can be explained i n two ways. Ei t h e r the breaker indices do not work i n the natural environment, and/or the breaker indices work but require measurements that take into account the v a r i a b i l i t y of the environment . For instance, the value of H 0/L 0 should represent the spectra of incoming values and the value of the slope should represent the complex, composite slopes. Izumiya and Isobe (1986) found that an equivalent slope can be defined as the mean slope i n the distance of Sd^ offshore of the breaking point, where d^ denotes the water depth at the breaking point. But, t h i s again i s a r e s u l t f or an impermeable beach. Also, using H 0/L Q as the independent v a r i a b l e may not be adequate. Obtaining the value of H 0/L 0 requires e i t h e r deep water sensors, which are d i f f i c u l t and expensive, or back c a l c u l a t i n g from nearshore conditions, which introduces c a l c u l a t i o n errors. Therefore, using the nearshore conditions as the independent variables ( i e . H^/d^, H b/gT 2, d^/gT 2) may represent a better method. The most probable cause of the discrepancy between observed and predicted breaking types i s that the indices are for plane beaches. Waves are forced to break on a beach that do not respond to the incoming wave conditions. Dalrymple et al.(1976) provide a graph that p l o t s the equilibrium slope against the nondimensional f a l l v e l o c i t y of the beach sediment, H Q/V fT where V f i s the f a l l v e l o c i t y . Therefore, combinations of H 0/L 0, 31 beach slope, and breaker types for plane beaches may be impossible to achieve on natural beaches since the laboratory slopes are not necessarily the equilibrium slope f o r the incoming waves. 2.4 E X P E R I M E N T A L D E S I G N A N D P R O C E D U R E S The objective of t h i s experiment i s to study the r e l a t i o n s h i p between the breaker type, beach slope, and deepwater wave steepness, and to compare these r e s u l t s with those found by other researchers. A model beach was b u i l t i n a wave flume that i s 28 meters long, 0.60 meters wide, and 0.70 meters deep. The waves were generated by a flap-type wave paddle capable of varying the wave height and period. The beach was r i g i d , wooden, and impermeable and sloped at 1:15. This slope was chosen to provide the greatest range of breaker types and nearshore conditions. The o v e r a l l setup i s shown i n figure 2.18. Wave and breaking conditions were varied by changing the depth of the constant depth section, the period, and the wave height. I n i t i a l l y , the depth and period are held constant while the wave height was changed. For each wave height, 10 to 15 minutes were given f o r the conditions i n the flume to become stable. The wave height was measured 2 meters offshore of the junction between the sloped beach and f l a t bed. The breaker type was recorded using the d e f i n i t i o n s found i n section 2.2. Video pictures of each run were made so that the breaking process, which i s a very quick process, could be analyzed at a slower speed. 32 Wave Vave Generator Filter 28n PLAN VIEW 0.6n SVL Rigid Plywood Beach 0,7n SIDE VIEW Figure 2,18 Overall flume se tup (not to scale) 33 Once the data was recorded, the deepwater wave steepness was back calculated using the intermediate depth conditions and l i n e a r wave theory. 2 . 5 E X P E R I M E N T A L R E S U L T S Results from experiments for wave breaking on a 1:15 slope are shown i n figure 2.19. The value of H Q/L Q i s back calculated from the intermediate depth conditions using l i n e a r wave theory. The breaking types are defined as i n section 2.2 . There appears to be a region of overlap where ei t h e r a plunging or s p i l l i n g breaker can occur. From the r e s u l t s , Galvin's l i m i t roughly corresponds to the smallest value of H 0/L 0 f o r which a s p i l l i n g breaker occurs. Iversen's and Patrick's l i m i t s roughly corresponds to the large s t value of H 0/L 0 for which plunging breakers occur. In section 2.3.2, the t r a n s i t i o n boundaries are assumed to be the l i m i t s of the zone. From laboratory experiments, t h i s assumption appears to be correct. Plunging breakers are also observed to occur for conditions that should produce surging breakers as predicted by Iversen and Patrick et a l . In fact, surging breakers could not be produced i n the se r i e s of experiments. The plunge-surge t r a n s i t i o n s of Iversen and Patrick et a l . are assumed to occur at the end of the plunge region (figures 2.5 and 2.7). These researchers may have had the same problem; surging breakers are d i f f i c u l t to produce on lower sloped beaches. I f t h i s i s the case, then the plunge-surge t r a n s i t i o n predicted by them w i l l be too high for lower sloped beaches. 34 Deepwater wave s teepness, H o / L o 0 . 0 7 0 0 . 0 6 0 0 . 0 5 0 -0 . 0 4 0 -0 . 0 3 0 -0 . 0 2 0 0 . 0 1 0 -0 . 0 0 0 O Spi l l ing breaker X Plunging breaker Plunge I v e r s e n ( i 9 5 3 ) - h P a t r i c k ( l 9 5 5 ) G a l v l n ( 1 9 6 8 ) 0 . 0 0 0 0 . 0 2 0 0 . 0 4 0 0 . 0 6 0 0 . 0 8 0 0 . 1 0 0 0 .120 0 . 1 4 0 Slope ( m ) Figure 2.19 Experimental r e s u l t s f o r Ho /Lo and the b r e a k e r type on the 1:15 slope 35 2 . 6 D I S C U S S I O N The r e s u l t s of Iversen (1953), Patrick et a l . (1955), and Galvin (1968) show s i g n i f i c a n t differences, e s p e c i a l l y for the plunge-surge t r a n s i t i o n . Up to two orders of magnitude separate the t r a n s i t i o n between the three breaker type groups. Some v a r i a b i l i t y i s expected since the t r a n s i t i o n between breaking types i s a continuous change occurring over a range of H 0/L Q. Pinpointing the change i n the breaking form i s d i f f i c u l t and i s affected by differences i n personal judgement. The researchers obtained two r e s u l t s f o r each slope condition: the type of breaking wave; and, the value of H Q/L 0 associated with the breaking wave. The value of H Q/L 0 depends upon the method of c a l c u l a t i o n . Galvin eliminates the e f f e c t of wave r e f l e c t i o n s and secondary waves by c a l c u l a t i n g H 0/L 0 using the displacement of the wave generator. Iversen and Patrick et a l . use measured wave heights, which are affected by r e f l e c t i o n s and secondary waves, to calcu l a t e H 0/L Q. Values of H 0/L 0 calculated by considering the generator displacement are larger than those calculated by considering measured wave heights. However, i t i s c l e a r l y shown that Iversen's r e s u l t s are always above Galvin's. Therefore, the differences between the researchers judgements must be the cause of the differences i n the r e s u l t s . Iversen and Patrick et a l . only provide the end of the plunging region for the beach slopes 1:20 and 1:50. The beginning of the surge region i s not given but i s assumed to occur at the end of the plunging region. This assumption i s reasonable since i t agrees with Galvin's findings. S t i l l , t h i s 36 assumption does add uncertainty to Iversen's and Patrick's plunge-surge t r a n s i t i o n and may give values of H 0/L Q larger than the actual values. Even though there are differences, experimental r e s u l t s showed that a reasonable p r e d i c t i o n of the breaker type can be made given the deepwater wave steepness and the beach slope, provided that the beach i s plane and impermeable. The breaking indices do not seem to be useful i n predicting conditions i n the natural environment. The conditions on a natural beach are complex and are effected by composite slopes, bars, steps, spectra of H 0 / L 0 , permeability, and non-uniform conditions. The simple models developed i n the laboratory should not be expected to give exact d e t a i l s of natural events, but rather show the general trends, trends that are observed i n nature. The key to the o v e r a l l problem of determining nearshore and breaking conditions from offshore observations l i e s i n devising a set of parameters that replace the terms " s p i l l i n g " , "plunging", "collapsing", and "surging". At present, a l l studies are at best semi-quantitative i n that graphs are presented with parameters such as H Q / L Q , m, and regions of s t r i c t l y d e s c r i p t i v e terms. Once the d e s c r i p t i v e terms are s u c c e s s f u l l y replaced by adequate quantitative parameters that are based upon a better understanding of the physical reasons for the d i f f e r e n t types of breaking, i t w i l l be possible to t r a n s f e r the incoming wave spectra to the breaking conditions. 37 C H A P T E R 3 ; W A V E H E I G H T A N D D E P T H A T B R E A K I N G 3 . 1 I N T R O D U C T I O N As a wave moves from deep to shallow water, i t begins to undergo a series of changes. These changes occur as the wave moves into depths where the bed begins to influence the wave which r e s u l t s i n wave shoaling. The wave height begins to increase, the wavelength decreases, and the wave speed decreases. The period, though, stays constant. Together, these changes eventually cause the wave to become unstable and break. At two extremes, the wave can be one of two forms, either an o s c i l l a t o r y wave or a s o l i t a r y wave. O s c i l l a t o r y waves are pri m a r i l y c o n t r o l l e d by the wave steepness and s o l i t a r y waves are p r i m a r i l y c o n t r o l l e d by the water depth. Breaking occurs when the control reaches a c r i t i c a l i n s t a b i l i t y . For o s c i l l a t o r y waves, i n s t a b i l i t y occurs when the wave becomes too steep such that water s p i l l s down the face of the wave. For s o l i t a r y waves, the i n s t a b i l i t y occurs when the depth becomes 38 small enough such that the v e l o c i t y of the crest of the wave i s s i g n i f i c a n t l y greater than the v e l o c i t y of the base of the wave causing the wave to plunge forward. In both cases, at breaking the crest of the wave tr a v e l s f a s t e r than the base of the wave. This i s the r e s u l t of breaking for o s c i l l a t o r y waves, but the cause of breaking for the s o l i t a r y wave. As a general rule of thumb, the r a t i o of the breaking height to the breaking depth i s assumed to be 0.83. There are large amounts of l i t e r a t u r e and various empirical c r i t e r i a which attempt to predict breaking and hence the value of the r a t i o . Experimental evidence shows that the breaking height-to-depth r a t i o v a r i e s over a large range of values, depending upon the beach slope and wave conditions. I f the mechanisms for d i f f e r e n t types of breaking waves are better understood, then p h y s i c a l l y based c r i t e r i a , such as the height-to-depth r a t i o , would a r i s e n a t u r a l l y . This chapter uses the behaviour of the breaking height-to-depth r a t i o as a clue to the mechanisms that cause breaking. 3 . 2 D E F I N I T I O N O F T H E B R E A K I N G H E I G H T A N D D E P T H The wave height and depth at the breaking point depend upon where the wave i s defined to break. Referring to the c l a s s i f i c a t i o n s given i n chapter two, a s p i l l i n g breaker i s located where the crest f i r s t becomes discontinuous. For a plunging breaker, the "breaking point i s located where the wave face f i r s t becomes v e r t i c a l . For a surging breaker, the point of breaking i s located where the maximum drawdown of the water from the previous wave occurs. 39 Figure 3.1 shows the variables that are used to describe the conditions at the breaking p o s i t i o n . The breaker height (H b) i s the difference between the maximum and minimum water surface elevation during the passage of one wave. For o s c i l l a t o r y waves three depths are possible: the s t i l l water depth (d s) at the breaker p o s i t i o n ; the mean water depth (d m) at the breaker p o s i t i o n ; and, the depth of the trough (d t) at the breaker p o s i t i o n . Because there are three possible breaking depths, the one that i s used to describe the breaking process must be c l e a r l y defined. The symbol d^ i s used to represent one of d s, d m, or d t . The depth d f a i s usually equated with d s and t h i s convention w i l l be used unless otherwise defined. If s o l i t a r y wave theory i s used to describe the shoaling waves, then d^ i s equated with d^ .. A s o l i t a r y wave i s a type of cnoidal wave that has an i n f i n i t e period and i n f i n i t e wavelength and propagates forward i n water of constant depth. The wave setdown at breaking i s the difference between the mean water depth and the s t i l l water depth (Longuet-Higgins and Stewart, 1964), _ S b = d m - d s (3.1) The setdown r e s u l t s from the flow of excess momentum produced by the shoreward movement of the waves and i s further discussed i n chapter four. The diff e r e n c e between the mean water depth, d^, and trough depth i s the trough amplitude, a t = d m - d t (3.2) 40 Figure 3.1 Var iab les f o r wave a t breaking point 41 3.2.1 The heiqht-to-depth r a t i o at breaking As a wave shoals, the trough between the crests f l a t t e n s and the wave height changes. The shape of the wave begins to approach the s o l i t a r y wave shape and, i n t h i s case, the maximum wave height w i l l begin to be controlled by the depth rather than by the wavelength, which i s the case for o s c i l l a t o r y waves. Many researchers have used s o l i t a r y wave theories to study the height-to-depth problem and the t h e o r e t i c a l r e s u l t s are shown in table 3.1. The ( H b / d b ) m a x values assume that the wave remains symmetrical to the point of breaking. However, actual experiments have shown that (H^/d^^x varies from 0.65 for steep waves on low slopes to about 1.25 for low steepness waves on steep slopes. Generally, f o r s p i l l i n g breakers ( H ^ / d ^ ) ^ ^ i s i n the range 0.65 - 0.85 (Iversen 1952, 1953; Galvin 1972; Weggel 1972) and i s approximately that value obtained from the highest s o l i t a r y wave theory. The good comparison between t h e o r e t i c a l and experimental r e s u l t s for s p i l l i n g breakers i s not s u r p r i s i n g since these breakers most c l o s e l y resemble the symmetrical s o l i t a r y waves assumed by the theory. For plunging breakers, H^/d^ increases and values as large as 1.3 f o r p e r i o d i c waves (Iversen 1952, 1953) and 3.0 f o r s o l i t a r y waves (Ippen and Kulin 1955; Camfield and Street 1969) have been measured. Weishar and Byrne (1978) studied films of 116 breaking waves on a natural beach. They note that the average value of H b / d j 3 i s 0.78. This value i s within the range estimated by other researchers, but they notice that larger valaues of H^/d^ 42 RESEARCHER DATE (Hb/db) max Boussinesq 1871 0.73 McCowan 1891 0.75 McCowan 1894 0,78 Gwyther 1900 0.83 Davies 1952 0,83 Packman 1952 1,03 Chappelear 1959 0,87 Laitone 1960 0.73 Lenau 1966 0,83 ( a f t e r Galv in 1972) Table 3,1 Maximum theore t ica l h e i g h t - t o - d e p t h ra t ios f o r sol i tary waves 43 occur for plunging waves than for non-plunging waves. As shown by the previous findings, t h i s should be expected. The Shore Protection Manual provides two graphs that demonstrate the r e l a t i o n between the breaker type, H^/d^, and H Q / L 0 (figures 3.2 and 3.3). Figure 3.2 shows Goda's (1970) empirically derived r e l a t i o n s h i p between H ^ / H Q and H 0 / L Q for several beach slopes. Figure 3.3 shows the r e l a t i o n s h i p between d B / H B and H b / g T 2 for various slopes. The curves i n figure 3.3 are given by, J u b 1 (3.3) H b b - ( a H b / g T 2 ) where a and b are functions of the beach slope and are approximated by, a = 43.75 ( 1 - e" 1 9 n ) (3,4) b = ! ' 5 6 , 1 "19.5m. ( 1 + e ) (3.5) By combining the two graphs, H^/d^ can be shown as a function of the deepwater steepness and the beach slope (figure 3.4). Included i n the figure are the three breaker regions. These occur i n the recognized sequence. For each slope, as H Q/L 0 decreases H^/d^ increases and the breaker type changes from s p i l l i n g to plunging to surging, agreeing with the previous discussion. On steeper slopes H^/d^ has a greater range of values and varies by large percentages between plunging and non-plunging breakers I t i s i n t e r e s t i n g to compare these r e s u l t s with the r e s u l t s found by Iversen (1952) for periodic waves. Figures 3.5 through 44 0.002 0.004 0.006 0.006 0.010 0.012 0.014 0.016 0.018 0.020 Figure 3,2 Breaker height index v e r s u s deepwater wave s teepness (SPM 1984) 0.0004 0 0006 0 001 0002 0 004 0 006 0 01 0 02 0 03 (ofttr Gotfo, 1970) Figure 3.3 Non-dimensional b r e a k e r depth v e r s u s breaking s t e e p n e s s (SPM 1984) 45 B e a c h s l o p e m = 1:50 —|— m = 1:30. m = 1:20 - B - m = 1:10 0 .001 "1 1 I I I I 0 .010 Hb /db i r 1.35 h 1.25 1.15 h 1 .05 h 0 . 9 5 0 . 8 5 I I I ' 0 . 7 5 0 . 1 0 0 Deepwater wave s teepness, H o / L o Figure 3.4 Dependence of Hb/db on deepwater wave s t e e p n e s s and beach slope 46 3.8 are plo t s of H Q/L Q versus H b/d b for 1:10, 1:20, 1:30, and 1:50 slopes. For values of H Q/L 0 l e s s than approximately 0.015, these findings agree with the predictions made by figure 3.4. In a l l cases, the value of H^/d^ increases f o r decreasing values of H 0/L 0 u n t i l H 0/L Q approaches 0.015. At t h i s value, H^/d^ begins to decrease for the 1:10 and 1:50 slopes. Whether t h i s occurs for the 1:20 and 1:30 slopes i s d i f f i c u l t to determine due to the large scatter of the data points f o r lower values of H 0/L 0, however, these curves do tend to f l a t t e n . The breaking mechanism for H Q/L 0 greater than 0.015 must be d i f f e r e n t than the mechanism occurring for H 0/L Q l e s s than 0.015. This i s further explored i n the following three sections by analyzing how the steepness, breaking depth, and beach slope a f f e c t breaking. 3 . 3 I N F L U E N C E O F T H E W A V E S T E E P N E S S O N B R E A K I N G The deepwater steepness, H 0/L Q, has been shown to be a factor i n determining how a wave w i l l break on a beach of a given slope (section 2.3). Deepwater waves are not affected by the presence of the bed and the maximum wave height i s lim i t e d by the wavelength. Michell (1893) determines the l i m i t i n g deepwater steepness to be, ( H 0 / L 0 ) n a x 1/7 (3,6) Breaking over the entir e range of depths i s covered by a formula proposed by Miche (1944), H/L = 0,141 tanh(27Td/L) (3.7) This formula serves as a useful engineering approximation of wave steepness i n that i t predicts the maximum steepness at 47 Hb /db 0 .001 1.25 h 1 .20 h 1 .15 h 1.10 h 1 .05 1 .00 0 . 9 5 h 0 . 9 0 h 0 . 8 5 h 0 . 8 0 0 . 7 5 h 0 . 7 0 "1 I—I I | | 0 . 6 5 0 . 1 0 0 Deepwater wave s teepness, H o / L o Figure 3,5 Plot of da ta ( Iversen 1952) f o r MO slope 48 Hb/db 1.20 0 .001 h 1 .15 h 1.10. h 1 .05 1.00 h 0 . 9 5 0 . 9 0 h 0 . 8 5 h 0 . 8 0 0 . 7 5 0 . 7 0 h 0 . 8 5 T I I I—I I | | 1 0 . 6 0 0 .010 0 . 1 0 0 Deepwater wave s teepness, H o / L o Figure 3,6 Plot of da ta ( Iversen 1952) for - 1:20 slope 49 Hb/db 0.001 1.20 1.15 h 1.1 o 1.05 h L O O h 0 . 9 5 h 0 . 9 0 h 0 . 8 5 0 . 8 0 h 0 . 7 5 h 0 . 7 0 h 0 . 6 5 0 . 6 0 0 . 1 0 0 Deepwater wave s teepness, H o / L o Fi 9 U r e 3 ' 7 P l , o t o f data ( Iversen 1952) f o r 1:30 slope 50 Hb /db 0.001 0 .010 Deepwater wave s teepness, H o / L o 1.20 1.15 h 1.10 1.05 h L O O 0 . 9 5 h 0 . 9 0 h 0 . 8 5 h 0 . 8 0 h 0 . 7 5 0 . 7 0 0 . 6 5 0 . 6 0 0 . 1 0 0 Figure 3,8 Plot of da ta ( Iversen 1952) f o r 1:50 slope 51 which a wave becomes unstable and then breaks. Figure 3.9 shows Iversen's r e s u l t s plotted against the wave steepness l i m i t described by Miche's formula. The data points f a l l within the intermediate depth range, according to l i n e a r wave theory. Because of t h i s , i t i s correct to assume that the bed i s beginning to a f f e c t the shoaling and breaking c h a r a c t e r i s t i c s . Since breaking i s no longer s t r i c t l y c o n t r o l l e d by steepness the data points are expected to show some degree of scatter about the l i m i t given by Miche. In the figure, the data points representing breaking on the 1:10 slope are more scattered and generally p l o t above Miche's l i m i t . As the slope decreases, the points become les s scattered with the breaking on the 1:50 slope being well approximated by Miche's l i m i t . I t seems that the gentler slope allows the wave to change slowly, i n respect to the shoaling process, and continue to maintain a p r o f i l e more consistent with a symmetrical s o l i t a r y wave. On steep beaches, the wave does not have enough time to respond to depth changes and so an unsymmetrical breaking form causes scatter about Miche's l i m i t . The shoaling c h a r a c t e r i s t i c s of o s c i l l a t o r y waves can be described by using l i n e a r wave theory. By equating the wave power per unit wave crest width i n deep and shallow water, the r a t i o of wave steepness to deepwater wave steepness can be calculated. The cal c u l a t i o n s are shown i n Appendix A and give the shoaling equation as, 1/2 coth(kd) 52 H/L 1 . 0 0 0 n • m = 1:10 + m = 1:20 0 m = 1:50 0 . 1 0 0 -0 . 0 1 0 -MICHE(1944) SHALLOW INTERMEDIATE DEPTH DEEP 0 . 0 0 1 0.01 Figure 3,9 i i i i i i r 0.10 d/L Comparison of experimental r e s u l t s with Miche's formula 1.00 Three assumptions are made for the c a l c u l a t i o n : the wave crests are p a r a l l e l to the beach such that no r e f r a c t i o n occurs; the slope i s close to being zero such that the properties of the wave at any depth are the same as the properties of an i d e n t i c a l wave i n a horizontal channel; and, the loss i n power i s n e g l i g i b l e . In p r i n c i p l e , turbulence, which represents a power loss, comes from two sources. Those are the bottom boundary layer and the surface breaker. At breaking, the contribution of the bottom boundary layer i s t o t a l l y outweighed by the d i s s i p a t i o n due to breaking, and i s consequently neglected. Up to breaking, though, f r i c t i o n occurs at the bed but w i l l have an i n s i g n i f i c a n t e f f e c t on the o v e r a l l r e s u l t s (Kamphuis 1975; Thornton and Guza 1983). Turbulence i s zero since the flow i s i r r o t a t i o n a l within the wave. As an o s c i l l a t o r y wave shoals, the value of kd decreases and the wave properties w i l l follow those described by the shoaling equation. Eventually the wave w i l l break which i s defined by Miche's formula. This assumes that breaking i s i n i t i a t e d by the wave at t a i n i n g c r i t i c a l steepness. Therefore, equation 3.8 i s v a l i d up to the point of breaking. By p l o t t i n g these equations on a single graph the conditions at breaking for constant values of H 0/L Q can be found. These conditions are at the points of in t e r s e c t i o n i n figure 3.10. Using the conditions at each i n t e r s e c t i o n point, Miche's formula can be transferred to figure 3.4, which gives the value of H^/djj as functions of beach slope and deepwater wave steepness. Figure 3.11 i s the p l o t with Miche's l i m i t included. Cle a r l y , Miche's c r i t e r i a represents the lowest l i m i t i n g 54 H/L Ho/Lo = 1:20 Ho/Lo = 1:100 Ho/Lo = 1 :200 0.001 ' ' 1 1 1 — ' ' ' ' ' • 001 0.10 d/L Figure 3,10 Miche's formula and the shoaling equation f o r s e l e c t e d values of Ho /Lo 55 Hb/db 0 .001 h 1 .25 h 1.1 5 h 1 .05 h 0 . 