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WAVE TRANSFORMATION BY BATHYMETRIC ANOMALIES WITH GRADUAL TRANSITIONS IN DEPTH AND RESULTING SHORELINE RESPONSE By CHRISTOPHER J. BENDER A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2003 ACKNOWLEDGMENTS I am truly blessed to have my wife and best friend, Kathryn, in my life. This completion of this work was made possible by her support, encouragement, and love. I am fortunate beyond measure to have her next to me as I walk life's journey. Many individuals at the University of Florida have made the completion of this work possible. Dr. Robert G. Dean has not only been my major advisor but also my mentor and friend during my studies at the University of Florida. I have gained many valuable lessons from his work ethic, his character, and his tireless quest for understanding the coastal environment. I am grateful to the other members of my committee (Dr. Daniel M. Hanes, Dr. Andrew Kennedy, Dr. Ulrich H. Kurzweg, Dr. Robert J. Thieke) for their instruction and involvement during the course of my doctoral studies. I wish to thank my family (especially Mom and Dad B, Mom and Dad Z, Caryn, John, and Kristin) and friends for their support in all my endeavors. Each of them has contributed to who I am today. An Alumni Fellowship granted by the University of Florida sponsored this study with partial support from the Bureau of Beaches and Wetland Resources of the State of Florida. TABLE OF CONTENTS page ACKNOW LEDGM ENTS ............................................................................................. ii LIST OF TABLES ....................................................................................................... vi LIST OF FIGURES .......................................................................................................... vii ABSTRACT......................................................................................................................... xix CHAPTER 1 INTRODUCTION AND M OTIVATION .......................................... ............... 1.1 M otivation................ ..................................................................................... 2 1.2 M odels Developed and Applications............................................. .............. 4 2 LITERATURE REVIEW ...................................................................................... 7 2.1 Case Studies .................................................................................................. 8 2.1.1 Grand Isle, Louisiana (1984).............................................. ............... 8 2.1.2 Anna M aria Key, Florida (1993)........................................ .......... ... 11 2.1.3 M artin County, Florida (1996) ........................................... .......... ... 15 2.2 Field Experiments ........................................................................................ 16 2.2.1 Price et al. (1978) .............................................................................. 17 2.2.2 Kojima et al. (1986)............................................................................. 17 2.3 Laboratory Experiments..... ............................. ...................................... 19 2.3.1 Horikawa et al. (1977)........................................................................ 19 2.3.2 W illiams (2002).................................................................................... 20 2.4 W ave Transformation .................................................................................. 23 2.4.1 Analytic M ethods .............................................................................. 23 2.4.1.1 2Dimensional methods ................................................ .....24 2.4.1.2 3Dimensional methods ........................................................34 2.4.2 Numerical M ethods ........................................................................... 40 2.5 Shoreline Response........................................................................................ 46 2.5.1 Longshore Transport Considerations .................................................. 46 2.5.2 Refraction M odels ............................................................................. 47 2.5.2.1 M otyka and W illis (1974)........................................................47 2.5.2.2 Horikawa et al. (1977) ...........................................................49 2.5.3 Refraction and Diffraction M odels................................................... 51 2.5.3.1 Gravens and Rosati (1994)....................................................51 2.5.3.2 Tang (2002)...................................................... ...................54 2.5.4 Refraction, Diffraction, and Reflection Models.................................. 54 2.5.4.1 Bender (2001) .............................................. ....................... 54 3 2DIMENSIONAL MODEL THEORY AND FORMULATION .........................59 3.1 Introduction .................................................................................................. 59 3.2 Step Method: Formulation and Solution..................................... .......... .... 60 3.2.1 Abrupt Transition ................................................... ........................ 63 3.2.2 Gradual Transition.............................................. ............................. 65 3.3 Slope Method: Formulation and Solution................................... ........... ... 67 3.3.1 Single Transition .................................................... ......................... 69 3.3.2 Trench or Shoal ..................................................... .......................... 72 3.4 Numerical Method: Formulation and Solution.............................................. 74 4 3DIMENSIONAL MODEL THEORY AND FORMULATION ..........................77 4.1 Introduction .................................................................................................. 77 4.2 Step Method: Formulation and Solution.................................... ........... ... 77 4.2.1 Abrupt Transition .................................................... ........................ 81 4.2.2 Gradual Transition............................................................................. 85 4.3 Exact Shallow Water Solution Method: Formulation and Solution ............... 88 5 2DIMENSIONAL MODEL RESULTS AND COMPARISONS.........................92 5.1 Introduction .................................................................................................. 92 5.2 Matching Condition Evaluation................................................................ 92 5.3 Wave Transformation ................................................................................ 95 5.3.1 Comparison of 2D Step Model to Numerical Model FUNWAVE 1.0. 97 5.4 Energy Reflection ........................................................................................ 99 5.4.1 Comparison to Previous Results....................................... ........... .... 99 5.4.2 Arbitrary Water Depth............... ........................................................... 101 5.4.3 L ong W aves........................................................................................... 116 6 3DIMENSIONAL MODEL RESULTS AND COMPARISONS.........................125 6.1 Introduction...................... ......................................................................... 125 6.2 Matching Condition Evaluation....................................................................... 126 6.3 Wave Height Modification .............................................................................. 127 6.3.1 Comparison of 3D Step Model and Analytic Shallow Water Exact M odel................................................. ................................................. 14 1 6.4 W ave Angle M odification.......................................................................... 144 6.5 Comparison of 3D Step Model to Numerical Models.................................... 147 6.5.1 3D Step Model Versus REF/DIF1................................................... 148 6.5.2 3D Step Model Versus 2D Fully Nonlinear Boussinesq Model......... 154 6.6 Comparison to Laboratory Data of Chawla and Kirby (1996) ...................... 156 6.7 Direction Averaged Wave Field Modification ........................................... 161 6.8 Energy R election ....................................................................................... 166 6.8.1 Comparison to Prior Results............................................................. 166 6.8.2 Effect of Transition Slope on Reflection........................................... 167 6.9 Analytic Nearshore Shoaling and Refraction Method................................... 171 6.9.1 Comparison of Analytic Method to REF/DIF1.................................... 176 6.9.2 W ave Averaged Results ......................................................................... 183 6.10 Shoreline Evolution M odel......................................................................... 185 6.10.1 Shoreline Change Estimates Shoreward of Bathymetric Anomalies. 186 6.10.2 Effect of Nearshore Form on Shoreline Change ................................. 197 6.10.3 Investigation of Boundary Conditions ........................................... 200 6.10.4 Investigation of Transition Slope on Shoreline Evolution................ 202 7 CONCLUSIONS AND DIRECTIONS FOR FUTURE STUDY ........................208 7.1 C onclusions................................................................................................ 208 7.2 Future W ork ................................................................................................. 2 11 APPENDIX A ANALYTIC WAVE ANGLE CALCULATION ........................................ ...213 B ANALYTIC FARFIELD APPROXIMATION OF ENERGY REFLECTION....215 C ANALYTIC NEARSHORE SHOALING AND REFRACTION METHOD........218 D ANALYTIC SHORELINE CHANGE THEORY AND CALCULATION...........222 R EFEREN C ES ..........................................................................................................228 BIOGRAPHICAL SKETCH ..................................................................................... 236 LIST OF TABLES Table page 21 Capabilities of selected nearshore wave models................................. ........... 42 41 Specifications for two bathymetries for exact solution method.............................91 LIST OF FIGURES Figure page 21 Aerial photograph showing salients shoreward of borrow area looking East to West along Grand Isle, Louisiana, in August, 1985..........................................9 22 Aerial photograph showing salients shoreward of borrow area along Grand Isle, Louisiana, in 1998 ...................................................... ............................ 10 23 Bathymetry off Anna Maria Key, Florida, showing location of borrow pit following beach nourishment project........................... .......................................13 24 Beach profile through borrow area at R26 in Anna Maria Key, Florida ..............14 25 Shoreline position for Anna Maria Key Project for different periods relative to A ugust, 1993 ............................................................................................... 14 26 Project area for Martin County beach nourishment project .................................15 27 Fouryear shoreline change for Martin County beach nourishment project: predicted versus survey data. ................................... .........................................16 28 Setup for laboratory experiment.............................................. ....................... 19 29 Results from laboratory experiment showing plan shape after two hours. ............20 210 Experiment sequence timeline for Williams laboratory experiments .................21 211 Volume change per unit length for first experiment..........................................22 212 Shifted even component of shoreline change for first experiment......................23 213 Reflection and transmission coefficients for linearly varying depth [hi/hill] and linearly varying breadth [b2/b 2]. ................................................... ..........26 214 Approximate reflection and transmission coefficients for the rectangular parallelepiped of length 8.86h0 in infinitely deep water......................................27 215 Reflection coefficient for a submerged obstacle. ................................................28 216 Transmission coefficient as a function of relative wavelength (h=10.1 cm, d=67.3 cm trench width =161.6 cm).................................................................30 217 Transmission coefficient as a function of relative wavelength for trapezoidal trench; setup shown in inset diagram....................................... ..31 218 Reflection coefficient for asymmetric trench and normally incident waves as a function ofKhl: h2/hl=2, h3/hl=0.5, L/hi=5; L = trench width .....................32 219 Transmission coefficient for symmetric trench, two angles of incidence: L/h1=10, h2/hi=2; L = trench width. ....................................................................32 220 Transmission coefficient as a function of relative trench depth; normal incidence: kIhi= 0.2: (a) L/hi=2; (b) L/hi=8, L = trench width...........................33 221 Total scattering cross section of vertical circular cylinder on bottom...................35 222 Contour plot of relative amplitude in and around pit for normal incidence; kl/d = 7/10, k2/h=7i/102, h/d=0.5, b/a =1, a/d=2, a = crossshore pit length, b = longshore pit length, h = water depth outside pit, d = depth inside pit, L2 = w avelength outside pit. ...................................... ..............................................37 223 Contour plot of diffraction coefficient in and around pit for normal incidence; a/L=l, b/L=0.5, d/h =3, kh=0.167 ...................................................39 224 Contour plot of diffraction coefficient around surfacepiercing breakwater for normal incidence; a/L=l, b/L=0.5, kh=0.167. ...............................................39 225 Maximum and minimum relative amplitudes for different koa, for normal incidence, a/b=6, a/d=nt, and d/h=2......................................................................40 226 Comparison of wave height profiles for selected models along transect parallel to shore located 9 m shoreward of shoal apex[*=experimental data].......44 227 Comparison of wave height profiles for selected models along transect perpendicular to shore and through shoal apex [*=experimental data]. ...............44 228 Calculated beach planform due to refraction after two years of prototype w aves for tw o pit depths. ...................................... ..............................................48 229 Calculated beach planform due to refraction over dredged hole after two years of prototype w aves. ................................... ............................. ...........50 230 Comparison of changes in beach plan shape for laboratory experiment and numerical model after two years of prototype waves..........................................51 231 Nearshore wave height transformation coefficients near borrow pit from R CPW A V E study............................................................................................... 53 232 Nearshore wave angles near borrow pit from RCPWAVE study; wave angles are relative to shore normal and are positive for westerly transport...........53 233 Reflection coefficients versus dimensionless pit diameter divided by wavelength inside and outside the pit; water depth = 2 m, pit depth = 4 m.........55 234 Shoreline evolution resulting from each transport term individually for transect located 80 m shoreward of a pit with a radius = 6 m, last time step indicated w ith [+] .............................................................................................57 235 Shoreline evolution using full transport equation and analytic solution model for transect located 80 m shoreward of a pit with a radius = 6 m, last tim e step indicated w ith [+]. ............................................................................ 58 31 Definition sketch for trench with vertical transitions...........................................63 32 Definition sketch for trench with stepped transitions...........................................66 33 Definition sketch for linear transition.................................................................70 34 Definition sketch for trench with sloped transitions.......................................73 41 Definition sketch for circular pit with abrupt depth transitions............................78 42 Definition sketch for boundary of abrupt depth transition...................................82 43 Definition sketch for boundaries of gradual depth transitions .............................85 44 Definition sketch for boundaries of exact shallow water solution method............90 51 Matching conditions with depth for magnitude of the horizontal velocity and velocity potential for trench with abrupt transitions and 16 evanescent m odes............................................................................................. .............. 94 52 Matching conditions with depth for phase of the horizontal velocity and velocity potential for trench with abrupt transitions and 16 evanescent m odes........................................................... ........................... .. ......................94 53 Relative amplitude for crosstrench transect for k1hl = 0.13; trench bathymetry included with slope = 0.1...............................................................95 54 Relative amplitude for crossshoal transect for k1h, = 0.22; shoal bathymetry included w ith slope = 0.05................................................................................ 96 55 Relative amplitude along crosstrench transect for Analytic Model and FUNWAVE 1D for klhi = 0.24; trench bathymetry included with slope 0 .1 ................................................................................................................. 9 8 56 Relative amplitude along crossshoal transect for Analytic Model and FUNWAVE 1D for k1hi = 0.24; shoal bathymetry included with upwave slope of 0.2 and downwave slope of 0.05.........................................................99 57 Comparison of reflection coefficients from step method and Kirby and Dalrymple (1983a Table 1) for symmetric trench with abrupt transitions and normal wave incidence: h3 = hi, h2/h1 = 3, W/hi = 10 ........................................ 100 58 Comparison of transmission coefficients from step method and Kirby and Dalrymple (1983a Table 1) for symmetric trench with abrupt transition and normal wave incidence: h3= hi, h2/h = 3, W/hi = 10 ........................................ 