CIES Army Coast. Ere Res. Ctr MR 80-6 WHO] DOCUMENT COLLECTION A Numerical Model for Predicting Shoreline Changes by Bernard Le Mehaute and Mills Soldate MISCELLANEOUS REPORT NO. 80-6 JULY 1980 ZeoB8PS OF 5p AON Ao \ distribution unlimited. Prepared for U.S. ARMY, CORPS OF ENGINEERS COASTAL ENGINEERING RESEARCH CENTER oC. Kingman Building 20% Fort Belvoir, Va. 22060 1058 Me co-G Reprint or republication of any of this material shall give appropriate credit to the U.S. Army Coastal Engineering Research Center. Limited free distribution within the United States of single copies of this publication has been made by this Center. Additional copies are available from: National Technical Information Service ATTN: Operations Division 5285 Port Royal Road Springfield, Virginia 22161 The findings in this report are not to be construed as an official Department of the Army position unless so designated by other authorized documents. oonaa HAP Ol MN q I) MN UNCLASSIFIED ——_—— EE SECURITY CLASSIFICATION OF THIS PAGE (When Data Entered) READ INSTRUCTIONS REPORT DOCUMENTATION PAGE BEFORE COMPLETING FORM T. REPORT NUMBER 2. GOVT ACCESSION NO. RECIPIENT'S CATALOG NUMBER MR 80-6 4. TITLE (and Subtitle) - TYPE OF REPORT & PERIOD COVERED Rae ih Report A NUMERICAL MODEL FOR PREDICTING SHORELINE CHANGES 6. PERFORMING ORG. REPORT NUMBER Tetra Tech Report No. TC-831 7. AUTHOR(s) 8. CONTRACT OR GRANT NUMBER(s) Bernard Le Mehaute and Mills Soldate DACW72-7T-C-0002 9. PERFORMING ORGANIZATION NAME AND ADDRESS 10. PROGRAM ELEMENT, PROJECT, TASK Tetra Tech Inc AREA & WORK UNIT NUMBERS ’ ° 630 North Rosemead Boulevard Pasadena, California 91107 11. CONTROLLING OFFICE NAME AND ADDRESS 12. REPORT DATE | Coastal Engineering Research Center 13. NUMBER OF PAGES Kingman Building, Fort Belvoir, Virginia 22060 14. MONITORING AGENCY NAME & ADDRESS(if different from Controlling Office) 15. SECURITY CLASS. (of this report) UNCLASSIFIED F31551 15a. DECLASSIFICATION/ DOWNGRADING SCHEDULE 16. DISTRIBUTION STATEMENT (of this Report) Approved for public release; distribution unlimited. DISTRIBUTION STATEMENT (of the abstract entered in Block 20, if different from Report) - SUPPLEMENTARY NOTES KEY WORDS (Continue on reverse side if necessary and identify by block number) Currents Numerical model Shoreline evolution Great Lakes Shore structures Wave diffraction Holland Harbor, Michigan Shoreline changes Wave refraction 20. ABSTRACT (Continue om reverse side if necessary and identify by block number) A mathematical model for long-term, three-dimensional shoreline evolution is developed. The combined effects of variations of sea level; wave refrac- tion and diffraction; loss of sand by density currents during storms, by rip currents, and by wind; bluff erosion and berm accretion; effects of manmade structures such as long groin or navigational structures; and beach nourish- ment are all taken into account. A computer program is developed with various subroutines which permit modification as the state-of-the-art progresses. The program is applied to a test _case at Holland Harbor, Michigan. FORM DD tgan 7a 1473 EDITION OF Tt NOV 65 1S OBSOLETE UNCLASSIFIED SECURITY CLASSIFICATION OF THIS PAGE (When Data Entered) mT catia Vili i i igh) i] ‘ , 8 i ui | Weis wiht: iad ELGN are ‘ (ey i Bye yr i iy if ‘ h } Pb hay et : i 0 AY ; it han: i) y ey) | SS ‘ ' ( ntl EF ' 4 : vn, a 1 . = Mi ; : f LA H J Ds i i ‘ ard j ii i i at I t * j 7 in PREFACE This report is published to provide coastal engineers with a mathematical modeling procedure for predicting shoreline evolution resulting from the con- struction of navigation and shore structures. The model is calibrated with a test case at Holland Harbor, Michigan. This report is a continuation of an investigation by Le Mehaute and Soldate (1977) to determine the feasibility of applying numerical models to real case situations. The work was carried out under the coastal structures program of the Coastal Engineering Research Center (CERC). The report was prepared by Bernard Le Mehaute and Mills Soldate, Tetra Tech, Inc., Pasadena, California, under CERC Contract No. DACW72-7T-C-0002. The authors acknowledge the assistance of Dr. J.R. Weggel, CERC, and C. Johnson, U.S. Army Engineer District, Chicago, for their pertinent comments during this investigation. Informal discussions with Dr. R. Dean, University of Delaware, were very constructive. The contributions of Dr. I. Collins, Dr. C. Sonu, and J. Nelson are also acknowledged, as well as A. Ashley's analysis of the input data. Dr. J.R. Weggel was the CERC contract monitor for the report, under the general supervision of N.E. Parker, Chief, Engineering Development Division. Comments on this publication are invited. Approved for publication in accordance with Public Law 166, 79th Congress, approved 31 July 1945, as supplemented by Public Law 172, 88th Congress, approved 7 November 1963. Colonel, Corps of Engineers Commander and Director CONTENTS Page CONVERSION FACTORS, U.S. CUSTOMARY TO METRIC (SI). ...... SHAMS) JANI) DIS TIN IITIONSS, 65.6110 6 6 bo 6 A oo bo 6.0 6 610 9 6 7 I INDE OIDONCHEHON 5 eo. BMG No Had Ola) 6! dio YONG og Oo GG lla o. oO. 66, 6 9 II THE OREM CALS DEVELO PMENMS ME wircimten Meanreiiirey retire irises hey irey ules ar Nile arene oeabal III INU IDM GG. Si 6 5 A ai fee Uahiusie “ay) cee cal Sea ee 1. Description and naistomy of Nevaleneion Project at Holland Ete nobel Mean aint GemcEMG MO SONU Gul G nM ECE MOON ECOG 20)0'.0 Bal Zo GOOMOMNONOME' Golo 6 66 (a oa 6 6 6 4 5 00 61619 0 oo 0) (OS See laktetOmaluMart ers italles kim oun ueWen reriuren ety ew etot Pcie tas Pau) Palla fatal < Yh verti aaa area ALTON eh ere(oneto)) Loven vaio ey inl ioulocateh Vow ida NOs a Dae Om OMEGA IME A A Gua laiiol tac. oo Soi labcbrollOraye G5 0 emcniteecuio A .o) 2S) 6. Littoral Drift Soe aimeree Sao. hee Sinuiseles., 66 0 0 a oo ES 7. Analysis of Aerial Photos... 6 Would 1G) oll so 0 0 4S 8. Comparisons of Profiles from 1945 ail 1975 SwaaresrS- ee ao. (2S a! Sorcerers (6 oo 6 oo 6 6 oO oO 0 6 6 0 16, 06 | oo) 5 IV CONCHLUSTONSWANDERE COMMEND AMON Sincere seco icles uu reuncimizen enn THe ESS) LITERATURE: GETED:0) 4 pita cannon iosnite usa heestuces ee ol eprep age at eae Gy) NPS COMUNE PROGR AG Sb gid odie 1h velo oldies (a) id) elo eo lo oo, 6 0 68 TABLES 1 History of shoreline positions at Holland, Michigan, from 1950 to 17S wacthy andawaithowtmllalkemlen ell (coms; e CtelOMSiipy.) venue pletlelt siete) Miele aon LY 2 Summary of cross-sectional changes and computed volumetric changes scoe NGS ging) IDPS SUES 6 6.6 66 6b 0 6 6 06 6 6 BG O10 6 oO 6 0 6 §M 3 Summary of shoreline accretion rates north of Holland Harbor GINS) Sake aN) enue etn ice laai san crrennl IMim Mn MENG Io \ducol co ooo jo Go SQ 4Sunnary of beach area’ Ghanses south) of Holland Harbor v5) 9) ae eS 5 SSO eyes sxoie Jeol iemyel oksdaese, g 6 6 6 oo 16. 6 oa oo 6 6 00 60 6. §9 FIGURES LY MCOASt aM ZONe r/o Ne inp icy. ellie!) ou, lovatsiyhceuanlsc eh monll-sitibet icouireel (emiacl) volte REN st Mo Met ican em mela 2; Her eh OL DINERO T abe TMy 4 oy Ley ley May well den viey deinieitliceiipeuian Wolk SOc ney ine] Mei vaveMi mt alte tame Sr) Chanigevot ibeach prosiwlieny matoe -uuainien emccd ach nenite | cliesiite: Vetitoireniom esi l/cih har atone mm 4 Changes of sand volume due to translation of one beach profile. .... 14 SOREL CESNOR WAVelreEnactdonssonway cum ved beach mri menisci no) nc mnrsnn wit TmneLes (Sj WWhisererceleealov PACINE: TNSIEERCSLOM Go 6 5 6 6 O10 Of do 61a a d10.0 00 0 o,0 20 7 Noeehesiom, tose Chiskirereealon Seth, 5 5566 66 0,8 6 60,010 010 of 0 6 6 o Bd 28 BY CONTENTS FIGURES--Continued Page Initial shoreline and incoming wave direction (A); shorelines at SUCCESSIVE wills Ore ies Aes Sites. MOMeS einel ZOMG (3) 6 6 6.46 06 5 «6 Za Shorelines at successive times of 5At, 10At, and 15At (A); shorelines KOTMEMME A NET (Be eat ne Wvenutel peers era eaincae 2 nes Initial shoreline and incoming wave direction (A); computed and minimal shorelines for finite-difference scheme of time 1At (B) ... 27 Transport function Q(a,) = cos a, sin za, for selected values of z.. 28 Layout of shoreline reaches characterized by different beach COMMONS 4s Tessa aula ere mi taal nC CU rains Vet ipas llear Megane sa pentivern er Ne" Cerys paltenicetalet Melee at MES Nearshomesbathymeitaayy a olelamcetl aac naire mrinan eile) elec ei cleey nena nS. Beach profile summaries) tom holland vHanbor sy 2). ). «= 1 20.) eee eee NOD HolstandsHarhoreneciairshonrembiatany ne Casveuemrcuitely ope carted ill chl a stent een-mneuinar an ee Conditions at Holland Harbor entrance before dredging in 1973 aniGeO7Aey Rs welts) Si ieluis 40 Wind rose for an average 12-month period and for an average 9-month Ce rree speTlod ror Souchexnlralce Ma chaiearlen) |) riicuen one -amCaye acini ume nnEraL Wave rose for an average 12-month period and for an assumed 9-month LESAGE) SILO! ieOEAe SOEs Iba WolelyyeinG 6 5 56605060000 0 Ail Wave height exceedance probabilities (9-month ice-free period) .... 42 Variations of probability of various wave heights with months of VELOVGH SVEN eS Ton We MRGe Meet esE No nbd ea ANN VEO M ON ic ateMmL ates Pein bg taal Baie MN taal Leh eNeM Uen, weala. aay Neem iWonemly lakes leweils simee 1205.6 6965656 556 60 oo ooo A Monthly variations of computed gross and net littoral drifts for Elo gel rani eaMakc hae oan ve) ar ieeetimnre nine) imac uel, eeinete st sumed eile Mathai cy ie er Salle ome i rial 37 /R SRE MEETS Shoreline positions at Holland, Michigan, in 1973, as determined by aerial photos. aes : 48 Smoothed average rates of shoreline erosion and accretion at Foliar dimen al Chioameensr nee e reid ie cect Vel ee epee Prem y icine ibsemlenaiice, Maco une ws, aie! recreate AC) Historical shorelines north of Holland, Michigan, from 1849 to 1945. . 