9 5 0 . 8 5 h 0 . 7 5 I | 0 . 6 5 0 . 1 0 0 Deepwater wave s teepness, Ho /Lo . Figure 3,11 Miche's limit 5 6 condition at breaking which means that i f a wave breaks s o l e l y from steepness e f f e c t s the value of H^/d^ w i l l be a minimum. For o s c i l l a t o r y waves, larger values of H b/d b are obtained because wave steepness increases f a s t e r than the wave height for a wave t r a v e l l i n g onto a beach. This i s shown by the shoaling equation and implies that an o s c i l l a t o r y wave that s t a r t s out steep enough i n deepwater w i l l grow i n steepness f a s t enough that steepness w i l l induce breaking i n water deeper than where breaking i s s o l e l y depth controlled. This early breaking suggests that high i n i t i a l steepness should be associated with larger breaking depths. I f so, then H^/d^ would be expected to decrease as H Q/L 0 increases for higher values of H 0/L 0. This i s supported by Iversen for H 0/L 0 greater than 0.015 (figure 3.5 to 3.8), but does not appear to be true for H 0/L 0 less than 0.015, as w i l l now be discussed. 3.4 I N F L U E N C E O F T H E D E P T H O N B R E A K I N G I t i s c l e a r from Iversen's findings that H^/d^ i s decreasing, or r e l a t i v e l y constant, f o r decreasing values of H 0/L 0 below 0.015. Presumably, a d i f f e r e n t breaking mechanism i s responsible f o r the change. The previous section assumed o s c i l l a t o r y waves broke because of steepness e f f e c t s . Waves that no longer shoal as o s c i l l a t o r y waves begin to appear as s o l i t a r y waves which break because of depth e f f e c t s . 57 An id e a l s o l i t a r y wave has a wavelength and a period that both approach i n f i n i t y . I t i s a wave of t r a n s l a t i o n that l i e s wholly above the s t i l l water l e v e l and i s co n t r o l l e d by the depth. The v e l o c i t y of such a wave i s given by, S.— = 1 + f ( H / d ) (3.9) VgcT Both the depth and the height-to-depth r a t i o are important. Breaking of such a wave can be explained by noting that as the depth increases the wave speed increases. The change of wave speed with depth w i l l be greater i n more shallow depths. Since there i s a s i g n i f i c a n t depth increase i n the wave i t s e l f , the top portion of the wave moves fas t e r that the bottom of the wave. When the s i t u a t i o n becomes unstable the wave w i l l break (Longuet-Higgins 1980). To become s o l i t a r y , waves with i n i t i a l low steepness must reach depths where the wave i s controlled by the depth. These waves must have an i n i t i a l low steepness since i t i s already shown that waves with high i n i t i a l steepness w i l l break due to steepness e f f e c t s . When s o l i t a r y waves are considered, the response of the wave to changing shoaling conditions i s important. Ideally, the response should be immediate, but i n r e a l i t y there w i l l be a lag i n the response. Figure 3.12 shows what occurs when a s o l i t a r y wave responds to depth changes. According to the figure, the slow response of the s o l i t a r y wave w i l l r e s u l t i n the wave height at breaking being reached further shoreward i n a smaller depth. Therefore, the t h e o r e t i c a l value of H^/d^ w i l l be less than that which i s measured. As the i n i t i a l wave steepness 58 W a v e H e i g h t I n c r e a s e — -B&» A c t u a l P o s i t i o n W a t e r D e p t h T h e o r e t i c a l D e c r e a s e { gure 3,12 Breaking response o f a so l i tary wave 59 increases, for low i n i t i a l values of H Q/L 0, the e f f e c t increases since the height of the incoming s o l i t a r y wave w i l l increase for increasing H Q/L 0. However, s o l i t a r y wave development requires time which can only be provided by shoaling on mild slopes. Therefore, one would expect decreasing values of H b/d b for decreasing values of H 0/L 0 on mild slopes. Iversen's r e s u l t s f o r the 1:30 and 1:50 slopes show t h i s as the trend. How can the r e s u l t s for the 1:10 slope be explained? Here the slope i s steep enough to rul e out the formation of a s o l i t a r y wave even though H 0/L 0 i s les s than 0.15. This i s probably best explained by the t r a n s l a t i o n of a pre-breaking o s c i l l a t o r y wave into shallow water on a steep slope. The response of H b/d b to changing H Q/L 0 i s the same r e s u l t as that j u s t described for the s o l i t a r y wave. 3 . 5 I N F L U E N C E O F T H E B E A C H S L O P E O N B R E A K I N G Beach slope a f f e c t s wave conditions by c o n t r o l l i n g the time and distance spent by the wave moving towards breaking. A greater period of time, measured i n wave periods, or, equivalently, a greater distance, measured i n wavelength, on the beach allows f r i c t i o n and nonlinear wave shape changes to have a greater e f f e c t . F r i c t i o n helps to reduce wave height (Thornton and Guza 1983) while nonlinear changes can eithe r increase or decrease the wave height. In Iversen's experiment, the breaker height reduces to a greater extent on f l a t t e r slopes (table 3.2, figure 3.13). However, the r e s u l t s of Ippen and Kulin indicate that for s o l i t a r y waves the reverse i s true. The equivalent breaker 60 SLDPE Ho/Lo=0,01 Ho/Lo=0,02 Hb/Ho 0.02 1,31 1,12 0,033 1,43 1,23 0.05 1,60 1,31 0,10 1,76 1,41 ( f r o m I v e r s e n 1952) Table 3,2 Slope e f f e c t on the breaking height f o r osc i l l a to ry waves SLDPE Ho/Lo=0,01 Hb/Ho 0,023 1,70 0,065 1,40' Table 3,3 Slope e f f e c t on the breaking height f o r so l i ta ry waves 61 0> Hb/Ho 0 .02 0 . 0 3 0 .04 0 . 0 5 Ho/Lo 0 . 0 6 0 .07 0 . 0 8 0 . 0 9 0.10 Figure 3,13 Hb/Ho v e r s u s Ho /Lo ( Iversen 1952) height index, Hb/HQ, increases with decreasing slope for constant values of the deepwater steepness (table 3.3). For o s c i l l a t o r y waves, the v a r i a t i o n of the breaking depth with deepwater wave height i s shown i n figure 3.14 and there does not appear to be a strong slope influence. However, as H 0/L Q increases, d b/H 0 decreases. The combined r e s u l t s of H b/d b versus H Q/L 0 has previously been shown i n section 3.2. These show a strong slope influence, which, from the above r e s u l t s , must only come from the influence of the slope on H b/H Q. Figure 3.15 compares the trends found for both o s c i l l a t o r y and s o l i t a r y waves. The bottom two graphs show that for a given slope the behaviour of the r a t i o Hj^/d^ for increasing values of H 0/L Q i s d i f f e r e n t f o r o s c i l l a t o r y and s o l i t a r y waves. The r a t i o H b/d b decreases for o s c i l l a t o r y waves and increases for s o l i t a r y waves. This finding explains the curves of H 0/L 0 versus H b/d b f o r the 1:20, 1:30, and 1:50 slopes given by Iversen In the three cases, each representing the ch a r a c t e r i s t i c s f o r a p a r t i c u l a r slope, the curve changes s i g n i f i c a n t l y at H 0/L Q equal to 0.015. This value of H 0/L Q marks the t r a n s i t i o n between breaking as a s o l i t a r y wave and breaking as an o s c i l l a t o r y wave. Also, regardless of the type of breaking, the value of H b/d b increases as the slope increases for a given H 0/L Q. 63 db/Ho CTl 3.00 • • 2.50 H 2.00 H 1.50 H 1.00 a m = 1:10 + m = 1:20 m = 1:50 • m = 1:30 • 4- + 0.50 0.00 0.00 0.02 0.04 0.06 Ho/Lo 0.08 0.10 Figure 3,14 clb/Ho v e r s u s Ho /Lo ( Iversen 1952) OSCILLATORY SOLITARY H b / H o INCREASING SLOPE H b / H o H o / L o INCREASING SLOPE H o / L o d b / H o d b / H o H o / L o H o / L o H b / d b H b / d b H o / L o H o / L o Figure 3,15 Trends f o r the ra t ios of osc i l la tory (Iversen 1952) and sol i tary (Ippen and Kulin 1955) waves 64-A 3.6 D I S C U S S I O N The wave height and depth at breaking are points of i n t e r e s t f o r researchers and engineers. The r u l e of thumb i s that the r a t i o of wave height-to-depth at breaking i s 0.83. However, as found by many researchers, the actual r a t i o can have a large range of values. Values as high as 1.3 to 3.0 are found fo r plunging and surging type breakers. S p i l l i n g breakers have values between 0.65 and 0.85. The r e s u l t s for s p i l l i n g breakers are c l o s e r to the values predicted t h e o r e t i c a l l y since the s p i l l i n g breaker most c l o s e l y resembles s o l i t a r y waves at breaking. The behaviour of the height-to-depth r a t i o , with respect to deepwater steepness, can be explained using the e f f e c t of wave steepness, depth, and beach slope. Figure 3.16 shows the graph of H b/d b versus H 0/L Q. I t i s divided into three regions where i n each region breaking i s i n i t i a t e d by d i f f e r e n t mechanism. In region one, wave steepness plays the major r o l e i n breaking process. The wave breaks when the wave reaches c r i t i c a l steepness The o s c i l l a t o r y wave obeys the shoaling equation and increases i n steepness f a s t e r than i n height. This causes the wave to break i n water deeper than i f the wave were s o l e l y c o n t r o l l e d by depth. This i s not to say that the depth does not take part i n the shoaling process, but that the breaking i s due to a l i m i t i n g wave steepness. For values of H 0/L 0 greater than 0.015, the H b/d b r a t i o increases as the deepwater wave steepness decreases. In region two, the beach slopes are low enough to allow s o l i t a r y waves to develop from incoming o s c i l l a t o r y waves. The 65 Hb/db Ho/Lo = 0,015 Figure 3,16 Relationship between the beach slope, deepwater wave s teepness , and breaking conditions 65-A slope must be low since a s o l i t a r y wave needs time and distance to form from an o s c i l l a t o r y wave. Breaking i s i n i t i a t e d by the difference between the forward v e l o c i t y of the crest and the base of the wave, the v e l o c i t y being proportional to the square root of the depth. Because the s o l i t a r y wave cannot respond immediately to changing depths, at breaking the wave moves into water that i s more shallow than i s necessary for breaking. Therefore, the r a t i o H b/d b increases as H 0/L Q increases because the wave height w i l l increase with increasing H 0/L Q. The t r a n s i t i o n between s o l i t a r y waves and o s c i l l a t o r y waves i s when H 0/L 0 equals 0.015. The development of s o l i t a r y waves i s r e s t r i c t e d to beaches with low slopes, so as the slope increases s o l i t a r y wave development ceases. This necessitates a slope above which s o l i t a r y waves are no longer able to develop. The approximate value of the t r a n s i t i o n slope, based upon Iversen's r e s u l t s , i s 1:25. In region three, because of the steep slope, a pre-breaking o s c i l l a t o r y wave moves into shallow water. The wave breaks from a combination of steepness and depth e f f e c t s . As i n the s o l i t a r y wave case, the r a t i o H b/d b increases as the deepwater steepness increases for values of H Q/L 0 less than 0.015. Because of the d i f f e r e n t breaking mechanism i n each region, s p e c i f i c breaking types predominate i n the three regions. In region one, o s c i l l a t o r y waves break from steepness e f f e c t s and are expected to s p i l l or plunge s l i g h t l y . In region two, the forward v e l o c i t y of the wave crest i s greater than the forward v e l o c i t y of the base causing s o l i t a r y waves to plunge. F i n a l l y , 66 i n region three, o s c i l l a t o r y waves t r a v e l into shallow water and are expected to plunge or surge. 67 C H A P T E R 4 ; F L O W D Y N A M I C S W I T H I N T H E S U R F Z O N E 4 . 1 I N T R O D U C T I O N From the point of view of f l u i d mechanics, the surf zone i s characterized by the i r r e v e r s i b l e and complete transformation of organized i r r o t a t i o n a l flow into motions of d i f f e r e n t types and d i f f e r e n t scales, including v o r t i c a l motions and turbulence. The wave goes through a series of changes which permit the surf zone to be broken into a number of regions where d i s t i n c t wave shapes and f l u i d flows occur. Figure 4.1 shows the regions which make up the surf zone. The outer region i s characterized by a rapid transformation of the wave shape from the i n i t i a l breaker shape to a (periodic) bore. The inner region i s characterized by the rather slow changes of the bore and the run-up region i s characterized by the fact that no surface r o l l e r e x i s t s . The previous two chapters discuss the conditions necessary for the surf zone to e x i s t . This chapter discusses the f l u i d 68 B r e a k i n g Po in t V S V L MV/L •LITER REGION -*G 5» INNER REGION , R U N - U P REGION — 5*-•*= • Figure 4,1 S u r f zone regions 69 motions produced within each region as a r e s u l t of wave breaking with p a r t i c u l a r emphasis on the transformation of i r r o t a t i o n a l wave motion into r o t a t i o n a l and ultimately turbulent flows. Both the mass transport v e l o c i t i e s and water l e v e l changes, which are r e s u l t s of these flows, are discussed. Experimental observations of f l u i d and p a r t i c l e motions, as well as careful measurements of water l e v e l changes, are used. 4 . 2 T R A N S I T I O N T O R O T A T I O N A L F L O W Two b a s i c a l l y d i f f e r e n t types of motion can be distinguished i n the evolution of breakers as considered here. The f i r s t i s e s s e n t i a l l y i r r o t a t i o n a l motion during wave steepening, overturning and j e t formation. This motion occurs outside the surf zone and w i l l not be treated here. The second type of motion i s that following j e t impingement, leading to the gradual development of a turbulent bore. These motions occur within the outer region and are considered f o r both s p i l l i n g and plunging breakers. Surging breakers are ignored because they do not e x h i b i t any r o t a t i o n a l tendencies except at the base of the wave face where the wave and beach meet. The i n i t i a l c h a r a c t e r i s t i c s of both s p i l l i n g and plunging breakers i s the formation of a j e t , though at much d i f f e r e n t scales. Careful observation reveals that these two breaking types can be further divided i n an e f f o r t to help describe the r e s u l t i n g f l u i d flows. These are symmetric s p i l l i n g breakers, symmetric plunging breakers, and asymmetric plunging breakers. The following descriptions of each w i l l describe the breaker 70 p r o f i l e and the f l u i d motions within the outer region of the surf zone. The symmetrical s p i l l i n g breaker i s no d i f f e r e n t from the t y p i c a l s p i l l i n g breaker since a l l s p i l l i n g breakers are symmetrical. The symmetry continues at and a f t e r breaking as the wave moves forward d i s s i p a t i n g energy. Breaking i s i n i t i a t e d when a small j e t of water appears at the crest creating turbulence. The turbulence gradually s l i d e s down the face of the wave creating a shear surface between the turbulent white water and the cl e a r water on the wave face. The shear layer generates a series of vo r t i c e s which are l e f t behind the crest as the wave t r a v e l s towards the beach. The t r a i n of vo r t i c e s move s l i g h t l y forward and elongate obliquely to the d i r e c t i o n of t r a v e l . Generally, the v o r t i c e s are confined to the region nearer the free surface (figure 4.2). Occasionally a vortex can st r e t c h to the bed. The strength and diameter of each subsequent vortex decreases u n t i l no more are created. This corresponds to the end of the outer region of the surf zone. The symmetrical plunging breaker acts as a s p i l l i n g breaker, but breaks as a plunging breaker. The j e t i s not yet large enough to plunge i n front of the wave face. Instead the j e t plunges into the face creating an i n i t i a l plunging vortex and white turbulent water on the wave face. The plunging j e t does not penetrate to the bed. The wave symmetry i s not affected and the breaker now looks and acts as the symmetric s p i l l i n g breaker as previously described. The important fact i s that even though the symmetrical s p i l l i n g and symmetrical 71 Figure 4,2 V o r t e x train produced by a symmetrical spilling breaker 72 plunging breakers have d i f f e r e n t i n i t i a l physical c h a r a c t e r i s t i c s , each has a s i m i l a r e f f e c t i n the creation of ro t a t i o n a l flow i n the outer region of the surf zone. The asymmetrical plunging breaker i s defined to break when the wave face becomes v e r t i c a l . A j e t , larger than i n the previous two cases, progressively extends from the crest u n t i l i t closes upon the forward slope of the wave. At t h i s point, a large plunging vortex i s formed with c i r c u l a t i o n around the cavit y i n the sense of wave advance. Observations show that the cavity quickly collapses while the a i r entrapped i n i t mixes with the water. The r e s u l t i s a region with v o r t i c a l motion and a high concentration of a i r bubbles which gradually r i s e to the surface. The i n i t i a l j e t may or may not reach the bed. However, i n eithe r case, the shape of the wave i s asymmetrical. When the t i p of the j e t s t r i k e s the forward free surface, the r e s u l t i n g splash can cause a wedge shaped amount of water to splash up. This c u r l s forward and forms another j e t which s t r i k e s the water below i t , and so on (figure 4.3). As many as f i v e successive v o r t i c e s can r e s u l t from t h i s process ( M i l l e r 1976, Basco 1985). The r e s u l t i s a sequence of two dimensional vortex structures r o t a t i n g i n the same d i r e c t i o n . Since they rotate i n the same d i r e c t i o n the regions between them have a high rate of d i s s i p a t i o n of the organized motion and the wave energy. Nadoaka (1986) has observed that the two dimensional structure of the horizontal v o r t i c e s breaks down through the formation of vo r t i c e s extending obliquely downward. These are found to originate i n the areas of maximum s t r a i n rate between 73 Figure 4,3 Asymmetrical plunging b r e a k e r and the splash-plunge cycle (Longuet-Higgins 1953) 74 the v o r t i c e s , and to be roughly oriented along the corresponding p r i n c i p a l axis. The i n i t i a l wave motion, or energy, i s transformed to r o t a t i o n a l flow which i s then dissipated by the shearing forces between the vortex structures. The bed can be d i r e c t l y affected by the v o r t i c a l structures and the plunging j e t . Generally, the vortex t r a i n formed by the symmetrical s p i l l i n g or the symmetrical plunging wave does not reach the bed. When they do there i s a noticeable increase i n the offshore a c t i v i t y of materials on the bed. The plunging j e t of the asymmetrical plunging breaker can penetrate d i r e c t l y to the bed and throw large amounts of sediment into suspension. The i n i t i a l plunging vortex and subsequent splash-plunge v o r t i c e s can also act on the bed and move material offshore due to t h e i r sense of rot a t i o n near the bed. 4 . 3 E S T A B L I S H E D T U R B U L E N T F L O W As time elapses a f t e r the i n i t i a l breaking, the large scale, ordered vortex motion i n the outer region of the surf zone degenerates into small scale motions with increasing disorder. At some stage the i d e n t i f i c a t i o n of coherent structures i s no longer possible. When t h i s occurs, the scales of motion can be treated as turbulence. The motions of a breaking wave i n the outer region show a s i m i l a r decrease i n scale and increase i n disorder. I d e n t i f i a b l e v o r t i c e s are no longer created. The inner region s t a r t s where the breaking wave has been transformed into a turbulent bore. The bore i s characterized by a steep, turbulent front with' an area of r e c i r c u l a t i n g flow between the crest and 75 the toe (figure 4.4). A continuous shearing motion occurs between the toe and the undisturbed inflowing water creating turbulence which spreads behind the bore and decays i n the wake. As a r e s u l t , there i s a d i r e c t and continuous transformation of wave motion into turbulent motion. 4 . 4 M A S S T R A N S P O R T Before wave breaking occurs, sediment i n the presence of progressive shallow water waves i s observed to be transported as bedload i n the d i r e c t i o n of wave propagation. This movement can be accounted f o r by the non-linear behaviour of shallow water waves. In the absence of a current the waves generate a mass transport v e l o c i t y and other higher order wave v e l o c i t i e s . Stokes (1847), assuming a perfect non-viscous f l u i d , was the f i r s t to show that i n a water wave the f l u i d p a r t i c l e s , apart from t h e i r o r b i t a l motion, have a second-order d r i f t v e l o c i t y . Longuet-Higgins (1953) made an important t h e o r e t i c a l contribution by explaining the observed movement of the surface water and bedload i n the d i r e c t i o n of wave propagation. The solutio n i s based on very low waves using laminar boundary layers at the bed and surface. Good agreement between theory and experiment i s obtained for values of kd between 0.9 and 1.5. Experiments f o r higher values of kd generally y i e l d a mass transport v e l o c i t y p r o f i l e resembling better the Stokes p r o f i l e . The n o n - l i n e a r i t i e s have also been extensively studied by Russel and Osorio (1958), A l l e n and Gibson (1959), L i u and Davis (1977), and Isaacson (1978). In a l l cases, the r e s u l t are 76 D i r e c t i o n o f t r a v e l ' S ^ e 4.4 Turbulent bore 77 s i m i l a r to the above findings and are for constant depth sections. Data from experiments generally indicate that the d r i f t v e l o c i t y i s i n the d i r e c t i o n of wave propagation near the bed, but against the wave for the section i n the middle of the water column. At the surface, the v e l o c i t y can be onshore or offshore (figure 4.5). For beaches the mass transport occurs on a sloping beach and t h e o r e t i c a l or experimental r e s u l t s for mass transport v e l o c i t i e s over gently sloping bottoms are very scarce. Bijker et al.