101 59 Setup for symmetric trenches with same depth and different bottom widths and transition slopes .............................................................................................102 510 Reflection coefficients versus k1hi for trenches with same depth and different bottom widths and transition slopes. Only onehalf of the symmetric trench crosssection is shown with slopes of 5000, 1, 0.2 and 0.1....103 511 Transmission coefficients versus klhl for trenches with same depth and different bottom widths and transition slopes. Only onehalf of the symmetric trench crosssection is shown with slopes of 5000, 1, 0.2 and 0.1. ... 104 512 Reflection coefficient versus the number of evanescent modes used for trenches with same depth and transition slopes of 5000, 1, and 0.1..................105 513 Reflection coefficient versus the number of steps for trenches with same depth and transition slopes of 5000, 1, and 0.1..........................................106 514 Reflection coefficients versus klhi for trenches with same bottom width and different depths and transition slopes. Only onehalf of the symmetric trench crosssection is shown with slopes of 5000, 1, 0.2 and 0.05 ..................107 515 Reflection coefficients versus k1ih for trenches with same top width and different depths and transition slopes. Only onehalf of the symmetric trench crosssection is shown with slopes of 5000, 5, 2, and 1. ........................108 516 Reflection coefficients versus k1hi for trenches with same depth and bottom width and different transition slopes. Only onehalf of the symmetric trench crosssection is shown with slopes of 5000, 0.2, 0.1, and 0.05 .........................109 517 Reflection coefficients versus klhl for shoals with same depth and different top widths and transition slopes. Only onehalf of the symmetric shoal crosssection is shown with slopes of 5000, 0.5, 0.2 and 0.05..........................110 518 Reflection coefficients versus k1hi for Gaussian trench with C1 = 2 m and C2 = 12 m and ho = 2 m. Only onehalf of the symmetric trench cross section is shown with 43 steps approximating the nonplanar slope .................11 519 Reflection coefficients versus k1hi for Gaussian shoal with C1 = 1 m and C2 = 8 m and ho = 2 m. Only onehalf of the symmetric shoal crosssection is shown with 23 steps approximating the nonplanar slope.................................112 520 Reflection coefficients versus k1hi for a symmetric abrupt transition trench, an asymmetric trench with sl = 1 and S2 = 0.1 and a mirror image of the asym m etric trench............................................................................................. 113 521 Reflection coefficients versus k1hi for an asymmetric abrupt transition trench (hi # h5), an asymmetric trench with gradual depth transitions (hi # h5 and sl = S2 = 0.2) and a mirror image of the asymmetric trench with s, = s2. ......114 522 Reflection coefficients versus k1hi for an asymmetric abrupt transition trench (h, # h5), an asymmetric trench with gradual depth transitions (hi # h5 and sl = 1 and s2 = 0.2) and a mirror image of the asymmetric trench with si S2 S .................................................................................................... ........... 115 523 Reflection coefficient versus the space step, dx, for trenches with same depth and different bottom width and transition slopes. Only onehalf of the symmetric trench crosssection is shown with slopes of 5000, 1 and 0.1...........117 524 Reflection coefficients versus k3h3 for three solution methods for the same depth trench case with transition slope equal to 5000. Only onehalf of the symmetric trench crosssection is shown........................................................... 118 525 Reflection coefficients versus k3h3 for three solution methods for same depth trench case with transition slope equal to 1. Only onehalf of the symmetric trench crosssection is shown....................................... ............... 19 526 Reflection coefficients versus k3h3 for three solution methods for same depth trench case with transition slope equal to 0.1. Only onehalf of the symmetric trench crosssection is shown...................................... ............ ... 119 527 Conservation of energy parameter versus k3h3 for three solution methods for same depth trench case with transition slope equal to 1. Only onehalf of the symmetric trench crosssection is shown. ..............................................120 528 Reflection coefficients versus klhi for step and numerical methods for Gaussian shoal (ho = 2 m, C1 = 1 m, C2 = 8 m). Only onehalf of the symmetric shoal crosssection is shown with 23 steps approximating the nonplanar slope ...................................................................................................12 1 529 Reflection coefficients versus k3h3 for step and numerical methods for Gaussian trench in shallow water (ho = 0.25 m, C1 = 0.2 m, C2 = 3 m). Only onehalf of the symmetric trench crosssection is shown with 23 steps approximating the nonplanar slope................................................... ............. 122 530 Reflection coefficients versus k3h3 for three solution methods for same depth trench case with symmetric abrupt transition trench and asymmetric trench with unequal transition slopes equal to 1 and 0.1.....................................123 531 Reflection coefficients versus k3h3 for three solution methods for same depth trench case with asymmetric abrupt transition trench and asymmetric trench with unequal transition slopes equal to 1 and 0.2...................................124 61 Matching conditions with depth for magnitude of the horizontal velocity and velocity potential for pit with abrupt transitions and 10 evanescent modes........127 62 Matching conditions with depth for phase of the horizontal velocity and velocity potential for pit with abrupt transitions and 10 evanescent modes........128 63 Contour plot of relative amplitude with k1h, = 0.24 for pit with transition slope = 0.1; crosssection of pit bathymetry through centerline included..........129 64 Relative amplitude for crossshore transect at Y = 0 with k1hi = 0.24 for pit with transition slope = 0.1; crosssection of pit bathymetry through centerline included. Note small reflection. ................................................130 65 Relative amplitude for longshore transect at X = 300 m with k1lh = 0.24 for pit with transition slope = 0.1; crosssection of pit bathymetry through centerline included............................................................................................... 130 66 Relative amplitude versus number of evanescent modes included for different Bessel function summations at two locations directly shoreward of a pit with abrupt transitions in depth...............................................................132 67 Relative amplitude versus number of steps approximating slope for different Bessel function summations for two pits with gradual transitions in depth........133 68 Contour plot of relative amplitude for shoal with k1hi = 0.29 and transition slope = 0.1; crosssection of shoal bathymetry through centerline included.......134 69 Relative amplitude for crossshore transect at Y = 0 for same depth pits for k1hi = 0.15; crosssection of pit bathymetries through centerline included with slopes of abrupt, 1, 0.2 and 0.07. ..............................................................135 610 Relative amplitude for crossshore transect at Y = 0 for same depth pits for klh, = 0.3; crosssection of pit bathymetries through centerline included with slopes of abrupt, 1, 0.2 and 0.07. ................................................. .........136 611 Relative amplitude for alongshore transect at X = 200 m for same depth pits for k1hi = 0.15; crosssection of pit bathymetries through centerline included with slopes of abrupt, 1, 0.2 and 0.07. ..................................... ...137 612 Relative amplitude for alongshore transect at X = 200 m for same depth pits for k1h = 0.3; crosssection of pit bathymetries through centerline included with slopes of abrupt, 1, 0.2 and 0.07. ............................................................... 138 613 Relative amplitude for crossshore transect located at Y = 0 for same depth shoals with k1hi = 0.15; crosssection of shoal bathymetries through centerline included with slopes abrupt, 1, 0.2 and 0.05.......................................139 614 Relative amplitude for alongshore transect located at X = 300 m for same depth shoals with khi = 0.15; crosssection of shoal bathymetries through centerline included abrupt, 1, 0.2 and 0.05.................................................140 615 Relative amplitude for crossshore transect at Y = 0 for pit with Gaussian transition slope for k1hi = 0.24; crosssection of pit bathymetry through centerline included. ..............................................................................................141 616 Relative amplitude for crossshore transect at Y = 0 for pit with for h = C/r in region of transition slope; crosssection of pit bathymetry through centerline included............................................................................................ 142 617 Relative amplitude for longshore transect at X = 100 m for pit with for h = C/r in region of transition slope; crosssection of pit bathymetry through centerline included ............................................................................................ 143 618 Relative amplitude for crossshore transect at Y = 0 for shoal with for h = C*r in region of transition slope; crosssection of shoal bathymetry through centerline included............................................................................................ 143 619 Relative amplitude for longshore transect at X = 100 m for shoal with for h = C*r in region of transition slope; crosssection of shoal bathymetry through centerline included............................................................................. 144 620 Contour plot of wave angles in degrees for pit with k1hi = 0.24 and transition slope = 0.1; crosssection of pit bathymetry through centerline included ................................................ ........... ................................................. 145 621 Contour plot of wave angles for Gaussian shoal with C1 =1 and C2 = 10 for kihi = 0.22; crosssection of shoal bathymetry through centerline included with 23 steps approximating slope.................................................................. 146 622 Wave angle for alongshore transect at X = 300 m for same depth pits for klhi = 0.15; crosssection of pit bathymetries through centerline included with slopes of abrupt, 1, 0.2 and 0.07. Negative angles indicate divergence of w ave rays ......................... .......................................................................... 147 623 Relative amplitude using 3D Step Model and REF/DIF1 for crossshore transect at Y = 0 for Gaussian pit with k1hi = 0.24; crosssection of pit bathymetry through centerline included. ........................................................149 624 Relative amplitude using 3D Step Model and REF/DIF1 for longshore transect at X = 100 m for Gaussian pit with klh1 = 0.24; crosssection of pit bathymetry through centerline included. .....................................................150 625 Relative amplitude using 3D Step Model and REF/DIF1 for longshore transect at X = 400 m for Gaussian pit with k1hi = 0.24; crosssection of pit bathymetry through centerline included. .................................................. ......151 626 Wave angle using 3D Step Model and REF/DIF1 for longshore transect at X = 100 m for Gaussian pit with k1lh = 0.24; crosssection of pit bathymetry through centerline included. .................................................. ......152 627 Relative amplitude using 3D Step Model and REF/DIF1 for crossshore transect at Y = 0 for pit with linear transitions in depth with k1ih = 0.24; crosssection of pit bathymetry through centerline included with slope = 0 .1 ............................................................................................................... .153 628 Relative amplitude using 3D Step Model and REF/DIF1 for longshore transect at X = 350 m for pit with linear transitions in depth with k1hi = 0.24; crosssection of pit bathymetry through centerline included with slope = 0.1. ........................................................................................................... 154 629 Relative amplitude using 3D Step Model and 2D fully nonlinear Boussinesq model for crossshore transect at Y = 0 for shoal with k hi = 0.32; crosssection of pit bathymetry through centerline included....................55 630 Experimental setup of Chawla and Kirby (1986) for shoal centered at (0,0) with data transects used in comparison shown ............................................... 157 631 Relative amplitude using 3D Step Model, FUNWAVE 2D and data from Chawla and Kirby (1996) for crossshore transect AA with k1hi = 1.89; crosssection of shoal bathymetry through centerline included.........................158 632 Relative amplitude using 3D Step Model, FUNWAVE 2D and data from Chawla and Kirby (1996) for longshore transect EE with kihl = 1.89; crosssection of shoal bathymetry through centerline included.........................159 633 Relative amplitude using 3D Step Model, FUNWAVE 2D and data from Chawla and Kirby (1996) for longshore transect DD with k1hi = 1.89; crosssection of shoal bathymetry through centerline included.........................160 634 Relative amplitude using 3D Step Model, FUNWAVE 2D and data from Chawla and Kirby (1996) for longshore transect BB with k1hi = 1.89; crosssection of shoal bathymetry through centerline included......................... 161 635 Relative amplitude averaged over incident direction (centered at 0 deg) for alongshore transect at X = 300 m for pit with klhl = 0.24; crosssection of pit bathymetries through centerline included with slope = 0.1.......................... 162 636 Wave angle averaged over incident direction (centered at 0 deg) for alongshore transect at X = 300 m for pit with klhl = 0.24 with bathymetry indicated in inset diagram of previous figure. ...........................................163 637 Relative amplitude averaged over incident direction (centered at 20 deg) for alongshore transect at X = 300 m for pit with k1hl = 0.24; crosssection of pit bathymetries through centerline included with slope = 0.1.......................... 165 638 Wave angle averaged over incident direction (centered at 20 deg) for alongshore transect at X = 300 m for pit with k1hl = 0.24 with bathymetry indicated in inset diagram of previous figure. ...........................................165 639 Reflection coefficient versus nondimensional diameter; comparison between shallow water transect method and farfield approximation method. ...167 640 Reflection coefficient versus k1hl based on farfield approximation and constant volume and depth pits; crosssection of pit bathymetries through centerline included with slopes of abrupt, 1, 0.2 and 0.07.................................168 641 Reflection coefficient versus kihl based on farfield approximation and constant volume bottom width pits; crosssection of pit bathymetries through centerline included with slopes of abrupt, 1, 0.2 and 0.05...................169 642 Reflection coefficient versus k1hi based on farfield approximation and constant volume and depth shoals; crosssection of shoal bathymetry through centerline included with slopes of abrupt, 1, 0.2 and 0.07....................170 643 Bathymetry for two nearshore regions used in analytic shoaling and refraction model: Pit 1; linear transition slopes with linear nearshore slope and Pit 2; Gaussian transition slopes with Equilibrium Beach form for nearshore slope................................................................................................ 172 644 Wave height and wave angle values at start of nearshore region and at breaking for longshore transect with bathymetry with H = 1 m and T = 12 s for Pit 1 ................................................ ....................................................... 173 645 Contour plot of wave height for Pit 1 in nearshore region with breaking location indicated with H = 1 m and T = 12 s.................................................174 646 Wave height and wave angle values at start of nearshore region and at breaking for longshore transect with bathymetry with H = 1 m and T = 12 s for Pit 2 bathym etry. ........................................ ................. ................ ...........175 647 Contour plot of wave height for Pit 2 in nearshore region with breaking location indicated with H = 1 m and T = 12 s.................................................176 648 Wave height and wave angle values at start of nearshore region from 3D Step Model and REF/DIF1 (linear) with H = 1 m and T = 12 s for Pit 1..........177 649 Wave height and wave angle values at h = 1.68 m (X = 616 m) from 3D Step Model and REF/DIF1 (linear) with H = 1 m and T = 12 s for Pit 1...........178 650 Wave height and wave angle values at h = 1.44 m (X = 628 m) from 3D Step Model and REF/DIF1 (linear) with H = 1 m and T = 12 s for Pit 1...........179 651 Wave height and wave angle values at start of nearshore region from 3D Step Model and REF/DIF1 (linear) with H = 1 m and T = 12 s for Pit 2...........180 652 Wave height and wave angle values at h = 2.04 m (X = 672 m) from 3D Step Model and REF/DIF1 (linear) with H = 1 m and T = 12 s for Pit 2...........181 653 Wave height and wave angle values at h = 1.6 m (X = 704 m) from 3D Step Model and REF/DIF1 (linear) with H = 1 m and T = 12 s for Pit 2...........182 654 Weighted wave averaged values of wave height and wave angle at start of nearshore region and at breaking for longshore transect with H = 1 m and T = 12 s and bathym etry for Pit 1.........................................................................184 655 Weighted wave averaged values of wave height and wave angle at start of nearshore region and at breaking for longshore transect with H = 1 m and T = 12 s and bathym etry for Pit 2.........................................................................184 656 Shoreline evolution for case Pit 1 with K1 = 0.77 and K2 = 0.4 for incident wave height of 1 m, wave period of 12 s, and time step of 50 s with shoreline advancement in the negative X direction; Pit 1 bathymetry along crossshore transect included. ...........................................................................187 657 Parameters for shoreline change after 1st time step showing shoreline position, and longshore transport terms for case Pit 1 with Ki = 0.77 and K2 = 0................................................................................................ .............. 