51 SuMmMaiyaOLesanda bud getmnorsehi oa Holellatyd pa Matchlgany cue mnonnr ini tn te lnnnre nD S) Historical shorelines immediately south of Holland, Michigan, between TSANS) eeiravele INCU toes o lay Lvaiiio: GERM (a enWia\ ual or colts Me: bpiiben nl ag coy al No oar eaeaan et yA Historical shorelines farther south of Holland, Michigan, between USVI a@mel ISH o 6 6 6 6 6 6 6 9 6 56.8 OOo 68 Oo a oo 8 oo 6 Oo a oo, S55 Conpurcecl .ckeita, wersms actual datas .o5 56 8 oo oo oot USD CONVERSION FACTORS, U.S. CUSTOMARY TO METRIC (SI) UNITS OF MEASUREMENT U.S. customary units of measurement used in this report can be converted to metric (SI) units as follows: Multiply by To obtain inches 25.4 millimeters 2.54 centimeters square inches 6.452 Square centimeters cubic inches 16.39 cubic centimeters feet 30.48 centimeters 0.3048 meters square feet 0.0929 square meters cubic feet 0.0283 cubic meters yards 0.9144 meters square yards 0.836 square meters cubic yards 0.7646 cubic meters miles 1.6093 kilometers square miles 259.0 hectares knots 1.852 kilometers per hour acres 0.4047 hectares foot-pounds 1.3558 newton meters millibars OMS lOme kilograms per square centimeter ounces 28.35 grams pounds 453.6 grams 0.4536 kilograms ton, long 1.0160 metric tons ton, short 0.9072 metric tons degrees (angle) 0.01745 radians Fahrenheit degrees 5/9 Celsius degrees or Kelvins! Ifo obtain Celsius (C) temperature readings from Fahrenheit (F) readings, use formula: Cr= (5/9) (E32) To obtain Kelvin (K) readings, use formula: K= (6/9) (E32) oer SYMBOLS AND DEFINITIONS proportionality constant (8 = aa) height of bluff (in case of erosion) or berm (in case of accretion) phase velocity at breaking deepwater phase velocity water depth at ye water depth breaking depth variation of sand volume due to horizontal displacement during time, dt variation of sand volume due to beach slope variation during time, dt group velocity at breaking deepwater wave group velocity gravity acceleration breaking wave height deepwater wave height diffraction coefficient refraction coefficient from deep water to the line of breaking inception percentage of silt time-constant parameter wavelength at breaking deepwater wavelength length of groin perpendicular to shore loss of sand by rip currents beach slope at shoreline horizontal axis parallel to the average initial beach profile horizontal axis perpendicular to x, oriented positively seaward vertical axis, positive upward longshore energy flux dimensionless littoral drift discharge littoral drift discharge in cubic yards per year loss of bluff volume (in case of erosion) due to silt loss of bluff volume (in case of erosion) due to silt SYMBOLS AND DEFINITIONS--Continued loss of sand by wind sand discharge in a direction perpendicular to the beach average bottom slope to depth of no motion horizontal displacement at depth, D characterizing slope variation GQ? shoreline wave period time volume of sand over a stretch of shoreline, Ax, unity shoreline coordinates dimensionless shoreline deepwater limit of the beach function of x, y, t defining the bottom topography shoreline elevation above a datum z = o iD 158 08) SLi deepwater wave angle with the ox-axis angle of breaking with the shoreline deepwater wave angle with the shoreline angle of the beach limit with the ox-axis in front of a groin horizontal displacement of the beach profile during time, dt function characterizing beach-slope variation as function of x function characterizing beach-slope variation as function of t function characterizing beach-slope variation with time due to rapid change of sea level function characterizing beach-slope variation with time due to manmade structures such as groin or navigation works a symptotic (in axium) value of Ay) when t > variation of wave direction by diffraction angle of wave ray with a perpendicular to shore in a diffraction zone water density A NUMERICAL MODEL FOR PREDICTING SHORELINE CHANGES by Bernard Le Mehaute and Mills Soldate I. INTRODUCTION This report establishes a mathematical model for shoreline evolution and calibrates the model with a test case at Holland Harbor, Michigan. Even though the mathematical model is general and could be applied to a number of situa- tions, the emphasis is on the Great Lakes, and more specifically, on shoreline evolution near navigation structures. An interim report by Le Mehaute and Soldate (1977) reported on the feasi- bility of applying existing mathematical models to real case situations. This report is a continuation of that first investigation. The present mathematical model includes many of the characteristics already covered in the literature. In addition, the model presents an integrated ap- proach on a large number of phenomena previously neglected. Its main purpose is to develop a practical numerical scheme which could be used to predict shore- line evolution. The model would then be able to point out the shortcomings in the present state of knowledge. Therefore, the mathematical model covers some aspects of shoreline evolution which cannot be quantified with the data obtained from the considered test case of Holland Harbor. The mathematical model then has to be regarded as a research guide for the future. One important aspect is the effect of sand size and density. It is well known that the rate of shoreline erosion and sediment loss is largely affected by these parameters. The fine sand is transported offshore, while larger size sand tends to proceed alongshore according to a littoral drift formula. This effect could not presently be quantified; therefore, it is introduced in the mathematical model as a constant. The present mathematical model can continuously be upgraded as the state- of-the-art progresses, and as the model is tested for a large number of cases. It is important to remember that the model deals with long-term shoreline evo- lution as defined in Le Mehaute and Soldate (1977). Short-term evolution must be considered as local perturbations which are superimposed on the presently defined topography. Three time scales of shoreline evolution which can be distinguished are (a) geological evolution over hundreds and thousands of years, (b) long-term evolution from year-to-year or decade, and (c) short-term or seasonal evolu- tion during a major storm. Associated with these time scales are distances or ranges of influence over which changes occur. The geological time scale deals, for instance, with the entire area of the Great Lakes. The long-term evolution deals with a more limited stretch of shoreline and range of influence; e.g., between two head- lands or between two harbor entrances. The short-term evolution deals with the intricacies of the surf zone circulation; e.g., summer-winter profiles, bar, rhythmic beach patterns, etc. For the problem under consideration, long-term evolution is of primary importance, the short-term evolution appearing as a superimposed perturbation on the general beach profile. Evolution of the coastline is characterized by low monotone variations or trends on which are superimposed short bursts of rapid development associated with storms. The primary cause of long-term evolution is water waves or wave-generated currents. Three phenomena intervene in the action which waves have on shore- line evolution: (a) Erosion of beach material by short-period seas versus accre- tion by longer period swells; (b) effect of water level changes on erosion; and (c) effect of breakwaters, groins, and other structures. Although mathematical modeling of shoreline evolution has inspired some research, it has received only limited attention from practicing engineers. The present methodology is based mainly on (a) the local experience of engi- neers who have a knowledge of their sectors, understand littoral processes, and have an inherent intuition of what should happen; and (b) movable-bed scale models that require extensive field data for their calibration. In the past, theorists have been dealing with idealized situations, rarely encountered in engineering practice. Mathematical modelers apparently have long been discouraged by the inherent complexity of the phenomena encountered in coastal morphology. The lack of well-accepted laws of sediment transport, offshore-onshore movement, and poor wave climate statistics have made the task of calibrating mathematical models very difficult. Considering the im- portance of determining the effect of construction of long groins and naviga- tion structures, and the progress made in determining wave climate and littoral drift, a mathematical approach now appears feasible. The complexity of beach phenomena could, to a large extent, be taken into account by a numerical mathematical scheme (instead of closed-form solutions), dividing space and time intervals into small elements in which the inherent complexity of the morphology could be taken into account. Furthermore, better knowledge of the wave climate, a necessary input, will allow a better calibra- tion of coastal constants (such as those in the littoral drift formula). In past investigations (Le Mehaute and Soldate, 1977), many important ef- fects have been neglected, such as combined effects of wave diffraction around littoral obstacles, change of sea level, height of berm and bluff, beach slope, etc. The present investigation attempts to include all the important factors associated with long-term shoreline evolution. In the case of the Great Lakes, importance must be given to variations in lake level. The coastal zone is de- fined by a three-dimensional bottom topography instead of a two-dimensional shoreline (or two lines) as in the previous cases. Mathematical modeling is essentially approached by: (a) Understanding the phenomenology of shoreline evolution quantitatively. (b) Calibrating, determining empirical constants, and separating various effects, such as due to change of lake level, navigation Structures, etc. (c) Predicting long-term future evolution. (d) Assessing the effect of future construction. The mathematical model presented in this report is just a first step in this direction. Much of the model may be modified after the application of the mathe- Matical model to other cases which could be used for further calibration. Theoretical developments for the model are presented in Section II. Section III describes the shoreline evolution recorded at Holland Harbor, Michigan, and compares the mathematical model with the test case investigated. Section IV provides recommendations and conclusions. An Appendix describes a computer program to investigate shoreline behavior. Ii. THEORETICAL DEVELOPMENTS Consideration is given to a coastal zone limited by boundaries at a small distance from the surf zone (Fig. 1). The bottom topography is defined in a three-coordinate system, oxyz, by a function zp = f(x,y,t) where the ox-axis is parallel to the average shoreline direction, the oy-axis is perpendicular seaward, and oz is positive upward from a fixed horizontal datum. The angle of the shoreline with the ox-axis is small. The shoreline is defined by y = y,, Z = Zg = Zph(x,yg,t) which also defines the sea level as a function of time. Waterline Yb Ys Ye Yq y Figure 1. Coastal zone. The deepwater limit of the beach is y = Ye: (This limit defines the con- tour line where the sand is no longer moved by wave action.) The water depth at y= y, is D,. It will be assumed that D, remains constant as sea level and beach profiles change. Therefore, dzg/dt = 3z,/odt. B is the height of the bluff in case of erosion, i.e., when dV¥o/at < 0, and) the jhetght of the berm incase) of accretion aes), when ioye/.ot On (Ealoeme)) EROSION ACCRETION Figure 2. Height of bluff or berm. The quantity of sand over a stretch of shoreline, Ax = unity and bounded by theydatums9z 5" 0) sy) —) 0, andiithelbeach) profile iz7 yy atetime sy stamps Y We] | ater) ae 0 Assume that, for some reason, the beach profile changes during an infini- tesimal amount of time, dt. A further assumption is that the initial beach profile which is considered at time, t = t;, could be the normal "equilibrium profile." (The equilibrium profile may never exist under varying prototype conditions (similarly two-dimensional waves never exist), but it is a conven- ient idealized concept which could be approached in two-dimensional wave tank experiments. In this case, it could be defined as the statistical long-term average beach profile which exists under a given wave climate. The model here is actually independent from this definition. ) The