(1974) studied the mass transport v e l o c i t y on a sloping beach and found that t h e o r e t i c a l considerations, based on l i n e a r wave theory, show that the slope w i l l have the greatest influence on the mass transport v e l o c i t i e s for r e l a t i v e l y long waves on steep impermeable slopes. The numerical values, however, remain r e l a t i v e l y small (influences l e s s than 20%). In addition, t h e i r experiments show that the bottom mass transport v e l o c i t i e s are more determined by the l o c a l depth than by the magnitude of the bottom slope. The bottom v e l o c i t i e s predicted by a horizontal bottom theories are too large f o r the sloping bottom case. As the roughness increases, the d r i f t v e l o c i t i e s change s l i g h t l y and considerably when a r i p p l e - l i k e roughness i s present (figure 4.6). Wang et a l . (1982) measured the d r i f t v e l o c i t y p r o f i l e s at the wave breaking point for waves on a plane impermeable beach. Their r e s u l t s are f o r s p i l l i n g , plunging, and t r a n s i t i o n a l breakers. The difference i n breaker type was i n i t i a l l y f e l t to influence the v e r t i c a l d r i f t v e l o c i t y p r o f i l e s at the breaking 78 D i r e c t i o n o f wave t r a v e l E3~ SWL Figure 4.5 Typical mass t r a n s p o r t velocity prof i les (Longuet-Higgins 1953) 79 profile nr. 10 11 12 13 1* 15 IS IT M 1* 20 03 O tondrough bottom oxp. nr. D-Sa rippltd bottom •ip nr. D-Sb initial wivt : nr. 5 T . 1.5 i. L • 275 m H • 016 m krt- IOi KbAo" 0.0*6 015 m | - —*« A* • MIS m ml. v«l«ctlf scat* profile on hor. bottom h . 0 411 h i 01] n mtasurod m.t.v. prof i le thtor prof i l t L H - Ih to ry ( a f t e r B i j k e r e t al.) Fiqure 4.6 E f f e c t of increasing beach roughness on the nass t r a n s p o r t velocity point; however, experimental r e s u l t s show that, i r r e s p e c t i v e of breaker type, the d r i f t v e l o c i t y i s onshore near the bed and at the surface, and offshore i n the mid-depth region. The v e l o c i t y p r o f i l e does tend to a more uniform d i s t r i b u t i o n (figure 4.7). The cause of t h i s may be attributed to the increased v e r t i c a l momentum tran s f e r by turbulence near the breaking point. So f a r the mass transport v e l o c i t y p r o f i l e has been described up to the point of breaking. Onshore of the breaking point, the mass transport v e l o c i t y p r o f i l e i s d i f f i c u l t to measure because of the turbulence created by breaking. Instead, q u a l i t a t i v e observations of the flow pattern can be made using a va r i e t y of flow indic a t o r s . The indicators used i n the present experiment are f i n e quartz sand, pea gravel (longest axis < 1.0 cm), crushed Bakelite, a c r y l i c cubes (side < 2.0 mm), and a c r y l i c spheres (6 mm diameter). The purpose of using a range of d i f f e r e n t materials was to attempt to investigate both suspended and bedload action. The experiment confirmed the well known s i t u a t i o n that offshore of the breaking zone, bedload moves onshore, whereas onshore of the breaking zone bedload moves offshore, converging towards a n u l l point (Appendix B). For a constant wave condition the n u l l point remained stationary and was positioned onshore of the breaking point and s l i g h t l y offshore of the f i r s t major vortex created by breaking, regardless of the breaking type. Conditions offshore of the n u l l point are f i r s t examined. Suspension of the a c r y l i c cubes occurs primarily at the n u l l point. Once the cubes are i n suspension they are l i f t e d further by the upward v e l o c i t i e s under a crest and are slowly moved 81 D i r e c t i o n o f wave t r a v e l B r e a k i n g po in t 82 offshore. Eventually , the cubes f a l l to the bed and are moved as bedload onshore to the n u l l point. The general path t r a v e l l e d by the cubes i s shown i n figure 4.8. Each upward movement indicates the presence of a crest and each downward movement the presence of a trough. The greatest upward movement occurs f o r the f i r s t few wave periods a f t e r the i n i t i a l suspension. Not a l l of the cubes follow the same path. Some are suspended e a r l i e r and some s e t t l e e a r l i e r . The s e t t l i n g usually occurs at the breaking point, e s p e c i a l l y for the plunging breakers. The suspension i s greater for the asymmetrical plunger since both the plunging vortex and the j e t promote suspension. The movement of the other p a r t i c l e s i s predominantly as bedload. However, when these p a r t i c l e s do go into suspension they usually remain i n suspension f o r l e s s than one wave period and are moved offshore. They return onshore to the n u l l point as bedload. The movement of the indicators suggests that a mass transport v e l o c i t y e x i s t s between the breaking point and n u l l point and i s consistent with the p r o f i l e at the breaking point: an onshore v e l o c i t y component near the bed and at the surface and an offshore component v e l o c i t y i n the middle section of the water column. Onshore of the n u l l point, the flow f i e l d i s d i f f e r e n t . The bedload i s transported offshore to the n u l l point and suspended p a r t i c l e s are transported onshore from the n u l l point. Suspension of the a c r y l i c cubes occurs at the n u l l point i n the same manner as previously described; however, instead of 83 Figure 4,8 T r a n s p o r t nechanisn o f f s h o r e of the null point i 84 t r a v e l l i n g offshore, the p a r t i c l e i s l i f t e d high enough such that the top h a l f of the f i r s t vortex forces the p a r t i c l e onshore (figure 4.9). The p a r t i c l e i s passed from one vortex to another and i s moved onshore . These v o r t i c e s are ei t h e r formed by the shear surface of symmetrical breaking waves or by splash-plunge cycles. The p a r t i c l e s f i n a l l y s e t t l e s when the vor t i c e s are no longer strong enough to keep them suspended. After the p a r t i c l e s e t t l e s , i t tr a v e l s as bedload back to the n u l l point. Once the p a r t i c l e begins to t r a v e l back to the n u l l point i t i s ra r e l y resuspended. The flow pattern i s e a s i l y traced by observing the movement of the a c r y l i c cubes, which are the only p a r t i c l e s moved onshore of the n u l l point. When any of the other materials are placed onshore of the n u l l point, they quickly are transported offshore to the n u l l point as bedload. However, i n some cases, the larger pieces i f pea gravel remain on the upper section of the beach and are not transported offshore u n t i l the wave attack i s increased. The general movement of the sediment i s shown i n figure 4.10a) which i s consistent with mass transport v e l o c i t i e s shown in figure 4.10b). S l i g h t l y onshore of the breaking point, the mass transport v e l o c i t y p r o f i l e s predicted by waves theories break down. Changing the c h a r a c t e r i s t i c s of the shoaling waves w i l l change the l o c a t i o n of the of the n u l l point. As the wave changes from s p i l l i n g to plunging to surging, the n u l l point moves onshore. From the motion of the n u l l point, the change i n the beach slope can be determined. Even though the n u l l point observations are for a plane 1:15 slope, any forward movement of 85 Nul l po in t Figure 4,9 T r a n s p o r t mechanism onshore o f the null point 86 a) b) B r e a k i n g po in t F i r s t m a j o r v o r t e x Nul l po in t B r e a k i n g po in t Nul l po in t Figure 4,10 a) Circulation cel ls on e i ther side o f the null point b) Mass t r a n s p o r t velocity prof i les in the s u r f zone 87 the n u l l point indicates a beach building process and any offshore movement indicates a beach erosion process. In t h i s respect, f o r experiments performed on an impermeable plane beach, the breaker conditions and sediment responses are forced to conform to the conditions imposed by the slope. Rather, for a natural beach, the beach and breaker conditions depend upon each other which eventually r e s u l t s i n the establishment of an equilibrium beach p r o f i l e . Therefore, the beach slope w i l l increase as the n u l l point moves up the beach, and w i l l decrease as the n u l l point moves offshore. 4 . 5 S E T U P I N T H E S U R F Z O N E Wave setup i s defined as the superelevation of the mean water l e v e l (MWL) caused by the wave action alone. The t o t a l setup needs to be better understood because i t might e i t h e r occur uniformily over the whole surf zone or i t may follow some other non-uniform d i s t r i b u t i o n . I f energy i s dissipated immediately at breaking, then uniform setup would be expected. But, i f energy i s dissipated uniformily, then a f a i r l y l i n e a r increase i n the setup would be expected. The k i n e t i c energy of the waves i s converted to quasi-steady p o t e n t i a l energy as the wave t r a v e l s up the beach. Observations of waves breaking on a 1:15 slope i n the laboratory indicate that there i s a decrease i n the water l e v e l to below the s t i l l water l e v e l (SWL) p r i o r to breaking, with a maximum setdown at approximately the breaking point. As the waves breaks the mean water l e v e l begins to r i s e above the SWL and eventually inter s e c t s with the shore. Similar 88 observations are found by S a v i l l e (1961). The variables defining wave setup are shown i n figure 4.11. Theoretical studies of setup caused by monochromatic waves have been c a r r i e d out by Longuet-Higgins and Stewart (I960, 1962, 1963, 1964), Bowen, Inman, and Simmons (1968), and Goda (1975). The major contribution to the understanding of setdown and setup i s provided by Longuet-Higgins and Stewart (1963). They develop a new term c a l l e d the rad i a t i o n stress which i s the att r i b u t e d to the momentum caused by the forward movement of the waves and i s written as, Sxx = (~~ + 4 - ) E (4-1) x x V sinh(2kd) 2 y v ; where E i s the energy of the wave and i s written as, E = p g H 2 / 8 (4.2) As a wave t r a v e l s from deep water to shallow water, the ra d i a t i o n stress changes from S x x=E/2 to S x x=3E/2. The wave height generally increases as the wave shoals. This being the case, the r a d i a t i o n stress w i l l l a r g e l y continue to increase up to the point of breaking. Since the t o t a l momentum remains constant up to breaking, ignoring any small losses which occur at the bed, the water depth w i l l drop, decreasing the momentum component associated with the water depth. Therefore, the point of maximum setdown should occur near the breaking point where the flow i s no longer i r r o t a t i o n a l . The computation of setup i s of p r a c t i c a l importance for a thorough design e f f o r t requiring water l e v e l estimations. In the next chapter, a model of the beach environment i s presented 89 B r e a k i n g po in t A S T o t a l s e t u p S e t u p S e t d o w n a t b r e a k i n g S j S e t d o w n f o r c o n s t a n t wave c o n d i t i o n s Figure 4,11 Definition s k e t c h of t o t a l se tup 90 which r e l i e s heavily of the setdown and setup and, therefore, requires an understanding of the setup process. The Shore Protection Manual presents a serie s of equations that allow the variables given i n figure 4.11 to be calculated. For setdown at breaking, 1 / 2 Ho 2 S = 9  T b The setup at the shore i s , S w = AS - S b (4.4) Equations 3.4, 3.5, and 3.6 define d b i n terms of the breaker height, period, and beach slope. These equations are repeated here f o r convenience b - < a H b / g T 2 ) where a and b are (approximately), a = 43.75 ( 1 - e" 1 9 n ) • * -19.5m. ( 1 + e ) Longuet-Higgins and Stewart (1963) have shown from analysis of S a v i l l e ' s data (1961) that the t o t a l setup for shallow water waves i s , AS = 0.15 d b (4.5) 91 Combining equations 3.4, 3.5, and 3.6 with equations 4.3, 4.4, and 4.5 gives, 1 /2 2 g Ho t where, d b = i i (4.7) 1.56 43.75(1 - e~ 1 9 m)H b 1 , - 1 9 . 5 m T 2 1 + e gT Figure 4.12 i s a p l o t of equation 4.4 i n terms of S w/H b versus H b/gT 2 for various slopes. 4.5.1 Experimental design and procedures The objective of t h i s experiment i s to determine the setdown and setup p r o f i l e for d i f f e r e n t wave and breaking conditions. The beach described i n section 2.4 i s modified to allow the surface p r o f i l e to be recorded. Twelve manometers are attached to the beach face at 61 cm i n t e r v a l s along the length of the beach. Each manometer i s made up of a pl e x i g l a s s pipe 2 5 mm i n diameter which i s connected to the beach face by 5 mm diameter f l e x i b l e tubing. The opening on the beach face i s 1.5 mm i n diameter. The manometers are designed to damp the pressure f l u c t u a t i o n associated with wave motion. This allows the mean water l e v e l above each manometer opening to be measured. The assumptions involved i n t r a n s l a t i n g t h i s measurement into the mean surface elevation have been considered i n d e t a i l by Longuet-Higgins and Stewart (1962). 92 Figure 4.12 Setup c u r v e s of Sw/Hb v e r s u s Hb/gT (SPM 1984) For several runs, the depth and wave height i n the constant depth section are held constant and the period i s varied. In other runs, the period and depth are held constant and the wave height i s varied. For each run, relevant wave data i s recorded and the setdown/setup p r o f i l e i s determined from the l e v e l s indicated by the damped manometers. For each wave condition, s u f f i c i e n t time i s allowed for the flume to reach steady state. 4.5.2 Experimental Results The data f o r 59 d i f f e r e n t wave conditions i s presented in Appendix C. From the r e s u l t s c o l l e c t e d i n the laboratory, the point of maximum setdown for s p i l l i n g breakers i s near the defined breaking point which i s located where white turbulent water i s f i r s t seen on the crest. For plunging breakers, maximum setdown occurs s l i g h t l y onshore of the defined breaking point and offshore of the f i r s t plunging vortex. This i s located where the j e t f i r s t penetrates the free water surface but does not yet produce any s i g n i f i c a n t r o t a t i o n a l flow. As the j e t penetrates i t i s also moving forward so that the plunging vortex i s formed s l i g h t l y onshore of the point where the j e t f i r s t penetrates the water surface. Bowen et a l . (1968) f i n d that the slope of the setup p r o f i l e i s a constant proportion of the beach slope. This trend i s seen i n the majority of the cases observed i n the laboratory. The setup i s approximately l i n e a r beyond the point of maximum setdown. Figure 4.13 shows the setup p r o f i l e f or run two which i s representative of most of the setup p r o f i l e s . 94 Figure 4,13 Setup prof i les f o r experimental run 2 Longuet-Higgins and Stewart (1963) have shown from an analysis of S a v i l l e ' s data (1961) that the t o t a l setup for pure shallow water waves where S x x=3E/2 i s , AS = 0.15 d b From the c o l l e c t e d r e s u l t s , the t o t a l setup for the 1:15 beach conditions i s found to be, AS = 0,31 d b ( 4 i 8 ) which i s twice the value found by Longuet-Higgins and Stewart. The difference indicates that the breaking waves are not s t r i c t l y shallow water waves and that, as Bowen et a l . found, the t o t a l setup i s dependent upon the beach slope. Comparison of the t o t a l setup for measured and calculated values, using equations 4.3 through 4.5, i s shown i n figure 4.14. In a l l cases the measured values of t o t a l setup are greater than the calculated values assuming shallow water wave behaviour. The value of d b i s not calculated using equation 4.7, but rather i s taken at the point of maximum setdown from the setup p r o f i l e s . The scat t e r of the measured t o t a l setup i s best seen i n figure 4.15. This i s a p l o t of the t o t a l setup versus the deepwater wave steepness and i t shows a d e f i n i t e trend that as the deepwater wave steepness increases, the t o t a l setup decreases. S p i l l i n g breakers, associated with higher values of H 0/L 0, break further offshore and tend to break over a greater distance. This releases a s i g n i f i c a n t amount of energy over the length of the surf zone as turbulence. The energy that i s not di s s i p a t e d as turbulence i s converted into p o t e n t i a l energy as 96 1 6 0 . 0 1 4 0 . 0 1 2 0 . 0 H 1 0 0 . 0 8 0 . 0 H 6 0 . 0 H 4 0 . 0 H 2 0 . 0 H 0 . 0 Calcu la ted Total Setup, S c (mm) 0 . 0 2 0 . 0 4 0 . 0 6 0 . 0 8 0 . 0 1 0 0 . 0 1 2 0 . 0 1 4 0 . 0 160 .0 Measured Total Setup, S (mm) Figure 4.14 Calculated t o t a l se tup v e r s u s measured t o t a l se tup f o r an experimental M5 slope 97 a) 1 6 0 . 0 1 4 0 . 0 1 2 0 . 0 100.0 H 80.0 H 6 0 . 0 H Total Setup, S (mm) 4 0 . 0 2 0 . 0 -i 0 .0 0 . 0 0 0 0 2 0 . 0 4 o . o e o . o a Deepwater S teepness , H o / L o 0.10 b) 1.20 1.00 Relative Setup, S / d b 0 . 8 0 h o.eo o.4o H 0 . 2 0 h 0 . 0 0 0 . 0 0 0 0 2 0 . 0 4 0 . 0 6 0 . 0 8 Deepwater S teepness , H o / L o 0.10 Figure 4,15 a) Tota l se tup v e r s u s deepwater s teepness b) Relative t o t a l se tup v e r s u s deepwater s teepness 98 setup. The components of the t o t a l setup f o r run 7, run 10, and run 11 are shown i n figure 4.16. For increasing period and decreasing deepwater steepness, the t o t a l setup increases. 4 . 6 DISCUSSION The flow dynamics within the surf zone are driven by the conversion of i r r o t a t i o n a l flow to r o t a t i o n a l flow. The rot a t i n g f l u i d generates turbulence which leads to the d i s s i p a t i o n of energy, but momentum i s conserved. Any of the i n i t i a l k i n e t i c energy that i s not dissipated as turbulaence i s converted into p o t e n t i a l energy as setup. S p i l l i n g and plunging breakers can be further divided into symmetrical s p i l l i n g , symmetrical plunging, and asymmetrical plunging breakers. Each of these has a d i f f e r e n t i n i t i a l motion, but the net f l u i d motions are s i m i l a r when observed over a number of wave periods. Regardless of the type of breaking wave, s i m i l a r flows are maintained i n the surf zone. Two c i r c u l a t i o n c e l l s are created on e i t h e r side of a point that exhibits no net motion, c a l l e d the n u l l point. Offshore of the n u l l point, the sediments t r a v e l s onshore as bedload and offshore as suspended load. This occurs between the breaking point and the n u l l point. Onshore of the n u l l point, the sediments t r a v e l onshore as suspended load and offshore as bedload. Changing the wave conditions only causes the breaking point and n u l l point to move ei t h e r onshore or offshore depending upon whether the wave attack decreases or increases. The two c i r c u l a t i o n c e l l s stretch and collapse y 99 SETDOWN, SETUP, TOTAL SETUP (mm) SETDOWN, SETUP, TOTAL SETUP (mm) b) M SETDOWN S E T U P GTJ TOTAL s e T u p • I n Run i a i Run 10.2 Run 10.8 Run 10.4 • SETDOWN, SETUP, TOTAL SETUP (mm) c) • i SETDOWN S3 S E T U P E3 TOTAL S E T U P • Figure 4,16 Components of t o t a l se tup f o r experimental run 7, run 10, and run 11 100 depending upon the locations of the breaking point and n u l l point. The setdown/setup p r o f i l e s for given wave conditions are e a s i l y determined by using damped manometers. In a l l cases a steady p r o f i l e develops. The shape of the p r o f i l e s d i f f e r l i t t l e between conditions and are r e l a t i v e l y l i n e a r . The t o t a l setup does increase as the deepwater steepness decreases or as the period increases The above observations are important because they show that the complexities of the surf zone when averaged over many wave periods can be reduced to a r e l a t i v e l y steady state. In chapter f i v e , the importance of these observations becomes apparent when a transport model i s developed from a control volume defined by the steady state. 101 CHAPTER 5; DEVELOPMENT OF THE BEACH FACE CONTROL VOLUME MODEL 5.1 INTRODUCTION The onshore-offshore p r o f i l e changes of a beach under wave attack have been described by many researchers. Some of the more recent work i s by Hattori and Kawamata (1980), H a t t o r i (1982), S t r i v e and Bajjtes (1984), Leont'ev (1985), and Nishimura and Sunamura (1986). Komar (1983) and Horikawa (1981) summarize other researcher's work. The majority of onshore-offshore p r o f i l e models r e l y on d i r e c t sediment transport measurements or the comparison of nondimensional parameters with experimentally established values. The method presented i n t h i s chapter takes a new approach and examines the beach p r o f i l e using a beach face control volume model (Quick 1989a, 1989b). The model r e l i e s on time-averaging of conditions within the surf zone. In the model, the onshore-offshore sediment transport and beach slope i s explained and defined i n terms of the basic 102 momentum balance and r e s u l t i n g pressure and shear d i s t r i b u t i o n along the beach face, as modified by beach permeability. The model uses the breaking point, the mean water l e v e l , and the beach face as the boundaries of the control volume. The r e l a t i o n s h i p s between the main variables are analyzed by observing the response of the main equation to varying wave attacks. 5.2 FORMULATION OF THE MODEL The s t a r t i n g point of t h i s model i s at the wave breaking point. I t i s here that complex r o t a t i o n a l motion begins and continues up the beach. The incoming wave momentum i s conserved at breaking and the i n i t i a l k i n e t i c energy i s ei t h e r dissipated as turbulence or converted to po t e n t i a l energy which i s seen as a change i n the water l e v e l from the s t i l l water l e v e l (SWL). The previous three chapters have described and quantified many of the changes that occur from the breaking incident waves through to the turbulent bore that runs up the beach face to eventually return as backrush. The whole process i s complex and dynamic, but, fortunately, can be s i m p l i f i e d by using time-averaging techniques. Time-averaging the free surface of the water waves re s u l t s i n the establishment of a mean water l e v e l (MWL). Shown i n figure 5.1, the MWL forms the top boundary, AC, of the beach face control volume. The boundary AB i s at the point of wave breaking. Conditions at the seaward boundary, AB, are defined by the hydrostatic pressure d i s t r i b u t i o n from the MWL together with the extra momentum flu x produced by the waves, M^. The 103 igure 5,1 The beach f a c e cont ro l volume and the f o r c e s acting upon it 104 time-averaged hydrostatic force and the time-averaged net shear stress act along the boundary BC. Permeability of the beach face modifies the forces acting on the beach face control volume by reducing the t o t a l setup and reducing the volume of backrush. These changes are important and w i l l be accounted for l a t e r since the development of the model i s better understood i f the beach i s f i r s t considered impermeable. 