188 658 Final shoreline planform for case Pit 1 with K1 = 0.77 and K2 = 0, 0.2, 0.4, and 0.77 for incident wave height of 1 m and T = 12 s, with shoreline advancement in the negative X direction; Pit 1 bathymetry along cross shore transect included.........................................................................................189 659 Change in shoreline position with modeling time at 4 longshore locations (Yp = 0, 100, 200, 300 m) for case Pit 1 with K1 = 0.77 and K2 = 0.4 with shoreline advancement in the negative X direction; Pit 1 bathymetry along crossshore transect included. ...........................................................................191 660 Comparison of wave height and wave angle values at 600 m and 600 m +/ 20 m for case of Pit 1 for T = 12 s with bathymetry and transect locations indicated in bottom plot. ........................................................... ...................192 661 Final shoreline planform for case Pit 2 with K1 = 0.77 and K2 = 0, 0.2, 0.4, and 0.77 for incident wave height of 1 m, wave period of 12 s, and time step of 50 s with shoreline advancement in the negative X direction; Pit 2 bathymetry along crossshore transect included.............................................. 193 662 Shoreline evolution for case Pit 2 with K1 = 0.77 and K2 = 0 for incident wave height of 1 m, wave period of 12 s, and time step of 50 s with shoreline advancement in the negative X direction; Pit 2 bathymetry along crossshore transect included. ...........................................................................194 663 Change in shoreline position with modeling time at 4 longshore locations (Yp = 0, 100, 200, 300 m) for case Pit 2 with Ki = 0.77 and K2 = 0 with shoreline advancement in the negative X direction; Pit 2 bathymetry along crossshore transect included. ........................................................................... 195 664 Final shoreline planform for case Shoal 1 with K1 = 0.77 and K2 = 0, 0.2, 0.4, and 0.77 for incident wave height of 1 m, wave period of 12 s, and time step of 50 s with shoreline advancement in the negative X direction; Shoal 1 bathymetry along crossshore transect included.............................................. 196 665 Shoreline evolution for case Shoal 1 with Ki = 0.77 and K2 = 0.2 for incident wave height of 1 m, wave period of 12 s, and time step of 50 s with shoreline advancement in the negative X direction; Shoal 1 bathymetry along crossshore transect included. .................................................................197 666 Final shoreline planform for case Pit 1 with linear nearshore slope and EBP form for T = 12 s and with Ki = 0.77 and K2 = 0 and 0.77 with shoreline advancement in the negative X direction; Pit 1 and Pit lb bathymetry along crossshore transect included. ......................................................................... 198 667 Final shoreline planform for case Shoal 1 (linear nearshore slope) and Shoal lb (Equilibrium Beach Profile) for T = 12 s and with K1 = 0.77 and K2 = 0 and 0.77 with shoreline advancement in the negative X direction; Shoal 1 and Shoal lb bathymetry along crossshore transect included..........................199 xvii 668 Final shoreline planform for case Pit 1 for T = 12 s and with K1 = 0.77 and K2 = 0.4 for two boundary conditions with shoreline advancement in the negative X direction; Pit 1 bathymetry along crossshore transect included.......201 669 Final shoreline planform for case Shoal 1 for T = 12 s and with K1 = 0.77 and K2 = 0.77 for two boundary conditions with shoreline advancement in the negative X direction; Shoal 1 bathymetry along crossshore transect include ed ................................................................................................................202 670 Final shoreline planform for constant volume pits for T = 12 s and with K1 = 0.77 and K2 = 0 with shoreline advancement in the negative X direction; crosssection of pit bathymetries through centerline included with slopes of abrupt, 1, 0.2, 0.07. ..............................................................................................203 671 Final shoreline planform for constant volume pits for T = 12 s and with K1 = 0.77 and K2 = 0 and 0.4 with shoreline advancement in the negative X direction; crosssection of pit bathymetries through centerline included with slopes of abrupt, 1, 0.2, 0.07. ............................................ ....... ...............205 672 Final shoreline planform for 5 periods for constant volume pits with K1 = 0.77 and K2 = 0 with shoreline advancement in the negative X direction; crosssection of pit bathymetry through centerline included with slope 0.2.......205 673 Maximum shoreline advancement and retreat versus period for constant volume pits with K1 = 0.77 and K2 = 0; crosssection of pit bathymetries through centerline included with slopes of abrupt, 1, 0.2, 0.07.........................206 674 Maximum shoreline advancement and retreat versus period for constant volume pits with K1 = 0.77 and K2 = 0.4; crosssection of pit bathymetries through centerline included with slopes of abrupt, 1, 0.2, 0.07.........................207 C1 Setup for analytic nearshore shoaling and refraction method.............................218 D1 Definition sketch for analytic shoreline change method showing shoreline and contours for initial location and after shoreline change..............................223 xviii Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy WAVE TRANSFORMATION BY BATHYMETRIC ANOMALIES WITH GRADUAL TRANSITIONS IN DEPTH AND RESULTING SHORELINE RESPONSE By Christopher J. Bender May 2003 Chair: Robert G. Dean Major Department: Civil and Coastal Engineering Analytic models for the propagation of linear water waves over a trench or shoal of finite width and infinite length (2D Step Model) and over an axisymmetric bathymetric anomaly (3D Step Model) are developed. The models are valid in a region of uniform depth containing a bathymetric anomaly of uniform depth with gradual transitions in depth allowed as a series of uniform depth steps approximating linear or nonlinear slopes. The velocity potential obtained determines the wave field in the domain. The 2D Step Model provides the complete wave field and, therefore, the reflection and transmission characteristics for depth anomalies of infinite length that are either symmetric or asymmetric. The 3D Step Model determines the wave transformation caused by the processes of wave refraction, diffraction and reflection. Using the known velocity potential an energy flux approach is used to calculate the wave angle and reflected energy at large distances from the anomaly. The 3D Step Model is joined with an analytic shoaling and refraction model (Analytic S/R Model) to extend the solution into the nearshore region. The Analytic S/R Model is employed to propagate the wave field up to the point of depth limited breaking. The breaking wave conditions are used to drive the longshore sediment transport and associated shoreline change using the Shoreline Change Model, which provides estimates of the equilibrium shoreline planform located landward of a bathymetric anomaly. Comparisons of the 2D Step Model to shallow water models developed in the study, previous studies, and a numerical model are made with good agreement demonstrated. The reflection coefficients found for symmetric and asymmetric trenches and shoals indicate multiple and no instances of complete transmission, respectively, with the solution independent of the incident wave direction for both cases. Comparisons to a shallow water model, numerical models, and experimental data verify the results of the 3D Step Model for several bathymetries, with the Analytic S/R Model verified by a numerical model for breaking wave conditions. Modeled equilibrium planforms landward of bathymetric anomalies indicate the importance of the longshore transport coefficients with either erosion or shoreline advancement possible for several cases presented. CHAPTER 1 INTRODUCTION AND MOTIVATION Irregular and unexpected shoreline planforms adjacent to nearshore borrow areas have increased awareness of the wave field modification caused by bathymetric anomalies such as borrow pits or shoals. When a wave field in a region of generally uniform depth encounters a bathymetric anomaly the wave field is modified through wave refraction, wave diffraction, wave reflection and wave dissipation, with the first three known collectively as scattering. These four wave transformation processes result in a modified wave field that propagates towards, and eventually impacts, the shoreline. The modified wave field will alter the longshore transport leading to a shoreline planform that is held to an artificial equilibrium. Case studies such as the beach nourishment projects at Grand Isle, Louisiana, and Anna Maria Key, Florida, have shown the possible effects of a nearshore borrow area on the adjacent shoreline planform. The ability to predict, and possibly design for, the equilibrium planform in the vicinity of a bathymetric anomaly requires a better understanding of both the wave and sediment processes near the anomaly. The focus of this report is to better understand the wave field modifications caused by bathymetric anomalies, of both twodimensional and threedimensional forms, with results providing the foundation for study on the sediment transport processes and shoreline changes induced. Several studies including field and laboratory scales have been conducted to investigate this issue. These studies examined the wave transformation over a bathymetric anomaly with the shoreline changes caused by the altered wave field. Earlier, dating back to the early 1900's, the focus was on the modification of a wave train encountering a change in bathymetry, which was solved using analytic methods. This early research included development of analytical solutions for bathymetric changes in the form of a step, or a pit, first of infinite length (in one horizontal dimension; 2D models), and, more recently, of finite dimensions (in two horizontal dimensions; 3D models). The complexity of the 3D models has advanced from a pit/shoal with vertical sidewalls and uniform depth surrounded by water of uniform depth, which can be solved analytically, to domains with arbitrary bathymetry that are solved using complex numerical schemes. Some models combine the calculation of the wave transformation and resulting shoreline change, whereas others perform the wave calculations separately and rely on a different program for shoreline evolution. 1.1 Motivation Changes in offshore bathymetry modify the local wave field, thus causing an equilibrium planform that may be altered significantly from the previous, relatively straight shoreline. Not only can a bathymetric change cause wave transformation, but also may change the sediment transport dynamics by drawing sediment into it from the nearshore or by intercepting the onshore movement of sediment. Knowledge of wave field modifications and the resulting effects on sediment transport and shoreline evolution is essential in the design of beach nourishment projects and other engineering activities that alter offshore bathymetry. Beach nourishment has become the preferred technique to address shoreline erosion. In most beach nourishment projects, the fill placed on the eroded beach is obtained from borrow areas located offshore of the nourishment site. The removal of large quantities of fill needed for most projects can result in substantial changes to the offshore bathymetry through the creation of borrow pits or by modifying existing shoals. The effect of the modified bathymetry in the borrow area on the wave field and the influence of the modified wave field on the shoreline can depend on the incident wave conditions, the nourishment sediment characteristics and some features of the borrow area including the location, size, shape and orientation. The large quantities of sediment used in beach nourishment projects combined with the increase in the number of projects constructed, and an increased industrial need for quality sediment have, in many areas, led to a shortage of quality offshore fill material located relatively near to the shore. This shortage has increased interest in the mining of sediment deposits located in Federal waters, which fall under the jurisdiction of the Minerals Management Service (MMS). Questions have been raised by the MMS regarding the potential effects on the shoreline of removing large quantities of sediment from borrow pits lying in Federal waters (Minerals Management Service, 2003). A better understanding of the effects of altering the offshore bathymetry is currently needed. The scattering processes of wave refraction, diffraction, and reflection modify the wave field in a complex manner dependent on the local wave and nearshore conditions. A more complete understanding and predictive capability of the effect of bathymetric changes to the wave field and the resulting shoreline modification leading to less impactive design of dredge pit geometries should be the goal of current research. 1.2 Models Developed and Applications A better understanding of the wave field near bathymetric anomalies can be obtained through models that more accurately represent their shapes and the local wave transformation processes. The models developed in this study extend previous analytic methods to better approximate the natural domain and extend the problem from the offshore region to the shoreline. By meshing a model constrained by a uniform depth requirement outside of the bathymetric anomaly to a nearshore model with a sloped bottom, linear waves can be propagated from the offshore over the anomaly and into the nearshore where shoaling and refraction lead to wave breaking and sediment transport. A longshore sediment transport model can then predict the shoreline changes resulting from the wave field modified by the bathymetric anomaly. Previous analytic 3D models and all of the previous 2D models, with the exception of Dean (1964) and Lee et al. (1981), have domains that contain abrupt transitions in depth (vertical sidewalls) for the bathymetric anomaly. A more realistic representation of natural bathymetric anomalies should allow for gradual transitions (sloped sidewalls). The focus of the present study is the propagation of water waves over a 2D (trench or shoal) or 3D (pit or shoal) bathymetric anomaly of more realistic geometry and the wave transformation they induce. Three solution methods are developed for a 2D domain with linear water waves and normal wave incidence: (1) the 2D step method, (2) the slope method and (3) a numerical method. The 2D step method is valid in arbitrary water depth while the slope method and the numerical method are valid only for shallow water conditions. The step method is an extension of the Takano (1960) solution as modified by Kirby and Dalrymple (1983a) that allows for a trench or shoal with "stepped" transitions that 5 approximate a specific slope or shape. The slope method is an extension of the Dean (1964) solution that allows for linear transitions between the changes in bathymetry for a trench or shoal creating regular or irregular trapezoids. The numerical method employs a backward spacestepping procedure for arbitrary (but shallow water) bathymetry with the transmitted wave specified. The 2D models are compared against each other, with the results of Kirby and Dalrymple (1983a) and with the numerical model FUNWAVE 1.0 [1D] (Kirby et al., 1998). For a 3D domain, an analytic solution to the wave field modification caused by bathymetric anomalies with sloped transitions in depth is developed. This solution is an extension of previous work for anomalies with abrupt depth transitions in regions of otherwise uniform depth that employs steps to approximate a gradual transition in depth. A shallow water analytic solution is also developed, which is valid for specific sidewall slope and pit size combinations. The 3D models are validated with the laboratory data of Chawla and Kirby (1996), the numerical models REF/DIF1 (Kirby and Dalrymple, 1994) and FUNWAVE 1.0 [2D] (Kirby et al., 1998), and the numerical model of Kennedy et al. (2000) and through direct comparisons. The application of the different models to realworld problems depends on the situation of interest. The study of 2D models can demonstrate the reflection caused by long trenches or shoals of finite width such as navigation channels and underwater breakwaters, respectively. The 3D models can be employed to study problems with variation in the longshore and nonoblique incidence. Wave related quantities such as energy flux and wave direction are calculated and indicate the influence of the bathymetric anomaly. The wave heights and directions for transects located shoreward of 6 the anomaly are combined with an analytic shoaling and refraction method to determine the wave propagation in a nearshore zone of arbitrary slope. The longshore transport and shoreline evolution are also calculated for the nearshore shoaledd and refracted) wave field that occurs shoreward of the anomaly. Through the methods developed in this study the wave transformation, energy reflection, longshore transport, and shoreline evolution induced by a 3D bathymetric anomaly with gradual transitions in depth can be investigated. CHAPTER 2 LITERATURE REVIEW Several methods have been employed to quantify the impact on the shoreline caused by changes in the offshore bathymetry including case studies, field experiments, analytical developments, numerical models, and laboratory studies. The intriguing behavior of the shoreline following beach nourishment projects at Grand Isle, Louisiana, Anna Maria Key, Florida, and Martin County, Florida, have led to questions and investigations regarding the impact of the significant offshore borrow areas present in each case. Field studies have been used to investigate the impact of offshore dredging in relatively deeper water to attempt to define a depth at which bathymetric changes will not induce significant wave transformation. Laboratory experiments have documented wave transformations caused by changes in the bathymetry and the resulting effects on the shoreline in controlled settings possible only in the laboratory. Solutions for wave transformation by changes in the bathymetry are outlined primarily in chronological order following the development from analytical solutions for long waves in one horizontal dimension (2D) through numerical models for arbitrary bathymetry that include many waverelated nearshore processes in 3 dimensions. Modeling of shoreline responses due to wave field modification from changes in offshore bathymetry is examined with models that include both wave field and shoreline changes and by coupling models that evaluate these processes independently. The wave transformation processes included in nearshore models are important factors in the capability to predict a salient leeward of a pit, the shoreline responses observed in the limited laboratory experiments and at Grand Isle, Louisiana. 