departure and modification from this initial beach profile can be characterized by (see Fig. 3): (a) A translation in the yz plane defined by an elementary vector of components. IV dD re ake where dD/dt is the rate of change of sea level. Note that this translation is independent from the beach profile and, in particular, if the beach profile normally exhibits a number of significant bar formations, under normal conditions the translation will reproduce this characteristic at the same water depth. (b) A perturbation characterizing the departure or variation from the initial profile. Since the rate of the vertical component 12 y,(t+dt) vg (t) Yq (t + dt) Yo (0) 1 Figure 3. Change of beach profile. of translation is dD/dt, the perturbation can be defined only by a horizontal displacement. At the deepwater limit, D,, the horizontal displacement characterizing the rate of change of the average beach slope is defined by 95, (x,t) ot so that The quantity of sand within the considered domain at time (t + dt) is vp V(t + dt) = ih apy (Epp pe \ cle). chy (GD) 0 and the variation Y, dz V 1 6 eas ae i = dy YY, and 0 are fixed limits) VI OZ OZ OZ Ge [es en dt 0 at Oy cle 9 Bee” eke where 9z,/0x is the variation of beach elevation along the ox-axis. Since the angle of the beach with the ox-axis is small 13 dZp OZp (2 bans aye m is the beach slope), Se Ovane i.e., the change in z occurring along the beach is small relative to change across profile and the beach profile variation over a distance dx is small but remains an infinitesimal of higher order than over a distance dy. For the same reason, the velocity of beach variation along the ox-axis dx/dt is small as compared to the beach variation along oy; therefore IZp Cbs IZp dy OxeMGit Oven ae and could be neglected. To evaluate the other terms it is simpler to refer to Figure 3. It is seen that the integral is equal to the sum of: (a) The change of volume of sand, dv/dt, due to the translation, dyg/dt, is the sum of the change of volume due to the horizontal translation (Fig. 4) (B + Cc) — (2) and the change of volume due to the vertical displacement (ve - yp) (3) where Vi is the ordinate characterizing the location of the bluff. c Aas Yb dD B _—— _-— _—— ~~ _-— 6 ———= Cc AY Ye Figure 4, Changes of sand volume due to translation of one beach profile. 14 the The sea level displacement, dD, being small, the difference in the quantity of sand represented by the two triangles ABC and A'B!C! in Figure 4 is considered as a difference of infinitesimals and there- fore is neglected. So, combining equations (2) and (3) dv 8Yg dD —_—_— = —-_—___-=— - 4 _—__ 4 ony oe SET Ole: Bb. ide (4) It is seen that when dV/dt = 0 Via cnaty b dD dyna = oe Vela BeSTDP ioe Ss where S is the average bottom slope. Note that this expression is independent of the beach profile and, therefore, implicitly takes into account bar formation. (b) The change of volume due to the perturbation and departure from the equilibrium profile. This departure is characterized by a change of slope. Therefore, the corresponding variation of sand volume (area AEG in Fig. 3) is Se ene) 3) t BTS aha on TOL eel in accordance with the previous assumption. Therefore, the total variation of sand volume, dV/dt, is (adding eqs. 4 and 5): dV _ 5 any. 5 oS qe Oo Ue) ae = Cle lee ay Mane (6) This variation of volume is due to the variation of littoral drift along ox-axis and the onshore-offshore motion. The following terms are included: (a) The discharge of sand leaving the beach per unit of width includes: (1) Qyw due to loss of sand by wind. (2) Qs 09D due to the quantity of silt contained in the bluff and which tends to move offshore by suspension. This loss occurs only in case of erosion (dy,/at < 0) and is equal to Qs =UKAB oye/ ot), where) Ka) iisiithespercentage) of silt ain the bluff. (3) QF due to the loss of sand from the beach by density current during a storm. QF is a function of the size dis- tribution and density of material. A beach of fine material (<0.1 tm) tends to erode more rapidly than a beach of coarse material (>1 mm). The coarse material tends to move along the shore while the fine sand moves offshore. 15 The determination of these three quantities is given from sand budget investigations. (b) A general term, M(x,t), expressing the local variation in the sand budget due to loss of sand by rip currents along groins, and to sudden dumping of sand in case of beach nourishment or flood. (c) The variation of littoral drift along the ox-axis which is re) OQa(S) > One 2 Cbg) So ae dx (7) Many formulas in literature sources express the rate of longshore transport, Q,, as a function of the incident wave energy along a straight shoreline. Long- shore transport is also a function of the sand characteristics (size distribu- tion and density), wave steepness, beach slope, etc. The form of this formulation on shoreline evolution is of paramount impor- tance. In particular, determination of the relative rate of sediment transpor- ted in suspension and by bedload is very important since this ratio influences the loss of sediment by rip currents. Such evaluation is beyond the state-of-the-art, and any improvement would require a major effort beyond the scope of the present investigation. Any im- provement in the longshore transport rate formula could eventually be introduced in the model at a later date. Therefore, it is assumed that the rate of sedi- ment transport is independent of density and size and depends solely on the longshore energy flux, P,, by the empirical relationship On) Shad, sO ID, (8) where Q, is in cubic yards per year. P,» is in foot-pounds per second per foot of shoreline and is expressed by the relationship (U.S. Army, Corps of Engineers, Coastal Engineering Research Center, 1977) 2 - OF BND ae Po 64 TASK) sin 2c, (9) where Kp = refraction coefficient from deep water to the line of breaking inception T = wave period H, = deepwater wave height a, = angle of the deepwater wave with the shoreline ap = angle of breaking with the shoreline Note that this equation expresses implicitly the rate of littoral drift along a straight shoreline in terms of breaking wave characteristics (except for the deepwater wave height, H,). The equation is assumed to hold in case 16 of gentle beach curvature. The refraction coefficient, Kp, amd angle, ap, can be determined as functions of the deepwater wave characteristics H,, T, a (or ap) and the angle of the shoreline at breaking, dy,/dx. In the case of a groin perpendicular to the shore, at x > - », the deep- water wave angle, a,, with bottom contours is equal to the angle a of the wave with the ox-axis since the shoreline has the same direction as the ox-axis; at x = 0 (i.e., at the groin), the shoreline becomes parallel to the incident wave crest very rapidly. Therefore, a, = o and i (10) 8X |x = 0 Qo = -tan™ In the general case, for any value of x oy. Be =il § 11 ao a + tan ARE ( ) The breaking wave characteristics of wave height, Hp, water depth dz, and the angle of breaking op, can be obtained from the deepwater wave charac- ieriisiGLes SHA ah, and a5.) a5) vse eiven by ‘equation (11)))an) texms) of a) ;and dy,/dx which takes into account the curvature of the shoreline. The following equations permit a determination of Hp, dp, and ap, provided the bottom contours are parallel along a wave ray (Le Mehaute, 1961), but could be curved along the shoreline (Fig. 5): PAR ALLEL CONTOURS 7 Figure 5. Effects of wave refractions on a curved beach. (a) Snell's law. ee ee 1 CO Sani: ST /25 (2) (b) Wavelength (dispersion relation). L 27d 2 = tan n (+) (13) O Db (c) Conservation of energy flux between wave orthogonals (G is the group velocity). 2 = H2 14 HS cos a,G, = He cos a,G, (14) (d) Breaking criteria. Hy, 21d — = 0.14 tanh (15) Lp Lp From these equations, it has been shown (Le Mehaute and Koh, 1967) that when a, remains smaller than 50°, ap could be approximated as O 9.25 + 5.5 Mo 16 =~ Jie a 5 — On =A cs (16) where aie lL. = @ 27 Therefore, the refraction coefficient ae COS Ap 1/2 A COS Ob could be written, inserting oa, as given by equation (11), and ap as given by equation (16) 2) cos(a - tan7! 9Y, ae Tee ae fe ee ee a Nd = (G7) fi ( ois: 2aH2 cos |(a - tan Ye/, ) (0.25 SS Oloc hy of, 2) ( Be 2 Now it is possible to formulate the variation of littoral drift over a distance Ax = unity. From equation (9) 2Q¢ ox ‘Db 2 dKp if 5 + AH52Kp ae Zap (18) = AH2K% 2 cos 2a ee oR Dia where On the other hand, since (see eq. 16) 0p 30p ( Ho ARTE PST, 0.25 + 5.5 —= (19) and by taking into account equation (11) lor 1 one —o = (20) a bie ( Ys/,.) iH Finally, by inserting equation (20) into equation (19) O25 = 5.5 1o/, 32 Jone) y t Lo 9 Ys (21) = 1 6/5 oo In case of wave diffraction, the wave height varies significantly along a wave crest. Then, the previous refraction coefficient, Kps has to be replaced by a combined diffraction-refraction coefficient, such as K,K,. (A diffraction current in opposite direction to a longshore current takes place at a distance from the end of the groin.) The variation 9Kp/dx is much larger than dKp/dx. Therefore, in analogy with equation (18) 3Q¢ dp OS 2 — = nix (2x2 cos 2a, =— + 2Kp == sin 20, (22) In a diffraction zone, az, is due to the sum of variation of shoreline direc- tion, tan7! dY¢/9x, and because of diffraction the rotation of the wave crest around the end of the groin, §& (Fig. 6). 6 is the angle which has the end of the groin as apex and extends from the limit of the shaded area to the con- Sidered location defined by x. Figure 6 shows that © = ajo - 8' and 6' = tan7! x/2, where 2% is the length of the groin. Therefore, = em! 28 Ss wenel & 23 apy waean = + A an (23) 1 Oeste Change Une er eL (24) OR rox et (x 2) and, differentiation equation (23) and inserting equation (24) 30 a*y SDAIN tens, San Sepp live Tce wale. (25) Q Se a C2) where dKp/ ox is the coefficient of variation of wave height due to combined effects of wave diffraction and wave refraction. It also includes the effect of diffraction current. An empirical formulation for determining the combined effect of diffraction and refraction is more suitable to quantitative analysis of a real sea spectrum 19 Figure 6. Diffraction zone notation. than more exact theories of wave diffraction which are valid for periodic waves over a horizontal bottom and are represented by a Fresnel integral. For this, it will be assumed that the energy travels laterally along a wave crest as well as along a wave ray. The lateral speed of propagation is assumed to be equal to the group velocity of a periodic wave of average period. (Actually, since the problem is confined to very shallow-water waves, and the longest waves of the spectrum diffract most, the limit of the diffraction zone is defined by an angle such as the velocity of propagation of wave energy along the crest and is simply VgD, where D is the water depth.) This lateral transmission of energy results in a decrease of wave energy from the exposed area to the shaded area, such that (Fig. 7) D H2dx = i H? ds (26) A 0 where s is the distance along a crest from the end of the groin. Figure 7 shows from simple geometrical relationships that Q V2 