5.2.1 Assumption of an Impermeable Beach Equating the components of the time-averaged momentum i n the horizontal d i r e c t i o n y i e l d s an estimate of the t o t a l setup. Experimental observations of setup show that i t i s f a i r l y l i n e a r over the surf zone (Appendix C) which agrees with observations of Bowen, Inman, and Simmons (1968) who f i n d that the setup i s l i n e a r l y r e l a t e d to the beach slope. Figure 5.2 shows a s i m p l i f i c a t i o n s of the forces acting on the beach face control volume assuming a uniformly sloped beach and a l i n e a r setup p r o f i l e . The t o t a l horizontal momentum balance for an impermeable beach becomes, A l i n e a r beach i s assumed for a number of reasons. F i r s t , f o r s i m p l i c i t y since a l i n e a r beach w i l l not be subject to any strong l o c a l accelerations associated with bars and r i p p l e s which could a f f e c t the pressure force represented i n figure 5.2 by the force t r i a n g l e at BC. I t can be argued that even for normal concave beach p r o f i l e s the l o c a l accelerations w i l l s t i l l 105 F i g u r e 5.2 S i m p l i f i e d beach f a c e c o n t r o l volume 106 be small enough to not a f f e c t the horizontal forces. Second, the extra momentum flux of the waves, M^ , w i l l be calculated using the rad i a t i o n stress theory developed by Longuet-Higgins and Stewart (1964). And f i n a l l y , the setup p r o f i l e s measured in chapter four are for a l i n e a r impermeable beach which w i l l be used to analyze the model. The value of the extra momentum flux, Mb, may be affected by r e f l e c t i o n s occurring from the beach face. For the 1:15 sloped beach used i n the laboratory, t e s t s indicate that r e f l e c t i o n s are about 2% of the incident wave height and are i n s i g n i f i c a n t f o r the conditions studied i n both the t o t a l setup experiment and the flow dynamic observations (Appendix D). However, i f the r e f l e c t i o n s are found to be s i g n i f i c a n t the r e f l e c t e d momentum must be included as an extra shoreward momentum fl u x acting on the control volume. In any event, decreases i n the beach slope or increases i n the incident wave steepness r e s u l t i n a reduction of r e f l e c t i o n s . The net shear force acts along the beach face of the control volume. This force depends upon the drag forces acting on the bed during uprush and backrush cycles. In fact, the d i r e c t i o n of the net shear force can be onshore, offshore, or even zero. Figure 5.3 shows the forces acting on a p a r t i c l e . Summing over a unit area of the beach the uprush and backrush drag during one wave cycle y i e l d s the net shear s t r e s s . The horiz o n t a l components of the l o c a l net shear stresses are integrated over the length of the beach face, BC, to determine the net horizontal shear force acting on the control volume. The p a r t i c l e weight does not contribute to the shear force, 107 i gure 5 .3 F o r c e s a c t i n g upon a par i : l a y i n g on the beach f a c e 108 however, i f the net shear stress i s zero the gravity force along the beach face w i l l s t i l l produce an offshore movement when the shear stress for the uprush or backrush are greater than the c r i t i c a l shear stress needed f o r the onset of motion. I t i s quite possible that the uprush shear stress and/or backrush shear stress can be les s than the c r i t i c a l shear stress needed for the onset of motion; however, for t h i s model the uprush and backrush shear stresses are assumed to be much greater than the c r i t i c a l shear stress. The c r i t i c a l shear stress then becomes i n s i g n i f i c a n t with respect to the net sediment motion. Under t h i s condition, the sediment movement w i l l be offshore for an onshore net shear stress smaller than the offshore gravity force. For an onshore net shear stress greater than the offshore gravity force, the sediment movement w i l l be onshore. An offshore net shear stress combines with the offshore gravity force to produce a stronger offshore sediment movement. Vigorous wave attack i s known to cause offshore movement of sediments, so that i t appears that such wave attack i s capable of high offshore shear stresses. For such a s i t u a t i o n , the onshore shear stress must be les s than the offshore shear stress, which i s d i f f i c u l t to explain unless the incoming wave can r i d e over top of the water already on the beach face, so that the uprush shear stress i s small. Figure 5.4 shows t h i s s i t u a t i o n and i t i s seen that the net shear stress acting on the beach face control volume i s onshore. Referring to equation 5.1, the second and t h i r d terms r e s u l t i n the c a l c u l a t i o n of the net pressure force, which must act i n the offshore d i r e c t i o n ( i e . the t h i r d term i s always 109 In i t i a I Wave B r e a k i n g R e g i o n o f U p r u s h R i d i n g D v e r B a c k r u s h F l o w B r e a k i n g Wave Mov i ng up B e a c h S h o r e w a r d S t r e s s P r e d o m i n a n t l y D f f s h o r e S t r e s s o f W a t e r on S e d i m e n t F i g u r e 5,4 A s p e c u l a t i v e e x p l a n a t i o n f o r an o f f s h o r e net shear s t r e s s no greater that the second for any setup). I f the net pressure i s offshore, which implies net offshore transport, then the wave setup and the offshore shear stress increase even further so as to keep the equation i n equilibrium. The fourth term i s the horizontal component of the time averaged net shear stress integrated over the length of the beach face control volume. The value of the extra momentum at the breaking point, Mb, i s impossible to measure and very d i f f i c u l t to d i r e c t l y c a l c u l a t e because no wave theories e x i s t that can account for the highly asymmetrical wave shapes commonly found at breaking. To solve the problem, a new control volume, the offshore control volume, i s defined seaward of the breaking point (figure 5.5) together with a new momentum at breaking, Mb' (Quick 1989b). The reason for using Mb' w i l l become cleare r once the momentum of the backrush flow i s included i n the analysis. The seaward boundary, DF, i s located at the point where the setdown begins to increase below the setdown associated with constant, uniform waves. The shoreward boundary, AB, i s at the breaking point, the MWL i s again the time-averaged surface boundary, and the bed forms the bottom boundary, BF. Along the bed, for s i m p l i c i t y , the shear stress i s assumed to be zero which i s only possible for small horizontal v e l o c i t i e s occurring at the bed. However, observations i n chapter four show that there i s onshore bedload transport offshore of the breaking point. This indicates that there i s a net shear stress component acting offshore on the offshore control volume. Neglecting i t s e f f e c t increases the onshore momentum flu x acting on the beach face control volume, 111 112 but does not i n t e r f e r e with the basic r e l a t i o n s h i p s between the variables of the beach face control volume. The momentum flux at the boundary DF, caused by the presence of the waves, i s Mj. Because of setdown an extra term i s added to Mb', giving, n D / p A r b l b = M, + pdy -F J \ B 0 P b d v pdy (5.2) Mj represents the incoming wave momentum and i s e a s i l y calculated using the ra d i a t i o n stress formula f i r s t proposed by Longuet-Higgins and Stewart (1964), M, = ( 2kd + J _ A E (5.3) 1 \ sinh(2kd) 2 / where E i s the wave energy and i s equal to 7H2/8. The conditions for Mj are measured at the seaward boundary, DF. For equation 5.3 to be v a l i d , the p o s i t i o n of the seaward boundary must be such that the MWL i s constant and the wave shape i s r e l a t i v e l y constant The e f f e c t of backrush on the momentum leaving the offshore control volume must be accounted for. Observations show that at the boundary AB the forward movement of the wave i s suppressed. This i s caused by the slowing e f f e c t of shoaling and the offshore flow of the backrush. As the forward edge of the offshore control volume i s stopped, a new wave crosses the offshore boundary, DF, and compresses into the offshore control volume. The backrush, i n e f f e c t , decreases the momentum ex i t i n g the offshore control volume. Therefore, the r e l a t i o n s h i p 113 between the momentum e x i t i n g the offshore control volume and entering the beach face control volume i s , <b = M b - M (5-4) where M R represents the momentum flux of the backrush. T y p i c a l l y , the period of the backrush, t b , i s about h a l f of the wave period and must be taken into account i n the f i n a l determination of MR. The offshore momentum flux of the backrushing water from the beach face, generated by the previous breaking wave,is, M = 1/T o p q u b d t (5.5) where q i s the backrush flow over some portion of the breaking depth, hdjjU^ (figure 5.4). The backrush v e l o c i t y i s represented by u b . The factor 1/T can be j u s t i f i e d by considering the impulse of the momentum flux, MjT, associated with the incoming wave and the subtraction of the impulse of the backrush, Mpt^, which occurs only for the in t e g r a l time 0 to t ^ . The factor T divides a l l the terms when reverting to momentum flux. By combining equations 5.2 through 5.5, the equation for the extra momentum flux entering the beach face control volume at the breaking point i s found to be, 2kd s inh(2kd) + J _ 2 E + FJ pdy -1/T P q u b d t B J p bd y + J pdy (5.6) 114 The t o t a l momentum balance for the beach face control volume of an impermeable beach i s obtained by combining equation 5.1 and equation 5.6. The f i n a l r e s u l t gives, 2kd 1 sinh(2kd) 2 1/T p q u b d t E + D pdy - P hdy + B pdy + B J Pb d y + pdy 7~0 cos Od L q 0 (5.7) where the f i r s t two l i n e s of terms are the momentum flux a t t r i b u t e d to the breaking waves entering the beach face control volume; the t h i r d l i n e i s the offshore momentum flu x caused by the t o t a l setup occurring over the beach face control volume; and, the f i n a l term i s the momentum flu x of the shear stresses acting along the bottom of the control volume. 5.2.2 Inclusion of the E f f e c t s of Permeability To model r e a l beaches, permeability must be included within the formulation of equation 5.7. During wave uprush and backrush, water w i l l i n f i l t r a t e into the beach causing a reduction i n both the t o t a l setup and the backrush flow. Any reduction i n the t o t a l setup causes a reduction i n the offshore stress acting on the sediments which implies an increase i n the offshore component of the shear force acting on the beach face control volume. Therefore, as a consequence of increasing beach 115 permeability, the offshore transport of sediment i s decreased and may be reversed. Beach permeability w i l l be used to explain the primary aspects of beach slope behaviour (Quick 1989a, 1989b). In equation 5.7, the f i f t h term, describing the backrush momentum flux, and seventh term, describing the offshore hydrostatic force on the beach face control volume, are affected. During the uprush and backrush, water i n f i l t r a t e s into the beach which reduces the impermeable beach backrush flow by the i n f i l t r a t i o n flow, q-j-. The backrush v e l o c i t y also changes and i s represented by u b I . The equation describing the backrush momentum flux, modified by permeability, i s , M R = 1 /T P ( q - q , ) u . d t O J b i Q I (5.8) I n f i l t r a t i o n modifies the s i x t h term of equation 5.7 because, by continuity, the i n f i l t r a t i o n , integrated over the resident time, t r , d i r e c t l y reduces the t o t a l setup and hence reduces the offshore hydrostatic force acting on the beach face. The change i n the hydrostatic force i s calculated by integrating the i n f i l t r a t e d volume along the beach length, L B, and i s given by, pdy > = j pdy ( -) I ' B J )q B J 0 « • t r p g q , d t d L B (5.9) where the subscript I i s for the setup with i n f i l t r a t i o n and the subscript 0 i s for the setup with no i n f i l t r a t i o n . The resident 116 time, t r , w i l l vary over the length of the beach and i s the t o t a l duration of the uprush and backrush at any given point. By incorporating equations 5.8 and 5.9 i n equation 5.7, the t o t a l momentum balance for the beach face control volume of a permeable beach i s given as, 2 k d , + -L- ) E + \ s inh(2kd) 2 / F , - 1/T I P ( q - q , )u b | dt + Pb dY B J (* D pdy -• c pdy c B P b dy + + B, pdy 0 B p gq , d t d L B #- c r o c o s O d L B = 0 (5.10) 5.3 DIFFERENT BEACHES SUBJECTED TO THE SAME WAVE ATTACK Consider two beaches, one composed of f i n e sand and the other composed of loose gravel. I f both are subjected to the same wave attack, the increased permeability of the gravel beach w i l l cause the t o t a l setup and the backrush flow to be less than the values f o r the sand beach. Therefore, as the permeability of the beach i s increased, the t o t a l setup decreases and the i n f i l t r a t i o n flow increases. In equation 5.10, an increase i n the i n f i l t r a t i o n flow causes an increase i n momentum flux reaching the beach and an increase i n the onshore force acting on the beach face control volume. These increases must be balanced by a decrease i n the f i n a l shear force term which can only be attained by an increase i n the beach slope and a 117 decrease i n the offshore shear stress acting on the sediments. Since the beach slope increases, the offshore gravity force acting on the sediments increases and must both compensate for the loss of offshore shear stress and balance the increase of the onshore force. The consequence i s that more permeable beach materials w i l l form steeper beaches. This i s consistent with the known behaviour of sand and gravel beaches. Fine sand forms very low sloped beaches whereas gravel forms steep beaches, sometimes at or near the angle of repose. In a l l cases, i n f i l t r a t i o n has been assumed to occur evenly over the whole length of the beach. This probably i s not the case. As shown i n figure 5.6, the water that i n f i l t r a t e s into the upper sections of the beach must percolate through the beach and a portion may e x f i l t r a t e at the lower sections. The e f f e c t of t h i s i s to cause a reduction of the time-averaged i n f i l t r a t i o n f o r points lower on the beach. Time-averaged e x f i l t r a t i o n may occur over the lowest section of the beach. Therefore, as the i n f i l t r a t i o n decreases and reverses to e x f i l t r a t i o n , the offshore shear stress w i l l increase causing a reduction i n the beach slope. The e f f e c t of e x f i l t r a t i o n i s to increase the net offshore shear stress, reducing the slope, because, at equilibrium, there must be a decrease i n the downslope gravity force to compensate for the increased offshore shear s t r e s s . This argument, therefore, explains the general tendency of beaches to e x h i b i t concave p r o f i l e s , and gravel beaches, which are more permeable, should, and i n f a c t do, e x h i b i t more concavity than less permeable sand beaches. 118 INFILTRATION Increasing o f f s h o r e shear s t r e s s ^ EXFILTRATIDN Infiltration, ex f i l t ra t ioa and beach concav 119 The r e l a t i o n s h i p between the beach slope and beach permeability for beaches subjected to the same wave attack can be analyzed using the control volume. Permeability of sands and gravels i s known to be a function of sediment s i z e . Hazen (1911) proposes a formula that r e l a t e s the permeability of a sediment to the square of the 10% f i n e r p a r t i c l e s i z e present i n the material, since the drainage that occurs i n a material i s con t r o l l e d by the p a r t i c l e s that f i l l the pores. The formula i s , K = 10 D 1 Q 2 ( 5 , U ) where K i s the permeability i n millimeters per second and D 1 0 i s the 10 % f i n e r diameter i n millimeters. The kinematic v i s c o s i t y , v , and the gravity force, g, should be included i n the formula to make i t dimensionally correct, so that, K * 1/1000 D ^ g / i , (5.12) The beach length, L B, assuming a l i n e a r dependency on both the wave height and beach slope, i s given as, L B o < H / s i n Q ( 5 j 3 ) The resident time of the water on the beach, t r , i s required to cal c u l a t e the volume of i n f i l t r a t i o n . The residence time w i l l depend on the length of the beach and the wave speed, which, for shallow water waves i s written as proportional to %/ gH, since the wave height at breaking i s related to the depth at breaking and should scale f o r depths higher on the beach. Therefore, 1 / 2 t r oc L B / ( g H ) ( 5 i 4 ) 120 and the volume of i n f i l t r a t i o n , V j , for the beach length, per unit width of the beach, i s , V , oc L B t r K _ d h _ ( 5 . 1 5 ) d L B where dh/dL B i s the hydraulic gradient that drives the flow and i s e a s i l y expressed as H/L B which, by equation 5.13, i s s i n O . Substituting the appropriate terms into equation 5.15, s i n © ( g h ) ' / 2 „ L B ( 5 - 1 5 ) The volume of i n f i l t r a t i o n represents a decrease i n the t o t a l wave setup and therefore represents an increase i n the net onshore, upslope force acting on the sediment layer and i s given by, 7 V , c o s O ( 5 . 1 7 ) This force must be balanced by a steepening on the beach such that the downslope, offshore gravity force, G, increases, and i s given by, G oc L B ( 7 S - 7 ) D M ( 5 i 1 8 ) where Efy i s the mean p a r t i c l e s i z e and LgDj^ represents the volume of mobile sediment on the beach face. For equilibrium, V , c o s 0 oc L B ( 7 S - 7 ) D M sin© (5.19) which can be s i m p l i f i e d to give the equation, ( q H ) 1 / 2 D/o = C D M tanO ( 5 . 2 0 ) v where the c o e f f i c i e n t C absorbs the s p e c i f i c weight terms and accounts for the pr o p o r t i o n a l i t y . The mean p a r t i c l e s i z e , DM, 121 depends upon the beach material grading curve and can be written i n terms of D 1 0, using a grading c o e f f i c i e n t , k g, D M = k g D 1 0 (5.21) ' For beach sand, the material i s usually well graded and grading c o e f f i c i e n t might be on the order of 2. As the beach material gets coarser, the grading c o e f f i c i e n t w i l l increase. The f i n a l form of equation 5.21, a f t e r s u b s t i t u t i o n for DM, i s then, 1 /2 (gH) D 1 0 = C kg tan© (5.22) v This equation represents a nondimensional formulation that can be used to c a l c u l a t e the change i n the beach slope for d i f f e r e n t beaches subjected to the same wave attack. To begin c a l c u l a t i o n s f o r a beach, the sediment si z e for D 1 0 and the grading c o e f f i c i e n t , kg, need to be obtained. For a given wave condition, and using the corresponding beach slope, the c o e f f i c i e n t C can be estimated by using equation 5.22. Once the values of C and kg are known for a known slope, then changes i n the beach slope for d i f f e r e n t wave conditions can be calculated. Figure 5.7 i s the p l o t of equation 5.22 for d i f f e r e n t values of kg and D 1 Q. This treatment represents the d i f f e r e n t equilibrium slopes that d i f f e r e n t beaches a t t a i n for a constant wave condition. Unfortunately, the v a r i a t i o n i n figure 5.7 appears to be too great when compared to the behaviour of r e a l beaches. One must remember that t h i s formulation i s based upon simple arguments. The r e a l s i t u a t i o n i s much more complex, e s p e c i a l l y for the i n f i l t r a t i o n flow. There are three p r i n c i p a l 122 MEAN PARTICLE SIZE (millimeters) Figure 5,7 Undamped solution f o r the same wave attacking d i f f e ren t beaches (equation 5.22) reasons for the overestimation of the changes i n the beach slope. F i r s t , the seepage flow w i l l change from laminar to turbulent flow as the sediment resistance increases and hence reduces the i n f i l t r a t i o n such that the slope does not have to increase as much to compensate for the increase i n the net onshore, upslope force acting on the sediment. A s i m i l a r resistance increasing mechanism w i l l be the entraining of a i r within the sediment matrix. This may occur during higher wave attacks or during the action of plunging breakers and splash-plunge cycles where large turbulent v o r t i c e s carry entrained a i r to the bed. The e f f e c t of the a i r i s s i m i l a r to that mentioned above. The a i r i n the sediment causes the i n f i l t r a t i o n flow to become turbulent thus increasing the flow resistance. F i n a l l y , the beach material i s not necessarily uniformly graded and often shows a v e r t i c a l gradation with coarser material at the surface and f i n e r material increasing with depth into the beach. I n f i l t r a t i n g water w i l l again meet increased resistance such that the slope w i l l not have to increase as much to balance the reduced onshore, upslope force acting on the sediments The i n t e r a c t i o n s of each of these e f f e c t s are complex and nonlinear and tend to decrease, or damp, the large change i n the beach slope predicted by the simple analysis for increasing sediment s i z e s . The beach does not f l a t t e n or steepen as much as the as the analysis predicts. To show the e f f e c t of these changes on the analysis, damping must be introduced into the procedure. In the o r i g i n a l analysis, the hydraulic gradient that produces the i n f i l t r a t i o n flow i s represented by dh/dL B and 124 i s simply expressed as H/L B which, by equation 5.13, i s sin© , Therefore, a higher gradient e x i s t s for points higher up the beach. To mimic damping, the gradient, dh/dL B, i s held constant. Although t h i s i s an o v e r s i m p l i f i c a t i o n , i t has the advantage of i l l u s t r a t i n g the e f f e c t of damping while maintaining the dimensional homogeneity of the analysis. Assuming dh/dL B to be constant, equation 5.22 becomes, 1 /2 (5.23) (gH) D 1 0 - C kg sin2© v cos© Figure 5.8 shows the curves f o r equation 5.2 3 for various values of D 1 Q and kg. The curves show a marked decrease i n the slope changes for coarser sediments. The general trend i s more consistent with observations of natural beaches and i t i s seen that gravel beaches can even reach the natural angle of repose, a common observation. In addition, selected r e s u l t s given by Dalrymple and Thompson (1976), i n the form of equilibrium slope versus nondimensional f a l l v e l o c i t y , are recalculated and shown on the graph. These points are dependent upon an assumed wave height of 2 meters, a period of 8 seconds, and converting the s e t t l i n g v e l o c i t y to an equivalent mean p a r t i c l e s i z e . I t i s important to remember that the mean p a r t i c l e s i z e i s calculated by kgD 1 0' each of which are the important factors i n the analysis. The points are very approximate, but show that the trend i s correct. This figure i s i l l u s t r a t i v e at t h i s time and requires c a r e f u l experimental values to confirm the v a l i d i t y of equation 5.23. 