2.1 Case Studies 2.1.1 Grand Isle, Louisiana (1984) The beach nourishment project at Grand Isle, Louisiana, provides one of the most interesting, and well publicized examples of an irregular planform resulting from the effects of a large borrow area lying directly offshore. One year after the nourishment project was completed, two large salients, flanked by areas of increased erosion, developed immediately shoreward of the offshore borrow area. Combe and Solieau (1987) provide a detailed account of the shoreline maintenance history at Grand Isle, Louisiana, specifications of the beach nourishment project that was completed in 1984, and details of the shoreline evolution in the two years following completion. The project required 2.1x106 m3 of sediment with approximately twice this amount dredged from an area lying 800 m from the shore (Combe and Soileau, 1987) in 4.6 m of water (Gravens and Rosati, 1994). The dredging resulted in a borrow pit that was "dumbbell" shaped in the planform with two outer lobes dredged to a depth of 6.1 m below the bed, connected by a channel of approximate 1,370 m length dredged to 3.1 m below the bed (Combe and Soileau, 1987). The salients seen in Figure 21 started to form during storm events that occurred during the winter and spring of 1984/85 (Combe and Soileau, 1987). By August 1985 the salients and associated areas of increased erosion were prominent features on the shoreline. An aerial survey of the area that was completed by the New Orleans District of the Army Corps of Engineers and the Coastal Engineering Research Center concluded that the size and location of the borrow area were such that its presence could affect the local wave climate (Combe and Soileau, 1987). Oblique aerial photography identified the diffraction of the wave field as a result of the borrow area (Combe and Soileau, 1987). The area of increased erosion near the salients was found to "affect 25% of the project length and amounted to about 8% of the neat project volume" (Combe and Soileau, 1987, pg. 1236). Figure 21: Aerial photograph showing salients shoreward of borrow area looking East to West along Grand Isle, Louisiana, in August, 1985 (Combe and Solieau, 1987). Three major hurricanes impacted the project area in the hurricane season following the project's completion, the first time that three hurricanes struck the Louisiana coastline in the same season (Combe and Soileau, 1987). While these storms did tremendous damage to the newly formed berm and caused large sediment losses, the location and size of the salients remained relatively unchanged. The salients have remained on the Grand Isle shoreline as shown by an aerial photograph from 1998 (Figure 22). It appears that the eastern salient has decreased in size while the western salient has remained the same size or even become larger. Figure 22: Aerial photograph showing salients shoreward of borrow area along Grand Isle, Louisiana, in 1998 (modified from Louisiana Oil Spill Coordinator's Office (LOSCO), 1999). A series of detached offshore breakwaters was constructed along the eastern part of Grand Isle in the 1990's, which terminate at the eastern salient and may have affected its shape. Bathymetric surveys taken through the borrow area in February 1985 and August 1986 revealed that the outer lobes had filled to about half their original depth and the channel connecting the lobes had reached the sea bed elevation (Combe and Soileau, 1987). Currently, the borrow area is reported to be completely filled by fine material (Combe, personal correspondence) which would have required the same approximate volume of sediment that was dredged for the initial placement. Although the origin of the sediment that has refilled the borrow pit is not known, it is reported to be finer than the sediment dredged for the nourishment project, indicating that the material did not originate from the project. While no longer a bathymetric anomaly, the borrow areas are reported to continue to modify the wave field as local shrimpers use the waters shoreward of the pit as a harbor to weather storms. The reason for the sheltering effect of the filled pit may be due to the energydissipating characteristic of the finer material that has filled the pit. 2.1.2 Anna Maria Key, Florida (1993) The 1993 beach nourishment project at Anna Maria Key, Florida, is another example of a project with a large borrow area lying offshore in relatively shallow water. The project placed 1.6x106 m3 of sediment along a 6.8 km segment (DNR Monuments R 12 to R35*) of the 11.6 km long barrier island (Dean et al., 1999). The borrow area for the project was approximately 3,050 m long and ranged from 490 to 790 m offshore in approximately 6 m of water (Dean et al., 1999). A planview of the bathymetry near the project including the borrow area is shown in Figure 23. A transect through the borrow area, indicated in the previous figure at Monument R26, is shown in Figure 24 and shows dredging to a depth of 3.1 m below the local seabed. This figure shows one pre project transect, a transect immediately following completion, and two postnourishment transects. The postnourishment transects indicate minimal infilling of the borrow pit. The "DNR Monuments" are permanent markers spaced at approximately 300 m along the Florida sandy beaches for surveying purposes The shoreline planform was found to show the greatest losses shoreward of the borrow area. Figure 25 shows the shoreline position relative to the August, 1993, data for seven different periods. A large area of negative shoreline change indicating erosion is found from DNR Monument numbers 25 to 34 for the July, 1997, and February, 1998, data. This area lies directly shoreward of the borrow area shown in Figure 23. The behavior of the shoreline directly leeward of the borrow area is seen to be the opposite of the Grand Isle, Louisiana, response where shoreline advancement occurred. Volume changes determined from profiles in the project area did not show large negative values near the southern end of the project. The difference between the shoreline and volume changes at the southern end of the project implies that the constructed profiles may have been steeper near the southern end of the project as compared to those near the northern end (Wang and Dean, 2001). The proximity of the borrow area to the shoreline is one possible contribution to the local erosion. Although the reason for the increased erosion in this area is not clear, it is interesting that the anomalous shoreline recession did not occur until the passage of Hurricanes Erin and Opal in August and October, 1995, respectively. Hurricane Opal was a category 4 hurricane with sustained winds of 67 m/s when it passed 600 km west of Manatee County (Liotta, 1999). A reported storm surge of 0.3 to 1.0 m, combined with the increased wind and wave action, resulted in overtopping of the beach berm, flooding of the back area of the project and transport of sediment to the back beach or offshore (Liotta, 1999). The average shoreline retreat for the project area was approximately 9.1 to 15.2 m, based on observations (Liotta, 1999). Passage Key Inlet 13 o R14 *R15 eR18 17 R18 19 *R20 OR21 03 22 q 23 24 25 *R26 27 28 T30 *R31 ,R32 .R33A *R34 S,,R35 36 *R37 *R38 \ R&R39 R40 41 Longboat Pass Figure 23: Bathymetry off Anna Maria Key, Florida, showing location of borrow pit following beach nourishment project (modified from Dean et al., 1999). 9.15 6.10 3.05 0 3.05 6.10 9.15  0 152.4 304.8 457.2 609.6 Distance from Monument (m Figure 24: Beach profile through borrow area at R26 in / (modified from Wang and Dean, 2001). 20 .. ........ ........ ..... ............ 7. , Longshre Exi ofBorriPit Sii 1 0 ............ ...... .... 40 .... ^ ..... ........ .. ... ... ......i.......  P Inist 5 10 15 20 25 30 DNR Moumrent No. Figure 25: Shoreline position for Anna Maria Key Project to August, 1993 (modified from Dean et al. 1999). 762.0 914.4 a Maria Key, Florida Mnna Maria Key, Florida for different periods relative 2.1.3 Martin County, Florida (1996) The Hutchinson Island beach nourishment project in Martin County, Florida was constructed in 1996 with the placement of approximately 1.1x106 m3 of sediment along 6.4 km of shoreline, between DNR Monuments R1 and R25 (Sumerell, 2000). The borrow area for this project was a shoal rising 4.9 m above the adjacent bed and lying 910 m offshore in 12.8 m of water (Sumerell, 2000). Figure 26 shows the borrow area location offshore of the southern end of the project area. An average of 3 m of sediment was dredged from the central portion of the shoal. N~il~w PHOJ( I I ArtANiIC OCEAN BORROW AREA Figure 26: Project area for Martin County beach nourishment project (Applied Technology and Management, 1998). The 3year and 4year postnourishment shoreline surveys show reasonable agreement with modeling conducted for the project, except at the southern end, near the borrow area (Sumerell, 2002). Figure 27 shows the predicted shoreline and the survey data for the 4year shoreline change. This case differs from the previous two as the borrow area did not create a pit but reduced the height of an offshore shoal. By lowering the height of the shoal the shoreline leeward of the borrow area was exposed to greater wave action, which is the opposite of the sheltering (through reflection) effect of an offshore pit. The borrow area, with its large extent and proximity to the project, is a possible reason for the higher than expected erosion at the southern end of the project. 30.5 m Predicted 15.2 Survey Data (December 99) E 0 3 15.2 30.5 45.7 Figure 27: Fouryear shoreline change for Martin County beach nourishment project: predicted versus survey data (modified from Sumerell, 2000). 2.2 Field Experiments Field studies have been conducted to examine the effects of offshore dredging on the coastal environment. The purposes of the these studies have varied and include the tendency of a dredged pit to induce sediment flows into it from the nearshore, the interception of sediment transport, and wave transformation effects of a newly dredged pit on the shoreline. 2.2.1 Price et al. (1978) Price et al. (1978) investigated the effect of offshore dredging on the coastline of England. The tendency of a dredge pit to cause a drawdown of sediment and to prevent the onshore movement of sediment was investigated. The study by Inman and Rusnak (1956) on the onshoreoffshore interchange of sand off La Jolla, California, was cited. This threeyear study found vertical bed elevation changes of only +/ 0.03 m at depths greater than 9 m. Based on the consideration that the wave conditions off the southern coast of England would be less energetic than off La Jolla, California, Price et al. (1978) concluded that beach drawdown at a depth greater than 10 m would not occur. A radioactive tracer experiment off Worthing, on the south coast of England, was performed to investigate the mobility of sediment at depths of 9, 12, 15, and 18 m. The 20month study found that at the 9 and 12 m contours there was a slight onshore movement of sediment and it was concluded that the movement of sediment beyond a depth contour of 18 m on the south coast of England would be negligible. Therefore, at these locations and in instances when the onshore movement of sediment seaward of the dredge area is a concern, dredging in water beyond 18 m depth below low water level was considered acceptable (Price et al., 1978). A numerical model of the shoreline change due to wave refraction over dredged holes was also employed in the study, the details of which will be examined later in Section 2.5.2.1. The model found that minimal wave refraction occurred for pits in depths greater than 14 m for wave conditions typical off the coast of England. 2.2.2 Koiima et al. (1986) The impact of dredging on the coastline of Japan was studied by Kojima et al. (1986). The wave climate as well as human activities (dredging, construction of structures) for areas with significant beach erosion and/or accretion was studied in an attempt to determine a link between offshore dredging and beach erosion. The study area was located offshore of the northern part of Kyushu Island. The wave climate study correlated yearly fluctuations in the beach erosion with the occurrence of both storm winds and severe waves and found that years with high frequencies of storm winds were likely to have high erosion rates. A second study component compared annual variations in offshore dredging with annual beach erosion rates and found strong correlation at some locations between erosion and the initiation of dredging although no consistent correlation was identified. Hydrographic surveys documented profile changes of dredged holes over a four year period. At depths less than 30 m, significant infilling of the holes was found, mainly from the shoreward side, indicating a possible interruption in the longshore and offshore sediment transport. This active zone extends to a much larger depth than found by Price et al. (1978) and by Inman and Rusnak (1956). The explanation by Kojima et al. is that although the active onshore/offshore region does not extend to 30 m, sediment from the ambient bed will fill the pit causing a change in the supply to the upper portion of the beach and an increase in the beach slope. Changes in the beach profiles at depths of 35 and 40 m were small, and the holes were not filled significantly. Another component of the study involved tracers and seabed level measurements to determine the depths at which sediment movement ceases. Underwater photographs and seabed elevation changes at fixed rods were taken at 5 m depth intervals over a period of 3 months during the winter season for two sites. The results demonstrated that sediment movement at depths up to 35 m could be significant. This depth was found to be slightly less than the average depth (maximum 49 m, minimum 20 m) for five proposed depth of closure equations using wave inputs with the highest energy (H = 4.58 m, T = 9.20 s) for the 3month study period. 2.3 Laboratory Experiments 2.3.1 Horikawa et al. (1977) Laboratory studies have been carried out to quantify wave field and nearshore modifications due to the presence of offshore pits. Horikawa et al. (1977) performed wave basin tests with a model of fixed offshore bathymetry and uniform depth containing a rectangular pit of uniform depth and a nearshore region composed of moveable lightweight sediments. The experimental arrangement is shown in Figure 28. The incident wave period and height were 0.41 s and 1.3 cm, respectively. With the pit covered, waves were run for 5.5 hours to obtain an equilibrium planform followed by wave exposure for three hours with the pit present. Shoreline measurements were conducted at 1 hour intervals to determine the pit induced changes. U 400 300 200 100 0 c ,,120 1 5 1o 100 30 C BEACH .60 1 SHORE INEni tO LS 0 (unit:cm) 20 400 300 200 100 0 Offshore distance Figure 28: Setup for laboratory experiment (Horikawa et al., 1977). The results of the experiment are presented in Figure 29. Almost all of the shoreline changes with the pit present occurred in the first two hours. At the still water level, a salient formed shoreward of the pit, flanked by two areas of erosion that generally extend to the sidewalls of the experiment; however, the depth contour at a water depth, h = 0.85 cm, also shown in Figure 29, shows only a slightly seaward displacement at the pit centerline. HOLE "" 30emc. or 20  h 0.5 cm 0 C. 0 Initial shoreline (h0) h 0 cm 0 20 40 6C 80 100 120 Longshore distance, X (cm) Figure 29: Results from laboratory experiment showing plan shape after two hours (Horikawa et al., 1977). 2.3.2 Williams (2002) Williams (2002) performed wave basin experiments similar to those of Horikawa et al. (1977). The experimental setup of a fixed bed model containing a pit with a moveable sand shoreline was constructed for similar trials by Bender (2001) and was a larger scale version of the Horikawa et al. (1977) arrangement. The Williams experimental procedure consisted of shoreline, bathymetric, and profile measurements after specified time intervals that comprised a complete experiment. Figure 210 shows the experiment progression sequence that was used. For analysis, the shoreline and volume measurements were made relative to the last measurements of the previous 6 hour phase. The conditions for the experiments were 6 cm waves with 1.35 s period and a depth of 15 cm in the constant depth region surrounding the pit. The pit was 80 cm long in the crossshore direction, 60 cm in the longshore direction and 12 cm deep relative to the adjacent bottom. CompLete Experiment Current Control Next Control Phases Phases Previous Test (covered pit) Current Test (covered pit) Phases Phases (open pit) (open pit) I I I II I I 6.0 7.5 9.0 0.0 1.5 3.0 6.0 7.5 9.0 12.0 1.5 3,0 6.0 (12.0) (0.0) Time Step (Hours) Figure 210: Experiment sequence timeline for Williams laboratory experiments (Williams, 2002) Shoreline and volume change results were obtained for three experiments. Figure 211 shows the volume change per unit length versus longshore distance for results with pit covered (control phase) and uncovered (test phase). The dashed line represents volume changes for a covered pit relative to the time step zero which concluded 6 hours of waves with the pit uncovered. The solid line shows the change with the pit uncovered relative to time step six when six hours of wave exposure with the pit covered ended. The volume change results show the model beach landward of the pit lost volume at almost every survey location during the period with the pit covered and experienced a gain in volume with the pit uncovered. S  150    It ~........ vQ.. ; "^..... 