OD ee ea PET Gp @, 25) 2 oA = -% tan (45° - a,) (27) oC) = Litany (4570+ oF) therefore, : ‘ an Uecann (4 eac)) us He ee ee H*dx (28) SO ley A) eh cae) ISO Say) 20 Figure 7. Notation for diffraction study. In the case where the long groin is in the previously defined wave diffrac- tion zone as in Figure 6, it is assumed that the wave energy which reaches the groin is absorbed by friction. It is also assumed that the combined effects of diffraction and refraction of a wave spectrum can simply be represented by a sinusoidal variation of wave height along the breaking line (Mobarek and Wiegel, 1966) (Fig. 7). Assuming that Kp(x) is of the form sin k(x - X5) as shown in Figure 7, Such as Kp at A= 0, Kp at C = 1, then to satisfy these conditions T 1 1 oe oe (ean CS Ss) En Gers a) (29) which after some simple trigonometric transformations can still be written k = 7 cos 2a (30) mats O to satisfy the equation expressing the conservation of energy flux, such as given by equation (29), then and Kp(x) = H(x)/H, is given by os (2 1/2 2 Kp (x) = (2 een) sin (ee ee bso & can CS> = 2031) (31) sin Ot Ay 2 | By differentiation equation (31) dKp (x) a 2 cos ae a Tm cos (20,) EI | NIA sin Gs AL 7 Cos (2a,) COS \ aa erm [ocacteecae ani (4S 201) (32) Inserting this value in the littoral drift equation (22) permits the mathemat- ical model to completed. In summary, the variation of volume of sand dV/dt given by equation (6) is equal to the sum of: (a) The loss of sand by wind Que (b) The loss of silt Qs = K,B dy,/at where K, is the percentage of silt in the bluff. (c) The loss of sand by density current during storn, Qf. and loss by rip current (or brought upon by nearby river). (d) The quantity of sand dredged or (at the opposite) deposited during beach nourishment. (e) The variation of littoral drift along the ox-axis is 9Q,/9x. In a refraction zone alone, 9Q,/dx is given by equation (18) where K,, is given by equation (17). op is given by equation (16) as a function of A>, and a, by equation (11), with respect to the ox-axis. Also, 9a,/9x is given by equation (21). In a diffraction zone 9Q,/3x is given by equation (22) instead of equa- tion (18) and the diffraction coefficient Kp by equation (31), and dKp/dx by equation (32) for a wave spectrum. ap is now given by equation (23) and dap / ax by equation (25). Since all the phenomenological equations have been established, it is more convenient to express them in dimensionless form. The ''general" equation ex- pressing the sand budget balance can still be written. (The loss terms have been dropped for simplicity and can easily be included if necessary.) Ys Gb) a ey Oe Golestehe We) aare ore Vb) mes a Do ae ae (33) For an analysis, this equation is considered in dimensionless form. Let e length B, + De where Bo is chosen later + At —————— (35 G34 Dae BS DD Q = 5 KAKs sin 2op (36) a2 The general equation is thus transformed to A ag Dy) a B+ Di. Me ee yh dd, 1 Dy, dAy nis KpKe sin 2a, Mo eo n~ The hats will be dropped from all variables at this point in all discus- Sions except those relating to observed results. Also, the subscript s for y and Q will be omitted. Hence, the general equation used in the following is DO AS B + De. dy 1 De day 9 Knkp sin 20, * (We - ¥p) = + 5 eS sO) Bo + De at dt 2 Bo Dende dX 2 Recall > _ COS A Rumcosmap where a5 = a + tan-! dy/ax and ap, = function of ap, as f(a.) (the function f depends only on H,/Lg as previously shown). Therefore, Ne 22 . 12 9 Kikp sin 202, = Ks cos a, Sin ap, Note ee cr ec) Bets ax COS % Sin ap = (cos A cos ap Jes = Sill Cy) Sala «p) aa (a5) aa The general equation (37) then becomes (after some rearrangements) BceaeD) 2 dy O C 1 a“y ae 2 SS PCy), Se oe eee Rei) (38) ot Be De Tae ( Lax) axe where By ap 1D) O Cc Ja, dD RES yait) = B+ Ds F (a,) yx Ye 5 Yp) dt D oK 1 C dAy D : Ss =— 9 7 By © A ce + 2Kp 5, COS Op Sin ap (39) and a(x) in the diffraction zone a The above equation is the general dimensionless form which gives the time- dependent sand budget. The general equation is nonlinear and appears to be impossible to solve ana- lytically. As it is also difficult to solve computationally in the most univer- sal case (where lake level, beach slope, and bluff-berm height vary as functions of x, y, t), several simple cases are described in detail which indicate the behavior of the equations. Although numerical results are presented, detailed descriptions of the numerical methods used are given in a later discussion. 23 The simplest examples are those of beaches having negligible or identically constant bluff-berm, uniform beach slope, and uniform lake level. In these instances, the general equation (38) reduces to 2 aK aa F(a) ieee oy 2Kp TES Ks GSN Cy 2 PGs) ee (40) ot : (C7, ) ox ox ox i+ ax subject to the appropriate boundary conditions. Suppose (as shown in Fig. 8) the beach is initially zero, bounded by a breakwater which is a complete littoral barrier, and subjected to waves having uniform positive deepwater direction. The boundary conditions are oy = 1S > TEIN Ou ke 0) OX e) avi Olay xd hls ax where L is a distance along ox, large enough so as not to be influenced by the groin. The shoreline is calculated for a fixed Hj, T, at intervals of equal At (shown in Fig. 8) for selected times. The general shape of these curves is Similar to the type formulations obtained by Pelnard-Considere (1956). INCOMING WAVE DIRECTION INCOMING WAVE DIRECTION INITIAL SHORELINE \ INITIAL SHORELINE Figure 8. Initial shoreline and incoming wave direction (A); shorelines at successive times of 1At, 2At, SAt, 10At, and 20At (B) (vertical scale exaggerated). 24 An extension of the above example is shown in Figure 9 where both sides of a breakwater are considered. The initial position of the shoreline and wave conditions in deep water are as before (Fig. 9). The boundary conditions become ox oY = O at x = -O (simplification of more exact condition) Oyen a — = -tana at x = +0 ox The uniform depth theory of Penny and Price (1952) is used as an approxi- mation (not substantiated) for diffraction about the end of the breakwater. The shoreline is calculated for various multiples of a fixed At (Fig. 9,A). INCOMING WAVE DIRECTION \ INITIAL SHORELINE INCOMING WAVE INCOMING WAVE DIRECTION DIRECTION \ INITIAL SHORELINE Figure 9. Shorelines at successive times of SAt, 10At, and 15At (A); shorelines for time 2At (B) (vertical scale exaggerated). 25 Allowing the wave direction to alternate between +a and -a at a fixed rate gives an interesting variant to the previous example. The boundary condi- tions are as before for a> 0. The results are shown in Figure 9(B) for the same values as the physical parameters used in the problem of Figure 9(A). Of interest is that the undulatory patterns of the shoreline in Figure 9(A) dis- appear in Figure 9(B). Hence, diffraction-induced undulations in a natural shoreline probably rarely appear since offshore wave climates are often multidirectional. The numerical scheme used to generate the preceding examples was based on the use of implicit finite differences. Such schemes, whether implicit or ex- plicit (or both), are commonly used to efficiently solve parabolic problems. However, even in the case where only refraction is considered (Fig. 8), the boundary condition numerically gives a solution which initially may not conserve mass, i.e., the integrated transport equation L -- I ydx = Q(L) may not be satisfied. Unfortunately, this feature is unavoidable for most such schemes (the ex- ceptions are discussed below) as the following demonstrates. Figure 10(A) shows an initially straight shoreline. In any finite-difference scheme, after one time increment At, the shoreline is bounded below by the solid shoreline in Figure 10(B). This shoreline has the least possible area, A, where A= as tan a (41) The conservation of mass equations requires ACQ (LS TAtcosmaksanwicn =A Thus, At must satisfy the inequality Ne > Hist D | (42) 2 Sin) Gp Cos= -a Since the accuracy (and in explicit schemes, stability as well) depends on the ratio } = At/Ax*, the above inequality places a lower bound on the accuracy of the solution which may be unacceptable in practice. The finite-difference form of the equation for the conservation of mass may be incorporated directly into the numerical scheme. In this case a solution exists which is similar to the previous case but shows a small erosion throughout the reach. For engineering applications, the primary quantity of interest is the amount of sand on a given shoreline. Then it is more important to conserve mass than to satisfy the shore- line boundary condition as written in the present form. The general equation is 26 a= INCOMING WAVE DIRECTION —— MINIMAL SHORELINE ——-—-— COMPUTED SHORELINE Ax 2Ax Ax 2Ax x Figure 10. Initial shoreline and incoming wave direction (A); computed and minimal shorelines for finite-difference scheme of time 1At (B)- now used to derive an equivalent equation for the transport Q which, although subject to similar numerical problems, will satisfy the transport boundary con- ditions exactly. In a situation where only refraction is important, the general equation then becomes OF, Ow dt Ox where Q is cos J Sin op, Ap 18 f(a), and a, is a - tan! dy/ox. Differen- tiating by x gives as i eS (43) The transport function Q can be considered as a function of o, which may be solved for a,, such as a = g(Q). Thus, the transport equation (43) becomes ane SO, Dior a ania ic) = op an(g(Q) - a) = aD therefore, BO) ls 37Q st Cosi (g(a) ao but 3g(Q _ dg(Q) 2Q at dol) 3c therefore, aQ__ cos? (g(Q) - a) 37Q ot dg(Q /ag ax? (44) 27 Assuming a solution for this equation is known, the shoreline y can be calcu- lated from the equation t y(t,» = y(o,x) + [ = (t,x) dt 9) In practice, the equation for Q is not solved in the above form. Implicit in the above formulation is the assumption that the function g exists. However, as shown in Figure 11, g is not single-valued if the maximum range of the angle a, is greater than approximately 41°. This difficulty may be removed by considering the equation for Q and y as a system subject to the boundary conditions for Q. Note that cos? (2(@) = 5) 40 j[, | (2 j L dg (Q)/aQ da, ox Hence, the equation for Q becomes a0) al) 1 92Q STE Ra EINE MRD // AULT NOE WARE (45 Oe cy is (2¥/ax) ax ) 0.6 z = 0.25+ 55H /Ly 0.5 eis os z = 0.75 g £ ae) is] 8 z = 0.50 0.2 z = 0.25 0.1 ao 10 20 30 40 50 INCOMING WAVE ANGLE (°) Figure 11. Transport function Q(a,) = cos a sin za, for selected values of z. 28 This, together with the equation dy/dt = 3Q/dx, is solved in a cyclic scheme. One possible method is the centered Crank-Nicolson type implicit- explicit scheme discussed below. Suppose y is given for all x at a given time, t, and that from t the wave climate is specified by the (constant) triple (a5H), 1). Let Ny ie mi EQ Ot Eh ( xd) = dag ee (8¥/ax)° At eres L(t,x) = an approximation to L(t,x) Integrating the Q equation gives At 92Q 97Q OG <2 Neyo) = OlGeno) mal L(t,x) aoe IGE av Att, x) aa (46) it ANE where OQ OCs hd = 206) = WGe= iss oS eee ee ee ee (47) ox Ax Integrating the y gives At [3Q IQ QB S ING559) = 37(E5) Sea] Oe (48) 2 \\ee aX t + At where 30 _ Q(x + Ax) = Qe = 4x) (49) OX 