125 36 34 32 h 30 28 -26 -24 -22 -20 -18 -16 -14 -12 10 8 6 h 4 T 1 I I I I I 1 1 1 I I I I 1 1 1—I I I D^Bmm, Real beaches may follow this type behaviour with k increasing for coarser sediments Dl0= O.lmm J ' i t i i i • 1 1 i i i i i i 1 Very approximate points from Dalrymple j i i — l i i I i 10 - i Figure 5.8 K)° 10 MEAN PARTICLE SIZE (millimeters) Damped solution f o r the same wave attacking d i f fe ren t beaches (equation 5,23) with s e l e c t e d data from Dalrymple and Thompson (1976) IO2 5.4 SAME BEACH SUBJECT TO A VARYING WAVE ATTACK Now consider the e f f e c t of a varying wave attack on the control volume varies with the wave height squared, where, When a beach i s i n equilibrium, equation 5.10 i s i n balance. The net incoming momentum flu x i s balanced by the offshore pressure terms and the beach permeability absorbs enough water to also a i d i n the balance. As the wave height increases, the beach i n f i l t r a t i o n cannot absorb the extra water u n t i l the beach f l a t t e n s out, e f f e c t i v e l y increasing the beach length and allowing more i n f i l t r a t i o n . Therefore, the wave setup increases and produces an increase i n the net offshore shear stress acting on the sediments, thus causing the beach to f l a t t e n u n t i l a new equilibrium i s reached. Since the slope i s le s s , the downslope gravity force i s reduced which indicates that the net onshore shear stress i s also reduced. Consequently, the slope reduces s l i g h t l y less than i s required by the increase i n the i n f i l t r a t i o n f or an increased wave attack. Referring to equation 5.10, any change i n the incoming momentum flu x w i l l be accompanied by a proportional change i n the i n f i l t r a t i o n volume. As a f i r s t approximation, the momentum flux, Mj, i s assumed to be proportional to the i n f i l t r a t i o n volume, V-r, given by equation 5.16 such that, same beach. The momentum flux, M I' delivered to the beach face M, oc H 2 (5.24) H 2 K d h M, <* oc (gH)' NV2 d L B (5.25) 127 where the permeability, K, has been used for s i m p l i c i t y . Two equations can be written assuming non-damped and damped s i t u a t i o n . F i r s t , using dh/dL B to be sin© , 1/2 (g H ) sin© = constant (5 2 6 ) K and using dh/dL B to be constant, ( L A 1 / 2 • 2 (gH) S i n z 0 = constant K ( 5 . 2 7 ) The value of K w i l l be constant for the same beach, therefore these equations can be reduced to, and, s ine oc H 1 / 2 (5.28) sinQ oc H 1 / 4 ( 5 . 2 9 ) Equation 5.28 appears to predict too much v a r i a t i o n , although 1 the general tendency i s correct as shown i n figure 5.9. Again, the more damped behaviour of equation 5.29 i s preferred. This equation shows that the beach slope w i l l decrease under increased wave attack, but w i l l be small since i t depends upon the fourth root of the wave height. For example, a 5^ " beach slope w i l l change to a new slope of 3..5*5 for a four f o l d increase i n wave height. Predictions of the change i n the equilibrium slope, using equation 5.29, are plotted i n figure 5.10 f o r a wave height increase from 1 meter to 2 meters for an 8 second wave period. Th equilibrium beach slope decreases, and decreases to a larger extent for coarser sediments. Also p l o t t e d are points taken from Dalrymple and Thompson (1976) for 128 1 1—I—I I I I T 1 1 1 I I I • I 1—I \p\ I I Beoch slope decreoses under ^ wdve dltock k=2.3 k»l.7 I I I 1 M i l l ' « 1 i i i i i J • • ' ' ' 10 -1 10° 10 MEAN PARTICLE SIZE (millimeters) I0! Figure 5.9 Undanped solution f o r d i f f e r e n t waves attacking the sane beach (equation 5.28) T 1 1 1 ) 1 1 1 1 1 1 I I I I 17 r-16 15 -14 -13 12 h I I 10 9 8 7 6 5 4 3 2 I O T 1—I TT ^ -From Dalrymple using H= lm» T=6sec. S is > ^ A 7 —^k*4.5 _. v Different sediments but same wave attack and based From Dalrymples on 3.5° slope graph (approximate when values based on D l f t=0l.k = l 7 H«2m,T=8sec.) j — - J » • ' • ' 1 ' 1 1 i i i i I i • • I i i l i 10 - i K>° 10 MEAN PARTICLE SIZE (millimeters) 10* Figure 5.10 Damped solution f o r d i f f e r e n t waves attacking the same beach (equation 5,29) with se lec ted data from Dalrymple and Thompson (1976) waves with heights and periods of 1 meter and 6 seconds, and 2 meters and 8 seconds. These show good agreement with predictions made by equation 5.29. The c o e f f i c i e n t C used i n equation 5.22 and equation 5.2 3 r e l a t e s the i n f i l t r a t i n g volume to the downslope gravity component f o r a given wave height. Combining equation 5.2 3 with equation 5.29, the v a r i a t i o n of C with respect to the wave height can be made. F i r s t introducing a new constant C into equation 5.29, sinQ oc cZpr^4 ( 5 . 3 0 ) and the equation for the constant C i s , C O S 0 1 /2 . N C = , , , \ 2 ' g ° i o H (5.31) where, for small beach slopes, cos0 can be assumed to equal unity. This shows that C varies l i n e a r l y with the wave height. An exactly s i m i l a r r e s u l t i s reached when equation 5.22 and equation 5.28 are used. 5.5 O B L I Q U E W A V E A T T A C K The whole of t h i s thesis has been directed towards understanding the processes waves and beaches go through for waves crests t r a v e l l i n g perpendicular to the beach. In r e a l i t y , waves often approach the beach obliquely. The analysis presented i n t h i s chapter could be used provided that the momentum flu x i s subdivided into onshore and longshore components. I f the beach i s st r a i g h t and there are no sediment 131 sources or sinks, then i t i s reasonable to expect that the longshore and onshore components are independent. Treatment of the onshore component of momentum flux has already been described i n t h i s chapter. The longshore component w i l l be subject to the same i n f i l t r a t i o n and e x f i l t r a t i o n e f f e c t s so that the longshore shear stresses w i l l decrease for higher points on the beach. The offshore gravity force, which plays a s i g n i f i c a n t role, for the onshore component, w i l l be absent, but w i l l be replaced by shear stresses produced by any longshore currents. The strength of the longshore sediment transport w i l l depend upon the product of the shear stresses and current v e l o c i t y throughout the surf zone, modified by the changes i n the shear stress caused by the beach permeability. The permeability w i l l tend to l i m i t longshore transport on the upper beach, and increase transport i n the e x f i l t r a t i o n zone of the lower beach. 5.6 DISCUSSION A method has been developed to analyze the changes of a beach. I t i s based upon time-averaging the complex motions within the surf zone. I f these time-averaged processes are e s s e n t i a l l y steady, as they appear to be as discussed i n chapter 4, then they are representative of the conditions within the surf zone and the analysis should be v a l i d . Using the mean water l e v e l , the breaking point, and the beach face as the boundaries of the beach face control volume, the momentum i s balanced i n the horizontal d i r e c t i o n and gives the basis of the 132 model. The t o t a l model i s described by equation 5.10 which includes the e f f e c t s of beach permeability. The model shows that onshore-offshore sediment transport i s p r i m a r i l y dependent upon the magnitude of the wave setup shoreward of the breaking point, and the permeability of the beach. These two factors control the magnitude of the net shear stress, which, when combined with the downslope component of the sediment weight, determines the magnitude and d i r e c t i o n of transport. The magnitude of the l o c a l sediment i s probably best represented by the integrated excess stream power, which i s the excess shear stress plus or minus the sediment weight component mu l t i p l i e d by the l o c a l v e l o c i t y . The excess shear stress i s the calculated shear stress minus the c r i t i c a l shear s t r e s s . Based upon t h i s model, simple r e l a t i o n s h i p s between the beach slope and sediment s i z e can be derived. Equation 5.23 applies to d i f f e r e n t beaches being attacked by i d e n t i c a l wave conditions. Figure 5.8 i s a p l o t of the equation and shows that the equation i s i n reasonable agreement with experimental r e s u l t s . The second r e l a t i o n s h i p i s given by equation 5.29 which states that the beach slope, sinO , i s proportional to the inverse of the fourth root of the breaking wave height. As shown i n figure 5.10, t h i s r e l a t i o n s h i p appears to be i n reasonable agreement with other findings. The exact power of the breaking wave height requires more analysis and c a r e f u l accounting of various n o n l i n e a r i t i e s of the beach environment. Several conclusions; based on these findings, can be made about beach observations. The f i r s t conclusion i s about the tendency of beaches to exhibit concave p r o f i l e s . According to 133 the theory, the offshore shear stress acting on the sediments increases and the beach slope decreases as the i n f i l t r a t i o n decreases. Therefore, for beaches that have i n f i l t r a t i o n over the upper sections and e x f i l t r a t i o n over the lower sections, the shear stress w i l l be greater for the lower sections. A larger shear stress i s associated with f l a t t e r slopes such that the i n f i l t r a t i o n - e x f i l t r a t i o n argument predicts a concave p r o f i l e . Gravel beaches, being f a r more permeable than sand beaches, are predicted, and i n fact are, steeper and more concave than sand beaches. The second conclusion i s on the behaviour of armored beaches. Beaches that have one or two layers of gravel or cobbles e x h i b i t slopes more consistent with sand beaches. This i s best explained by considering the permeability of the beach, since the cobble or gravel layer i s underlaid by sand, i t w i l l be the permeability of the sand that controls the beach response to the incoming wave conditions. As already mentioned, the offshore shear stress acting on the sediments i s larger for less permeable materials. The sand, being l e s s permeable than the upper gravel layer, causes the whole beach to act as a sand beach. As the gravel layer increases i n thickness, i n f i l t r a t i o n increases and the offshore shear stress decreases. The beach begins to steepen. The f i n a l conclusion concerns the difference i n the behaviour of a well graded beach material and a uniform beach material. The lower permeability of the graded material w i l l promote an increased shear stress which moves the sediment offshore and reduces the slope. This leads to an i n t e r e s t i n g 134 observation that i f a beach i s a r t i f i c i a l l y protected by adding a material that increases the grading, the beach i s now more susceptible to erosion since the permeability i s decreased. The new combined material w i l l be moved more aggressively offshore and the breaking point may move further onshore i f enough material i s moved offshore. This would be p a r t i c u l a r l y true for the addition of a well graded sand to a gravel beach. The time-averaged method of analysis of onshore-offshore behaviour of beaches answers many questions, but also raises others. An important question i s the r e l a t i o n between the d i s t r i b u t i o n and magnitude of the setup and the volumes of water delivered to the beach, as a function of beach permeability. Since the wave setup and beach permeability are found to be the two important parameters c o n t r o l l i n g the shear stress and beach slope, c a r e f u l experiments and measurements are needed to confirm these conclusions. Also, the influence of random waves and the v a l i d i t y of time-averaging of random waves needs to be further studied. F i n a l l y , as t h i s model i s adapted to oblique wave attack, the independence of longshore and onshore momentum flux and sediment transport needs to be addressed. 135 CHAPTER 6; TESTING OF THE BEACH FACE CONTROL VOLUME MODEL 6.1 INTRODUCTION To t e s t whether the models developed i n the previous chapter give reasonable r e s u l t s , an analysis of a plane impermeable beach i s performed. The shear stress i s calculated using equation 5.7, the model of an impermeable beach. For each wave condition, the shear stress i s the maximum value that can be calculated. As discussed i n the previous chapter, the offshore shear stress increases as the beach permeability decreases. Since the beach i s impermeable, the calculated shear stress i s the largest possible for the given conditions. In t h i s chapter the equations developed i n chapter 5 are used to ca l c u l a t e the shear stress for measured setup conditions, and investigate the s e n s i t i v i t y of the model and the consequent requirements for the accuracy of the experimental measurements. 136 6.2 MODEL FOR AN IMPERMEABLE BEACH The equation that models an impermeable beach i s repeated here for convenience. 2kd + 1 V sinh(2kd) 2 - V T p q u . d t ) ' + + p b dy -bJ B r C c pdy + J r o c o s O d L B = 0 A r> B Pb dY + fJ pdy Each part of the equation must be evaluated i n terms of the varia b l e s that define the offshore and beach face control volumes (figure 5.2 and figure 5.3). The f i r s t term represents the momentum flux passing through the boundary DF of the offshore control volume. The energy, E, i s proportional to the wave height squared. The wave height, H, i s measured at the boundary and the value of kd i s i t e r a t i v e l y c alculated using the equation, 2 / . / T 2 > kd 4 T T z (d /gT z ) (1/tanh(kd)) (6.1) where d i s the depth to the SWL at the boundary DF. The next three terms represent the increase i n the momentum caused by the setdown that occurs along the length of the offshore control volume. Assuming the setdown to be l i n e a r , and 137 knowing the slope to be 1:15, the three in t e g r a l s are equivalent to, \y (d + d b - s, ) ( s b - S,) (6.2) The f i f t h term represents the offshore momentum flux associated with the backrush, which i s subtracted from the momentum flux entering the beach face control volume. Calcula t i o n of the in t e g r a l depends upon the backrush period, t b , the backrush depth, and the backrush v e l o c i t y , u^. Interestingly, these terms are found to be approximately constant for the wave condition that are studied. The backrush period i s t y p i c a l l y about h a l f of the wave period. The backrush depth i s 0.85 of the breaking depth, and the backrush v e l o c i t y i s 0.33 m/s. I t should be cautioned that the backrush v e l o c i t y was measured a f t e r the series of runs was completed. Instead of generating every condition again, the backrush v e l o c i t y was measured for the maximum and minimum conditions of each depth. The measured v e l o c i t y d i f f e r e d very l i t t l e from 0.3 3 m/s. Therefore, using these conditions and assumptions, the backrush momentum f l u x i s equal to, - j p (0.85 d b ) (0.33) 2 (6.3) The net pressure force acting on the beach face control volume i s represented by the next two in t e g r a l s . This force w i l l act offshore for any setup, and assuming a l i n e a r setup, i s equal to, - } r d b A S (6-4) 138 F i n a l l y , the net shear force acting along the bottom of the beach face control volume can be expressed as, r 0 (d b + AS)/ tanO ( 6 . 5 ) This assumes that the net shear stress acts onshore on the control volume, offshore on the sediments, and i s constant along the length of the beach. Combining the f i v e terms and rearranging to solve for the shear stress, the equation i s , ~ j y (d + d b - S, ) ( S b - S, ) + j p (0.85 d b ) ( 0 . 3 3 ) 2 + ± 7 d A S I t q n Q - ( 6 . 6 ) 2 ' ) ( d b + A S ) 6.3 RESULTS Data i s c o l l e c t e d for 59 d i f f e r e n t wave conditions and i s shown i n Appendix C . The water depth varies between 40.0 cm and 49.3 cm. The wave height at boundary DF varies between 10.5 cm and 24.0 cm, and the wave period varies between 1.29 seconds and 3.44 seconds. In runs one to eleven, the wave period i s increased f o r r e l a t i v e l y constant wave heights and water depths. For runs twelve to f i f t e e n , the wave height i s changed for a constant wave period. Table 6.1 shows the value of each term i n equation 6.6 as well as the calculated net shear stress. A p o s i t i v e value for 139 Table 6,1 • The ca lcu la ted net shear s t r e s s using the data co l lec ted f o r each wave condition RUN MOMENTUM SHEAR TERM 1 TERM 2 TERM 3 TERM 4 TERM 5 dBR=e.8Sdb ub=0.33 i / s N / i N / i N / i N / i N / i N / i A 2 1.3 -74.03 0.00 9.56 57.73 0.253 -1.78 1.1 -76.91 0.00 8.68 55.64 0.269 -3.38 1.2 -80.85 -3.41 6.58 98.35 0.235 4.87 1.4 -83.92 -7.29 9.32 151.07 0.188 13.02 2.4 -17.72 -6.23 10.02 35.57 0.267 5.77 2.3 -21.72 -6.91 9.97 42.81 0.260 6.29 2.2 -23.68 0.00 10.07 45.34 0.256 8.13 2.1 -26.36 -15.06 5.B3 34.58 0.367 -0.37 3.1 -49.35 -14.05 7.85 68.67 0.264 3.47 4.5 -14.93 -5.62 9.29 39.59 0.277 7.84 4.4 -15.59 -4.04 7.42 36.17 0.323 7.74 4.3 -17.35 -1.86 13.12 63.69 0.202 11.66 4.2 -18.73 -4.94 5.52 30.37 0.389 4.76 4.1 -18.86 -3.72 7.37 47.79 0.302 9.86 5.1 -26.70 -8.51 13.01 61.06 0.205 7.96 5.2 -29.28 -5.27 7.37 36.86 0.323 3.13 5.3 -33.01 -1.75 11.19 58.09 0.229 7.92 5.4 -34.24 3.70 13.09 73.52 0.199 11.13 5.5 -35.71 -0.66 9.28 49.66 0.266 6.00 £.1 -38.55 -5.57 11.07 61.69 0.229 6.54 6.2 -42.20 -3.49 11.12 61.27 0.228 6.09 6.3 -41.74 -5.84 14.88 90.68 0.176 10.20 6.4 -46.15 -4.44 13.02 82.79 8.195 8.83 7.1 -47.31 -7.78 14.86 7B.73 0.180 6.92 7.2 -56.54 -2.94 12.96 75.95 0.199 5.85 7.3 -58.09 -3.32 13.01 86.56 0.194 7.40 8.1 -17.49 -5.50 8.11 38.93 0.302 7.27 8.2 -28.22 -6.72 8.10 40.77 0.300 6.57 8.3 -22.08 -3.06 8.15 42.73 0.296 7.61 8.4 -23.75 -2.46 8.20 44.15 0.292 7.64 8.5 -27.54 -8.84 6.17 41.22 0.339 3.74 9.1 -26.73 -6.94 11.93 49.32 0.225 6.19 9.2 -28.48 -10.07 8.05 43.95 0.296 3.98 9.3 -31.46 -2.07 11.90 58.26 0.220 8.05 9.4 -36.28 -0.69 11.89 86.22 0.205 12.53 9.5 -38.36 -12.20 6.08 66.98 0.283 6.37 10.1 -36.55 -1.64 10.10 43.35 0.258 3.93 10.2 -36.92 -4.90 10.05 52.74 0.250 5.24 10.3 -45.00 -3.28 10.10 71.17 0.234 7.73 10.4 -48.65 0.00 15.79 91.21 0.168 9.83 10.5 -50.50 -9.14 8.03 76.97 0.253 6.41 11.4 -52.42 -3.26 10.03 49.43 0.253 8.96 11.3 -56.07 -7.69 15.58 84.23 0.172 6.20 11.2 -60.34 -3.84 15.61 119.09 0.163 11.49 11.1 -67.91 -13.78 7.75 79.28 0.253 1.35 140 Table 6,1 RUN HOHENTUH SHEAR TERM 1 TERM 2 TERM 3 TERH 4 TERM 5 dBR=B.B5db ub=8.33 a/s N / i N / l N / i N / i N / i N / V 2 12.1 -74.83 -3.37 13.42 188.82 8.182 8.16 12.2 -77.22 -18.59 6.52 82.18 8.257 8.23 12.3 -38.60 -17.83 6.33 57.71 8.299 2.52 12.4 -35.34 -2.93 9.26 75.54 8.241 11.28 13.1 -37.83 -5.86 8.38 54.24 8.266 5.19 13.2 -67.83 -6.89 18.89 76.99 8.238 3.21 14.1 -68.77 -11.13 14.35 76.79 8.185 2.08 14.2 -31.23 -3.13 8.84 44.97 8.279 5.42 14.3 -38.83 -11.88 8.84 47.78 8.275 4.87 14.4 -59.82 1.66 18.64 64.87 8.232 4.82 15.1 -47.79 -5.26 11.25 74.49 8.218 7.13 15.2 -24.77 -34.82 9.44 52.83 8.268 8.49 15.3 -68.53 -25.25 15.88 89.79 8.175 1.93 15.4 -15.73 -28.88 8.42 46.42 0.285 2.94 141 the net shear stress indicates that i t acts onshore on the control volume and offshore on the beach sediments A l l except three net shear stress values are p o s i t i v e , i n d i c a t i n g that the average net shear stress on an impermeable beach i s generally offshore, as i s expected based upon observations discussed i n chapter 4. The magnitude of the calculated net shear stress range from 0.23 N/m2 to 13.02 N/m2. The majority of the stresses f a l l i n the range 1.0 N/m2 to 6.0 N/m2. Figure 6.1 i s used to determine the maximum sized p a r t i c l e a given shear stress can move. This figure i s based upon Shields(1936) entrainment function which i s rearranged to p l o t as shear stress versus jthe material diameter. Using t h i s as a guide, the calculated net shear stresses are capable of moving materials that range i n s i z e from 0.01 mm to 14 mm with the majority f a l l i n g between 1.5 mm to 7mm. Pea gravel (longest axis < 10 mm) that was dropped on the upper length of the plane beach was usually moved offshore to the breaking point. In a few cases, the pea gravel remained on the upper beach. The pea gravel seems to be the largest material that can be transported f o r any wave condition that could be generated on the plane 1:15 beach. Because the model predicts p a r t i c l e s i z e s consistent with t h i s observation the approach the model uses i n analyzing onshore/offshore transport i s supported. Figure 6.2 and figure 6.3 show the calculated net shear stress p l o t t e d against the wave height for two d i f f e r e n t depths. The wave height i s taken at the boundary DF and i s held r e l a t i v e l y constant f o r each run For runs 4 to 7 and 8 to 11, the depths are 45.0 cm and 47.5 cm, respectively. The period i s 142 Figure 6,1 Shear s t r e s s required f o r a moveable bed, based upon Shields entrapment function Calculated Shear Stress (N/rrT2) 14 - i • . . 12 H 10H 6 H 2 H 0.002 0.004 0.006 0.008 H/gT2 0.01 Run 3 Run 4 Run 5 -B- Run 6 -X- Run 7 Run 15 T 0.012 Figure 6.2 Calculated net shear s t r e s s v e r s u s H / g T 2 depth = 47,5 cm 0.014 Calculated Shear Stress (N/rrT2) Run 8 - f - Run 9 Run 10 -B- Run 11 -X- Run 14 0.002 0.004 0.006 H/C/T2 0.008 0.01 0.012 Figure 6.3 Calculated net shear s t r e s s v e r s u s H / q T 2 depth = 45,0 cm increased for each wave height. Increasing the period e f f e c t i v e l y increases the wavelength causing the breaker type to change from s p i l l i n g to plunging which occurs for run 5 through run 10. The calculated net shear stress, plotted against H/gT2, decreases as the wave s p i l l s , but begins to increase as the wave begins to plunge. The calculated net shear stress continues to increase u n t i l i t peaks and then decreases again. This behaviour can be explained by examining the e f f e c t the breaking depth and t o t a l setup have on the f i v e terms i n equation 6.6. As the breaking wave changes from s p i l l i n g to plunging, caused by an increasing period, the breaking point moves onshore reducing the breaking depth. For a 1:15 slope, t h i s reduction can be s i g n i f i c a n t . Recalling figure 3.4, the increase i n the r a t i o H b/d b i s large for a small decrease i n H 0/L Q. The large increase i n H b/d b can be attributed to a decrease i n the breaking depth. Any decrease i n the breaking depth has a strong e f f e c t on equation 6.6 by reducing the values of the backrush momentum flux and offshore pressure force, term 3 and term 4 respectively. Also, as the period increases, the value of kd decreases such that sinh(kd) approaches kd causing the incoming wave momentum flu x to increase, or term 1 to become more negative. As term 1 gets more negative and terms 3 and 4 get smaller the net shear stress acting onshore on the control volume must decrease to maintain equilibrium. The net shear stress i s able to decrease and s t i l l maintain equilibrium because the onshore momentum flux, which i s les s than the offshore momentum flux, i s increasing as the offshore momentum flux i s decreasing. 