'E 50 0 ScE 1 O 120 100 80 60 40 >20 0 20 40 60 A'0 100 120 1 0 1 20 40 6018e )100 12010 150 ~ r    4100 '  Longshore Position (cm) S 0 to 6 Hours 6 to 12 Hours (pit covered) (pit present) Figure 211: Volume change per unit length tor thirst expenment (Williams, 2002). The net volume change for the first complete experiment (control and test phase) was approximately 2500 cm3. Different net volume changes were found for the three experiments. However, similar volume change per unit length results were found in all three experiments indicating a positive volumetric relationship between the presence of the pit and the landward beach. The shoreline change results showed shoreline retreat, relative to Time Step 0.0, in the lee of the borrow pit during the control phase (pit covered) for all three experiments with the greatest retreat at or near the centerline of the borrow pit. All three experiments showed shoreline advancement in the lee of the borrow pit with the pit uncovered (test phase). With the magnitude of the largest advancement being almost equal to the largest retreat in each experiment, it was concluded that, under the conditions tested, the presence of the borrow pit resulted in shoreline advancement for the area shoreward of the borrow pit (Williams, 2002). An evenodd analysis was applied to the shoreline and volume change results in an attempt to isolate the effect of the borrow pit. The even function was assumed to represent the changes due solely to the presence of the borrow pit. The even components were adjusted to obtain equal positive and negative areas, which were not obtained using the laboratory data. For each experiment, the shifted even results shoreward of the pit showed positive values during the test phase for both the shoreline and volume changes with negative values during the control phase. The shifted even component of shoreline change for the first experiment is shown in Figure 212. These results further verify the earlier findings concerning the effect of the pit. 0 0 Ur U) w Longshore Position (cm) * 0 to 6 Hours 6 to 12 Hours (pit covered) (pit present) Figure 212: Shifted even component of shoreline change for first experiment (Williams, 2002). 2.4 Wave Transformation 2.4.1 Analytic Methods There is a long history of the application of analytic methods to determine wave field modifications by bathymetric changes. Early research centered on the effect on normally incident long waves of an infinite step, trench or shoal of uniform depth in an otherwise uniform depth domain. More complex models were later developed to remove the long wave restriction, add oblique incident waves and allow for the presence of a . ..... .... .. .. t 5    9 10 t0 120 100 ; 60 20 0 20 60 100 120 1 1   current. More recently, many different techniques have been developed to obtain solutions for domains containing pits or shoals of finite extent. Some of these models focused solely on the wave field modifications, while others of varying complexity examined both the wave field modifications and the resulting shoreline impact. 2.4.1.1 2Dimensional methods By matching surface displacement and mass flux normal to the change in bathymetry Lamb (1932) was one of the first to develop a long wave approximation for the reflection and transmission of a normally incident wave at a finite step. Bartholomeauz (1958) performed a more thorough analysis of the finite depth step problem and found that the Lamb solution gave correct results for the reflection and transmission coefficients for lowest order (kh) where k is the wave number and h is the water depth prior to the step. Sretenskii (1950) investigated oblique waves over a step between finite and infinite water depths assuming the wavelength to be large compared to the finite depth. An extensive survey of early theoretical work on surface waves including obstacle problems is found in Wehausen and Laitone (1960). Kreisel (1949) developed a method that conformally mapped a domain containing certain obstacles of finite dimensions into a rectangular strip. The reflection from obstacles at the surface and on the bottom were considered. Kreisel (1949) presents a proof demonstrating that the reflection coefficient is independent of the incident wave direction for a symmetric (upwave and downwave depth equal) or asymmetric obstacle (trench or shoal) in arbitrary water depth. Jolas (1960) studied the reflection and transmission of water waves of arbitrary relative depth over a long submerged rectangular parallelepiped (sill) and performed an experiment to document the wave transformation. To solve the case of normal wave incidence and arbitrary relative depth over a sill or a fixed obstacle at the surface Takano (1960) used an eigenfunction expansion of the velocity potentials in each constant depth region and matched them at the region boundaries. The set of linear integral equations was solved for a truncated series. A laboratory experiment was also conducted in this study. Dean (1964) investigated long wave modification by linear transitions. The linear transitions included both horizontal and vertical changes. The formulation allowed for many domains including a step, either up or down, and converging or diverging linear transitions with a sloped wall. A proposed solution was defined with planewaves of unknown amplitude and phase for the incident and reflected waves with the transmitted wave specified. Wave forms, both transmitted and reflected, were represented by Bessel functions in the region of linear variation in depth and/or width. The unknown coefficients were obtained through matching the values and gradients of the water surfaces at the ends of the transitions. Analytic expressions were found for the reflection and transmission coefficients. The results indicate that the reflection and transmission coefficients depend on the relative depth and/or width and a dimensionless parameter containing the transition slope, the wavelength and the depth or width (Figure 213). In Figure 213 the parameter Z, = 4' for the case of linearly varying depth and L, S, Z, = for linearly varying breadth where I indicates the region upwave of the LS, transition, Sv is equal to the depth gradient, and SH is equal to onehalf the breadth gradient. These solutions were shown to converge to those of Lamb (1932) for the case of an abrupt transition (ZI=0). ]/ ,^ I JI __/I t L' 44' "fc!t i  001 002 005 0.1 02 05 I 2 5 10 20 50 100 200 S00 100 Value of hi/hiI b2 /b III Figure 213: Reflection and transmission coefficients for linearly varying depth [hi/hill] and linearly varying breadth [bi2/ b,12] (modified from Dean, 1964). Newman (1965a) studied wave transformation due to normally incident waves on a single step between regions of finite and infinite water depth with an integralequation approach. This problem was also examined by Miles (1967) who developed a plane wave solution for unrestricted kh values using a variational approach (Schwinger & Saxon 1968), which for this case essentially solves a single equation instead of a series of equations (up to 80 in Newman's solution) as in the integral equation approach. The difference between the results for the two solution methods was within 5 percent for all kh values (Miles, 1967). Newman (1965b) examined the propagation of water waves past long obstacles. The problem was solved by constructing a domain with two steps placed "back to back" and applying the solutions of Newman (1965a). Complete transmission was found for certain water depth and pit length combinations; a result proved by Kriesel (1949). Figure 214 shows the reflection coefficient, Kr, and the transmission coefficient, Kt, versus Kooho where K, is the wave number in the infinitely deep portion before the obstacle and ho is the depth over the obstacle. The experimental results of Takano (1960) are included for comparison. It is evident that the Takano experimental data included energy losses. Numerical results of Newman (1965b) 0'7 o Takano (1960) experimental results 0.6 i05 Kr 0 0.2 0 06 08 10 12 14 1 1.8 24, Figure 214: Approximate reflection and transmission coefficients for the rectangular parallelepiped of length 8.86ho in infinitely deep water (Newman, 1965b). The variational approach was applied by Mei and Black (1969) to investigate the scattering of surface waves by rectangular obstacles. For a submerged obstacle, complete transmission was found for certain kho values where ho is the depth over the obstacle. A comparison of the results of Mei and Black (1969) and those of Newman (1965b) is shown in Figure 215, which presents the reflection coefficient versus kho for a submerged obstacle. Data from the Jolas (1960) experiment are also included on the plot and compared to the results of Mei and Black (1969) for a specific i / ho, where is the halflength of the obstacle. [Mei and Black (1969)], o [data from Jolas (1960) experiment] O'S ( / ho =4.43, h/ho= 2.78) [Mei and Black (1969)],  [Newman i i \ ii 0 02 0 02 04 0 6 08 10 12 14 16 kho Figure 215: Reflection coefficient for a submerged obstacle (Mei and Black, 1969). Black and Mei (1970) applied the variational approach to examine the radiation caused by oscillating bodies and the disturbance caused by an object in a wave field. Two domains were used for both submerged and semiimmersed (surface) bodies: the first domain was in Cartesian coordinates, with one vertical and one horizontal dimension, for horizontal cylinders of rectangular cross section and the second domain was in cylindrical coordinates, for vertical cylinders of circular section. The second domain allowed for objects with two horizontal dimensions to be studied for the first time (see Section 2.4.1.2.). Black et al. (1971) used the variational formulation to study the radiation due to the oscillation of small bodies and the scattering induced by fixed bodies. Black et al. demonstrated the scattering caused by a fixed object in a single figure; see Black and Mei (1970) for further results. Lassiter (1972) used complementary variational integrals to solve the problem of normally incident waves on an infinite trench where the depth on the two sides of the trench may be different (the asymmetric case). The symmetric infinite trench problem was studied by Lee and Ayer (1981), who employed a transform method. The fluid domain was divided into two regions, one an infinite uniform depth domain and the other a rectangular region representing the trench below the uniform seabed level. The transmission coefficient for the trench is shown in Figure 216 with the theoretical results plotted along with data from a laboratory experiment conducted as part of the study. Results from a boundary integral method used to compare with the theoretical results are also plotted. The results show six of an infinite number of relative wavelengths where complete transmission (Kt=I) will occur, a result that had been found in prior studies (Newman (1965b), Mei and Black (1969)). The laboratory data show the general trend of the theoretical results, with some variation due to energy losses and reflections from the tank walls and ends. Lee et al. (1981) proposed a boundary integral method for the propagation of waves over a prismatic trench of arbitrary shape, which was used for comparison to selected results in Lee and Ayer (1981). The solution was found by matching the unknown normal derivative of the potential at the boundary of the two regions. A comparison to previous results for trenches with vertical sidewalls was conducted with good agreement. A case with bathymetry containing gradual transitions in depth was shown in a plot of the transmission coefficient for a trapezoidal trench (Figure 217). Note that the complete dimensions of the trapezoidal trench are not specified in the inset diagram, making direct comparison to the results impossible. So00 095  0.90 S [Numerical Solution] 085 S [Experimental Results] i x [Boundary Integral Method] S080 0.75 0.70 .. 0 005 010 0 15 020 025 Depth to wavelength ratio (h/X) Figure 216: Transmission coefficient as a function of relative wavelength (h=l0.O cm, d=67.3 cm, trench width =161.6 cm) (modified from Lee and Ayer, 1981). Miles (1982) solved for the diffraction by an infinite trench for obliquely incident long waves. The solution method for normally incident waves used a procedure developed by Kreisel (1949) that conformally mapped a domain containing certain obstacles of finite dimensions into a rectangular strip. To add the capability of solving for obliquely incident waves, Miles used the variational formulation of Mei and Black (1969). C cOr o'.. .0. 0 0. 0 . 00 so C C U.90 * t r 15.2 cm 0 03." C,8,5 i .......... 161.6 cm   0.5 1 0. 15 0.20 C. 2 Figure 217: Transmission coefficient as a function of relative wavelength for trapezoidal trench; setup shown in inset diagram (modified from Lee et al., 1981). The problem of obliquely incident waves over an asymmetric trench was solved by Kirby and Dalrymple (1983a) using a modified form of Takano's (1960) method. Figure 218 compares the reflection coefficient for the numerical solution for normally incident waves and the results of Lassiter (1972). The results from a boundary integral method used to provide verification are included. Differences in the results of Kirby and Dalrymple and those of Lassiter are evident. Lee and Ayer (1981,[see their Figure 2]) also demonstrated differences in their results and those ofLassiter (1972). The effect of oblique incidence is shown in Figure 219 where the reflection and transmission coefficients for two angles of incidence are plotted. 0.5 04 0.7 0.8 0. 0. 0.3 0.4 0.5 0.6 Kh, Figure 218: Reflection coefficient for asymmetric trench and normally incident waves as a function of KhI: h2/hl=2, h3/h1=0.5, L/hi=5; L = trench width (Kirby and Dalrymple, 1983a). /  [Numerical solution, 01 = 0 deg] K0.9\ / [Numerical solution, 1 =45 deg] khi Figure 219: Transmission coefficient for symmetric trench, two angles of incidence: L/hi=10, h2/hl=2; L = trench width (modified from Kirby and Dalrymple, 1983a). This study also investigated the planewave approximation and the longwave limit, which allowed for comparison to Miles (1982). Figure 220 shows transmission coefficients with the results of the numerical solution, the long wave solution, and values from the Miles (1982) solution, which is only valid for small kh values in each region. For the first case, with a small relative trench width, the numerical results from Kirby and [Kirby and Dalrymple (1983a)] , [Lassiter (1972, Fig. 7)] [Boundary Integral Method] ' ... "xx.,, *~ \': ,,, ' 0Ii Dalrymple compare well with the results using the Miles (1982) method and the plane wave solution is seen to deviate from these. For the case of a relative trench length equal to eight, the numerical results differ from the planewave solution, which diverge from the values using Miles (1982) for this case where the assumptions are violated. 1.000__ DO, Q)' 0.994 1.00 [Long Wave Solution] * [Numerical Solution] [Miles (1982) Solution] h2/hi Figure 220: Transmission coefficient as a function of relative trench depth; normal incidence: klhl= 0.2: (a) L/hi=2; (b) L/hi=8, L = trench width. (Kirby and Dalrymple, 1983a). The difference in scales between the two plots is noted. An extension of this study is found in Kirby et al. (1987) where the effects of currents flowing along the trench are included. The presence of an ambient current was found to significantly alter the reflection and transmission coefficients for waves over a trench compared to the no current case. Adverse currents and following currents made a trench less reflective and more reflective, respectively (Kirby et al., 1987). 2.4.1.2 3Dimensional methods Extending the infinite trench and step solutions (one horizontal dimension) to a domain with variation in the longshore (twohorizontal dimensions) is a natural progression allowing for the more realistic case of wave transformation by a finite object or depth anomaly to be studied. Changes in bathymetry can cause changes in wave height and direction through the four wave transformation processes noted earlier. Some of the twodimensional models study only the wave transformation, while others use the modified wave field to determine the impact of a pit or shoal on the shoreline. Several models use only a few equations or matching conditions on the boundary of the pit or shoal to determine the wave field and in some cases the impact on the shoreline in a simple domain containing a pit or shoal. Other, much more complex and complete models and program packages have been developed to solve numerically for the wave field over a complex bathymetry, which may contain pits and/or shoals. Both types of models can provide insight into the effect of a pit or shoal on the local wave field and the resulting impact on the shoreline. The wave transformation in a threedimensional domain was investigated in a study by Black and Mei (1970), which solved for the radially symmetric case of a submerged or floating circular cylinder in cylindrical coordinates. A series of Bessel functions was used for the incident and reflected waves, as well as for the solution over the shoal with modified Bessel functions representing the evanescent modes. As mentioned previously in the 2D section, a variational approach was used and both the radiation by oscillating bodies and the disturbance caused by a fixed body were studied. The focus of the fixed body component of the study was the total scattering cross section, Q, which is equal to the width between two wave rays within which the normally incident wave energy flux would be equal to that scattered by the obstacle and the differential scattering crosssection, which shows the angular distribution of the scattered energy (Black and Mei, 1970). Figure 221 shows the total scattering cross section for a circular cylinder at the seabed for three ratios of cylinder radius (a) to depth over the cylinder (h). 20 a/h=3 16 12 0.8 a/h=2 04  a/h=l 0 1 2 3 4 5 6 ka Figure 221: Total scattering cross section of vertical circular cylinder on bottom (modified from Black et al., 1971). Williams (1990) developed a numerical solution for the modification of long waves by a rectangular pit using Green's second identity and appropriate Green's functions in each region that comprise the domain. This formulation accounts for the diffraction, refraction and reflection caused by the pit. The domain for this method consists of a uniform depth region containing a rectangular pit of uniform depth with vertical sides. The solution requires discretizing the pit boundary into a finite number of points at which the velocity potential and the derivative of the velocity potential normal to the boundary must be determined. Applying matching conditions for the pressure and mass flux across the boundary results in a system of equations amenable to matrix solution techniques. Knowledge of the potential and derivative of the potential at each point on the pit boundary allows determination of the velocity potential solution anywhere in the fluid domain. The effect of a pit on the wave field is shown in a contour plot of the relative amplitude in Figure 222. A partial standing wave pattern of increased and decreased relative amplitude is seen seaward of the pit with a shadow zone of decreased wave amplitude landward of the pit flanked by two areas of increased relative amplitude. McDougal et al. (1996) applied the method of Williams (1990) to the case of a domain with multiple pits. The first part of the study reinvestigated the influence of a single pit on the wave field for various pit geometries. A comparison of the wave field in the presence of a pit versus a surface piercing structure is presented in Figures 223 and 224, which present contour plots of the transformation coefficient, K, (equal to relative amplitude) that contain the characteristics discussed in the last paragraph. For this case with the pit depth equal to 3 times the water depth a greater sheltering effect is found (K = 0.4) landward of the pit than for the case of the full depth breakwater. An analysis of the effect of various pit characteristics on the minimum value of K found in the domain was also performed. The dimensionless pit width, a/L, (a = cross shore dimension, L = wave length outside pit) was found to increase the distance to the region where K < 0.5 behind the pit and the value of K was found to decrease and then become approximately constant as a/L increases. The minimum values of the transformation coefficient for a wide pit are much lower than those values found in Lee and Ayer (1981) and Kirby and Dalrymple (1983a), which may be explained by the refraction divergence that occurs behind the pit in the 2D case (McDougal et al., 1996). :4 1 I. .T i i i I K i I 1 i i .. i i .i. :...:.. . ,., .' : : . ..... ....... S. ..1 107*. . .. ..:. . . . .: ... . ( .... : ,. ... : . .... ...........:... ........ :................... 1.09786 _ 'i " : P S= n 1::,::::::::1 M::::::: 5, = 1, 064. aji j. j. i ii !: i : I: I.i i: i i i i tLLI i.jL _. I.I I'^i^': ^':!'^ ;:i:.;^^ longshore pit length, h = water depth outside pit, d = depth inside pit, L2 = wavelength outside pit. (Williams, 1990). The effect of the dimensionless pit length, b/L, indicates that K decreases as b/L increases to 1 with a change in the trend, and an increase in K, from b/L = 0.55 to b/L = 0.65. Increasing the dimensionless pit depth, d/L, was found to decrease the minimum value of K with a decreasing rate. The incident wave angle was not found to significantly alter the magnitude or the location of the minimum K value, although the width of the shadow zone changes with incident angle. For the case of multiple pits, it was found that placement of one pit in the shadow zone of a more seaward pit was most effective in reducing the wave height. However, adding a third pit did not produce significant wave height reduction as compared to the twopit results. Williams and Vazquez (1991) removed the long wave restriction of Williams (1990) and applied the Green's function solution method outside of the pit. This solution was matched to a Fourier expansion solution inside the pit with matching conditions at the pit boundary. Once again the pit boundary must be discretized into a finite number of points and a matrix solution for the resulting series of equations was used. Removing the shallow water restriction allowed for many new cases to be studied, as the wave conditions approach deep water, the influence of the pit diminishes. A plot of the minimum and maximum relative amplitude found in the domain versus the dimensionless pit length (the wave number outside of the pit times the crossshore pit dimension, koa) is shown in Figure 225. The maximum and minimum relative amplitudes in Figure 225 are seen to occur near koa = 2n or when L = a and then approach unity as the dimensionless pit length increases. The reason that the extreme values do not occur exactly at koa = 27r is explained by Williams and Vazquez (1991) as due to diffraction effects near the pit modifying the wave characteristics. Figure 223: Contour plot of diffraction coefficient in and around pit for normal incidence; a/L=l, b/L=0.5, d/h =3, kh=0.167 (McDougal et al., 1996). Figure 224: Contour plot of diffraction coefficient around surfacepiercing breakwater for normal incidence; a/L=l, b/L=0.5, kh=0.167 (McDougal et al., 1996). 1K, ,n M Min "4 0.4 0 2 4 6 0 10 12 14 1s 1i 20 koa Figure 225: Maximum and minimum relative amplitudes for different koa, for normal incidence, a/b=6, a/d=7r, and d/h=2. (modified from Williams and Vasquez, 1991). 2.4.2 Numerical Methods The previous threedimensional solutions, while accounting for most of the wave transformation processes caused by a pit, are simple in their representation of the bathymetry and their neglect of many waverelated processes including energy dissipation. Berkhoff (1972) developed a formulation for the 3dimensional propagation of waves over an arbitrary bottom in a vertically integrated form that reduced the problem to twodimensions. This solution is known as the mild slope equation and different forms of the solution have been developed into parabolic (Radder, 1979), hyperbolic, and elliptic (Berkhoff et al., 1982) models of wave propagation, which vary in their approximations and solution techniques. Numerical methods allow solution for wave propagation over an arbitrary bathymetry. Some examples of the parabolic and elliptic models are RCPWAVE (Ebersole et al., 1986), REF/DIF1 (Kirby and Dalrymple, 1994), and MIKE 21's EMS Module (Danish Hydraulics Institute, 1998). Other models such as SWAN (Holthuijsen et al., 2000) and STWAVE (Smith et al., 2001) model wave transformation in the nearshore zone using the waveaction balance equation. These models provide the capability to model wave transformation over complicated bathymetries and may include processes such as bottom friction, nonlinear interaction, breaking, wavecurrent interaction, windwave growth, and white capping to better simulate the nearshore zone. An extensive review of any of the models is beyond the scope of this paper; however, a brief outline of the capabilities of some of the models is presented in Table 1. Maa et al. (2000) provides a comparison of six numerical models. Two parabolic models are examined: RCPWAVE and REF/DIF1. RCPWAVE employs a parabolic approximation of the elliptic mild slope equation and assumes irrotationality of the wave phase gradient. REF/DIF1 extends the mild slope equation by including nonlinearity and wavecurrent interaction (Kirby and Dalrymple, 1983b; Kirby, 1986). Of the four other models included, two are defined by Maa et al. (2000) as based on the transient mild slope equation (Copeland, 1985; Madsen and Larson, 1987) and two are classified as elliptic mild slope equations (Berkhoff et al., 1982) models. The transient mild slope equation models presented are Mike 21's EMS Module and the PMH Model (Hsu and Wen, in review). The elliptic mild slope equation models use different solution techniques with the RDE Model (Maa and Hwung, 1997; Maa et al. 1998a) applying a special Gaussian elimination method and the PBCG Model employing a Preconditioned Biconjugate Gradient method (Maa et al., 1998b). Table 21: Capabilities of selected nearshore wave models. RCPWAVE REF/DIF1 Mike 21 EMS STWAVE SWAN 3rd Gen. Elliptic Mild Parabolic Parabolic Elliptic Mild Conservation Conservation Solution Slope Equation Mild Slope Mild Slope of Wave of Wave Method (Berkhoff et al. Equation Equation Action Action (1972) Phase Averaged Resolved Resolved Averaged Averaged No No Spectral No Yes Yes Use REF/DIFS Use NSW unit Shoaling Yes Yes Yes Yes Yes Refraction Yes Yes Yes Yes Yes Yes Yes Yes No Diffraction No (SmallAngle) (WideAngle) (Total) (Smoothing) Yes Yes Yes Reflection No No (Forward only) (Total) (Specular) Stable Energy Stable Energy Bore Model: Depth limited: Bore Model: Breaking Flux: Dally et Flux: Dally et al. Battjes & Miche (1951) Battjes & Janssen al. (1985) (1985) Janssen (1978) criterion (1978) White Komen et al. (1984), No No No Resio (1987) Janssen (1991), capping Komen et al. (1994) Dalrymple et al. Quadratic Hasselmann et al. Bottom (1984) both Friction Law, (1973), Collins No No Friction laminar and Dingemans (1972), Madsen et turbulent BBL (1983) al. (1988) Currents No Yes No Yes Yes Cavaleri & MalanotteRizzoli Wind No No No Resio (1988) (1981), Snyder et al., (1981), Janssen et al. (1989, 1991) Availability Commercial Free Commercial Free Free A table in Maa et al. (2000) provides a comparison of the capabilities of the six models. A second table summarizes the computation time, memory required and, where required, the number of iterations for a test case of monochromatic waves over a shoal on an incline; the Berkhoffet al. (1982) shoal. The parabolic approximation solutions of REF/DIF and RCPWAVE required significantly less memory (up to 10 times less) and computation time (up to 70 times less) than the elliptic models, which is expected due to the solution techniques and approximations contained in the parabolic models. The required computation times and memory requirements for the transient mild slope equation models were found to be intermediate to the other two methods. Wave height and direction were calculated in the test case domain for each model. The models based on the transient mild slope equation and the elliptic mild slope equation were found to produce almost equivalent values of the wave height and direction. The parabolic approximation models were found to have different values, with RCPWAVE showing different wave heights and directions behind the shoal and REF/DIF showing good wave height agreement with the other methods, but no change in the wave direction behind the shoal. Plots of the computed wave heights for the six models and experimental data along one transect taken perpendicular to the shoreline and one transect parallel to the shoreline are shown in Figures 226 and 227. Only four results are plotted because the RDE model, the PMH model and PBCG model produced almost identical results. The wave directions found with REF/DIF1 in Maa et al. (2000) were found to be in error by Grassa and Flores (2001), who demonstrated that a second order parabolic 44 model, equivalent to REF/DIF1 was able to reproduce the wave direction field behind a shoal such as in the Berkhoff et al. (1982) experiment. 5 10 Longshore (m) Figure 226: Comparison of wave height profiles for selected models along transect parallel to shore located 9 m shoreward of shoal apex [=experimental data] (Maa et al., 2000). 15 4 8 10 12 14 Cross Shore (m) Figure 227: Comparison of wave height profiles for selected models along transect perpendicular to shore and through shoal apex [*=experimental data] (Maa et al., 2000). 3, 1  ~ Application of numerical models to the problem of potential impact on the shoreline caused by changes to the offshore bathymetry was conducted by Maa and Hobbs (1998) and Maa et al. (2001). In Maa and Hobbs (1998) the impact on the coast due to the dredging of an offshore shoal near Sandbridge, Virginia was investigated using RCPWAVE. National Data Buoy Center (NDBC) data from an offshore station and bathymetric data for the area were used to examine several cases with different wave events and directions. The resulting wave heights, directions, and sediment transport at the shoreline were compared. The sediment transport was calculated using the formulation of Gourlay (1982), which contains two terms, one driven by the breaking wave angle and one driven by the gradient in the breaking wave height in the longshore direction. Section 2.5.4 provides a more detailed examination of the longshore transport equation with two terms. The study found that the proposed dredging would have little impact on the shoreline for the cases investigated. Later, Maa et al. (2001) revisited the problem of dredging at the Sandbridge Shoal by examining the impact on the shoreline caused by three different borrow pit configurations. RCPWAVE was used to model the wave transformation over the shoal and in the nearshore zone. The focus was on the breaking wave height; wave direction at breaking was not considered. The changes in the breaking wave height modulation (BHM) along the shore after three dredging phases were compared to the results found for the original bathymetry and favorable or unfavorable assessments were provided for ensuing impact on the shoreline. The study concluded that there could be significant differences in the wave conditions, revealed by variations in the BHM along the shoreline depending on the location and extent of the offshore dredging. Regions outside the inner surf zone have also been studied through application of nearshore wave models. Jachec and Bosma (2001) used the numerical model REF/DIFS (a spectral version of REF/DIF1) to study borrow pit recovery time for seven borrow areas located on the inner continental shelf off New Jersey. The input wave conditions were obtained from Wave Information Study (WIS) data with nearshore bathymetry for the existing conditions and also different dredging scenarios. Changes in the wave induced bottom velocity were obtained from the wave height and direction changes determined by REF/DIFS. The waveinduced bottom velocities were coupled with ambient nearbottom currents to determine the sediment transport and then recovery times of the borrow areas. The recovery times from the numerical modeling were the same order of magnitude as recovery times estimated from two independent data sets of seafloor change rates offshore of New Jersey. 2.5 Shoreline Response 2.5.1 Longshore Transport Considerations The previous discussion on one and twodimensional models focused first on simple and complex methods of determining the wave transformation caused by changes in the offshore bathymetry and then applications that determined the changes to the wave height, direction and even longshore transport at the shoreline. However, none of the applications were intended to determine the change in shoreline planforms due to an anomaly or a change in the offshore bathymetry. With wave heights and directions specified along the shoreline, sediment transport can be calculated and, based on the gradients in longshore transport, the changes in shoreline position can be determined. The longshore transport can be driven by two terms as was discussed previously in the review of Maa and Hobbs (1998). In most situations where the offshore bathymetry is somewhat uniform, the magnitude and direction of the longshore transport will depend mostly on the wave height and angle at breaking as the longshore gradient in the breaking wave height will be small. In areas with irregular bathymetry or in the presence of structures, the transformation of the wave field can lead to areas of wave focusing and defocusing resulting in considerable longshore gradients in the wave height. Longshore transport equations containing a transport term driven by the breaking wave angle and another driven by the longshore gradient in the wave height can be found in Bakker (1971), Ozasa and Brampton (1980), who cite the formulation of Bakker (1971), Gourlay (1982), Kraus and Harikai(1983), and Kraus (1983). While the value of the coefficient for the transport term driven by the gradient in the wave height is not well established, the potential contribution of this term is significant. It is shown later that under steady conditions the diffusive nature of the angledriven transport term is required to modify the wave height gradient transport term in order to generate an equilibrium planform when the two terms are both active. 2.5.2 Refraction Models 2.5.2.1 Motyka and Willis (1974) Motyka and Willis (1974) were one of the first to apply a numerical model to predict shoreline changes due to altered offshore bathymetry. The model only included the effect of refraction caused by offshore pits for idealized sand beaches representative of those found on the English Channel or North Sea coast of England. A simplified version of the Abemethy and Gilbert (1975) wave refraction model was used to determine the transformation of uniform deep water waves over the nearshore bathymetry. The breaking wave height and direction were calculated and used to determine the sediment transport and combined with the continuity equation to predict 48 shoreline change. The longshore transport was calculated using the Scripps Equation as modified by Komar (1969): 0.045 Q= pg H C, sin(2ab) (21) Ys where Q is the volume rate of longshore transport, ys is the submerged unit weight of the beach material, p is the density of the fluid, Hb is the breaking wave height, Cg is the group velocity at breaking, and ab is the angle of the breaking wave relative to the shoreline. This form of the Scripps Equation combines the transport and porosity coefficients into one term; the values used for either parameter was not stated. This process was repeated to account for shoreline evolution with time. Figure 228 shows a comparison of the predicted shorelines for the equivalent of two years of waves over 1 m and 4 m deep pits with a longshore extent of 880 m and a crossshore extent of 305 m. The detailed pit geometries were not specified. 40  S30 WATER DEPTH (mi DISTANCE OFFSHORE (m) S17 08 2740 20 1762 3050 0 ro 4J 10  0O S30  Pit depth = 4 m 0o 50 100 1500 26O 20oo 3600 3o LO400 5o00 DISTANCE ALONG SHOREm PLANSHAPE OF BEACH DUE TO REFRACTION OVER DREOGED HOLE, 2740m OFFSHORE Figure 228: Calculated beach planform due to refraction after two years of prototype waves for two pit depths (modified from Motyka and Willis, 1974). The model determined that erosion occurs shoreward of a pit, with adjacent areas of accretion. For the wave conditions used, stability was found after an equivalent period of two years. During the runs, "storm" waves (short period and large wave height) were found to cause larger shoreline changes than the "normal" waves with longer periods and smaller heights, which actually reduced the erosion caused by the storm waves. The erosion shoreward of the pits is shown in Figure 228 with more erosion occurring for the deeper pit. 2.5.2.2 Horikawa et al. (1977) Horikawa et al. (1977) developed a mathematical model for shoreline changes due to offshore pits. The model applies a refraction program and the following equation for the longshore sediment transport: 0.77pg Q 77g H 2 Cg sin(2ab) (22) 16(p, p)(1 b) where X is the porosity of the sediment. Equation 22 is identified as the Scripps Equation in Horikawa et al. (1977); however, to match the Scripps Equation and for a dimensionally correct expression, the g term in the numerator should be removed. A model by Sasaki (1975) for diffraction behind breakwaters was modified to account for refraction only. The model computes successive points along the wave ray paths. Interpolation for the depth and slope is used along the ray path with an iteration procedure to calculate each successive point. The wave conditions were selected to be typical of the Eastern Japan coast facing the Pacific Ocean. Several pit dimensions and pit locations were used with the longshore dimension of the pit ranging from 2 km to 4 km, a crossshore length of 2 km, pit depth of 3 m and water depths at the pit from 20 m to 50 m. For the configurations modeled, accretion was found directly shoreward of the pit, flanked by areas of erosion. The magnitude of the accretion behind the pit and the erosion in the adjacent areas were found to increase with increasing longshore pit length and for pits located closer to shore. The shoreline planform for a model after the equivalent of 2 years of waves is shown in Figure 229 with a salient directly shoreward of the pit. 2 k hole w"lef depth: 40 m S!