2Ax These equations are solved numerically, subject to the appropriate boundary conditions, in steps using the cyclic algorithm: (a) Wee 16Ge 2 Wes) S Ges Wee, (b) Calculate Q(t + At,x) Vx subject to the appropriate boundary conditions. (c) Calculate y(t + At,x) Vx; calculate L(t + At,x) and set this equal to L(t + At,x); then calculate new Q. (d) If new Q compares with old Q, stop; if not go to step (c). Tests with this scheme have shown that it converges to its limit after one application of step (c). This method can easily be modified to solve the equation where both dif- fraction and variations in lake level are allowed; i.e., m is the slope (refer to eqs. 37 to 40). OB 1 0 = ED cos a, sin ap mars (50) (28) For convenience, let Q = coS dp sin ap Ty ce ee Q = K2Q (51) As before, referring to equations (43) and (44) CON “heyy apace a uals (CQ) OQ es ule as ot ox dt cco hie COSS Ce (O)Mena) dOMN ots icoscmcs(@) momo more Also, 2Q _ 2 && oND 0 Be OD from equation (51). The second term in each of the above two equations is negligible in physi- cal situations of usual interest where the distance between the shoreline and the tip of the breakwater is large compared to the distance the shoreline changes during a time At. Therefore, the transport equation becomes aq a2Q — = eos (BOQ) > &) == (52) at dE /y9 ox and oY. 2 eR ly, (53) ot ox mdt This system is solved using the same type of algorithm as used previously. In the present situation where only refraction is important, several approx- imations are possible which produce problems having analytic solutions. The most direct approximation, and essentially the assumption of Penard-Considere (1956), is to approximate a2y p) i ung (2 cos A cos ap - sin 9 sin of) ——7> 7-5 (54) We CVax) at where z = 0.25 + 5.5 H,/Lo (see eq. 16), subject to the boundary conditions OL ox = anino tad i ° by where a is aconstant. For the standard breakwater problem, the most logical choice for this constant is given by Z 1 = ———;— = |(Z cos a, cos op - Sin a, Sin a,)——77 a (55) eee yan aa ( Oo b O Or, BAY a since the shoreline in this case is principally governed by its behavior at the breakwater. This problem (defined by eq. 55) has the solution Vat 2 VAeent) = 2) canoe lems /4at V0 K - tan a x erfc{— == (56) Gr which is the same as that of Penard-Considere except that the constant a has been changed. This problem, however, doesn't conserve mass since co 3 P : een y(x,t) dx = z sin a cos a sin za cos a ae 46 When this approximation is used in the transport equation for Q, the problem becomes 3Q 2 Ns cia dt ax? subject to the boundary conditions Q(x = 0) = 0 Oise 2) GS G Sin oH, which has the solution cone Q(x,t) = cos a sin ap erf leas) (57) Integrating the equation, dy _ 2Q ot ox gives VR 2 x x = i Moai? (ere ene | es woPe ae y(X%,t) = cos o sin Oy [2 e = Sree ze) (58) which is of the same form as the previous solution (eq. 56). III. INPUT DATA 1. Description and History of Navigation Project at Holland Harbor. The input data, which are pertinent only to the present investigation, deal exclusively with the area near Holland Harbor, Michigan (U.S. Army Engineer District, Detroit, 1975). Construction at Holland Harbor began about 1856 when the city of Holland cut through a narrow tongue of land between Lake Michigan and Lake Macatawa (Fig. 12). The present dimensions of the navigation project were established in 1909. The existing navigation structures have been constructed, 3| LAKE MICHIGAN REACH 4 REACH 3. REACH REACH 1 ° tS SF ee BAV4UI9bI BAV WL 3NI7T ALNNOD ‘ay 1$3993919V3 LEGEND am 28-29x——_ 1944-——. 7U.S.6.8. 1000 C) 1000___ 2000 3000 FT Figure 12. Layout of shoreline reaches characterized by different beach conditions. reconstructed, and repaired by segments during different periods over the past 117 years. In general, the north side shows accretion while the south side shows accretion from the breakwater to about 1,200 feet south and then general erosion farther south. The following areas (Fig. 12) appear to be affected by the navigation struc- tures as evidenced by aerial photos, condition surveys, and plat maps for the period of record (1849 to 1945): (a) North of Holland Harbor (Eaglecrest Road to the North Pier of Holland Harbor). It is considered that the accretion fillet north of the piers has been relatively stable since 1933. After that date, the predominantly southward-moving littoral drift has been diverted lake- ward resulting in rising nearshore elevations north of the breakwater and deposition of material in the entrance channel. This reach of shoreline, about 5,000 feet in length, is characterized by increasing accretion from north to south (Fig. 12, reach 1). (b) South of Holland Harbor. (1) Holland Harbor entrance channel to a point 200 feet south of the Ottawa-Allegan County line (Fig. 12, reach 2). Reach 2 extends about 2,000 feet south of Holland Harbor, and consists of a sand beach about 50 feet wide backed by low sand dunes. The shoreline adjacent to the south pier accreted at the rate of 9.6 feet per year from 1871 to 1944, a total of 700 feet. This area of accretion diminishes progres- Sively southward from the piers to a point 200 feet south of the county line where the shoreline before major construction (1871) coincides with that of the 1929 and 1944 surveys. (2) From 200 feet south of the Ottawa-Allegan County line to 147th Avenue (Fig. 12, reach 3). This reach is about 2,400 feet in length and is characterized by high, stabilized sand dunes (up to 120 feet above low water datum) which are being undercut by wave action. ge (3) From 147th Avenue to a point approximately 1,900 feet south of 146th Avenue (Fig. 12, reach 4). Reach 4, about 4,500 feet long, is characterized by eroding sand dunes up to 225 feet high. Figure 12 shows that erosion is greater at the northern end of the reach and decreases progressively southward. At the southern end of the reach the shoreline before construction of the navigation structures coin- cides closely with that of the 1929 and 1944 surveys. Erosion appears to be greatest at a point about 800 feet south of 147th Avenue where the shoreline receded about 220 feet from 1866 to 1945, or an average of 3 feet per year. The rates of movement of the shoreline near the harbor have decreased (1945-73) but the trends of accretion on the north side and erosion on the south side have continued. The present lower rates of shoreline movement are due to several factors: (a) The rate of shoreline movement due to the navigation structures decreases with time. (b) Local interests have built more shore protection structures as their property was threatened. The consequence of these structures is to reduce the apparent rate of erosion locally, but this is accomplished at the expense of steepening of the offshore bottom profiles and in- creased erosion downdrift. The shoreline protective structures now extend almost continuously from 1,000 feet to about 4,600 feet south of the harbor. (c) The bluffs, currently under erosive wave attack between 5,000 and 7,000 feet south of the harbor, are very high so that erosion rates measured in feet per year are low but the corresponding volume rates (in cubic yards per foot) are high. The shorelines apparently have become comparatively stabilized since about 1933. 2. Geomorphology. Figure 13 shows the nearshore bathymetry determined by a survey in April 1973. Beach profiles from this survey and one in May 1945 are summarized in Figure 14. Three or four lines of longshore bars are prominent most of the time and occur in water depths to approximately 30 feet. The first bar, in depths of 1 to 4 feet, can change rapidly from storm to storm and is often broken into short segments from 200 to 1,000 feet long by rip channels. Adja- cent to the breakwaters, the rip channels tend to elongate lakeward as far as the second bar. The second and third bars, occurring in depths of 6 to 14 feet, appear to be affected only by larger storm waves and are relatively continuous along the shore. Aerial photos show that these bars tend to lose their promi- nence as they approach the breakwaters from both sides, suggesting that pre- vailing erosion exists across the bar crests near the breakwaters. Tie fourth bar occurs at depths of 15 to 20 feet about 1,500 feet offshore and is charac- terized by crescentic crests displaying wavelengths between 3,000 and 5,000 feet. A crescentic fourth bar with an average wavelength of about 5,000 feet existed north of Holland Harbor, but none existed on the south side. Although the general bathymetry follows the coastline, significant departure from this trend occurs in a region offshore of the Holland Harbor breakwaters (Fig. 15). The 24-, 27-, and 29-foot-depth contours meander sharply off the 33 “(SL6T ‘2T0TI0q ‘3OTIISTC TeouTSug Away *S* WOLF) TOGteYH pueTTOH “AxzoUAYyIeG eLOYSIVeN “ET eAndTYy Be T | F i ; 7 | 5 z z 3 0008 ¢ + Gan 213NI7 fan : a) | Sa a ee eS SS SS ee: = Se = ~ Saat == 30009 34 HO,LAND APR 73 SEC 111 ea € ; Lan 18 873 0 1a howe ELevano® —— ora, --- ReOG 7e00 ©3000 8200-3400 G00 800 FOOU FrOO PAC “oc CCC‘ CC FEET Beach profile summaries for Holland Harbor (from U.S. Army Engineer Figure 14. District, Detroit, 1975). 59) HOLLAND APR 73 SEC 111 " ' {eo a: sre 6 reo FEET ° 20 “200 0 200 400 600 800 00 1200 00 00 00 2000 7700 7400 2600 7000 3000 S200 34a0 FEET 1 1 © 100 «0 600 800 WOO 1700 00 00 1800 7000 2700 7400 2600 72800 3000 HOO B00 cla (wD in 87Re 1D 3 0 600 800 00 FD 00 1600 1900 OOD ROC r400 7400 FeoD SOOO 3200 B400 FEEy Figure 14. Beach profile summaries for Holland Harbor (from U.S. Army Engineer District, Detroit, 1975) .