146 After the breaker has changed to a plunging breaker, the net shear stress begins to increase for increasing period. The data shows that the breaking depth increases and the t o t a l setup begins to r a p i d l y increase. Therefore, terms 3 and 4 begin to increase at a rate faster than term 1 as i t gets more negative. The r e s u l t i s that the net shear stress acting onshore on the control volume increases to maintain equilibrium. Eventually, the breaking depth begins to decrease and, as previously explained, t h i s causes a reduction i n the net shear stress acting onshore on the control volume . Figure 6.4 shows the e f f e c t of changing wave height on the value of the net shear stress. As the wave height increases, the incoming momentum flux increases as the square of the wave height and the t o t a l setup increases. Again, to maintain equilibrium i n equation 6.6, the net shear stress must decrease 6 . 4 SENSITIVITY OF THE MODEL Many of the measurements taken i n the experiments have an element of uncertainty associated with them. In many cases, the water l e v e l i n the manometers o s c i l l a t e d 1 mm about the MWL. When the measured setdown or setup i s only a few millimeters i t s e l f , the o s c i l l a t i o n can cause a s i g n i f i c a n t error i n the f i n a l measurement. This i s also true f o r the measured wave height, depths, and flow v e l o c i t i e s . To ca l c u l a t e the s e n s i t i v i t y of the model, the p a r t i a l d e r i v a t i v e of the shear stress (equation 6.6) with respect to each of the variables i s required. Using the data for each run, table 6.2 shows the s e n s i t i v i t y of the shear stress for small 147 Table 6.2 Sensitivity of the impermeable beach model RUN SHEAR Sensi t iv i ty Analysis u p n - o . o j u u ub=8.33 i / s dSHST/dH dSHST/dSI dSHST/dSb dSHST/dSw dSHST/db dSHST/dub dSHST/di dSHST N / i A 2 N / i A 2 / u N / i A 2 / u N / i A 2 / n N / I a 2 / I I N / i A 2 /a i ( N / i A 2 ) / ( i / s ) ( N / i A 2) /deg N / i A 2 1.3 -1.70 -0.170 0.747 -0.484 0.263 0.089 0.015 -0.448 2.232 1.1 -3.38 -0.188 0.769 -0.508 0.261 0.106 0.014 -0.890 2.450 1.2 4.87 -0.173 0.619 -0.470 0.147 0.155 0.009 1.279 2.312 1.4 13.02 -0.144 0.551 -0.400 0.149 0.111 0.011 3.422 2.421 2.4 5.77 -0.074 8.926 -0.663 0.260 0.031 0.016 1.517 2.416 2.3 6.29 -0.088 0.902 -0.649 0.251 0.037 0.016 1.654 2.445 2.2 8.13 -0.097 0.888 -0.646 0.242 0.034 0.016 2.139 2.532 2.1 -0.37 -0.152 1.113 -0.875 0.228 0.111 0.013 -0.097 2.893 3.1 3.47 -0.151 0.831 -0.619 0.206 0.100 0.013 0.912 2.466 4.5 7.84 -0.076 0.917 -0.675 0.240 0.033 0.016 2.860 2.516 4.4 7.74 -8.095 1.806 -0.788 0.217 8.848 0.015 2.036 2.777 4.3 11.66 -0.066 0.752 -0.506 0.246 0.019 0.016 3.066 2.330 4.2 4.76 -0.136 1.135 -0.932 B.2IB 8.886 0.013 1.251 3.072 4.1 9.86 -0.109 0.938 -0.745 0.191 0.058 0.014 2.592 2.811 5.1 7.96 -0.073 0.760 -0.50B 0.258 8.827 1. BIG 2.094 2.187 5.2 3.13 -0.125 1.803 -0.764 0.237 8.072 0.014 0.822 2.662 5.3 7.92 -0.100 0.804 -0.559 0.245 8.038 0.016 2.081 2.368 5.4 11.13 -0.090 0.734 -0.492 0.242 0.029 0.016 2.926 2.352 5.5 6.00 -0.126 0.876 •8.638 0.237 8.854 0.015 1.577 2.518 6.1 6.54 -0.108 8.798 -8.550 0.246 0.045 0.015 1.720 2.302 6.2 6.09 -0.189 0.797 -8.548 8.248 0.847 0.815 1.601 2.302 6.3 10.20 -0.086 8.686 -0.434 0.250 0.030 0.016 2.681 2.196 6.4 8.83 -0.104 8.723 -8.478 8.244 0.039 B.B15 2.322 2.268 7.1 6.92 -0.085 8.701 -8.435 0.264 0.032 0.016 1.818 2.057 7.2 5.85 -8.112 8.733 -8.476 8.256 0.045 8.816 1.538 2.165 7.3 7.48 -0.115 8.717 -8.478 0.246 0.046 0.015 1.945 2.224 8.1 7.27 -0.884 0.927 -8.697 8.227 0.046 B.B15 1.911 2.558 8.2 6.57 -0.097 8.919 -8.688 0.228 0.851 0.015 1.727 2.551 8.3 7.61 -8.185 8.906 -8.683 8.221 0.050 0.015 2.000 2.599 8.4 7.64 -0.112 8.899 -8.677 8.221 0.052 0.015 2.010 2.611 8.5 3.74 -8.144 8.973 -8.765 8.283 0.096 0.013 0.983 2.740 9.1 6.19 -0.088 8.781 -B.516 8.263 0.038 0.016 1.629 2.162 9.2 3.98 -8.114 8.987 -8.668 8.235 0.066 0.014 1.846 2.468 9.3 8.05 -0.093 8.768 -0.509 8.251 0.033 0.016 2.117 2.259 9.4 12.53 -8.097 8.788 -8.488 8.228 0.048 0.015 3.294 2.417 9.5 6.37 -0.143 8.810 -0.649 8.155 0.125 0.010 1.676 2.618 18.1 3.93 -8.187 8.844 -8.583 0.261 8.847 0.016 1.034 2.280 10.2 5.24 -0.104 0.819 -0.570 0.247 0.051 0.015 1.378 2.299 10.3 7.73 -8.119 8.768 -8.543 0.223 0.859 0.014 2.031 2.387 10.4 9.83 -0.092 0.653 -0.395 0.257 0.028 0.016 2.585 2.123 18.5 6.41 -0.145 8.773 -8.579 0.191 8.896 0.B12 1.685 2.482 11.4 0.96 -0.131 8.827 -0.560 0.266 0.065 0.815 0.252 2.192 11.3 6.20 -0.097 8.663 -8.393 0.268 8.033 B.B16 1.630 1.975 11.2 11.49 -0.099 0.627 -0.385 0.241 0.036 0.015 3.020 2.197 11.1 1.35 -0.170 8.762 -8.554 0.202 0.120 8.B12 0.355 2.316 148 Table 6,2 RUN SHEAR Sens i t iv i ty Analysis uon-D.gjuu ub--fl.33 i /s dSHST/dH dSHST/dSI dSHST/dSb dSHST/dSv dSHST/db dSHST/dub dSHST/di dSHST l l / i * 2 N / i A 2 / a i N / I A 2 / M N / i A 2 / n N / I A 2 / M N / I A 2 / M ( N / i A 2 ) / ( i / s ) (N / i A 2 ) /deg N / i A 2 1 2 . 1 8 . 1 6 - 0 . 1 2 2 0 . 6 1 4 - 0 . 3 7 7 8 . 2 3 6 0 . 0 5 4 0 . 0 1 5 2 . 1 4 4 2 . 0 9 7 1 2 . 2 8 . 2 3 - 8 . 1 8 4 0 . 6 8 4 - 8 . 5 0 3 8 . 1 7 6 0 . 1 5 6 8 . 8 1 0 0 . 8 6 0 2 . 2 2 6 1 2 . 3 2 . 5 2 - 0 . 1 5 2 0 . 7 9 4 - 8 . 5 9 5 8 . 1 8 9 0 . 1 1 9 0 . 8 1 1 0 . 6 6 1 2 . 3 8 9 1 2 . 4 1 1 . 2 0 - 0 . 1 1 2 0 . 7 0 6 - 0 . 5 8 9 8 . 1 9 6 0 . 0 6 8 8 . 8 1 4 2 . 9 4 4 2 . 4 3 1 1 3 . 1 5 . 1 9 - 0 . 1 3 0 0 . 7 8 0 - 8 . 5 4 6 8 . 2 3 2 0 . 0 6 3 8 . 8 1 4 1 . 3 6 4 2 . 3 1 6 1 3 . 2 3 . 2 1 - 0 . 1 5 0 0 . 7 0 2 - 8 . 4 6 5 8 . 2 3 5 0 . 0 7 9 8 . 0 1 4 0 . 8 4 3 2 . 1 4 4 1 4 . 1 2 . 0 8 - 0 . 1 1 1 0 . 6 8 9 - 8 . 4 1 1 8 . 2 7 5 0 . 0 4 6 8 . 8 1 6 0 . 5 4 6 1 . 8 7 6 1 4 . 2 5 . 4 2 - 0 . 1 1 2 0 . 8 7 4 - 8 . 6 3 4 0 . 2 3 9 0 . 0 5 5 0 . 0 1 5 1 . 4 2 6 2 . 4 5 8 1 4 . 3 4 . 8 7 - 8 . 1 2 1 0 . 8 7 1 - 0 . 6 2 5 8 . 2 4 1 0 . 8 6 0 8 . 0 1 5 1 . 0 7 1 2 . 4 8 5 1 4 . 4 4 . 8 2 - 0 . 1 4 2 8 . 7 6 9 - 8 . 5 2 2 0 . 2 4 8 0 . 0 6 3 0 . 0 1 5 1 . 8 5 8 2 . 2 6 4 1 5 . 1 7 . 1 3 - 8 . 1 1 9 0 . 7 6 7 - 0 . 5 2 9 8 . 2 3 7 0 . 0 5 2 0 . 0 1 5 1 . 8 7 6 2 . 3 3 6 1 5 . 2 8 . 4 9 - 0 . 1 0 2 8 . 8 7 7 - 8 . 6 8 5 0 . 2 5 9 0 . 0 6 3 0 . 0 1 5 8 . 1 2 9 2 . 1 8 1 1 5 . 3 1 . 9 3 - 8 . 1 0 8 8 . 6 8 6 - 8 . 4 8 7 8 . 2 7 3 0 . 0 4 6 0 . 8 1 6 0 . 5 0 7 1 . 8 3 4 1 5 . 4 2 . 9 4 - 0 . 0 7 8 0 . 9 2 4 - 8 . 6 7 8 0 . 2 4 2 0 . 0 6 1 0 . 0 1 5 0 . 7 7 2 2 . 3 3 8 149 Calculated Shear Stress (N/rrT2) 1 2 -• 1 2 4 RUN d(cm) H(cm) T(s) 10 -• 12.1 12.2 12.3 1 2.4 40.00 40.00 40.00 40.00 22.1 1 21.50 15.20 15.20 2.00 2.75 2.75 2.00 • 1 3.1 1 3.2 40.40 40.40 15.40 20.50 2.28 2.00 8 -15.1 • 12.1 + 14.1 1 4.2 14.3 1 4.4 45.00 45.00 45.00 45.00 23.00 15.50 14.00 19.50 1.56 1.56 2.30 2.30 6 -13.1. 14.2 + 15.1 15.2 15.3 15.4 47.30 47.30 47.30 47.30 1 7.50 1 2.60 24.00 1 1.60 2.30 2.30 1.43 1.43 4 -14.3 _j_ " K 4 . 4 2 -12.3 • 15.2^/ 15.4 13.2 12.2 -h 14.1 15.3 0 • - | 1 1 l I i 0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 H / g T 2 Figure 6,4 Calcula ted net shear s t r e s s v e r s u s H/gT f o r chanqinq wave conditions errors i n the measurements. The maximum possible variance of the calculated value of the shear stress i s shown i n the f i n a l column. This c a l c u l a t i o n i s based upon the following accuracies: for the wave height, H, 2.5 mm; for the constant setdown, S j , 1 mm; for the setdown at breaking, S^, 1 mm; for the setup, S w, 1 mm; for the depth at breaking, d b, 2.5 mm; for the backrush v e l o c i t y , u b, 0.03 m/s; and, for the beach slope, m, 0.2 degrees. From the r e s u l t s , the model appears to be se n s i t i v e to small inaccuracies i n the measurements. The l e a s t s i g n i f i c a n t of the variables i s the backrush v e l o c i t y and the most s i g n i f i c a n t v a r i able i s the constant setdown. This seems reasonable since the contribution of the backrush momentum flux i s r e l a t i v e l y small, but the constant setdown, S j , i s an important variable c o n t r o l l i n g the momentum flux entering the beach face control volume. The maximum possible variance of the shear stress assumes that a l l the errors add together. The actual net shear stress f a l l s within the range given by calculated value plus or minus the maximum variance. The r e s u l t s show that the model and the cal c u l a t i o n s for the net shear stress are s e n s i t i v e to small inaccuracies i n the experimental measurements. The t o t a l variance generally ranges between +/-2.1 N/m2, regardless of the magnitude of the calculated net shear stress. There does not appear to be any re l a t i o n s h i p between the magnitude of the calculated net shear stress and the t o t a l variance. However, large calculated net shear stress values are r e l a t i v e l y less affected by small errors i n the measurements. 151 6.5 DISCUSSION From the above findings, several general observations can be drawn. For the plane impermeable beach, the calculated net shear stress acting on the beach i s generally offshore. This agrees with the findings i n chapter four where the c i r c u l a t i o n c e l l onshore of the breaking point rotates i n such a d i r e c t i o n as to cause net offshore transport along the beach face. Observations i n chapter four also show that pea gravel (longest axis < 10 mm) i s the largest sediment transported offshore. Occasionally, the pea gravel remained on the upper beach. This suggests that the upper l i m i t of the actual net shear stress acting on the beach i s able to move material 10 mm i n diameter. According to Shields entrainment function, the shear stress required to move such a sediment i s equal to or greater than 12.0 N/m . The model i n many cases predicted net shear stresses i n t h i s range. This i s not saying that i n every condition pea gravel i s moved offshore, but that when i t i s moved offshore a c e r t a i n net shear stress i s required which the model can predict. This suggests that the control volume approach used to create the model i s v a l i d . In chapter four, the breaker type did not appear to s i g n i f i c a n t l y influence the general pattern of flows or the setdown/setup p r o f i l e s , but the small change i n the magnitude can influence the calculated net shear stresses. For a constant depth and wave height and for an increasing period, the net shear stress decreases for s p i l l i n g breakers, but begins to increase as the breaker transforms to a plunging breaker. This i s explained by the large decreases i n the breaking depth 152 occurring up to the s p i l l - p l u n g e t r a n s i t i o n and the e f f e c t t h i s has upon the model. As the breaking depth decreases the offshore pressure force reduces which requires that the net shear stress acting offshore on the sediments must reduce to maintain equilibrium. As the period continues to increase, the wave breaks as a plunging breaker but moves closer to the plunge-surge t r a n s i t i o n . Because of t h i s , the t o t a l setup begins to r a p i d l y increase which causes the offshore pressure force, term 4, to also increase. The net shear stress must now increase to maintain equilibrium. The magnitude of the net shear stress acting on a beach face depends heavily upon the breaking depth and t o t a l setup and how these variables influence the offshore pressure acting on the control volume. When the wave height i s increased, for a constant period, the net shear stress decreases. I t i s thought that i f the net shear stress decreases the beach length must increase to keep the control volume i n equilibrium. However, t h i s i s only found i n about h a l f of the wave conditions. The model appears to be s e n s i t i v e to small changes i n the measurements. I t i s d i f f i c u l t to obtain highly accurate measurements i n the surf zone. Water l e v e l s are continually changing. To e s t a b l i s h the mean water l e v e l , the waves are time-averaged using damped manometers attached to the beach face. Even though these are damped the water l e v e l i n the manometer s t i l l o s c i l l a t e a few millimeters. This requires that the MWL be assumed to be at h a l f the distance between the upper and lower points of o s c i l l a t i o n , which i s not necessarily the 153 case. When the measurements themselves are only tens of millimeters i n length the actual value of the net shear stress can f a l l within a r e l a t i v e l y large range about the calculated value. The findings put the calculated values of the net shear stress i n doubt, but t h e i r magnitude s t i l l seem reasonable. I t i s important to remember that the findings are for the impermeable form of the model. These findings may or may not apply to the permeable form of the model; however, for s i m i l a r wave conditions, the offshore net shear stress acting on the sediments for an impermeable beach i s expected to be greater than that found on a permeable beach. As permeability i s included i n the model the beach i t s e l f i s able to respond to the changing wave conditions which adds s u b s t a n t i a l l y to the complexity of the model. For t h i s study, the r e s u l t s from the simple beach are s t i l l d i f f i c u l t to in t e r p r e t . Some trends are mentioned here but should only be accepted to occur on the single beach studied. The equation modelling the impermeable beach has f i v e terms which are made up of many variab l e s . For t h i s reason the re l a t i o n s h i p between them i s very complex. Trying to extend these findings to other beaches i s cautioned and requires further experimentation. However, the re s u l t s are promising and do lend v a l i d i t y to the model. 154 CHAPTER 7; SUMMARY AND CONCLUSIONS This thesis has presented a discussion of the events occurring i n the surf zone and the development of a model describing the onshore/offshore sediment transport i n terms of the forces acting on a beach. Before a model could be formulated, a better understanding of the surf zone was required. The following i s a summary for onshore/offshore wave conditions. The most v i s i b l e aspect of the surf zone are the d i f f e r e n t ways i n which a wave can break. These are s p i l l i n g , plunging, collapsing, and surging breakers. Using the available l i t e r a t u r e , the re l a t i o n s h i p between the breaking type, deepwater wave steepness, H 0/L Q, and the beach slope, m, was studied. Separate findings were compared and differences were explained i n terms of the d i s s i m i l a r i t i e s between the procedures, c a l c u l a t i o n methods, and d e f i n i t i o n s . Variations between the r e s u l t s were mainly caused f o r two reasons: each researcher had a d i f f e r e n t i n t e r p r e t a t i o n of the same d e f i n i t i o n 155 of the breaking types and breaking waves exhibit natural v a r i a b i l i t y to conditions within the surf zone. Even though there were differences, experimental r e s u l t s showed that a reasonable prediction of breaker type can be made given the deepwater wave steepness and the beach slope, provided that the beach i s plane and impermeable. From the study of the wave height and depth at breaking, the governing conditions of wave breaking were determined. Waves break because they reach some c r i t i c a l i n s t a b i l i t y . For o s c i l l a t o r y waves the i n s t a b i l i t y occurs because the wave becomes too steep causing the wave to s p i l l down i t s face. For s o l i t a r y waves the i n s t a b i l i t y occurs because the depth becomes small enough such that the v e l o c i t y of the crest of the wave i s s i g n i f i c a n t l y greater than the v e l o c i t y of the base of the wave, causing the wave to plunge forward. In both cases, the crest v e l o c i t y i s greater than the v e l o c i t y of the base of the breaking wave. This i s the.result of breaking for an o s c i l l a t o r y wave, but the cause of breaking for a s o l i t a r y wave. The value of the height-to-depth r a t i o defines where the wave breaks. For waves breaking from steepness e f f e c t s , the value of Hjy'djj increases as deepwater steepness decreases because the wave steepness increases fa s t e r than the wave height, as shown by the shoaling equation (equation 3.8). This was found to occur on most slopes for a deepwater steepness greater than approximately 0.015. For waves breaking from depth e f f e c t s , the o s c i l l a t o r y wave has enough time to change into a s o l i t a r y wave. For t h i s s i t u a t i o n , the value of H^/d^ decreases as deepwater steepness decreases because the s o l i t a r y wave 156 responds slowly to changing depth conditions and tr a v e l s into water more shallow than i s necessary for breaking. This was found to occur on low slopes for a deepwater steepness less than 0.015. A t h i r d breaking condition was found to occur on slopes steeper than 1:25 and for deepwater steepnesses l e s s than 0.015. In t h i s case, the beach slope i s steep enough such that an o s c i l l a t o r y wave, not being able to change into a s o l i t a r y wave, tr a v e l s into shallow water and i s forced to break by some combination of shoaling and depth e f f e c t s . For t h i s s i t u a t i o n , the value of H b / d b decreases as H 0 / L 0 decreases below 0.015. For a wave with any value of H Q / L Q the value of H b / d b increases as the slope get steeper. Observations of the flow dynamics within the surf zone provided information for the development of the transport model. An important observation was of two c i r c u l a t i o n c e l l s that developed when the surf zone was viewed over a large number of wave periods. These rotated i n opposite d i r e c t i o n s such that sediment was moved along the bed to a common point between them, the n u l l point. Regardless of the type of breaker, s p i l l i n g or plunging, these c i r c u l a t i o n c e l l s always formed and always c i r c u l a t e d such that both moved bed material to the n u l l point. The n u l l point occurred j u s t onshore of the breaking point and j u s t offshore of the f i r s t vortex caused by breaking. Movement of the n u l l point was caused by changes i n the wave attack. As the deepwater steepness was increased both the breaking point and the n u l l point moves offshore. Decreasing the deepwater steepness moved the point onshore. The c i r c u l a t i o n c e l l s 157 stretched and contracted, following the movement of the n u l l point. Wave dynamics outside and within the surf zone cause a difference between the mean water l e v e l and the s t i l l water l e v e l . The setdown/setup p r o f i l e was measured for many d i f f e r e n t wave and depth conditions. The purpose of t h i s was to determine i f there were differences between the setdown and setup for s p i l l i n g and plunging breakers. In a l l cases, offshore of the breaking point, the mean water l e v e l i s below the s t i l l water l e v e l . Onshore of the breaking point the mean water l e v e l remains below the s t i l l water l e v e l and reaches a point of maximum setdown. For s p i l l i n g breakers, maximum setdown i s at the breaking point. For plunging breakers, maximum setdown occurs at the n u l l point. Further onshore, the mean water l e v e l eventually r i s e s above the s t i l l water l e v e l and reaches maximum setup at the top of the beach. I t was found that as the deepwater wave steepness increases, waves tend to s p i l l rather than plunge and the t o t a l setup decreases. S p i l l i n g breakers, being associated with higher values of H 0/L Q, broke further offshore than plunging breakers and tended to break over a longer distance. This releases a s i g n i f i c a n t amount of energy as turbulence and the remaining energy i s turned into p o t e n t i a l energy as setup. Therefore, i t was found that for constant depth and wave height, the t o t a l setup was greater for plunging breakers than for s p i l l i n g breakers. The ind i v i d u a l setdown/setup p r o f i l e s , with respect to t h e i r shape, d i f f e r e d l i t t l e . 