(depth:3m E AFTER 2 YEARS I 2 km I 0 0 7.3 m 2 accr tion 0 05m OS 5 4 3 2 t 0 1 2 3 4 5 Longshore distance from center of dredged hole (km) Figure 229: Calculated beach planform due to refraction over dredged hole after two years of prototype waves (Horikawa et al., 1977). Although Horikawa et al. state that good qualitative agreement was found with Motyka and Willis (1974), the results were the opposite with Horikawa et al. and Motyka and Willis (1974) predicting accretion and erosion shoreward of a pit, respectively. The proposed reason for the accretion given in Horikawa et al. was that sand accumulates behind the pit due to the quiet water caused by the decrease in wave action behind the pit. However, a model that considers only refraction caused by a pit and only includes a transport term dependent on the breaking wave angle would have wave rays that diverge over the pit and cause sand to be transported away from the area behind the pit, resulting in erosion. The two models used different refraction programs and basically the same transport equation with Horikawa et al. having a pit that was 2 or 3 times as large, lying in deeper water and with longer period incoming waves. The refraction grid was 250 m square in the Horikawa et al. model and 176 m square in Motyka and Willis. Regardless of the differing results from Motyka and Willis, the mathematical model results of Horikawa et al. follow the trend of the lab results contained in that study showing accretion behind a pit (Figure 230); however, the aforementioned anomalous prediction of accretion considering only wave refraction remains. 2 Predicted 0 20 40 60 Longshore distance from center of dredged hole ( cm ) Figure 230: Comparison of changes in beach plan shape for laboratory experiment and numerical model after two years of prototype waves (Horikawa et al., 1977). 2.5.3 Refraction and Diffraction Models 2.5.3.1 Gravens and Rosati (1994) Gravens and Rosati (1994) performed a numerical study of the salients and a set of offshore breakwaters at Grand Isle, Louisiana (Figures 21 and 22). Of particular interest is the analysis and interpretation of the impact on the wave field and the resulting influence on the shoreline, of the "dumbbell" shaped planform borrow area located close to shore. The report employs two numerical models to determine the change in the shoreline caused by the presence of the offshore pits: a wave transformation numerical model (RCPWAVE) and a shoreline change model (GENESIS (Hanson, 1987, Hanson, 1989)) using the wave heights from the wave transformation model. RCPWAVE was used to calculate the wave heights and directions from the nominal 12.8 m contour to the nominal 4.3 m contour along the entire length of the island for 3 different input conditions. Figures 231 and 232 show the wave height transformation coefficients and wave angles near the pit (centered about alongshore coordinate 130). Significant changes in the wave height and direction are found near the offshore borrow area. The shadow zone centered at Cell 130 suggests the presence of one large offshore pit as opposed to the "dumbbell" shaped borrow pit for the project described in Combe and Soileau (1987). The shoreline changes were calculated using a longshore transport equation with two terms; one driven by the breaking wave angle, and one driven by the longshore gradient in the breaking wave height. Each of these terms includes a dimensionless transport coefficient. In order for GENESIS to produce a salient leeward of the borrow pit, an unrealistically large value for the transport coefficient associated with the gradient in the breaking wave height (K2 = 2.4) was needed, whereas 0.77 is the normal upper limit. While a single salient was modeled after applying the large K2 value, the development of two salients leeward of the borrow pit, as shown in Figures 21 and 22, did not occur. The nearshore bathymetric data used in the modeling was from surveys taken in 1990 and 1992. Significant infilling of the borrow pit occurred prior to the surveys in 1990 and 1992; however, details of how the pit filled over this time period are not known. 100 110 120 130 140 Alakmpml Cordii (lld sp g 100 ft) 150 160 Figure 231: Nearshore wave height transformation coefficients near borrow pit from RCPWAVE study (modified from Gravens and Rosati, 1994). 120 130 140 Alkgulbrm Coordma"r (c0ll "pciu 100 ft) Figure 232: Nearshore wave angles near borrow pit from RCPWAVE study; wave angles are relative to shore normal and are positive for westerly transport (modified from Gravens and Rosati, 1994). The authors proposed that the salient was formed by the refractive divergence of the wave field created by the borrow pit that resulted in a region of low energy directly shoreward of the borrow area and regions of increased energy bordering the area. The gradient in the wave energy will result in a circulation pattern where sediment mobilized in the highenergy zone is carried into the low energy zone. For GENESIS to recreate this circulation pattern K2 must be large enough to allow the second transport term to dominate over the first transport term. 2.5.3.2 Tang (2002) Tang (2002) employed RCPWAVE and a shoreline modeling program to evaluate the shoreline evolution leeward of an offshore pit. The modeling was only able to generate embayments in the lee of the offshore pits using accepted values for the transport coefficients. This indicates that wave reflection and/or dissipation are important wave transformation processes that must be included when modeling shoreline evolution in areas with bathymetric anomalies. 2.5.4 Refraction, Diffraction, and Reflection Models 2.5.4.1 Bender (2001) A study by Bender (2001) extended the numerical solution of Williams (1990) for the transformation of long waves by a pit to determine the energy reflection and shoreline changes caused by offshore pits and shoals. An analytic solution was also developed for the radially symmetric case of a pit following the form of Black and Mei (1970). The processes of wave refraction, wave diffraction, and wave reflection are included in the model formulations, however, wave dissipation is not. Both the numerical and analytic solutions provide values of the complex velocity potential at any point, which allows determination of quantities such as velocity and pressure. The amount of reflected energy was calculated by comparing the energy flux through a transect perpendicular to the incident wave field extending to the pit center to the energy flux through the same transect with no pit present (Figure 233). The amount of energy reflected was found to be significant and dependent on the dimensionless pit diameter and other parameters. Subsequently a new method has been developed which allows the reflected energy to be calculated using a farfield approximation with good agreement between the two methods. Pit Diameter/Wavelength(inside pit,d) 0. 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 + (radius = 6 m) 0.24 o o (radius = 12m) (radius = 25 m) 0.22 x (radius = 30 m) <> (radius = 75 m) 0.2 o 50.18 x  S0.16 + I +o o A0.14 + 0 S 0.1 0 0 0.5 1 1.5 2 2.5 3 Pit Diameter/Wavelength(outside pit,h) Figure 233: Reflection coefficients versus dimensionless pit diameter divided by wavelength inside and outside the pit; water depth = 2 m, pit depth = 4 m (Bender, 2001). The shoreline changes caused by the pit were calculated using a simple model that considers continuity principles and the longshore transport equation with values of the wave height and direction determined along a transect representing the shoreline. A nearshore slope and no nearshore refraction were assumed. The impact on the shoreline was modeled by determining the wave heights and directions along an initially straight shoreline, then calculating the transport and resulting shoreline changes. After updating the shoreline positions, the transport, resulting shoreline changes, and updated shoreline positions were recalculated for a set number of iterations after which the wave transformation was recalculated with the new bathymetry and values of the wave height and direction were updated at the modified breaker line. The impact on the shoreline was found to be highly dependent on the transport coefficients. Considering transport driven only by the breaking wave angle and wave height, erosion was found to occur directly leeward of the pit flanked by two areas of accretion as in Motyka and Willis (1974). Following an initial advancement directly shoreward of the pit, erosion occurs and an equilibrium shape was reached. Examining only the effect of the second transport term (driven by the longshore gradient in the wave height) accretion was found directly shoreward of the pit, with no equilibrium planform achieved. Figure 234 shows the shoreline evolution for each transport term. Including both transport terms with the same transport coefficients resulted in a shoreline with accretion directly shoreward of the pit that was able to reach an equilibrium state (Figure 235). The twoterm transport equation used to determine the shoreline in Figure 235 is KIH2.5 sin(Oab)cos(Oab) K2H cos(9ab)dH Q (23) 8(s 1X p) 8(s 1X1 p)tan(m) dy where Hb is the breaking wave height, g is gravity, K is the breaking index, 0is the shoreline orientation, a is the breaking wave angle, m is the beach slope, s andp are the specific gravity and porosity of the sediment, respectively, and K1 and K2 are sediment transport coefficients, which were set equal to 0.77 for the results presented here. A review of the development of Eq. 23 is contained in Appendix D. First Transport Term (wave angle) 200 150 100 50 0 S50 100 150 Erosion 200 79.5 80 80.5 81 Shoreline Position (m) Second Transport Term (dHbldx) 40 60 80 100 120 Shoreline Position (m) Figure 234: Shoreline evolution resulting from each transport term individually for transect located 80 m shoreward of a pit with a radius = 6 m, last time step indicated with [+] (modified from Bender and Dean, 2001). In application the wave height and angle calculations were limited to a uniform depth region; therefore no nearshore shoaling and refraction was performed to resolve the depth limited breaking values. The wave height and angle values at the transect location representing the shoreline were assumed to be the values at breaking; this limitation is removed in the model developed in Section 6.10. In these figures the water depth and pit depth were 2 m and 4 m, respectively, the period was 10 s, the incident wave height was 1 m, and averaging over 5 wave directions was used to smooth out the longshore variation in the wave height at large distances from the pit. The time step was 120 s and 10 iterations of shoreline change were calculated between wave height and direction updates for a total modeling time of 48 hours. The diffusive nature of the angledriven transport term is seen to modify the much larger wave height gradient transport term in order to generate an equilibrium planform when the two terms are used together. Comparison of these results with those described earlier establishes the significance of wave reflection and the second transport term on the equilibrium planform. 50 0 Q 0 S 50so 10 100 Full Transport Equation (both terms) Erosion 150 F 200 78 78.5 79 79.5 80 81 Shoreline Position (m) 0.5 81 81.5 Figure 235: Shoreline evolution using full transport equation and analytic solution model for transect located 80 m shoreward of a pit with a radius = 6 m, last time step indicated with [+] (modified from Bender and Dean, 2001). I ' CHAPTER 3 2DIMENSIONAL MODEL THEORY AND FORMULATION 3.1 Introduction The reflection and transmission of normally incident waves by twodimensional trenches and shoals of finite width with sloped transitions between the depth changes are studied. Prior twodimensional studies, outlined in Chapter 2, have all investigated the interaction of water waves with changes in bathymetry that have featured domains with an abrupt transition, with the exception of Dean (1964) and Lee et al. (1981). A more realistic representation of natural trenches and shoals should allow for gradual transitions (sloped sidewalls). The focus of the 2dimensional study is the propagation of water waves over a 2D trench or shoal of more realistic geometry. This will extend the study of Dean (1964) that investigated long wave modification by a sloped step and Lee et al. (1981), which did not directly address the effect of the transition slope on the reflection and transmission coefficients. Three solution methods are developed for linear water waves: (1) the step method, (2) the slope method and (3) a numerical method. The step method is valid in arbitrary water depth while the slope method and the numerical method are valid only for shallow water conditions. The step method is an extension of the Takano (1960) eigenfunction expansion solution as modified by Kirby and Dalrymple (1983a) that allows for a trench or shoal in arbitrary depth with "stepped" transitions that approximate a specific slope or shape. The slope method extends the long wave solution of Dean (1964) that allows for linear transitions between the changes in bathymetry for a trench or shoal creating regular or irregular trapezoids. The numerical method employs a backward spacestepping procedure for arbitrary (but shallow water) bathymetry with the transmitted wave specified. 3.2 Step Method: Formulation and Solution The twodimensional motion of monochromatic, smallamplitude water waves in an inviscid and irrotational fluid of arbitrary depth is investigated. The waves are normally incident and propagate in an infinitely long channel containing a two dimensional obstacle (trench or shoal) of finite width. Details of the fluid domain and the formulation of the solution vary depending on the case studied: abrupt transition or gradual transition with the slope approximated by the step method. The step method is an extension of the Takano (1960) formulation for the propagation of waves over a rectangular sill. The eigenfunction expansion method of Takano (1960) was extended in Kirby and Dalrymple (1983a) to allow for oblique wave incidence and again in 1987 by Kirby et al. to include the effects of currents along the trench. In the present formulation the method of Takano, as formulated in Kirby and Dalrymple (1983a), was generally followed for normal wave incidence. The solution starts with the definition of a velocity potential: j (x,z,t)= qj (x,z)e' (j= 1 J) (31) where j indicates the region, J is the total number of regions (3 for the case of a trench or sill with an abrupt transition), and a is the angular frequency. The velocity potential must satisfy the Laplace Equation: a + a b(XZ) = 0 (32) a9x 2 Z 2 the freesurface boundary condition: 84(x,z) 2 a )+ = 0 (33) az g and the condition of no flow normal to any solid boundary: 9<^(x,z) a (x 0 (34) on The velocity potential must also satisfy radiation conditions at large Ix. The boundary value problem defined by Eq. 32, the boundary conditions of Eqs. 33 and 34, and the radiation condition can be solved with a solution in each region of the form + cosh[k1(h1 + z) eiL +B cos[, (h ,)]e (Xx) (x,z) = A cosh[kjh ] eik(x) + B,, cos (=l (h + e coshkhTc e n=i (j = J), (n = oo) (35) In the previous equation A1+ is the incident wave amplitude coefficient, A( is the reflected wave amplitude coefficient and AJ+ is the transmitted wave amplitude coefficient. The coefficient B is an amplitude function for the evanescent modes (n = 1 + oo) at the boundaries, which are standing waves that decay exponentially with distance from the boundary. The values of the wave numbers of the propagating modes, kj, are determined from the dispersion relation: 2 = gkj tanh(khj) j = 1 J) (36) and the wave numbers for the evanescent modes, K ,n are found from a2 = gK,, tan(K,h) (j= 1 J), (n = 1 oo) (37) In each region a complete set of orthogonal equations over the depth is formed by Eqs. 3 5 to 37. To gain the full solution, matching conditions are applied at each boundary between adjacent regions. The matching conditions ensure continuity of pressure: S= j (x= xJ), (j = 1 J1) (38) and continuity of horizontal velocity normal to the vertical boundaries: = (x = x), j= J1) (39) ax ax The matching conditions are applied over the vertical plane between the two regions: (hi z < 0) if h, < hi+, or ( hi+, < z < 0) if hj > h+1. In order to form a solution, one wave form in the domain must be specified, usually the incident or the transmitted wave. Knowing the value of the incident, reflected and transmitted wave amplitudes, the reflection and transmission coefficients can be calculated from KR = a (310) a! a, cosh(k,h,) K, = c (311) a, cosh(k,h,) where the cosh terms account for the change in depth at the upwave and downwave ends of the trench/shoal for the asymmetric case. A convenient check of the solution is to apply conservation of energy considerations: K' + a] =1 (312) R a, nk, 1 63 where nj is the ratio of the group velocity to the wave celerity: 2k.h n si =( 1+j (313) 7 2 sinh(2k h.)J 3.2.1 Abrupt Transition The solution of Takano (1960) for an elevated sill and that of Kirby and Dalrymple (1983a) for a trench are valid for abrupt transitions (vertical walls) between the regions of different depth. For these cases the domain is divided into three regions (J = 3) and the matching conditions are applied over the two boundaries between the regions. The definition sketch for the case of a trench with vertical transitions is shown in Figure 31 where W is the width of the trench. z Nl X Xi X2 Region 1 Region 2 1 Region 3 *c W  Figure 31: Definition sketch for trench with vertical transitions. Takano constructed a solution to the elevated sill problem by applying the matching conditions [Eqs. 38 and 39] for a truncated series (n =1 > N) of eigenfunction expansions of the form in Eq. 35. Applying the matching conditions results in a truncated set of independent integral equations each of which is multiplied by the appropriate eigenfunction; cosh[kj(hj+z)] or cos[Kj,n(hj+z)]. The proper eigenfunction to use depends on whether the boundary results in a "step down" or a "step up"; thereby making the form of the solution for an elevated sill different than that of a trench. With one wave form specified, the orthogonal properties of the eigenfunctions result in 4N+4 unknown coefficients and a closed problem. By applying the matching conditions at the boundary between Regions 1 and 2 (x = x ), 2N+2 integral equations are constructed. For the case of a trench with vertical transitions (Figure 31) the resulting equations are of the form o 0 S((x1,, z) cosh[k,(h, + z)]dz = 2(I, ) ,cosh[k, (h, + z)]dz (314) h, h, 0 0 fo,(x,,z)cos[K,,,(h, + z)]dz = f2 (x1,z)cos[KI,,(h + z)]dz (n = 1 N) (315) hA hA 0J^ (x,,z)cosh[k2(h2 + z)dz = (x,, z) cosh[k2(h + z)]dz Ax Aax hi h, (316) = (x,,z) cosh[k2 (h2 + z)]dz h2 J (x z) cosK2, (h2+ z)]dz = (, z)cos[K2, (h2 + z)ldz h, (n = 1 N)(317) (x z) cos K2,n (h2 + z)]dz h2 The limits of integration for the right hand side in Eqs. 316 and 317 are shifted from hi to h2 as there is no contribution to the horizontal velocity for ( h2 < z < h,) at x = xl and ( h < z < h3) at x = x2, for this case. In Eqs. 314 and 315 the limits of integration for the pressure considerations are ( h, < z < 0) at x = x, and ( h3 z < 0) at x = x2. At the boundary between Regions 2 and 3 the remaining 2N+2 equations are developed. For the case of a trench the downwave boundary is a "step up", which requires different eigenfunctions to be used and changes the limits of integration from the case of the "step down" at the upwave boundary [Eqs. 314 to 317]. 