--Continued 36 “(SZ6T ‘2T01290q £39TIISTG TooUTSUq AWAY *S*p WoTZ) ATJOWAYIEG STOUSTeSU TOGIeH pueTIOH “ST oansty Si harbor entrance, delineating a ridgelike formation which extends obliquely lakeward from north to south. Deeper contours to the southwest of this forma- tion suggest that it may extend as deep as 50 feet. The ridge has a maximum height of approximately 7 feet, and occurs due west of the harbor entrance. Between the ridge and the harbor entrance is a steep trough about 34 feet deep. A lack of survey data before the harbor construction prevents a reliable con- clusion as to whether or not this underwater ridge was caused by the presence of the breakwaters. However, the possibility cannot be ruled out that it was formed by the littoral drift arriving predominantly from the north and flushed lakeward by the breakwater. This material, unable to return shoreward due to the absence of significant swell activity in this region, could have accumulated to form such a formation on the offshore bed over a long period. Implications of this interpretation are significant because it may mean that little, if any, bypassing material across the harbor entrance could be expected to reach the southern shore of Holland Harbor. Many of the dunes reach a height of 150 feet, and in a few places exceed 200 feet above the lake level. The highest dunes are confined to a belt about 1 mile wide but lower ones occur for a width of several miles near Holland. Dune building by wind action is still active. North of Holland Harbor some low dunes are actively migrating inland or leeward from the bluff onto rela- tively level upland. In this area the dunes are 40 to 120 feet high. At nu- merous places, slumping of vegetation is evident on the bluff slopes. South of Holland the dunes are higher, frequently more than 200 feet above lake level. The dunes are well vegetated except for occasional blowouts on larger dunes. At many places, dunes descend directly to the lake or to a narrow strip of beach up to approximately 50 feet in width. The only area near Holland Harbor where a substantial beach exists is about 2,000 feet immediately north and about 500 feet immediately south of the breakwaters. Maximum beach width in this area is about 500 feet on the north side and 150 feet on the south side, indicating the predominance of littoral drift from the north. Berm development is gener- ally poor for many miles of shoreline north of Holland. Berms develop more fre- quently south of Holland, growing to an average height of 1 foot in the swash zone. These berms are usually truncated at positions of a recessed shoreline where a rip current has gouged a deep channel across the surf zone. 3. Littoral Materials. The dominant littoral material which comprises dunes, beaches, and nearshore lake bottom is glacial sediment belonging to a fine sand category (less than 0.42 millimeter in median diameter). Scattered pebbles are found both on the dunes and the lake bottom. Sand-sized material is dominantly quartz {80 to 85 percent) with up to 12 to 15 percent feldspar. Heavy materials represent only ZEON Se PeXcenit. a. North of Holland Harbor. An analysis of bluff, foreshore (above low water datum) and bottom samples indicates that the majority of these sands belong to the fine sand category. (The Unified Soil Classification System defines fine sand as ranging in size from 0.42 to 0.074 millimeter (1.25 to 3.75 phi), and medium sand as ranging from 0.42 to 2.0 millimeters or 1.25 to -1.00 phi). North of the harbor, the average bluff and foreshore sand consists of 88.0 percent fine sand and 11.7 percent medium sand; the average lake bottom sand consists of 95.0 percent fine sand and 5.0 percent medium sand. A general 38 trend exists for the sand size to become progressively smaller from foreshore to offshore. Average median diameters are 0.34 millimeter at the foreshore, 0.28 millimeter at depths of 5 and 10 feet, 0.23 millimeter at depths of 15 and 20 feet, and 0.18 millimeter at depths of 25 and 30 feet. Dune and bluff sand is somewhat finer than the foreshore sand, but coarser than the lake bottom sand. There is little systematic variation in sand size with distances from the breakwater. Sorting ranges between 0.28 and 0.40 phi, indicating very good sorting; sorting varies little between dune, foreshore, and the 5-foot depth and becomes poorer toward deeper water, reaching approximately 0.40 phi at depths of 20 and 30 feet. b. South of Holland Harbor. The majority of the sands from the dune, the bluff, the foreshore, and the lake bottom on the south side of Holland belong to the fine sand category. Generally, only minor differences exist between average sands from the north and the south sides of Holland. The average bluff and foreshore sand consists of 85.1 percent fine sand (finer than 0.42 milli- meter) and 14.9 percent medium sand. The average lake bottom sand consists of 94.7 percent fine sand and 5.3 percent medium sand. Median diameters decrease from foreshore toward offshore, about 0.25 millimeter at depths of 5 and 10 feet, about 0.20 millimeter at depths of 15 and 20 feet, and about 0.16 milli- meter at depths of 25 and 30 feet. Dune and bluff sand is finer than the fore- shore sand but coarser than the lake bottom sand. The lake bottom sands are distinctly coarser immediately south of the harbor and also along lines 11 and 12 (Fig. 13), about 6,000 feet south of the harbor. Sorting displays a wider scatter than on the north side of Holland, ranging between 0.3 and 0.5 phi. As on the north side, sorting becomes poorer from foreshore toward offshore. The principal sources of littoral material are the beach and the bluff. Generally along eastern Lake Michigan, and particularly near Holland Harbor, the supply of littoral material from major inland water runoff is negligible. For this reason, the long-term geologic trend appears to be for sediment from shore erosion to fill in the lake. c. Shoaling and Dredging. Shoaling and dredging records indicate that the annual shoaling in the entrance channel of Holland Harbor averaged about 25,000 cubic yards for the period 1965-70. The pattern of accretion shows some yearly variation and indicates that material encroaches into the entrance area from both north and south. In addition, the relative amount of drift from north and south appears to vary from year to year. Figure 16 shows conditions at the harbor entrance before dredging in 1973 and 1974. 4. Meteorology. The dominant factors are winds, waves, and water level variations. The winds and waves are directly responsible for sediment movements, and fluctua- tions of water levels separate the regimes of the two areas affected, i.e., the backshore above the waterline and the foreshore below the waterline. a. Winds. Winds are a dominant force affecting the Holland Harbor area; they produce a number of major effects by (a) generating a force for waves, (b) causing lake level changes, and (c) transporting sand across the beaches, particularly the finer sediment sizes. Wind data (available from ship reports) over the southern half of Lake Michigan (south of 44° N.) show that more than 39 DEPTH IN FEET. ALL DEPTHS ARE REFERRED TO LOW WATER DATUM 576.8 ft. ABOVE 1.G.L.D. 1955 MAR. 1973 MAR. 1974 Figure 16. Conditions at Holland Harbor entrance before dredging in 1973 and 1974 (from U.S. Army Engineer District, Detroit, 1975). 21,000 observations were taken from 1963 to 1973. These data have been sum- marized for a 12-month period (Fig. 17,A) and for a 9-month ice-free period (Fig. 17,B). A 3-month ice period would correspond to a severe winter. b. Waves. Sources of wave data for Lake Michigan were (a) Summary of Synoptic Meteorological Observations (SSMO) (National Oceanic and Atmospheric Administration (NOAA), 1963-73); (b) Saville (1953); (c) Cole and Hilfiker (1970); and (d) Littoral Environment Observation (LEO) program data from the U.S. Army Coastal Engineering Research Center (CERC). Figure 18 shows deepwater wave roses for an average 12-month period and for an assumed 9-month ice-free period, respectively, in southern Lake Michigan. Comparison of these wave and wind roses reveals a close agreement with the wind and wave statistics. Dividing wave heights into two groups (i.e., by heights lower and higher than 4.1 feet), the low waves occur most frequently from the south, while the high waves occur predominantly from the north. Figure 19 shows exceedance probabilities of deepwater waves based on SSMO and hindcast data of Saville (1953) and Cole and Hilfiker (1970). Only those waves which occur during a 9-month ice-free period and are traveling shoreward are included in the statistics. The three statistics generally agree quite well for wave heights up to about 8 feet. For wave heights larger than 8 feet (which occur only about 1 percent of the time), a discrepancy between the three is evident. The SSMO data show a less frequent occurrence of higher waves, but 4 (oe) 10% 10 % NW. NE NE 1 Pron i Oe {| ly - 7. ” “ g ‘J / Lx ; rs Oy \ a ES eh LES KE oc, 03 ,§ = toy, SESS w Se Ie Sa 100 S PERIOD (~, - aN SSS —— ae Z WSL SRE! s LW Lf aan ap VS / bs \\\ SSS, 5% a vy SW: SE 1Q%o s WIND SPEED 2 (7) 0-10 kn 1-21 kn Sone) 22-33 kn WSIGESET 34-47kn Figure 17. Wind rose for an average 12-month period (A) and for an average 9-month ice-free period (B) for southern Lake Michigan (data from SSMO observations, 1963-73). WAVE HEIGHT Co) < 0.25 meter GEE] 0.25 -1.25 Figure 18. 1.25-2.25 GEEZER 2.25 < ( <0.82 foot ) (082-410 " ) (410-738 " ) (7.38< mi) Wave rose for an average 12-month period (A) and for an assumed 9-month ice-free period (B) for southern Lake Michigan (data from SSMO observations, 1963-73). WAVE HEIGHT, (ft) 100 EXCEEDANCE PROBABILITY, ( pct.) 0.1 —— SSMO (1963-73) —-— SAVILLE (1953) -+—+ COLE and HILFIKER (1970) 0.01 Figure 19, Wave height exceedance probabilities (9-month ice-free period). this may be partly due to the fact that ships tend to avoid storms. This may also be less true for the higher windspeeds; hence, the frequency of occurrence of high waves may tend to be overestimated. The disagreement between Saville!'s and Cole and Hilfiker's hindcast data probably results because Saville's data are based on coastal winds which are generally weaker than those on the deep- water lake surface. Figure 20 summarizes the probability of occurrence of various wave heights as a function of the month of the year. 100 1.64 ft OR HIGHER 3.28 ft OR HIGHER 4.92 ft OR HIGHER - PROBABILITY (Pct) cé,) oO c JAN FEB MAF APR MAY JUNE JULY AUG SEPT OCT NOV DEC JAN Figure 20. Variations of probability of various wave heights with months of the year. 42 5. Hydrology. a. Water Level Variations. Figure 21 summarizes the observed mean monthly lake levels since 1920. Records are available back to 1860. b. Currents. Currents in the Holland area are variable. They generally appear to take a northward set during the summer and a southward set during the winter in response to the prevailing wins although reversals are frequent. There also appears to be evidence of a rotational current system in Lake Michigan which is not very well understood. In the immediate nearshore area it is likely that the currents are wave induced and will propagate alongshore in the direction of the dominant waves at the time. Inshore of the first bar, rip currents are extremely well developed and can add a significant amount of offshore water movement to the general longshore current. Intensity of the rip currents is evident from a number of rip channels which are gouged 1 to 2 feet below the adjacent ground. Longshore currents approaching Holland Harbor are deflected lakeward along its breakwaters. Aerial photos indicate that the de- flected current, after bypassing the harbor entrance, will move a considerable distance lakeward before turning parallel to and toward the shore off the down- drift coast. c. Ice. Floating ice is common in the Holland Harbor area during the winter months. Ice accumulates on the beach by wind and wave action and re- portedly often extends in a solid mass from the shoreline to more than 1,000 feet offshore. Wave heights are substantially decreased within an ice field and ice along the shore blocks wave action to some extent. However, the action of ice can accelerate erosion processes; e.g., ice being pushed upon the beach will loosen consolidated beach and bluff material and some of the sediment be- comes embedded in the ice at the shore and over the longshore bars to be carried elsewhere if the ice drifts away during the spring thaw. The average ice season at Holland Harbor extends from late December to late March. 