158 Using the above observations, i t was concluded that the surf zone could be time-averaged to produce a r e l a t i v e l y steady picture of a very complex environment. The beach face control volume as defined by Quick (1989a), i s bounded by the beach face, the mean water l e v e l , and a v e r t i c a l plane at the breaking point. When the beach i s i n equilibrium for a given wave condition the forces acting on the control volume are also i n equilibrium. Since momentum i s conserved at breaking, the momentum flux passing into the control volume, the pressure forces acting on the control volume, and the shear stress acting on the beach face, as modified by beach permeability, can be equated to form the basis of the model. The following were observed from the modelling r e s u l t s and the experimental study using the impermeable form of the model: a) The magnitude of the net offshore shear stress acting along the face of the beach i s primarily dependent upon the t o t a l setup and the beach permeability. Increasing the permeability reduces the net offshore shear stress and vic e versa. The difference between gravel and sand beaches i s explained using t h i s argument. For a given wave attack, gravel beaches, being more permeable than sand beaches, w i l l have a smaller offshore shear stress acting on the face. Therefore, the gravel can form steeper slopes and s t i l l remain stable, as compared to sand. Concave beach p r o f i l e s can be explained by the e f f e c t i n f i l t r a t i o n and e x f i l t r a t i o n have on the shear s t r e s s . I n f i l t r a t i o n reduces the shear stress while e x f i l t r a t i o n increases the shear stress. I n f i l t r a t i o n , associated with 159 steeper slopes, occurs on the upper section of the beach where the runup percolates into the beach face. This water must e x f i l t r a t e on the lower section of the beach which increases the net offshore shear stress and reduces the slope (see figure 5.6). The steep upper slope and smaller lower slope form a concave beach p r o f i l e . The concavity of a gravel beach i s greater than a sand beach because of the gravels greater permeability. b) Based on the model, simple r e l a t i o n s h i p s between the beach slope and p a r t i c l e s i z e were calculated. Equation 5.2 3 predicts the change i n the slope for d i f f e r e n t beaches subject to the same wave attack. Damping of the flow into the beach was shown to be an important factor i n the behaviour of the slope. As the damping increases the change i n the slope decreases for changing wave conditions. Damping occurs when the i n f i l t r a t i n g water meets greater flow resistance and could be caused by a t r a n s i t i o n from laminar to turbulent flow within the matrix of the sediment. This suggests that gravel beaches experience larger slope changes than sand beaches fo r a change i n the wave conditions. The second r e l a t i o n s h i p i s given by equation 5.29 which states that the beach slope, s i n , i s proportional to the inverse of the fourth root of the breaking wave height. Smaller slopes are expected for beaches attacked by larger waves. This i s reasonable but the exact power requires more analysis and c a r e f u l accounting of the n o n - l i n e a r i t i e s of the beach environment. c) Experimental r e s u l t s using the impermeable form of the model and the setup data c o l l e c t e d i n chapter four suggest that 160 the shear stress acting on the beach face i s always offshore. This agrees with the observations of the flow dynamics within the surf zone and the rotation of the nearshore c i r c u l a t i o n c e l l also mentioned i n chapter four. An increase i n the beach slope for s p i l l i n g or plunging waves requires that the breaking point and the n u l l point move onshore since no onshore bedload transport occurs to steepen the slope a f t e r the breaking point. To move the n u l l point onshore, the deepwater wave steepness must be decreased. d) The magnitudes of the calculated net shear stresses were reasonable. The offshore net shear stresses ranged i n value from 0.23 N/m2 to 13.02 N/m2. The average net shear stress i s approximately 6.50 N/m2. Using Shields entrainment function, the average net shear stress i s found to be able to move material about 7.5 mm i n diameter. Considering that the larger pieces of pea gravel (longest axis < 10 mm) were sometimes observed to remain on the upper section of the beach, the calculated shear stresses lend v a l i d i t y to the model. e) For a constant wave height and an increasing wave period, the calculated net shear stress acting offshore on the sediments decreases for s p i l l i n g breakers, but begins to increase once the breaker turns into a plunging breaker. This i s explained by the large decrease i n the breaking depth that occurs when a s p i l l i n g breaker becomes a plunging breaker (see figure 3.4). As the breaking depth decreases the pressure forces acting on the control volume change which causes the offshore shear stress to decrease. 161 f) The impermeable form of the model i s s e n s i t i v e to small changes i n the measurements. Measuring the conditions i n the surf zone i s very d i f f i c u l t . Even though the mean water l e v e l was measured using damped manometers, small fluctuations s t i l l occurred. The accuracy of the measurements were to within one millimeter. Considering that many of the measurements are only millimeters i n magnitude the s e n s i t i v i t y of the model i s not surp r i s i n g . Assuming that a l l the measurements are incorrect by the maximum error, the average calculated net shear stress i s in error by plus or minus 2.4 N/m2. The observations, studies, and experiments i n t h i s thesis were p r i n c i p a l l y conducted on plane impermeable beaches. Unfortunately, the advantages gained by reducing the complexity of the beach environment are balanced by the disadvantages of loosing some of the r e a l i t y of the processes. For example, incident waves change the form of the beach which i n turn changes the form of the incident waves. This i s l o s t by using a plane impermeable beach. 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On the highest water waves of permanent type, B u l l e t i n of Disaster Prevention Research I n s t i t u t e , Kyoto Univ., Vol. 18, p. 1. 167 Appendix A Shoaling Charac t e r i s t i c s of an O s c i l l a t o r y Wave Using Linear Wave Theory 168 1) Equate wave power, P = P C l - the over bar denotes energy f l u x per u n i t wave c r e s t - assumes no r e f r a c t i o n 2) In termedia te and s h a l l o w water c o n d i t i o n s , P = nEc r 2kd -| n = 0.511 + s i n h ( 2 k d ) J § = g H 2 / 8 3) Deepwater c o n d i t i o n s , Po = n E 0 c 0 n Q = 0 . 5 E 0 = 9 H o 2 / 8 4) E q u a t i n g 2) and 3 ) , 2kd s i n h ( 2 k d ) r 2 k d 1 0 .5 1 + g H 2 / 8 c = 0 .5 g H 0 2 / 8 c, L s i n h ( 2 k d ) J 2kd , 1 + H 2 c = H Q 2 C q s i n h ( 2 k d ) J H / H Q = ( c c / c ) l / ; 5) Remember t h a t , r s i n h ( 2 k d ) , 1 / : L2kd + s i n h ( 2 k d ) J c / c 0 = L / L 0 tanh(kd) 169 6) S u b s t i t u t i n g 5) i n t o 4) r s inh(2kd> -.1/2 H /H r i = (coth(kd) ) 1 / 2 : L2kd + s i n h ( 2 k d ) J r sinh(2kd:> H/H C ) L / L o = L / L o (coth (kd)) 1 / 2  L2kd + s i n h ( 2 k d ) J H/L r sinh<2kd> , 1 / 2 = < c o t h < k d > ) 3 / 2  L2kd + s i n h ( 2 k d ) J H 0 / L 0 7) Us ing the h y p e r b o l i c i d e n t i t i e s , c o s h 2 * ' = 0 .5<cosh(2x) + 1) s inh(2x> = 2s inh(x )cosh<x) 8) The f i n a l r e s u l t i s , H/L r 2 c o s h 2 k d , 1 / 2 = coth(kd) L2kd + s inh(2kd>J Ho/Lo 170 Appendix B Observations of P a r t i c l e Motions Under breaking Waves 171 OBSERVATIONS OF PARTICLE NOTIONS UNDER BREAKING HAVES Nave Data: Condition Water Have Nave Depth Period Height ( C I ) (s) ( C I ) 1 47.0 1.72 20.7 2 48.0 1.72 17.0 3 47.0 1.72 13.2 4 47.0 1.72 20.7 5 47.0 1.72 24.3 6 47.0 1.32 20.0 7 47.0 1.32 16.6 8 47.0 1.32 11.0 9 40.0 2.26 20.8 IB 40.0 2.26 19.4 11 40.0 2.26 13.7 172 Condition 1 -Plunging breaker; i n i t i a l j e t does not penetrate to the bed; subsequent splash-plunge cycles (3-5) are generated. The turbulence and v o r t i c i t y generated from these does reach the bed eventually; d e f i n i t e side wall influence. Bedload -pea gravel(pg); sand(s); bakelite(b); a c r y l i c spheres(as); a c r y l i c cubes(ac) Suspended - a c r y l i c cubes(ac) -pg,s: form a small r i p p l e j u s t a f t e r the i n i t i a l plunge point and the before the f i r s t splash-plunge point, -b: t r a v e l s back and fort h over the r i p p l e . -as: t r a v e l s back and fort h over the r i p p l e sometimes jumping of f the r i p p l e when moving on or offshore. -ac: t r a v e l s as both bedload ans suspended load. The bedload t r a v e l s forward to the r i p p l e , i s put into suspension and i s held f o r up to 10 waves. The suspended p a r t i c l e s are slowly c a r r i e d offshore to approximately the point of breaking ( where the wave face becomes v e r t i c a l ) where, by t h i s time they s e t t l e to t r a v e l as bedload back to the r i p p l e . Offshore of the breaking point - a l l material moves forward to the r i p p l e as bedload only, -some suspended material moves further offshore to s e t t l e . Onshore of the breaking point - a l l material placed close to the top of the beach moves offshore to the r i p p l e as bedload only. -any of the ac that are l i f t e d into the region of the v o r t i c e s remain there and t r a v e l onshore u n t i l the v o r t i c e s can no longer keep them suspended and they t r a v e l back to the r i p p l e as bedload. 173 Condition 2 -The wave height has been reduced thereby decreasing the deepwater steepness. -Breaking moves further up the beach. -The wave i s s t i l l a plunging breaker; the i n i t i a l j e t does not s t r i k e the bed; 2-3 splash-plunge cycles; waves are regular with no secondary influences; further v o r t i c e s generated by the splash-plunge cycle reach the bed. Bedload pg,s,b,as,ac Suspended ac -pg,s: form a small r i p p l e ; remains r e l a t i v e l y stationary just a f t e r the i n i t i a l plunge point and before the "vortex wall", -b: moves back and forth across the r i p p l e ; p e r i o d i c a l l y gets suspended of the crest of the r i p p l e but does not go very far. -as: moves as bedload across the r i p p l e . -ac: moves both as suspended load and bedload; the ac tend to be suspended near the r i p p l e e i t h e r by the increase i n the v e l o c i t y near the bed of by the reduced pressure as the wave crest approaches; a c i r c u l a t i o n of ac occurs with the forward movement of the ac as bedload and the offshore movement as suspended load. Onshore of the vortex wall -bedload i s transported offshore to the r i p p l e and suspended load i s transported onshore. -suspended load i s suspended at the r i p p l e and i s l i f t e d high enough to be trapped i n the vor t i c e s near the surface to be ca r r i e d onshore. -some pg was put onshore of the vortex wall and some of i t , the larger pieces, did not move offshore as bedload, but remained. 174 Condition 3 -The wave height has been reduced thereby decreasing the deepwater steepness. -Breaking moves further up the beach. -The breaker i s plunging; the j e t s t i l l does not penetrate to the bed; the large vortex forms the vortex wall and the sediment does not go past i t as bedload. Bedload pg,s,b,as,ac Suspended ac -pg,s: again a small r i p p l e i s formed; t h i s time d i r e c t l y under the plunging point of the i n i t i a l plunger. Some pg placed onshore of the vortex wall remained and did not t r a v e l offshore as bedload, but only for the larger pieces, -b: t r a v e l s on equal sides of the r i p p l e as bedload. -ac: move as before; suspension occurs primarily at the r i p p l e , are c a r r i e d offshore where they s e t t l e and t r a v e l as bed load back to the r i p p l e ; the higher the p a r t i c l e i s suspended the further offshore i t t r a v e l s . I f i t i s i n i t i a l l y suspended high enough there i s a chance i t w i l l be c a r r i e d onshore being held up by a series of v o r t i c e s created by various splash-plunge cycles, and then t r a v e l offshore as bedload to the r i p p l e ( n u l l p o int). 175 Discussion on Conditions 1. 2. and 3 As the wave steepness was decreased the breaking point moved onshore as i s expected since, i f the rough ru l e of thumb i s used, Hu/db i s approximately 0.75, then the depth must decrease at the breaking point, hence the point moves onshore. In a l l cases, the j e t did not penetrate to the bed. An argument may be made that i f the j e t does not penetrate then the plunging breaker i s r e a l l y a s p i l l i n g type breaker from the point of view of the sediment. This i s and important d i s t i n c t i o n to make since for the same breaker type you can get d i f f e r e n t transport regimes. As the wave height decreases and the breaking depth decreases the bedload sediments were more prone to going into suspension. This i s probably due to the bed being under more d i r e c t influence of the turbulence generated by the breaking process because of the reducing depth. The v o r t i c e s generated by the j e t and the splash-plunge cycles .did not influence the offshore transport of a l l material offshore to the r i p p l e (except i n some instances where a larger piece of pea gravel would remain). Any of the a c r y l i c cubes that made i t past the f i r s t vortex did so as suspended load. 176 Condition 4 -Same as condition 1. -**The conditions are reversible, going from 1 to 2 to 3 and then back to 1 (ie condition 4) produce the same sediment c h a r a c t e r i s t i c s as those that o r i g i n a l l y existed i n condition 1. Condition 5 -Plunging breaker; j e t does not penetrate to the bed; forms a large vortex a f t e r i t plunges. -The plunging wave i s stronger and forms a very strong i n i t i a l plunger vortex; t h i s vortex reaches the bed and remains r e l a t i v e l y stationary; obvious splash-plunge cycles are set up (-3) . Bedload pg,s,b,as,ac Suspended as,ac,b -pg,s: forms a r i p p l e j u s t offshore of the plunging point; active movement of the pg and s within 24 cm wide zone; i s not spread out but remains i n one group. -b: t r a v e l s back and forth over the r i p p l e and i s occasionally suspended. -as: gets c a r r i e d quite high i n the water column as i t i s suspended o f f the crest. -ac: suspended as i n the other conditions. -Overall the sediments are much more active with only the pg and s remaining on the bed. 177 Condition 6 - S p i l l i n g breakers Bedload pg,s,b,as,ac Suspended ac -pg: larger angular pieces at the break point. Onshore of the vortex v a i l -No p a r t i c l e s are transported onshore of the r i p p l e . 178 Condition 7 -Small plunging breaker; j e t does not penetrate to the bed; splash-plunge cycles are set up. Bedload pg,s,b,as,ac Suspended ac -pg: small r i p p l e j u s t onshore of the plunge point; intermixed; some of the larger pieces of pg put onshore of the vortex wall remained. -b: moves as bedload on onshore side of the r i p p l e ; does advance over the r i p p l e with the advance of the crest, -as: Moves as bedload; occasionally being l i f t e d into suspension. -ac: became suspended mainly at the r i p p l e and t r a v e l back to the breaking point and s e t t l e ; t r a v e l as bedload back to the r i p p l e . -Same transport as previous conditions 1, 2, 3 for points onshore and offshore regions. 179 Condition 8 -Small scale plunging; j e t does not penetrate to the bed; splash-plunge cycles (2-3) . -Vortices reach the bed onshore of the f i r s t vortex. Bedload pg,s,b,as,ac Suspended ac,as,b -pg,s: again the r i p p l e i s formed, but t h i s time i t i s formed offshore of the plunge point. -b: moves over the r i p p l e occasionally going into suspension at the r i p p l e . -as: moves over the r i p p l e occasionally going into suspension at the r i p p l e . -ac: same as before however appears to be a l o t more suspension. 180 Condition 9 -Large plunging wave; j e t penetrates to the bed; large plunging vortex i s created which, acts on the bed; becomes deformed because of the bed; very c l e a r splash-plunge cycle (3-4). Bedload/Suspended A l l of the p a r t i c l e s do both to some degree; p a r t i c l e s that are suspended t r a v e l further offshore than the point of suspension except when the plunger vortex traps them and c a r r i e s them forward; these ac are usually high i n the water column so the plunging vortex c a r r i e s them forward. -pg,s: the p a r t i c l e s are moving i n a chaotic fashion with the sand and gravel being the lea s t moveable. -b,as: both move as the pg,s but further on each side of the pg,s; e a s i l y suspended. -ac: present i n the whole breaker area equal amounts suspended and on the bed. 181 Condition 10 -The wave height has ben reduced and a plunging breaker s t i l l occurs; the j e t does not plunge as v i o l e n t l y and does not appear to impact on the bed; the splash-plunge cycle s t i l l e x i s t s but are not as pronounced. -The sediment i s not moved as v i o l e n t l y as previous condition; the l i m i t s are unmoved; the breaking point has moved onshore, as expected. -The general pattern of sediment remains unchanged. -pg,s: form a r i p p l e near the plunge point. 182 Condition 11 -Wave height has been reduced and a plunging breaker occurs; the j e t does not penetrate to the bed; the plunger vortex i s formed but i s not strong and does not e f f e c t the bed. Bedload pg,s,b,as,ac Suspended  ac.as.b -the o v e r a l l impression are re s u l t s s i m i l a r to conditions l , 2, 3 -pg,s: form a r i p p l e j u s t onshore of the plunge point. -as,b: move over the r i p p l e and go into suspension sometimes, -ac: t r a v e l as suspended load offshore and are put into suspension near the r i p p l e ; bedload moves onshore; onshore of the f i r s t vortex the bedload i s transported offshore and the suspended load i s transported onshore. 183 Appendix C Wave Setdown/Setup Data 184 Beach Slope and Manometers WAVE SETDOWN/SETUP DATA FOR 1:15 IMPERMEABLE SLOPE RUN DEPTH HEIGHT? PERIOD BREAKER CONSTANT BREAKING SETUP TOTAL DEPTH© DEPTH TYPE SETDOWN SETDOWN SETUP BREAKING d H T SI Sb Sv A S db CD CI sec • • IB aa aa ca 1.3 48.80 22.00 2.88 pl/sp 4.5 4.5 52.5 57.B 2B.65 1.1 48.00 22.00 2.24 pi 4.5 4.5 56.0 68.5 18.75 1.2 40.00 22.00 2.75 sp/pl 7.7 9.0 132.8 141.8 14.22 1.4 40.00 22.00 3.50 pi 6.5 9.0 144.0 153.0 20.13 2.4 49.30 12.70 1.34 pi 3.7 5.5 2B.B 33.5 21.65 2.3 49.38 12.80 1.68 pi 4.5 6.5 34.0 40.5 21.55 2.2 49.30 12.50 2.17 pi 4.5 4.5 38.8 42.5 21.75 2.1 49.38 12.70 2.76 pi 5.8 10.0 46.0 56.8 12.59 3.1 47.38 17.30 2.81 pi 6.8 10.5 72.8 82.5 16.97 4.5 47.50 10.85 1.55 pi 1.5 3.2 37.0 48.2 20.88 4.4 47.58 18.57 1.79 pi 1.7 3.0 43.8 46.8 16.83 4.3 47.50 10.67 2.16 pi 1.3 1.8 44.8 45.8 28.35 4.2 47.50 18.75 2.62 pi 1.7 3.4 48.5 51.9 11.93 4.1 47.50 10.50 3.44 pi 3.8 4.2 57.8 61.2 15.92 5.1 47.58 15.09 1.41 sp 2.8 4.3 48.8 44.3 28.18 5.2 47.50 15.12 1.57 pl/sp 2.5 4.2 43.8 47.2 15.92 5.3 47.50 15.18 1.88 pl/sp 2.5 3.0 46.8 49.8 24.17 5.4 47.50 15.03 2.13 pi 3.5 2.5 50.5 53.0 28.28 5.5 47.58 15.88 2.43 pi 3.3 3.5 47.8 58.5 28.85 6.1 47.50 17.61 1.51 sp 4.8 5.6 47.8 52.6 23.91 6.2 47.58 17.73 1.68 sp/pl 4.8 5.8 47.8 52.8 24.82 6.3 47.50 17.07 1.88 pi 3.8 4.5 53.8 57.5 32.15 6.4 47.50 17.37 2.18 pi 2.8 4.8 56.8 68.8 28.13 7.1 47.50 19.96 1.43 pl/sp 3 . 8 5.8 45.8 58.8 32.10 7.2 47.50 20.18 1.80 pl/sp 4.5 5.3 50.0 55.3 28.80 7.3 47.50 19.65 2.89 pi 3.4 4.3 58.5 62.8 28.18 B . l 45.88 12.58 1.29 pl/sp 2 . 8 3.8 41.5 45.3 17.52 8.2 45.00 12.58 1.55 sp 1.8 4.8 43.5 47.5 17.58 8.3 45.80 12.47 1.88 pi 2.8 3.8 46.5 49.5 17.68 8.4 45.00 12.37 2.21 pi 1.8 1.8 49.0 58.8 17.72 8.5 45.00 12.98 2.64 pi 1.9 5.8 58.0 63.8 13.34 9.1 45.00 15.86 1.38 sp/pl 8.5 2.5 36.5 39.8 25.78 9.2 45.00 14.80 1.56 sp/pl 1.7 5.8 46.5 51.5 17.48 9.3 45.00 14.84 1.82 pi 2.6 3.2 43.0 46.2 25.71 9.4 45.08 15.38 2.28 pi 2.7 2.9 65.5 68.4 25.78 9.5 45.00 15.24 2.76 pi 2.7 7.8 97.0 184.8 13.13 18.1 45.80 17.61 1.38 sp 1.8 1.5 39.8 48.5 21.82 10.2 45.88 17.78 1.38 sp 1.8 2.5 47.8 49.5 21.72 18.3 45.00 17.78 1.84 pi 8.5 1.5 65.8 66.5 21.82 10.4 45.88 17.76 2.17 pi 1.5 1.5 53.0 54.5 34.12 18.5 45.00 17.68 2.61 pi 2.5 5.5 85 . 8 98.5 17.34 11.4 45.88 20.32 1.51 pi 2.8 3.0 43.5 46.5 21.67 11.3 45.08 19.96 1.77 pi 3.8 5.8 46.8 51.8 33.67 11.2 45.80 19.78 2.17 pi 3.5 4.5 67.5 72.8 33.72 11.1 45.80 28.18 2.89 pi 6.9 11.5 85 . 8 96.5 16.75 186 RUN DEPTH HEIGHTS PERIOD BREAKER CONSTANT BREAKING SETUP TOTAL DEPTHS DEPTH TYPE SETDOWN SETDOWN SETUP BREAKING d H T SI Sb 5w A 3 db CI CI sec • i •• •• •• CI 12.1 40.190 22.00 2.00 Pi 2.5 3.5 73.0 76.5 29.00 12.2 40.00 21.50 2.75 Pi 1.0 5.0 114.0 119.0 14.08 12.3 40.00 15.20 2.75 Pi 2.5 9.0 77.0 86.0 13.68 12.4 40.00 15.20 2.00 Pi 3.0 4.0 73.0 77.0 20.00 13.1 40.40 15.40 2.28 Pi 1.0 3.0 54.0 57.0 19.40 13.2 40.40 20.50 2.28 Pi 1.0 3.0 69.0 72.0 21.80 14.1 45.00 23.00 1.56 Pi 3.5 6.5 44.0 50.5 31.00 14.2 45.00 15.50 1.56 Pi 3.0 4.0 44.0 48.0 19.10 14.3 45.00 14.00 2.30 pl 0.0 3.5 47.5 51.0 19.18 14.4 45.00 19.58 2.30 Pi 3.0 2.5 55.0 57.5 23.00 15.1 47.30 17.50 2.30 Pl 1.0 2.5 60.0 62.5 24.30 15.2 47.30 12.60 2.30 Pl 1.0 11.5 40.5 52.0 20.40 15.3 47.30 24.00 1.43 Pl 5.0 11.5 45.0 56.5 32.40 15.4 47.30 11.50 1.43 Pl 2.5 11.5 40.5 52.0 18.28 "NOTE: 1) THE VALUE OF db IS FOR THE DEPTH TO THE MHL AT THE POINT OF MAXIMUM SETDOWN 2) THE CALCULATED TOTAL SETUP IS 8.15db ( FROM SPM ) 3) THE VALUE OF kd IS CALCULATED BY ITERATION USING THE EQUATION led = 4pi A2 * d/gTA2 » l/tanh(kd) 4) THE VALUE OF Sbc=(gA0.5 • HoA2 * T)/(64pi • dbA1.5) ( FROM SPM ) 5) THE VALUE OF S«c-Sc-Sbc 187 RUN CALC. d/gTA2 TOTAL SETUP •0 1.3 31. B 0.81019 1.1 28.1 B.BB813 1.2 21.3 B.B8539 1.4 38.2 8.B8333 2.4 32.5 8.82799 2.3 32.3 8.01781 2.2 32.6 8.B1867 2.1 18.9 0.00668 3.1 25.5 0.00611 4.5 38.1 0.02015 4.4 24. B 0.01511 4.3 42.5 0.01038 4.2 17.9 0.00705 4.1 23.9 0.00409 5.1 42.2 0.02435 5.2 23.9 0.01964 5.3 36.3 0.01370 5.4 42.4 0.01067 5.5 38.1 0.0082B 6.1 35.9 0.02124 6.2 36. B 0.01716 6.3 48.2 0.01370 6.4 42.2 0.01019 7.1 48.2 0.02368 7.2 42.B 0.01494 7.3 42.2 0.01108 8.1 26.3 0.02757 8.2 26.3 8.01909 8.3 26.4 0.01416 8.4 26.6 0.00939 8.5 2B.B 0.00658 9.1 38.7 0.02409 9.2 26.1 0.B18B5 9.3 38.6 0.01385 9.4 38.6 0.00948 9.5 19.7 0.00602 IB. 1 32.7 8.024B9 18.2 32.6 0.02409 18.3 32.7 B.01355 18.4 51.2 0.00974 IB.5 26. B 0.00673 11.4 32.5 0.02012 11.3 5B.5 0.01464 11.2 58.6 0.00974 11.1 25.1 0.00549 =2Pid/L H/Ho Ho c i 0.6801 0.98358 22.4 0.5985 1.01927 21.6 0.4784 1.09854 20.0 8.3706 1.21398 18.1 1.2842 0.91390 13.9 0.9501 0.92454 13.8 0.6984 0.97706 12.8 0.5335 1.85734 12.0 0.5115 1.07268 16.1 1.0288 8.91793 11.8 0.8580 0.93719 11.3 0.6872 0.98099 10.9 0.5535 1.04455 10.3 0.4131 1.16168 9.0 1.1676 0.91314 16.5 1.0118 0.91908 16.5 8.8086 0.94672 16.0 0.6984 8.97706 15.4 0.6015 1.01774 14.7 1.0646 0.91595 19.2 0.9281 0.92703 19.1 0.8086 0.94672 18.8 0.6800 0.98362 17.7 1.1453 0.91343 21.9 0.8522 0.93820 21.4 0.7137 0.97197 20.2 1.2733 0.91369 13.8 8.9933 8.92050 13.6 0.8248 0.94336 13.2 0.6492 0.99574 12.4 0.5329 1.05774 12.3 1.1587 8.91324 16.5 8.9852 B.92118 16.1 8.8138 0.94562 15.7 0.6525 0.99437 15.4 0.5078 1.07539 14.2 1.1588 0.91324 19.3 1.1588 B.91324 19.4 B.S033 0.94788 18.7 8.6628 0.99020 17.9 0.5395 1.05340 16.7 1.0275 0.91801 22.1 0.8417 0.94010 21.2 0.6628 0.99020 20.0 0.4831 1.09465 18.4 CALC. CALC. SETDOWN SETUP db/S Sbc Swc • 9 19 16.6 14.4 3.623 20.0 8.1 3.099 32.0 -10.7 1.009 19.8 10.4 1.316 4.0 28.5 6.463 5.0 27.3 5.321 5.5 27.2 5.118 13.9 5.0. 