0 0 J2 (x2,z)cosh[k3(h + z)]dz = (x2, z) cosh[k3(h, + z)]dz (318) h3 h3 0 0 (x2 2,z)cos[K3,n(h3 + z)]dz= 3(x2,z)cos[K3,,(h3 + z)]dz (n=1 N) (319) h3 h3 z) cosh[k2(h2 + z)]dz = (x2, z) cosh[k2 (h + z)]dz (320) h2I h3 S (x2, z) cos[K2, (h2 + z)]dz = (2 (x,z)cos[K2,,(h, + z)dz (n =1 N)(321) ax h ax At each boundary the appropriate evanescent mode contributions from the other boundary must be included in the matching conditions. The resulting set of simultaneous equations may be solved as a linear matrix equation. The value of N (number of non propagating modes) must be large enough to ensure convergence of the solution. Kirby and Dalrymple (1983a) found that N = 16 provided adequate convergence for most values of klh. 3.2.2 Gradual Transition The step method is an extension of the work by Takano (1960) and Kirby and Dalrymple (1983a) that allows for a domain with a trench or sill with gradual transitions (sloped sidewalls) between regions. Instead of having a "step down" and then "step up" as in the Kirby and Dalrymple solution for a trench or the reverse for Takano's solution for an elevated sill, in the step method a series of steps either up or down are connected by a constant depth region followed by a series of steps in the other direction. A sketch of a domain with a stepped trench is shown in Figure 32. In this method, as in the case of a trench or a sill, a domain with J regions will contain J1 steps and boundaries.  lR  X XI X2 X3 X4 X5 I I h, h2 z4 5 h Region 1 (R1) R2 R3 Region 4 RS Region 6 Figure 32: Definition sketch for trench with stepped transitions. Each region will have a specified depth and each boundary between regions will have a specified x location where the matching conditions must be applied. At each boundary the matching conditions are applied and depend on whether the boundary is a "step up" or a "step down." With the incident wave specified, a set of equations with 2(J1)N+2(J1) unknown coefficients is formed. The resulting integral equations are of the form: for(j =1 > J 1) if( hj > hj ) at x = xj then the boundary is a "step down"; 0 0 fJ(x,,z)cosh[k h +z)] dz = ft (x ,z)cosh[k,(hi +z)]dz (322) h, h, 0 0 f (xj,z)cos[Kj, (hj + z)]dz = Jj+,(xj,z)cos[K,, (hj + z)]dz (n=1 ) N)(323) h, hi o a< 0 (, \ + d J (x. z) cosh[kj,(,,h + z)dz = al(x z)cosh[k+,,(h+,, +z dz (324) h hi 0i Q ji I / \1 J C (x ,z)cos[Ki,,, +(h + + z)]dz = J (xz)cos[K+,n (h+, +z)dz h, AI (n = 1 + N) (325) if ( hi < h ) at x = xj then the boundary is a "step up"; o 0 i (x, z) cosh[kj, (ih, + z)]dz = i,+, (xj, z) cosh[k ,, (hj + z)]dz (326) h+) hA+I 0 0 Ji (xj, z) cos[Kj+,n (h+,, + z)dz = J+ (x., z) cos[Kj+.,, (h, + z)]dz hj+l hl+h (n =1  N) (327) Sx(x j, z)cosh[k (h, + z)]dz = J ( z) cosh[kj (h. + z)]dz (328) ax ax J (x, z) cos[K ,,(hj + z] dz = J l(xj,z)cos[K,, (j z) dz h hj+ (n = 1 N) (329) At each boundary (xj) the appropriate evanescent mode contributions from the adjacent boundaries (xjli, xj+1) must be included in the matching conditions. The resulting set of simultaneous equations is solved as a linear matrix equation with the value of N large enough to ensure convergence of the solution. 3.3 Slope Method: Formulation and Solution The slope method is an extension of the analytic solution by Dean (1964) for long wave modification by linear transitions. Linear transitions in the channel width, depth, and both width and depth were studied. The solution of Dean (1964) is valid for one linear transition in depth and/or width, which in the case of a change in depth allowed for an infinite step, either up or down, to be studied. In the slope method a domain with two linear transitions allows the study of obstacles of finite width with sloped transitions. The long wave formulation of Dean (1964) for a linear transition in depth was followed. By combining the equations of continuity and motion the governing equation of the water surface for long wave motion in a channel of variable crosssection can be developed. The continuity equation is a conservation of mass statement requiring that the net influx of fluid into a region during a time, At, must be equal to a related rise in the water surface, rl. For a channel of uniform width, b, this can be expressed as [Q(x) Q(x + Ax)]At = bAx[rq(t + At) r(t)] (330) where Q(x) and Q(x+Ax) are the volume rates of flow into and out of the control volume, respectively. The volume flow rate for the uniform channel can be expressed as the product of the cross sectional width, A, and the horizontal velocity, u, in the channel: Q=Au (331) By substituting Eq. 331 into Eq. 330 and expanding the appropriate terms in their Taylor series while neglecting higher order terms, Eq. 330 can be rewritten as (Au) = b (332) ax Ot The hydrostatic pressure equation is combined with the linearized form of Euler's equation of motion to develop the equation of motion for small amplitude, long waves. The pressure field, p(x,y,t), for the hydrostatic conditions under long waves is p(x,z,t) = pg[i(x,t) z] (333) Euler's equation of motion in the x direction for no body forces and linearized motion is 1 9p 6u Sap = u (334) p Ox at The equation of motion for small amplitude, long waves follows from combining Eqs. 3 33 and 334: 0 g Ou (335) ax at The governing equation is developed by differentiating the continuity equation [Eq. 332] with respect to t: (Au)= b   A = b (336) at 8x at ax at at2 and inserting the equation of motion [Eq. 335] into the resulting equation, Eq. 336 yields the result g A = b (337) ax r ax t2 Eq. 337 is valid for any small amplitude, long wave form and expresses r1 as a function of distance and time. Eq. 337 can be further simplified under the assumption of simple harmonic motion: r7(x,t) = 77, (x)ei(o+a) (338) where a is the phase angle. Eq. 337 can now be written as g abha7 +x r =O0 (339) b 9x ax where the subscript r l(x) has been dropped and the substitution, A = bh, was made. 3.3.1 Single Transition The case of a channel of uniform width with an infinitely long step either up or down was a specific case solved in Dean (1964). The definition sketch for a "step down" is shown in Figure 33. The three regions in Figure 33 have the following depths: Region 1, x < xj; h = h, (340) x Region 2, x < x < x2; h = h3 (341) x2 Region 3, x > x,; h = h3 (342) Region 1 Region 2 Region 3 x] x x2 Figure 33: Definition sketch for linear transition. For the regions of uniform depth, Eq. 339 simplifies to 2X2 which has the solution for rl of cos(kx) and sin(kx) where k = and X is the wave A length. The most general solution ofrl(x,t) from Eq. 343 is r(x, t) = B, cos(kx t + a ) + B, cos(kx + at + a2) (344) The wave form of Eq. 344 consists of two progressive waves of unknown amplitude and phase: an incident wave traveling in the positive x direction and a reflected wave traveling in the negative x direction. For the region of linearly varying depth, Eq. 341 is inserted in Eq. 339 resulting in a Bessel equation of zero order: 2x + r+ 7 = 0 (345) x ax Ox2 Ox where S= x2 (346) gh, The solutions ofrl(x) for Eq. 345 are q(x) = Jo(2p 12x1'2) and Yo(2P1/2x' 2) (347) where Jo and Yo are zeroorder Bessel functions of the first and second kind, respectively. From Eq. 347 the solutions for r(x,t) in Region 2 follow q(x,t)= BJo(2fl/2x1/2)cos(crt+a3)+ BYo(21/2x1/2)sin(at + a3) (348) + B4J(2fl12x1/2)cos(t + a4) B4 Y(2P/212x'2)sin(Tt + a4) The wave system of Eq. 348 consists of two waves of unknown amplitude and phase; one wave propagating in the positive x direction (B3) and the other in the negative x direction (B4). The problem described by Figure 33 and Eqs. 344 and 348 contains eight unknowns: Bl+4 and a(X14. Solution to the problem is obtained by applying matching conditions at the two boundaries between the three regions. The conditions match the water surface and the gradient of the water surface: 77 = 77,, at x = x (j = 1,2) (349) atx = x (j = 1,2) (350) Ox Ox Eqs. 349 and 350 result in eight equations (four complex equations), four from 7rC setting a t = 0 and four from setting at = which can be solved for the eight unknowns as a linear matrix equation. 3.3.2 Trench or Shoal The slope method is an extension of the Dean (1964) solution that allows for a domain with a trench or a sill with sloped transitions. Two linear transitions are connected by a constant depth region by placing two solutions from Dean (1964) "back to back." A trench/sill with sloped side walls can be formed by placing a "step down" upwave/downwave of a "step up." The definition sketch for the case of a trench is shown in Figure 34. In the slope method the depths are defined as follows Region 1, x < x,; h = h, (351) Region 2, x < x < x2; h = h + s, (x x,) (352) Region 3, x < x < x3; h = h3 (353) Region 4, x3 < x < x4; h=h3s (xx3) (354) Region 5, x > x4; h = h5 (355) where hi, h3, hs, si, s2, and W are specified. With the new definition for the depth in regions 2 and 4, the definition of the coefficient P in Eq. 345 changes to f = and gsI 2 p =  in regions 2 and 4, respectively. gs2 W  Figure 34: Definition sketch for trench with sloped transitions. The matching conditions of Eqs. 349 and 350 are applied at the four boundaries between the regions. With the transmitted wave specified and by setting a t = 0 and a t = for each matching condition a set of 16 independent equations is developed. 2 Using standard matrix techniques the eight unknown amplitudes and eight unknown phases can then be determined. The reflection and transmission coefficients can be determined from KR = aRand KT = a' (356), (357) a, a, Conservation of energy arguments in the shallow water region require K'+ K[ ] = 1 (358) This method can be extended to the representation of long wave interaction with any depth transition form represented by a series of line segments. 3.4 Numerical Method: Formulation and Solution A numerical method was developed to determine the long wave transformation caused by a trench or shoal of arbitrary, but shallow water bathymetry. A transmitted wave form in a region of constant depth downwave of the depth anomaly is the specified input to the problem. Numerical methods are used to space step the wave form backwards over the trench or shoal and then into a region of constant depth upwave of the depth anomaly where two wave forms exist; an incident wave and a reflected wave. As in the long wave solution of Section 3, the continuity equation and the equation of motion are employed to develop the governing equation for the problem. The continuity equation and the equation of motion in the x direction are written in a slightly different form than in Eqs. 332 and 335 of Section 3.3: = (359) at ax gh aq (360) ax at Taking the derivative of Eq. 359 with respect to t and the derivative of Eq. 360 with respect to x results in the governing equation for this method: Sgh a27g dh= 0 (361) 9at2 x2 dx ax where the depth, h, is a function of x and ri may be written as a function of x and t: 77 = qr(x)e"' (362) Inserting the form of r in Eq. 362 into the governing equation of Eq. 361 casts the equation in a different form (equivalent to Eq. 339) gh a28r(x) g dh ar(x) Cr(x)+ + 2 2 d=0 (363) a2 x2 2 d ax Central differences are used to perform the backward space stepping of the numerical method. (x) F(x + Ax) 2F(x)+ F(x x) (364) Ax2 F F(x + Ax) F(x x) FA'(x)= (365) 2Ax Inserting the forms of the central differences into Eq. 363 for rJ results in (x)+gh 77(x+ Ax) 27()+ (xAx) g dh 77(x+Ax)(xAx) (366) a 2 Ax2 a2 dx 2Ax For the backward space stepping calculation, Eq. 366 can be rearranged ) gh g dh 1 [ 2gh Ax)2 2 dx 2Ax 1 2x2 77(x Ax) = d2Ax (367) gh g dh 1 C.2Ax2 O2 dx 2Ax To initiate the calculation, values of r(x) and rl(x+Ax) must be specified in the constant depth region downwave from the depth anomaly. If the starting point of the calculation is taken as x = 0 then the initial values may be written as H r7(0) = (368) 77(Ax) = H [cos(kAx) i sin(kAx)] (369) The solution upwave of the depth anomaly comprises of an incident and reflected wave. The form of the incident and reflected waves are specified as H1 r], = cos(kx ot ,) (370) 2 HR qR = cs(kx + o't eR) (371) 2 where the E's are arbitrary phases. At each location upwave of the depth anomaly the total water surface elevation will be the sum of the two individual components: H1 77T = /i + 77R = cos(kx E,) cos(ot) + sin(kx e,) sin(ot) 2 + HR cos(k s ) cos(Ot) sin(kx eR ) sin(Ot) 2 = cos(ot)[ cos(hk ,) + cos(k c8) 2 2 I + sin(ot) sin(kx e,) Hr sin(kx R) 2 2 = VI2 +2 cos(O C) 6 = tan'j (372) Using several trigonometric identities, Eq. 372 can be reduced further to the form 7Tr = JH + H +2HH, cos(2kx e, ) cos(ot ) (373) which is found to have maximum and minimum values of rTrax = (H, +H,) (374) 2 r7m = H, HR (375) Eqs. 374 and 375 are used to determine the values of Hi and HR upwave of the trench/shoal, and allow calculation of the reflection and transmission coefficients. CHAPTER 4 3DIMENSIONAL MODEL THEORY AND FORMULATION 4.1 Introduction The threedimensional motion of monochromatic, smallamplitude water waves in an inviscid and irrotational fluid of arbitrary depth is investigated. The waves propagate in an infinitely long, uniform depth domain containing a threedimensional axisymmetric anomaly (pit or shoal) of finite extent. The addition of the second horizontal dimension provides many new, and more practical, possibilities for study compared to the 2D model domains, which excluded longshore variation. Two different models are developed for the 3D domains that contain linear transitions in depth. The analytic step method is an extension of Bender (2001) that determines the wave transformation in arbitrary water depth for domains with gradual transitions in depth that are approximated by a series of steps of uniform depth. The exact analytic model solves the wave transformation in shallow water for specific bathymetries that reduce the governing equations to known forms. 4.2 Step Method: Formulation and Solution The step method for a threedimensional domain is an extension of the Bender (2001) formulation for the propagation of waves past a circular anomaly with abrupt transitions. This method allowed oblique wave incidence, but was limited to the shallow water region. Following Bender (2001) with significant changes in notation the governing equations for the threedimensional models are developed. Details of the fluid domain and the formulation of the solution vary depending on the case studied: abrupt transition in depth between regions or gradual transition with the slope approximated by the step method. The definition sketch for the case of a circular pit with an abrupt depth transition is shown in Figure 41. r (r,0) Region 2 ^^_ {^ ^<.  Region 1 f , Figure 41: Definition sketch for circular pit with abrupt depth transitions. The domain is divided into regions with the bathymetric anomaly and its projection comprising Regions 2+Ns+l where Ns is the number of steps approximating the depth transition slope and the rest of the domain, of depth hi, in Region 1. For the case of an abrupt transition in depth the bathymetric anomaly occurs in Region 2 of uniform depth h2, where abrupt is defined as one step either down or up. For the case of a gradual depth transition the bathymetric anomaly will be divided into subregions with the depth in each subregion equal to hj for each step j = 2 > N, +1. The solution starts with the definition of a velocity potential in cylindrical coordinates that is valid in each Region j: 0j = Re( j(r,0,z)e') (j = + N,) (41) where w is the wave frequency. Linear wave theory is employed and Laplace's solution in cylindrical coordinates is valid: V 2 o+ + + = 0 (42) r 2 r ar r2a02 (z2 where the free surface boundary condition is a2 = 0 (43) az g and the bottom boundary condition is taken as S= 0 (44) az at z = hi in Region 1 or z = hj in Region 2. Separation of variables is used to solve the equations with the velocity potential given the form Q(r,O,z) = R(r)O(O)Z(z) (45) A valid solution for the dependency with depth is Z(z) = cosh(kj(hj+z)) (46) where kj is the wave number in the appropriate region and hj is constant within each region. Inserting Eqs. 45 and 46 into the Laplace equation gives 1 1 R"O + R' + 2E"R + ROk2 = 0 (47) r r If the form of the dependence with 0 is assumed to be (O)= cos(mO) (48) then Eq. 47 may be reduced to r2R" +rR' +R(k2r2 m2)= 0 (49) which is a standard Bessel equation with solutions Jm(kr), Ym(kr), and Hm(kr). The dependency of the solution on 0 cancels out of Eq. 49; a result of the separation of variables approach. The wave number is determined from the dispersion relation: a2 = gkj tanh(khj) (410) The standard Bessel solutions and the wave numbers of Eq. 410 represent the plane progressive wave component of the solution. Another solution is found when the dependence with depth is defined as Z(z) = cos(Kj(hj+z)) (411) Inserting Eqs. 45, 48 and 411 into the Laplace Equation gives r2R" +rR' R(K2r2 + m2)= 0 (412) which is a modified Bessel equation with solutions Km(Kr) and Im(Kr) where K is the wave number for the evanescent modes obtained using c = gKt,, tan(Kcih) (n = 1 + oo) (413) where n indicates the number of the evanescent mode. The modified Bessel solutions and the wave numbers of Eq. 413 represent the evanescent mode solutions, which decay with distance from each interface between regions. The evanescent terms are included to account for the distortion of the plane wave near the interface (Black and Mei, 1970) and to extend the range of the solution into the arbitrary depth region. 
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