6. Littoral Drift Estimate from Wave Statistics. Littoral drift has been computed from the available wave statistics dis- cussed earlier. The governing relationship used was (U.S. Army, Corps of Engineers, Coastal Engineering Research Center, 1977) Qy= 01000133 RER where Q is the potential littoral drift in cubic yards per day, and E the longshore momentum flux in feet per pound per day per foot of beach. The re- lationship between of, and a, is determined from linear theory in assuming that the bottom contours are straight and parallel. The number of waves per day for each wave height period-direction class is given by _ BS Ie be Sanu Os Sa where p is the probability of occurrence of that wave class. Computations were made using the wave statistics from SSMO data, Saville (1953), and Cole and Hilfiker (1970). The duration of computation is for the 9-month ice-free period from late March to late December. 43 *(SL6I *}T0130q ¢ JOTIZSTG To9uTsug Away S NM WOlF) QOZ6T 9duTS STOAST oyeT ATYJUOW UPA ‘TZ eins ty 44 All of the littoral drift computations show a large gross drift of 300,000 to 500,000 cubic yards, defined as the sum of all north and south movement, regardless of direction, whereas the net drift is much smaller. Individual yearly predictions show a wide variability in net littoral drift quantities. The littoral drift is predicted to be from south to north with a mean value of more than 265,000 cubic yards. The SSMO data and Coles and Hilfiker's wave statistics lead to the prediction of 61,000 and 76,000 cubic yards, respectively, from north to south. This direction is consistent with the observed growth of the fillets on each side of the harbor, the north side showing a much greater accretion. Figure 22 summarizes the monthly fluctuations of littoral drift based on the SSMO data. The figure shows that the monthly drift is large during late fall when the activity of extratropical cyclones intensifies in these regions, and small during the summer months, especially in June, when wave heights are generally smaller. MONTHLY GROSS DRIFT x10" (yd?/mo) MONTHLY NET DRIFT s & io 0 JAN BEDE MAR APR MAY JUNE UG SEPY OCT NOV DEC es iI‘ JULY ee yj ted 10 / 7 ° 251210 imal AVG. ICE-FRE® PERIOD Y AVG. ICE PERIOL Figure 22. Monthly variations of eo of computed gross and net littoral drifts for Holland, Michigan. 7. Analysis of Aerial Photos. A sequence of eight sets of aerial photos of Holland Harbor taken over the interval 27 July 1950 to 5 October 1973 was analyzed to assess the long-term evolutionary development of the shoreline. A regression analysis was verformed to correct the shoreline for lake level variations and to determine the long- term erosion-accretion rates along the coast. 45 Table 1 presents shoreline positions (with and without lake level correc- tions) as a function of distance along the coast, as determined from the aerial photos. The corrected shoreline positions are referenced to a common lake level of 578.5 feet, as determined by a regression against lake level. Figure 23 Shows the 1973 Holland Harbor shoreline. The origin of the coastline coordinate is taken at the harbor, positive in the north direction. The shoreline position was measured at 290-foot increments along the coast. A regression of shoreline position against lake level and time provided an estimate of the long-term evolutionary trend of the shoreline at each station along the coast. The regression study revealed general erosion extending south of the harbor and accretion immediately north of the harbor. Over an 8,410- foot span south of the harbor, the average beach erosion rate was 0.75 foot per year. Over a 4,060-foot span immediately north of the harbor, the average accretion rate was 1.65 feet per year. A span of 10,585 feet, starting 4,930 feet north of the harbor, had an average erosion rate of 1.28 feet per year. These evolutionary trends of the shoreline include natural effects as well as any effects which can be attributed to the harbor. Figure 24 shows the evolutionary smoothed trend (defined as the average accretion or erosion rate between 1950 and 1973) along the coast near Holland Harbor. Erosion dominates south of the harbor, accretion immediately north, and erosion again farther north of the harbor. The smoothed curve in Figure 20 demonstrates that the accretion resulting from the harbor extends about 5,000 feet north of the harbor. These estimates of the evolutionary trend of the shoreline are subject to large annual and spatial fluctuations. 8. Comparisons of Profiles from 1945 and 1975 Surveys. The surveys of May 1945 and April 1973 (lake levels at 578.4 and at 580.1 feet, International Great Lakes Datum (IGLD), respectively) permit a comparison of profile changes from which sediment volume changes can be estimated. Since the actual profile locations in the two surveys did not coincide (in general), the 1945 survey was interpolated on the lines of the 1973 survey to allow a comparison. Table 2 summarizes the cross-sectional area changes and the volume computations in two parts (above and below low water datum). The last colum in the table gives the erosion or accretion rates determined from the aerial photos. The average volumetric accretion rate on the north side (from ground survey data) is 2.13 cubic yards per year per foot of shoreline for 1,950 feet (450 to 2,400 feet north of the breakwater) compared with the average shoreline accre- tion rate of 1.92 feet per year for this same region determined from Figure 20. The average volumetric erosion rate on the south side of the harbor from the survey comparisons is 1.41 cubic yards per year per foot of beach, whereas the values from Figure 20 for the south stretch (470 to 5,550 feet south) give 0.82 foot per year as an average rate of shoreline erosion. The agreement on the north side indicates that the approximation of 1 square foot of beach per cubic yard is reasonably valid, whereas on the south side this is less true. The dis- crepancy on the south side is related to the bluff erosion contribution. The main erosion rate below the low water datum level is only 0.16 cubic yard per year per foot. Hence, the bluffs contribute at least an additional 1.25 cubic yards per year per foot, which exceeds the conditions for the approximation of 1 square foot of beach to correspond to 1 cubic yard of material. 46 Table 1. History of shoreline positions at Holland, Michigan, from 1950 to 1973, with and without lake level corrections. | Coastal position (ft) 27 June 1950 Distance lakeward from base line (ft) [11 oct. 1955] 8 June 1960 [19 Oct, 1962 Uncorrected lake levels 5 Sept. 1967 [9 May 1965] § Oct, 1973] f 0.00 1219.79 1088.80 1113.36 1146.11 1188.68 1187.04 1178.86 1146. 32 -290.00 982.38 949.38 982.38 982.38 998.78 966.01 941.45 921.15 \ -580.00 851,40 835.02 851.40 851.40 884.14 835.02 835.02 757.39 -870.00 769.53 736.78 769.53 766.26 785.90 744.97 783.16 716.45 -1160.00 720.41 671.29 687.67 687.67 705.68 671.29 687.67 655.04 -1450,00 671.29 630. 36 622.17 630.36 654.92 632.00 627,09 624.33 -1740.00 630.36 638.55 613.99 613.99 632,00 618.90 622.17 624,33 -2030.00 597.61 605. 80 605.80 597.61 $99.25 605.80 $89.43 583.39 -2320.00 $73.05 $07.56 $23.94 $15.78 $40.31 523.94 $15.75 $11.75 -2610.00 $23.94 474.82 491,19 483.00 499,38 491.19 474.82 460.57 -2900.00 483.00 433. 88 425.70 442.07 442,07 442.07 435.52 388.93 -3190.00 450.26 392.95 376.58 425.70 409.32 442.07 422.42 317.28 &} -3480.00 412.60 327.46 327.46 384.77 343.83 360.21 392.95 307.05 i= 2 -3770.00 376.58 319.27 311.09 376.58 343.83 327.46 343.83 296.81 ts -4060.00 376.58 343.83 335.65 376.58 353.66 330.73 352.02 327.52 ove -4350.00 376.58 352.02 343.83 376.58 360.21 356.93 335.65 286.58 2 2| -4640.00 373.30 343.83 343.83 386.40 409. 32 386.40 348.74 266.11 ea -4930.00 363.48 340.56 343.83 392.95 401.14 381.49 343.83 276.34 -5220.00 376.58 376.58 368.39 376.58 376.58 376.58 352.02 307.05 -$510.00 384.77 360.21 368.39 384.77 396.23 392.95 368.39 276,34 -5800.00 392.95 360.21 343.83 376.58 402.78 409. 32 376.58 307.05 -6090.00 412.60 343.83 376.58 384.77 409, 32 409.32 368.39 307.05 - 6380.00 425.70 343.83 368.39 384.77 425.70 392.95 376.58 276. 34 -6670.00 419.15 352.02 343.83 392.95 474.82 409.32 392.95 286.58 -6960.00 399.50 360.21 352.02 389.68 419.15 417.51 417.51 276.34 -7250.00 389.68 368.39 376.58 392.95 373.30 409.32 409.32 307.05 -7540.00 376.58 368.39 352.02 368.39 360.21 396.23 376.58 307.05 -7830.00 343.83 368.39 327.46 360.21 376.58 379.85 345.83 286.58 -8120.00 327.46 360.21 297.99 327.46 286.53 343.83 320.91 245.64 -8410.00 327.46 343.83 294.71 311.09 283.25 327.46 294.71 255.87 -870C6.00 327.46 1498.13 311.09 294.71 270.15 294.71 278.34 245.64 580.00 1473.57 1399. 89 1555.43 1449.01 1612.74 1540.70 1542.34 1498.40 870.00 1342.59 1309.84 1391.70 1358.96 1424.45 1358.96 1375.33 1310.08 &#; 1160.00 1288.28 1236.16 1318.03 1303.29 1375.33 1290.19 1304.93 1228.20 3B] 1450.00 1238.16 1170.67 1219.79 1236.16 1309.84 1224.70 1260.72 1136.08 $&&| 1740.00 1203.42 1129.74 1187.04 1195.23 1260.72 1208.33 1219.79 1166.79 3*| 2030.00 1178.86 1090.44 1154.30 1195.23 1211.60 1196. 87 1211.60 1115.61 2 2 2320.00 1146.11 1096.99 1105.18 1162.48 1244.35 1195.23 1195.23 1043.97 Ea} 2610.00 1113.36 1096.99 1088.80 1137.92 1227.97 1155.93 1162.48 1043.97 2900.00 1074.07 1064.24 1090.44 1105.18 1146.11 1146.11 1137.92 1023.50 3190.00 1031.50 1018.13 1064.24 1083.89 __1126.46 1096.99 1080.62 1003. Corrected lake levels 0.00 1211.21 1096.31 1113.36 1150.40 1161.86 1178.46 1189.23 ~290.00 974.29 956,72 982.38 986.43 973.46 957.91 961.62 580.00 827.42 847.25 851.40 858.38 840.47 821.05 827.27 870.00 761. 86 743.50 769.53 770.09 761.93 737.30 754.81 -1160.00 714.08 676.86 687.67 690.84 685.81 664.94 679.72 686.83 -1450.00 667.60 633.59 622.17 632.21 643.37 628.30 622.47 642.81 1740.00 629.99 638.87 613.99 614.17 630. 84 618.53 621.71 626.19 -2030.00 $95.91 607.29 605.80 598.47 593.93 604.10 $87.30 $91.91 -2320.00 $68.75 $11.33 $23.94 $17.90 $26.85 $19.63 $10.37 $33.28 -2610.00 518.61 479.47 491.19 485.66 482.75 485.87 468.17 487.18 -2900.00 475.24 440.68 425.70 445.95 417.80 434.30 425.81 427.77 -3190.00 436.12 405.33 376.58 432.77 365.13 427.93 404.75 387.99 5 -3480.00 404.36 334.67 327.46 388.89 318.07 351.96 382.65 348.27 AG -3770.00 370.06 324.98 311.09 379.84 323.45 320.94 335.68 329.42 23 -4060.00 373.38 346.63 335.65 378.18 343.67 327.54 343.50 8 a -4350.00 367.70 359.79 343.83 381.02 332.45 348.05 331.00 ae -4640.00 256.78 358.29 343.83 394.67 357.68 369.88 348.74 & |] -4930.00 349.45 352.84 343.83 399.97 357.28 367.46 346.51 -5220.00 368.52 383.63 368.39 380.61 351.40 368.52 347.33 -S510.00 369.84 373.26 368.39 392.23 349.59 378.03 350.96 -5800,00 380.42 373.17 343.83 382.84 363.62 396.80 369.70 6090.00 399.05 355.69 376.58 391.54 366.98 395.78 374.80 6380.00 406.77 3609.40 368.39 394.23 366.55 374.02 370.99 -6670.00 397.14 371.28 343.83 403.96 406.03 387.31 396.64 6960.00 380.26 377.04 352.02 399.30 359.02 398.27 372.55 -7250.00 378.92 377.80 376.58 398.33 339.69 398.57 360.83 ~7540.00 368.26 378.