2.248 16.3 9.2 2.857 3.7 26.4 4.995 5.5 18.5 3.485 2.6 39.9 6.190 10.5 7.4 2.299 6.9 17.0 2.601 4.0 38.1 6.343 10.4 13.5 3.373 6.3 29.9 4.933 5.2 37.2 5.336 9.2 20.9 3.970 7.4 28.4 4.546 8.1 27.9 4.619 5.2 43.0 5.591 7.1 35.1 4.688 5.8 42.3 6.42B 8.7 33.3 5.B63 8.9 33.2 4.475 5.2 21.1 3.868 6.1 2B.2 3.684 6.6 19.8 3.556 7.1 19.5 3.488 12.7 7.3 2.117 4.5 34.2 6.610 8.6 17.5 3.379 5.4 33.2 5.565 6.2 32.3 3.757 18.1 1.5 1.263 7.8 24.9 5.388 8.0 24.6 4.388 9.8 22.9 3.281 5.5 45.7 6.261 15.7 10.3 1.916 11.4 21.1 4.660 6.4 44.1 6.602 6.9 43.7 4.683 22.3 2.8 1.736 db/Ho Lo=gTA2/2 Ho/Lo c i 0.923 624.524 0.036 0.869 783.403 0.028 0.710 1188.741 0.017 1.111 1912.685 8.009 1.558 288.349 8.050 1.557 440.664 0.831 1.700 735.285 0.017 1.048 1189.343 0.818 1.052 1232.826 8.013 1.699 375.105 0.032 1.421 5B0.259 0.023 2.606 728.445 0.015 1.159 1071.746 0.010 1.761 1847.592 0.005 1.700 310.404 0.053 0.968 384.847 0.043 1.507 551.829 0.029 1.838 788.351 0.022 1.360 921.938 0.016 1.244 355.994 0.054 1.256 440.664 0.043 1.783 551.829 0.033 1.593 741.997 0.024 1.469 319.272 0.068 1.307 505.864 0.042 1.390 681.996 0.030 1.272 259.818 0.053 1.289 375.105 0.036 1.331 505.864 0.026 1.426 762.559 0.016 1.087 1088.171 0.011 1.563 297.336 0.055 1.083 379.968 0.042 1.638 517.168 0.030 1.670 755.674 0.020 8.927 1189.343 0.012 1.132 297.336 0.065 1.121 297.336 0.065 1.169 528.597 0.835 1.902 735.205 0.024 1.038 1063.588 8.816 0.979 355.994 0.062 1.586 489.143 8.043 1.688 735.205 0.027 0.909 1304.022 0.014 188 RUN CALC. d/gTA2 kd=2Pid/L H/Ho Ho CALC. CALC. TOTAL SETDOWN SETUP db/S db/Ho Lo=gT"2/2 Ho/Lo SETUP Sbc Sue • • CI • • • • cc 12.1 43.5 8.81819 0.6800 8.98362 22.4 10.0 33.5 3.791 1.297 624.524 0.036 12.2 21.1 8.88539 0.4783 1.89862 19.6 31.1 -9.9 1.183 0.719 1188.741 0.817 12.3 28.5 8.88539 0.4783 1.09862 13.8 16.2 4.3 1.591 0.989 1180.741 0.012 12.4 38.8 B.B1819 0.6B8B 0.98362 15.5 8.3 21.7 2.597 1.294 624.524 0.025 13.1 29.1 0.08732 0.5901 1.02365 15.0 9.4 19.7 3.484 1.290 811.631 0.019 13.2 32.7 0.00792 0.5901 1.B2365 20.0 14.0 18.7 3.028 1.089 811.631 0.025 14.1 46.5 0.01885 0.9853 0.92117 25.0 8.8 37.7 6.139 1.242 379.968 0.066 14.2 28.7 0.01885 0.9853 B.92117 16.8 8.2 20.4 3.979 1.135 379.960 0.844 14.3 28.7 0.00867 0.6206 1.00843 13.9 8.3 20.4 3.745 1.376 825.933 0.017 14.4 34.5 0.00867 0.6206 1.BB843 19.3 12.1 22.4 4.000 1.189 825.933 0.023 15.1 36.5 8.00911 0.6382 1.80045 17.5 9.2 27.3 3.888 1.389 825.933 0.021 15.2 38.6 0.00911 0.6382 1.00045 12.6 6.2 24.4 3.923 1.620 825.933 0.015 15.3 48.6 0.02358 1.1420 0.91349 26.3 8.3 40.3 5.735 1.233 319.272 0.082 15.4 27.3 0.02358 1.1428 0.91349 12.6 4.5 22.8 3.500 1.446 319.272 0.039 189 SETUP EXPERIMENT I 1 1) Hater depth at tonoteter 8: 48.8 c i 2) Have data: vave height leasured at lanoieter 8 Runt Height Period Breaker H (ci) T (sec) type 1.1 22.88 2.24 pi 1.2 22.88 2.75 sp/pl 1.3 22.88 2.88 pl/sp 1.4 22.88 3.58 pi 3) Ver t ica l displacement of NHL froa SHL: aeasureients in • i l l i i e t e r s Runt Manometer 8 1 2 3 4 5 6 7 8 9 18 11 1.1 -4.5 -4.5 -1.5 -2.7 -1.5 18.7 28.8 38.8 37.8 47.8 1.2 -7.7 -9.8 -6.8 -8.8 -7.7 -7.8 8.5 27.8 38.5 54.8 87.5 1.3 -4.5 -4.5 -4.8 -2.7 1.8 8.2 17.5 26.5 34.8 45.8 1.4 -5.8 -6.5 -7.5 -9.8 -7.2 1.8 8.5 18.8 31.8 51.8 87.5 190 SETUP EXPERIMENT t 2 1) Hater depth at lonoieter I : 49.5 c i 2) Have data: vave height leasured at lanoieter 0 Runt Height Period Breaker H (ci) T (sec) type 2.1 12.70 2.7G Pi 2.2 12.50 2.17 Pi 2.3 12.80 1.68 Pi 2.4 12.70 1.34 Pi 3) Vert ical displacement of HHL fro* SHL: leasureients in l i l l i i e t e r s Runt Hanoieter 8 1 2 3 4 5 6 7 8 9 10 11 12 2.1 -5.5 -10.0 -8.5 -18.0 -10.0 -11.0 -1B.0 -1.5 -5.0 7.0 18.5 34.8 2.2 -3.7 -4.5 -3.5 -4.0 -4.5 -4.5 -3.5 -2.7 7.3 12.7 23.0 34.5 2.3 -4.0 -4.5 -4.5 -5.0 -6.0 -6.5 -6.0 -6.2 2.5 12.7 23.8 31.5 2.4 -3.7 -4.0 -3.5 -4.5 -5.0 -5.5 -5.0 -4.5 4.5 12.3 18.7 26.4 4) Location of defined breaking: displacement left or right of lanoieter Runt Location (ci) 2.1 15.5--8 2.2 33 .8 -7 2.3 7--12.B 2.4 1.5-7 5) Location of f i r s t u j o r turbulance: displacement left of r ight of lanoieter Runt Location (ci) 2.1 8-25.8 2.2 7 -16 .5 2.3 8-18.8 2.4 8-12.0 1 9 1 SETUP EXPERIMENT t 3 1) Hater depth at tonometer 8: 47.3 c i 2) Have data: vave height leasured at manometer 8 Runt Height Period Breaker H (ci) T (sec) type 3.1 17.38 2.B1 pi 3) Ver t ica l displacement of NHL f ro i SHL: leasureients in l i l l i i e t e r s Runt Hanoieter 8 1 2 3 4 5 6 7 8 9 IB 11 12 3.1 -6.8 -18.5 -8.8 -18.5 -18.5 -18.5 -11.7 -9.5 14.5 23.8 26.8 54.8 4) Location of defined breaking: displacement left or right of lanoieter Runt Location (ci) 3.1 2 . 8 - 7 5) Location of f i r s t la jor turbulance: displacement left of right of lanoieter Runt Location (ci) 3.1 8 .8 -8 192 SETUP EXPERIMENT t 4 1) Hater depth at lonoieter I : 47.5 ci 2) Have data: wave height measured at manometer 0 Runt Height Period Breaker H (cm) T (sec) type 4.1 10.58 3.44 Pi 4.2 10.75 2.62 Pi 4.3 10.67 2.16 Pi 4.4 10.57 1.79 Pi 4.5 18.85 1.55 Pi 3) Vert ical displacement of HHL from SUL: measurements in millimeters Runt Manometer 8 1 2 3 4 5 6 7 8 9 10 11 12 4.1 -3.0 -4.0 -4.0 -4.2 -3.6 -4.8 -4.2 -3.9 0.8 9.7 28.8 45.5 4.2 -1.7 -2.5 -2.5 -1.8 -2.0 -2.6 -2.8 -3.4 4.8 14.5 27.5 45.5 4.3 -1.3 -1.5 -1.5 -1.8 -1.2 -1.6 -1.5 0.0 6.9 13.5 26.0 43.0 4.4 -1.7 -2.0 -2.5 -1.8 -2.1 -2.9 -3.8 -2.8 3.9 14.5 26.0 41.5 4.5 -1.5 -1.8 -2.5 -1.8 -2.1 -3.2 -3.1 -2.7 2.2 14.5 25.0 4) Location of defined breaking: displacement left or right of manometer Runt location (cm) 4.1 4 . 5 - 7 4.2 7-30.0 4.3 10.0-7 4.4 7-21.0 4.5 13.5-8 5) Location of f i r s t major turbulance: displacement left of right of manometer Runt Location (cm) 4.1 8 .5 -8 4.2 8-20.0 4.3 8 -1 .0 4.4 8 -8 .5 4.5 8-37.5 193 SETUP EXPERIMENT I 5 I) Hater depth at lonoieter fl: 47.5 2) Nave data: wave height leasered at lanoieter fl Runt Height Period Breaker H (ci) T (sec) type 5.1 15.89 1.41 S P 5.2 15.12 4.57 pl/sp 5.3 15.18 1.88 pl/sp 5.4 15.B3 2.13 Pl 5.5 15.88 2.43 Pl 3) Vert ical displacement of NHL f ro i SHL: leasureients i n l i l l i i e t e r s Runt Hanoieter 8 1 2 3 4 5 6 7 8 9 IB 11 12 5.1 -2.B -2.5 -3.4 -4.3 -3.6 -4.1 -3.2 2.8 11.8 18.5 26.3 4B.B 5.2 -2.5 -3.1 -3.B -3.7 -4.8 -4.1 -4.2 3.4 13.5 21.6 38.5 42.8 5.3 -2.5 -4.B -3.1 -3.8 -3.B -2.7 -8.8 7.8 15.5 23.5 32.9 44.8 5.4 -3.5 -3.5 -2.8 -2.5 -2.4 -2.8 -2.8 6.5 13.8 22.5 33.5 46.5 5.5 -3.3 -4.2 -3.1 -3.5 -3.5 -3.5 -3.2 5.5 13.5 21.5 3B.3 44.5 4) Location of defined breaking: displacement lef t or right of lanoieter Runt location (ci) 5.1 5-38.8 5.2 2 . 8 - 6 5.3 29 .5 -5 5.4 5-25.5 5.5 6 - 2 . 8 5) Location of f i r s t la jor turbulance: displacement lef t of right of lanoieter Runt Location (ci) 5.1 7--7.B 5.2 7-25.8 5.3 23 .B-7 5.4 7-18.8 5.5 7--21.5 194 SETUP EXPERIMENT I 6 1) Hater depth at monometer 0: 47.5 c i 2) Have data: wave height measured at manometer 0 Runt Height Period Breaker H (cm) T (sec) type 6.1 17.61 1.51 SP 6.2 17.73 1.68 sp/pl 6.3 17.07 1.88 Pi 6.4 17.37 2.18 Pi 3) Vert ical displacement of HHl from SHL: measurements in millimeters Runt Nanometer 0 1 2 3 4 5 6 7 8 9 10 11 12 6.1 -4.0 -4.5 -5.0 -5.B -5.6 -5.5 -4.0 7.5 17.0 26.0 33.2 44.5 6.2 -4.0 -4.5 -5.0 -4.5 -4.5 -3.5 0.0 7.6 7.6 24.8 33.5 45.0 6.3 -3.0 -3.0 -4.5 -3.8 -3.8 -2.5 1.5 11.8 11.8 26.5 35.0 48.0 6.4 -2.8 -3.5 -3.0 -4.0 -3.0 -2.0 2.5 11.3 11.3 25.0 34.5 49.0 4) Location of defined breaking: displacement lef t or right of manometer Runt Location (cm) 6.1 2 . 0 - 5 6.2 31 .5 -5 6.3 5 - 3 . 0 6.4 8 .0 -5 5) Location of f i r s t major turbulance: displacement left of right of manometer Runt Location (cm) 6.1 3 . 0 - 7 6.2 17.5-7 6.3 31 .0 -7 6.4 6-11 .0 195 SETUP EXPERIMENT I 7 1) Mater depth at monometer B: 47.5 c i 2) Have data: wave height measured at lanoieter B Runl Height Period 8reaker H (ci) T (sec) type 7.1 19.96 1.43 pl/sp 7.2 28.10 1.8B p l / p l 7.3 19.65 2.B9 Pl 3) Ver t ica l displacement of NHL from SHL: measurements in millimeters Runl Manometer B l 2 3 4 5 6 7 8 9 IB 11 12 7.1 -3.B -4.6 -5.B -4.5 -4.5 -4.3 3.B 15.8 2B.B 28.B 33.B 43.8 7.2 -4.5 -6.8 -5,B -5.3 -4.3 -4.3 1.8 12.8 2B.B 28.8 36.5 47.8 7.3 -3.4 -4.8 -4.8 -4.3 -3.8 -3.B 5.B 11.5 21.5 31.8 39.8 51.5 4) location of defined breaking: displacement left or right of manometer Runl location (cm) 7.1 4-19.8 7.2 4--6.B 7.3 4-13 .8 5) Location of f i r s t major turbulance: displacement lef t of r ight of manometer Runl Location (cm) 7.1 6-14.8 7.2 6--1B.B 7.3 34 .8 -7 196 SETUP EXPERIMENT I 8 1) Hater depth at sonometer B: 45.B c i 2) Have data: vave height measured at manometer B Runt Height Period Breaker H (cm) T (sec) type 8.1 12.58 1.29 pl/sp 8.2 12.5B 1.55 sp 8.3 12.47 1.8B pi 8.4 12.37 2.21 pi 8.5 12.98 2.64 pi 3) Vert ical displacement of HHL from SHL: measurements in millimeters Runt Nanometer B 1 2 3 4 5 6 7 8 9 18 11 12 8.1 -2.8 -2.B -2.8 -2.6 -2.7 -3.8 -2.2 4.8 13.8 17.5 31.5 8.2 -1.8 -2.B -2.8 -3.8 -2.8 -4.8 -2.6 4.8 14.1 22.8 34.5 8.3 -2.8 -2.8 -2.8 -2.5 -2.8 -3.8 -2.1 5.B 14.3 24.8 37.5 8.4 -1.8 -1.5 -1.8 -1.5 -1.5 -1.8 -8.5 9.2 16.8 23.5 39.5 8.5 -1.9 -3.5 -3.8 -3.3 -5.8 -5.B 2.7 13.3 26.8 43.5 75.5 4) Location of defined breaking: displacement left or right of manometer Runt Location (cm) 8.1 2B.B--6 8.2 9 .8 -6 8.3 24 .8-6 8.4 I3--6 8.5 6-11.8 5) Location of f i r s t major turbulance: displacement left of right of manometer Runt Location (cm) 8.1 7 -7 .5 8.2 7-11.8 8.3 7-12.8 8.4 7-15.8 8.5 7-31.8 197 SETUP EXPERIMENT I 9 1) Hater depth at lonoieter 8: 45.B c i 2) Have data: wave height leasured at lanoieter B Runl Height Period Breaker H (ci) T (sec) type 15.86 1.38 sp/pl 14.80 1.56 sp/pl 14.84 1.82 pi 15.38 2.2B pi 15.24 2.76 pi 3) Vert ical displacement of HHL froi SHI: leasureients in l i l l i i e t e r s Runl Hanoieter B 1 2 3 4 5 6 7 8 9 18 11 12 9.1 -8.5 -l.B -2.8 -2.5 -2.8 -2.8 2.3 18.5 17.8 23.8 32.5 9.2 -1.7 -2.5 -2.5 -2.7 -3.5 -5.8 -2.1 8.8 19.8 28.8 39.5 9.3 -2.6 -3.2 -3.2 -3.2 -2.7 -1.7 2.8 3.1 18.5 27.5 38. B 9.4 -2.7 -2.5 -1.7 -2.9 -1.7 2.7 3.8 14.5 19.5 28.5 48.5 8.8 9.5 -2.7 -5.B -3.5 -4.5 -5.8 -6.5 -7.B 3.5 18.B 29.8 47.5 77.8 4) Location of defined breaking: displaceient left or right of lanoieter Runt Location (ci) 9.1 36 .8-5 9.2 5-14.8 9.3 28 .8 -5 9.4 6 . 8 - 5 9.5 31 .8-7 5) Location of f i r s t lajor turbulance: displaceient left of right of lanoieter Runl Location (ci) 9.1 28 .8 -7 9.2 7-18.8 9.3 33 .8-7 9.4 6-13.8 9.5 7-33.8 9.1 9.2 9.3 9.4 9.5 198 SETUP EXPERIMENT t 18 1) Hater depth at tonometer 0: 45.8 c i 2) Have data: wave height measured at lanoaeter 0 Runt Height Period Breaker H (ci) T (sec) type 10.1 17.61 1.38 sp 10.2 17.70 1.61 sp 18.3 17.70 1.84 pl 18.4 17.76 2.17 pl 10.5 17.68 2.61 pl 3) Vert ical displacement of NHL f ro i SHL: teasureients i n millimeters Runl Nanoieter 8 1 2 3 4 5 6 7 8 9 18 11 12 18.1 -1.8 -2.8 -1.2 -1.5 -1.5 B.B 6.8 13.5 21.8 26.8 35.8 18.2 -1.8 -2.0 -2.2 -2.5 -2.5 -1.8 8.8 17.5 26.8 32.5 48.5 18.3 -4.5 -2.8 -1.5 -1.5 -1.5 3.8 12.8 18.5 25.5 33.8 42.5 6.5 18.4 -1.5 -0.5 0.0 0.0 0.0 1.5 9.5 19.B 24.5 33.5 45.B 18.5 -2.5 -3.5 -3.5 -4.5 -5.5 -5.5 8.8 11.5 23.5 37.8 49.5 72.8 4) Location of defined breaking: displacement lef t or right of lanoieter Runl Location (ci) 18.1 29.B--4 18.2 32.B--4 18.3 - 4 -18.4 4--17.B 18.5 5-32.0 5) Location of f i r s t lajor turbulance: displacement left of right of lanoieter Runl Location (ci) 18.1 5--12.B 18.2 5 .8 -6 18.3 21 .8-6 1B.4 18.B--6 18.5 22 .8 -7 199 SETUP EXPERIMENT I 11 1) Water depth at monometer 0; 45 2) Have data: wave height measured at manometer 0 Runt Height Period Breaker H (cm) T (sec) type 11.1 20.18 2.89 Pl 11.2 19.78 2.17 Pl 11.3 19.96 1.77 Pl 11.4 20.32 1.51 Pl 3) Ver t ica l displacement of MHL from SHI: measurements in millimeters Runt Manometer 0 1 2 3 4 5 6 7 B 9 18 11 12 11.1 -6.9 -7.0 -8.8 -6.5 -8.8 -8.8 -11.5 -9.5 7.5 25.8 39.8 53.5 74.5 11.2 -3.5 -3.5 -4.5 -4.0 -2.5 -2.5 5.2 18.8 19.5 25.8 34.5 44.5 67.8 11.3 -3.0 -4.0 -5.8 -4.8 -3.5 -2.5 8.8 12.5 28.8 26.5 34.8 42.8 11.4 -2.0 -2.2 -2.5 -3.0 -2.5 -3.8 0.0 11.8 21.5 27.5 34.0 41.5 4) Location of defined breaking: displacement left or right of manometer Runt Location (cm) 1.1 14.8--6 11.2 6 .8 -4 11.3 4 -9 .8 11.4 4--16.B 5) Location of f i r s t major turbulance: displacement lef t of right of manometer Runt Location (cm) 7-23.8 5-22.8 5 - 6 . 8 3.B--6 200 11.1 11.2 11.3 11.4 SETUP EXPERIMENT t 12 1) Hater depth at lonoieter 8: 48 2) Have data: wave height leasured at lanoieter 8 Runt Height Period Breaker H (ci) T (sec) type 12.1 22.88 2.88 Pi 12.2 21.58 2.75 Pi 12.3 15.28 2.75 Pi 12.4 15.28 2.88 pl 3) Vert ical displaceient of NHL f ro i SHL: leasureients in l i l l i i e t e r s Runt Hanoieter 8 1 2 3 4 5 6 7 8 9 IB 11 12.1 -2.5 -3.4 -3.2 -2.8 -2.B B.B 11.8 21.5 3B.B 48. B 79.8 B.B 12.2 B.B 8.8 -1.8 -1.8 -3.8 -3.8 -5.8 12.B 31.B 43.8 51.8 78.8 12.3 -2.5 -5.B -5.5 -4.5 -5.8 -6.0 -8.8 7.8 18.B 26.5 48.5 66.8 12.4 -4.8 -3.8 -4.8 -3.5 -4.8 -2.8 B.B 18.5 21.5 32.8 19.5 B.B 4) Location of defined breaking: displaceient lef t or right of lanoieter Runt Location (ci) 12.1 3--1B.3 12.2 5-38.2 12.3 6-18.3 12.4 4-61.8 5) Location of f i r s t lajor turbulance: displaceient le f t of right of lanoieter Runt Location (ci) 201 SETUP EXPERIMENT I 13 1) Mater d e p t h a t monoaeter 0: 40.4 ca 2) Wave d a t a : wave h e i g h t measured a t a a n o s e t e r 0 R u n l H e i g h t P e r i o d B r e a k e r H ( c i ) T ( s e c ) t y p e 13.1 15.40 2.28 p l 13.2 20.50 2.28 p l 3) V e r t i c a l d i s p l a c e m e n t o f MWL from SHL: measurements i n a i l l i a e t e r s R u n l M a n o s e t e r 0 1 2 3 4 5 6 7 8 9 IB 11 ' 12 13.1 8.8 -3.8 -1.8 -1.0 -2,0 0.0 3.2 12.5 19.5 35.8 39.0 13.2 8.0 -1.5 -3.0 0.0 0.0 1.5 9.5 21.0 26.5 26.0 14.5 0.8 4) L o c a t i o n o f d e f i n e d b r e a k i n g : d i s p l a c e a e n t l e f t o r r i g h t o f e a n o i e t e r Runt L o c a t i o n ( c s ) 13.1 4 - 1 8 . 3 13.2 3 - 4 5 . 5 5) B r e a k i n g and b a c k r u s h d a t a : R u n l B r e a k e r B r e a k e r B a c k r u s h B a c k r u s h h e i g h t d e p t h v e l o c i t y d e p t h H ( c n ) db ( c a ) ub ( i / s ) uBR ( c a ) 13.1 23.0 20.0 0.33 17.0 13.2 23.0 22.0 0.37 18.3 202 SETUP EXPERIMENT I 14 1) H a t e r d e p t h a t a o n o i e t e r 0: 45.0 ce 2) Have d a t a : wave h e i g h t measured a t nanometer 0 R u n l H e i g h t P e r i o d B r e a k e r H (ca) T ( s e c ) t y p e 14.1 23.00 1.5b p i 14.2 15.50 1.56 p i 14.3 14.00 2.30 p i 14.4 19.50 2.30 p i 3) V e r t i c a l d i s p l a c e i e n t o f NHL f r o a SWL: a e a s u r e a e n t s i n a i l l i a e t e r s R u n l M a n o t e t e r 12 0 1 2 3 4 5 6 7 8 9 10 11 14.1 -3.8 -6.5 -5.0 -6.2 -5.0 -3.8 4.5 13.0 20.5 28.5 31.5 32.7 14.2 -3.0 -4.8 -4.5 -4.0 -4.0 -4.0 -3.6 0.0 9.5 18.5 26.5 32.6 14.3 0.0 -2.2 0.0 8.0 0.0 -2.5 -3.0 4.0 9.5 19.0 23.5 36.8 14.4 -2.0 -4.8 -4.5 0.0 -2.0 -0.5 -1.0 10.5 19.5 23.5 30.5 39.0 4) L o c a t i o n o f d e f i n e d b r e a k i n g : d i s p l a c e m e n t l e f t o r r i g h t o f a a n o a e t e r R u n l L o c a t i o n ( c a ) 14.1 2 - 4 2 . 0 14.2 5 - 3 6 . 8 14.3 5 - 3 6 . 0 14.4 4 - 4 2 . 0 5) B r e a k i n g and b a c k r u s h d a t a : R u n l B r e a k e r B r e a k e r B a c k r u s h B a c k r u s h h e i g h t d e p t h v e l o c i t y d e p t h H ( c a ) db ( c a ) ub ( a / s ) uBR ( c a ) 14.1 14.2 14.3 14.4 23.0 17.5 18.0 22.5 30.5 19.0 20.0 22.5 8.30 0.33 8.31 0.34 26.5 16.5 16.7 18.5 203 SETUP EXPERIMENT I 15 1) H a t e r d e p t h a t a o n o a e t e r 0: 47.3 cn 2) Have d a t a : vave h e i g h t measured a t t a n o a e t e r 0 R u n l H e i g h t P e r i o d B r e a k e r H ( » ) T ( s e c ) t y p e 15.1 17.50 2.30 P l 15.2 12.60 2.30 P l 15.3 24.00 1.43 P l 15.4 11.50 1.43 P l 3) V e r t i c a l d i s p l a c e m e n t o f MWL f r o s SWL: measurements i n n i l l i s e t e r s R u n l H a n o n e t e r 0 1 2 3 4 5 6 7 8 9 10 11 12 15.1 0.0 1.0 1.0 0.0 0.0 0.0 2.5 3.0 12.8 20.0 25.5 31.0 49.0 15.2 0.0 -1.0 0.0 -0.5 -2.5 0.0 -3.0 0.0 3.5 12.5 18.5 23.0 15.3 -5.0 -7.5 -7.5 -6.5 -6.8 -4.0 0.0 12.0 18.0 24.5 38.5 31.0 45.0 15.4 -2.5 -2.5 -2.5 -2.5 -4.0 -3.5 -5.0 -4.0 -3.0 9.5 19.5 21.5 4) L o c a t i o n o f d e f i n e d b r e a k i n g : d i s p l a c e n e n t l e f t o r r i g h t o f a a n o a e t e r R u n l L o c a t i o n ( c i ) 14.1 - 5 -14.2 - 6 -14.3 - 3 -14.4 3 0 . 0 - 7 5) B r e a k i n g and b a c k r u s h d a t a : R u n l B r e a k e r B r e a k e r B a c k r u s h B a c k r u s h h e i g h t d e p t h v e l o c i t y d e p t h H ( c a ) db ( c a ) ub ( a / s ) uBR ( c a ) 15.1 15.2 15.3 15.4 22.0 18.5 24.0 14.5 24.0 20.5 33.5 17.5 0.34 0.32 0.34 0.27 19.3 16.6 27.0 13.1 204 There are no pages 205 and 206. Appendix D Wave Reflection Data 207 REFLECTION TESTS FOR A PLANE, IMPERMEABLE, 1:15 SLOPED BEACH SURF REFLECTION REFLECTED RUN DEPTH PERIOD Haax Hiin Hi Lo SIMILARITY COEFFICIENT WAVEHEIGHT (c«) (sec) (ca) (ca) (ca) (ca) PARAMETER * t f Hr » * (ca) 1 47.5 2.60 IB.08 8.74 9.41 1055.44 0.71 0.054 0.51 1 <• 47.5 2.00 14.10 13.17 13.64 624.52 0.45 0.021 0.29 3 47.5 1.74 19.98 18.88 19.35 472.78 0.33 0.010 0.19 4 47.5 1.52 19.83 17.00 18.02 368.72 0.30 0.010 0.18 5 47.5 1.39 9.18 8.28 8.65 301.66 0.39 0.815 0.13 6 . 42.5 2.44 18.54 17.43 17.99 929.54 0.48 0.023 0.41 7 42.5 1.80 18.68 16.58 17.59 505.86 8.36 0.018 8.32 8 42.5 1.40 19.31 17.44 18.38 306.02 0.27 0.010 0.18 9 42.5 1.33 11.94 11.00 11.47 276.18 0.33 0.010 0.11 IB 40.0 2.72 28.72 15.84 18.28 1155.12 0.53 0.023 0.42 t Surf S i a i l a r i t y Paraaeter - 1.0/((cot(beach slope))(Hi/Lo)) *t Reflection coefficient obtained froa figure 2-65, p.2-118, SPM «»The height of the reflected wave i s given by (REFLECTION COEFFICIENT)*Hi 208 Appendix E Laboratory Study of Breakers (Iversen 1952) 209 IVERSEN,H.H. (1952). LABORATORY STUDY OF BREAKERS. NATIONAL BUREAU OF STANDARDS, CIRCULAR 521, pp.9-32. SUMMARY OF DATA - BEACH SLOPE 1:11 : = r r r = r r : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : = r : : : : : : DEEP STILL CREST BACKHASH STAGNA- BACKHASH CREST WAVE-RUN HATER PERIOD HAVE HATER BREAKER DEPTH HEI6HT DEPTH TION VELOCITY LENGTH HAVE T HEIGHT* DEPTH* HEIGHT AT AT AT LOCATION Hb/db AT AT STEEP- Hi (at Hi) Hb BREAKING BREAKIN6 BREAKING Is BREAKING BREAKIN6 NESS! d i db Yb dBH VBH Vc lb=Vc»T Ho/Lo seconds feet feet feet feet feet feet feet ft/sec ft/sec 25 8.8797 1.88 8.488 2.33 8.35 8.45 8.72 8.27 8.53 - 8.778 3.55 3.558 5 8.8774 1.88 8.391 2.38 8.48 8.41 8.75 - - - 8.976 - NA 4 8.8774 1.88 8.391 2.38 8.48 8.41 8.75 - - - 8.976 - NA 17 8.8614 8.80 8.200 2.23 8.21 8.21 8.44 8.14 - - 1.000 2.74 2.192 18 8.8581 8.92 8.258 2.23 8.26 8.33 8.52 8.18 - - 8.788 3.18 2.852 16 8.8288 1.11 0.168 2.23 8.22 8.22 0.39 8.18 - - 1.000 3.48 3.774 22 8.8286 1.51 8.228 2.23 8.37 8.38 8.56 8.18 8.46 8.148 1.233 4.65 7.822 18 8.8167 1.27 0.129 2.17 8.22 8.18 8.38 - - - 1.222 - NA 145 8.8165 1.73 8.231 2.25 8.36 8.32 8.68 - - - 1.125 - NA 7 8.8158 1.26 0.114 2.17 8.19 8.16 8.32 - - - 1.188 -• NA 8 8.8125 1.45 8.123 2.17 8.28 8.18 8.32 - - - 1.111 - NA 27 8.8112 1.26 8.885 2.15 0.16 8.14 8.27 0.88 - 8.150 1.143 - NA 2 8.8876 1.98 8.148 2.24 8.31 8.38 8.51 - - - 1.833 - NA 23 8.8871 1.98 8.131 2.23 8.29 8.25 0.46 8.12 8.33 0.133 1.168 3.55 7.829 28 8.8854 2.18 8.113 2.22 8.23 8.28 8.47 - - - 8.821 - NA 24 8.0838 2.58 0.111 2.23 8.24 8.24 0.38 8.12 8.26 - 1.888 3.45 8.625 t Lo f ro i 5.12TA2; Ho f ro i Hi and di /Lo using s t a l l amplitude theory. + Constant depth portion of the channel 210 SUMMARY OF DATA - BEACH SLOPE 1:21 :zzz -.zzzzzzz: zzzzzzzzz : : r : : : : : : : : : : : : : : : : = = = = = r = r : : : : : : : : : : : : = = = z z : tzzzzzzz : : : : : : : : DEEP STILL CREST BACKHASH STAGNA- BACKHASH CREST WAVE-IUN WATER PERIOD HAVE HATER BREAKER DEPTH HEIGHT DEPTH TION VELOCITY LENGTH NAVE T HEIGHT* DEPTH* HEIGHT AT AT AT LOCATION Hb/db AT AT STEEP- Hi (at Hi) Hb BREAKING BREAKIN6 BREAKIN6 Xs BREAKIN6 BREAKIN6 NESSi d i db Yb dBH VBK Vc Lb=Vc»T Ho/Lo seconds feet feet feet feet feet feet feet ft/sec ft/sec 36 8.0767 8.74 8.214 1.55 8.198 0.29 0.42 B.2B 0.33 0.60 8.655 3.48 2.516 46 0.0730 1.84 8.383 1.75 8.358 0.54 0.79 8.35 - 0.79 B.648 4.35 4.524 45 8.8488 1.15 8.385 1.68 8.318 0.39 0.61 B.23 - - 8.795 - NA 37 0.8488 0.93 8.286 1.58 8.218 8.27 0.44 8.19 0.33 0.640 8.778 3.17 2.948 31 8.8368 1.40 8.33B l.BB 8.42B 0.53 0.86 8.33 - - 8.792 - NA 40 8.8368 1.03 8.185 1.58 8.188 8.25 0.38 0.18 0.32 0.60 8.728 3.28 3.296 42 8.8358 1.26 8.268 1.57 8.338 0.34 0.59 8.24 8.51 8.758 B.971 3.13 3.944 44 8.8298 1.33 8.238 1.68 8.388 8.34 0.56 0.24 0.48 0.65 8.882 2.68 3.458 34 8.8288 1.50 8.298 1.68 B.48B 0.46 0.76 - - - 0.87B - NA 39 0.8278 1.12 8.165 1.58 8.198 8.23 B.38 0.16 0.28 0.648 8.826 2.58 2.888 43 B.B22B 1.41 8.282 1.56 B.27B 0.33 0.53 0.22 0.47 8.58 B.818 3.28 4.512 38 8.8228 1.17 8.145 1.58 8.288 8.21 B.36 0.12 - - 8.952 - NA 3 B.B218 1.59 8.256 1.6B B.400 0.48 8.77 0.30 - B.833 - NA 29 8.8138 1.89 8.225 1.57 8.388 8.44 8.72 0.28 - - 8.864 - NA 47 0.8138 1.67 8.178 1.51 0.270 0.29 8.49 0.18 0.38 8.74 8.931 3.88 5.018 35 8.8138 1.34 8.118 1.5B 8.148 8.16 8.27 - - - 8.875 - NA 41 8.8883 1.55 8.894 1.47 0.150 0.18 8.29 0.12 0.25 8.64 8.833 3.B3 4.696 48 8.8879 1.93 8.144 1.49 8.258 8.25 8.45 8.16 0.41 8.68 1.888 2.9B 5.597 28 B.BB76 2.24 8.193 1.57 0.360 0.39 8.65 0.22 - - 8.923 - NA : : : : ===::::: : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : IZZZZZZZZZ : : : : : : : : : • Lo f r o i 5.12TA2; Ho f ro i Hi and di /Lo using s i a l l amplitude theory. * Constant depth portion of the channel 211 SUMMARY OF DATA - BEACH SLOPE 1:33 t z z : : z z z z z z z : : z z z z z z z : : : s z z s z z : : z s z : z : x z z : z z z s z s z z i i z z z s z z z z z z x z z z x z s r s : s z z z z z x z z : z z x x x z z DEEP STILL tUN HATER PERIOD HAVE HATER BREAKER DEPTH HAVE T HEIGHT* DEPTH* HEI6HT AT STEEP- Hi (at Hi) Hb BREAKIN6 Hb/db Hb/Ho db/Ho NESS! d i db Ho/Lo seconds feet feet feet feet 8 B.B665 1.85 8.356 1.65 8.358 8.365 0.959 0.932 8.971 9 B.B3S3 1.24 8.255 1.58 8.275 8.365 0.753 0.989 1.312 4 8.8214 1.46 8.214 1.55 8.285 8.350 0.814 1.219 1.497 IB 8.8138 1.49 8.144 1.44 8.225 8.278 0.833 1.433 1.728 3 8.BB99 1.87 8.169 1.58 8.262 0.373 0.702 1.477 2.182 11 B.BB93 1.68 8.112 1.48 8.175 8.268 0.673 1.434 2.131 5 8.8884 2.83 8.173 1.49 8.253 0.225 1.124 1.426 1.268 7 8.8888 2.37 8.238 1.64 8.415 8.518 8.814 1.882 2.215 12 8.8874 1.79 8.115 1.48 B.18B 0.260 0.692 1.481 2.140 14 B.8BS2 2.18 8.115 1.44 8.215 8.275 0.782 1.829 2.340 6 8.BB43 2.67 8.164 1.52 8.298 0.370 0.784 1.846 2.355 2 B.N42 2.29 8.116 1.43 8.238 B.28B 0.821 2.838 2.481 16 8.8835 2.52 8.117 1.43 8.288 0.265 0.755 1.756 2.326 IS 8.8827 2.52 8.893 1.42 8.198 8.230 0.826 2.162 2.618 1 8.8825 2.65 8.897 1.41 8.188 0.244 0.738 2.BB1 2.712 * Lo f ro i 5.12T"2; Ho f r o i Hi and di/Lo using s i a l l aipl i tude theory. • Constant depth portion of the channel * 212 SUMMARY OF DATA - BEACH SLOPE 1:50 zssz s s s s s s s : ========= ========= ========= •========= s r s r r r r z s : : : : : : : : : : : : : r : r : : r s r r s s = r i : : : : : : : : : DEEP STILL CREST BACKHASH STAGNA- BACKHASH CREST WAVE-RUN HATER PERIOD HAVE HATER BREAKER " DEPTH HEI6HT DEPTH TION VELOCITY LENGTH HAVE T HEIGHT* DEPTH* HEI6HT AT AT AT LOCATION Hb/db AT AT STEEP- Hi (at Hi) Hb BREAKING BREAKIN6 BREAKING Xs BREAKIN6 BREAKIN6 NESS* d i db Yb dBH VBH Vc Lb=Vc»T Ho/Lo seconds feet feet feet feet feet feet feet ft/sec ft/sec 62 8.8987 8.81 8.381 1.54 8.258 - 8.555 8.292 - - ERR - NA 58 8.0718 1.88 8.348 1.54 8.383 8.483 8.633 8.322 - - 8.752 3.88 3.888 61 8.8786 8.98 8.279 1.54 8.222 8.326 8.497 8.261 8.398 8.18 8.681 3.98 3.518 74 8.8584 8.95 8.222 1.54 8.191 8.228 8.394 8.183 - - 8.838 2.75 2.613 59 8.8474 1.8B 8.238 1.54 B.218 - 8.455 8.236 - - ERR - NA 78 0.8465 1.13 0.282 1.54 8.297 8.350 8.585 8.292 8.578 8.16 8.849 2.88 3.164 63 8.8376 1.17 8.243 1.54 B.274 8.321 8.554 8.274 0.490 8.11 8.854 3.58 4.895 68 8.0376 1.88 0.182 1.54 8.185 - - - - - ERR - NA 73 8.8385 1.38 8.241 1.54 8.248 8.328 8.538 8.262 - - 8.756 3a 58 4.558 77 0.0223 1.35 0.198 1.54 8.199 8.231 8.486 8.191 - - B.861 3.88 4.858 68 0.8198 1.62 8.228 1.54 8.268 8.386 8.522 8.251 - - 8.876 3.58 5.678 66 0.0130 1.74 8.198 1.54 8.283 8.334 8.565 8.262 8.468 8.19 B.847 2.85 4.959 71 0.0092 2.88 8.188 1.54 8.288 B.222 8.486 8.175 - - 8.937 3.75 7.588 78 0.0074 2.43 8.232 1.54 8.353 - - - • - - ERR - NA 83 0.8874 1.98 8.129 1.54 8.181 8.212 8.362 8.173 - - 8.854 NA 81 0.0065 2.65 8.243 1.54 8.398 8.513 0.848 8.436 8.688 8.11 8.776 2.38 6.895 88 0.0065 2.25 8.168 1.54 8.217 - 8.517 8.276 - - ERR 3.58 7.875 82 0.0849 2.65 8.186 1.54 8.328 8.422 8.691 8.368 8.578 8.070 8.758 - NA * Lo f ro i 5.12TA2| Ho f ro i Hi and di /Lo using s t a l l amplitude theory. * Constant depth portion of the channel 213 

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