€7 352.02 372.55 334.20 387.91 48.66 7830.00 333.79 377.18 327.46 368.23 345.21 369.82 336.77 -8120.00 320.83 366.00 297.99 330.77 265.82 337.21 278.77 ~8410.00 323.07 347.67 294.71 313.28 269.54 323.07 277.82 $80.00 1462.02 1508.24 1555.44 1454.79 1876.63 1529.14 1556.18 870.00 1331.98 1409.17 1391.71 1364.26 1391.30 148.35 1363.12 «| 1160.00 1270.90 1322.42 1318.03 1310.48 1330.39 1275.81 1300.10 S| 1450.00 1217.10 1282.84 1219.79 1245.69 1250.28 1205.64 1231.38 $5 1740.00 1192.73 1180.02 1187.04 1200.87 1227.33 1197.64 1220.21 3+} 2030.00 1165.84 1141.13 1154.30 1201.74 1170.91 1183.85 1180.72 ™ &! 2320.00 1121.49 1111.99 1105.18 1174.79 1167.40 1170.64 1167.08 3 & 2610.00 1092.75 1115.03 1088.80 1148.23 1163.55 1135.32 1147.05 2900.00 1053.97 1110.21 1090.44 1112.73 1098.91 1131.01 1099.02 3190.00 1018.51 1075.61 1064.25 1090.39 1085.86 1084.00 1067.99 3480.00 | _1020.92 1028.68 1023.31 1085.09 1048.59 1016.01 1043.63 \Determined by aerial photos. 47 ee —— NITY INID JDNVYLND YOBEVH °o ° ° ° o ° °o ° ° ° ° bod oO v e % (44) JYOHSI4O 48 ° wn S fe) Py p O oS joe ce Es} 4 4 o 3 a rw ae) : : 3 2 = = s o PS) o isc) n ey ise) ™~ a So fs) SI ey a 0 n S 3 60 “cd = us Shee. = z ~ (S w re a ° g p 3 yn “ = fe) “4 p “od wn ° a, ® 5 3 oH 3 o 0 4 e) iS) wn Figure 23. \ ee ears ee cw se es = BNITUI NID JINVYULNS YORYVH ” r) - ° 7 GV 00) MOL 3ED9V wOIsOUR ————— —es 49 22 (247 00) 13,000 10,000 5,000 25,000 (ft) N Smoothed average rates of shoreline erosion and accretion (feet per year) at Holland, Michigan. Figure 24. Table 2. Summary of cross-sectional changes and computed volumetric changes for 1945 and 1973 surveys. ! [ Above LD { ___below LWD lee observed shoreline Line ||Distance |} Cross section | Avg. rate of }/ Cross section | Avg. rate of || Total mean v | accretion- i g ' g ol. | erosion No. || between area change vol. change area change | vol. change change rate change rate profiles (1945-73) (1950-73) (ft) (£2), (yd?/yr/ft) || (ft?) | (ya /yr/ ft) (yd3/yr/ft) (ft/yr) 1 2,298 1,528 z 1,150 +1.47 +1.10 +2.57 +2.0 2 -78 134 800 +0.84 +0.65 +1.49 7113) 3 1,347 855 Harbor 4 2,144 928 740 +1.16 +0.30 +1.46 -0.3 § 391 -479 880 +0.17 -0.53 -0. 36 -0.6 6 642 -324 1,210 -0.24 -0.72 -0.96 -0.9 7 821 -761 605 -0.91 -2.80 -3.71 -1.0 8 -562 -3,480 830 -1.59 -3.39 -4.98 -1.2 8) -1,848 -1,660 815 +0.39 -0.88 -0.49 -0.9 One| +2,431 I 305 | poet a ee Nee ee ee | | 2 = lL = accretion; - = erosion As demonstrated previously in the analysis of the aerial surveys, the har- bor influence on the north side extends about 4,200 feet. Farther north it is assumed that the observed erosion rate will be close to the natural erosion rate for this area of the Lake Michigan coast. At 4,200 to 15,400 feet to the north of Holland Harbor, the average erosion rate during the 1950-73 period (from aerial photos) was 1.28 feet per year per foot of shoreline. If the approximation of 1 cubic yard of volume per square foot of beach is used, the corresponding sediment volume loss is 1.28 cubic yards per year per foot of shoreline plus the erosion rate of the dunes. The bluffs which have an average height of about 40 feet in this area would contribute an additional sediment loss of 40 x 1.28/27 = 1.90 cubic yards per year per foot. Hence, the total sediment loss to the offshore in this area is estimated to be averaging 1.90 + 1.28 = 3.18 cubic yards per year per foot of shoreline during the past 23 years. This rate of loss is considered to represent the natural rate of loss for littoral sediments in the Holland area due to all factors except the navi- gation structure. It must be remembered that the erosion rate of 3.18 cubic yards per year per foot of shoreline is an average rate for the past 23 years (1950-73). Since lake levels have been much higher during the past few years, the present erosion rate must be expected to be higher than the average and therefore will probably exceed 3.18 cubic yards per year per foot of beach. 9. Sediment Budget. a. North Side. An independent evaluation of the net littoral drift has been obtained from various surveys as summarized below in which a sand budget analysis is made for the north side of Holland Harbor. The shoreline changes shown in Figure 25 provide information for an area measurement of accretion. An approximation for volumetric accretion can subsequently be made from the relationship, wherein 1 square foot of change in beach surface area equals 1 cubic yard of beach material. This relationship has previously been shown to be reasonably valid for the north side. Subsequent corrections may be neces- sary because of the dune-building phenomenon. 50 HOLLAND BUILT 187-73 DREOGING LIMITS —@ ; Fur fe iG 1? 9: fol) D0 1) ee Eh bd (ZU ins en - 50 Co oo re aS ae S71, Le on ec ea ee co < Ost ocr ou os t er al 921 Historical shorelines north of Holland, Michigan, between 1849 and 1945 (from U.S. Army Engineer District, Detroit, 1975). Figure 25. S| The area changes corresponding to the waterline shown in Figure 21 were determined for a length of shoreline extending 2,550 feet north of the north breakwater. These areas are summarized in Table 3. A correction is applied for the variation in lake level by the relationship Correction in 1,000 square feet = sp ~ take Tevet ot rference ~ 2,580 which follows the assumption of an average beach slope of 1 on 45 for this area as determined from the 1945 surveys. Table 3. Summary of shoreline accretion rates north of Holland Harbor (1849-1945). Measured area from | Lake level Lake level Area Net area Rate breakwater to (N.Y. datum) difference | correction 2,550 feet north (1,000 ft? (ft) (ft) (1,000 £t2) | (1,000 ft?) | (1,000 £t2/yr) The results of the computations (Table 3) indicate rates of accretion vary- ing from 3,400 to 51,900 cubic yards per year if the general rule of 1 square foot of beach area corresponds to 1 cubic yard of beach material is applied. These accretion rates should be interpreted according to the history of the construction. The first breakwaters were constructed in 1856-60 and trapped most, if not all, of the littoral drift. Since the accretion of 51,900 square feet per year during the period 1856-71 is an average, then the corresponding littoral drift rate must have exceeded 51,900 cubic yards per year (it must be assumed that by 1871 some of the littoral transport was passing around the break- water heads into the navigation channel). The accretion was slowed to 3,400 square feet per year from 1871 until the harbor breakwater extensions were started in 1906. Therefore, the littoral transport passing around the north breakwater must have been quite high from 1871 to 1906; also, the building of new dunes by wind action probably occurred during this period. From 1906 to 1933 the accretion averaged 13,700 square feet per year; after 1933 this rate dropped slightly to 13,400 square feet per year. However, some inland dune building could still be taking place. Measurements from the 1945 survey show an accumulated volume of 137,400 cubic yards above elevation 585 (1945 datum) on the north side of Holland Harbor. This volume would only account ror about 1,900 cubic yards per year from 1871 to 1945 but windblown sand losses farther inland would increase this. The detailed analysis of aerial photos taken from 1950 to 1973 indicates an average accretion rate of about 1.65 feet per year per foot of shoreline for the first 4,000 feet of shoreline north of the harbor, a total of 6,600 square feet per year. The accretion rate based on a comparison of the 1945 and 1973 hydrographic surveys for profiles 1, 2, and 3 in Figure 10 was about 4,800 52 cubic yards per year for the first 2,550 feet north of the breakwater. The incoming littoral drift supply must be providing the observed accretion since there are expected offshore losses of 3.18 cubic yards per year per foot of shoreline for the first 4,000 feet of shoreline north of the harbor, and material is apparently being accumulated in the dunes in Holland State Park. To summarize the present situation, it is believed that about 61,000 cubic yards of littoral material is being transported toward Holland Harbor from the north. The disposition of this material is summarized in Figure 26. About 12,700 cubic yards per year is lost offshore (3.18 x 4,000), 6,600 cubic yards per year is accumulating on the beach face on the north side, and about 1,900 cubic yards per year is being accumulated in the dunes of Holland State Park. Although the exact amount is unknown, an estimated 1,300 cubic yards per year of sand is lost inland. Hence, it is calculated that about 38,500 cubic yards per year of sand arrives at the harbor entrance but at the most 25,000 cubic yards is trapped. It is likely that some of the 25,000 cubic yards dredged in the harbor entrance comes from the south side; therefore, at least 13,500 cubic yards per year is lost offshore. Consequently, it is concluded that the harbor structures cause a loss of materials from the littoral region of at least 13,500 cubic yards per year. In addition, the 25,000 cubic yards per year that is dredged is dumped offshore. The overall result is that the south side of the harbor is being starved of 61,000 cubic yards per year. 3 13,500 yd~ lost offshore a 25, 000: deposited ins RagGoU> entronce channel yd 37 fos? offshore 4 Total 12,700 yas lost offshore ae wan Total 9,800 yd > to State Beach Park 61,000 yd 2 Incoming littoral delft ee Total 1,900 yd 3 to dunes Y HOLLAND STATE PARK 3.18 yd 374 eroded from shoreline Wet bio taay 3 Total 1,300 yd to Inland 4,000ft Figure 26. Summary of sand budget north of Holland, Michigan. b. South Side. Figures 27 and 28 provide information for an area measure- ment of shoreline accretion and erosion. The beach area changes are summarized in Table 4 by areas north and south of a section located 1,200 feet south of the south breakwater (an arbitrarily selected station which appears to correspond to a stationary shoreline point for the period 1856-1945). The table summarizes the corrections in lake level variation about an average offshore beach slope of 1 on 24.5 for the first 1,200 feet south, and an average beach slope of 1 53 MACATAWA 2G rs IAT ITS IDS 02 — Historical shorelines immediately south of Holland, Michigan, between 1849 and 1945 (from U. Engineer District, Detroit, 1975). Figure 27. Army . S 54 Figure 28. Historical shorelines farther south of Holland, Michigan, between 1871 and 1945 (from U.S. Army Engineer District, Detroit, 1975). 55 Table 4. Summary of beach area changes south of Holland Harbor. Measured area Lake level Lake level Area Net area Rate (N.Y. datum) difference | correction (1,000 ft?) (ft) (£t) (1,000 ft?) | (1,000 ft?) | (1,000 £t2/yr) Breakwater to 1,200 feet south on 12.1 for a length from 1,200 feet to 3,600 feet south of the breakwater. The values shown in Table 4 can be added for a representative length of shore- line of 3,600 feet to yield the result that an accretion rate as high as 48,500 Square feet per year in the period 1856 to 1871 decreased to a rate of about 6,000 square feet per year from 1871 to 1906, then increased again to 20,500 square feet per year following construction of the breakwater extensions after 1906, and finally showed an average erosion rate of 8,000 square feet per year from 1933 to 1945. This last value corresponds to an erosion rate of 0.75 foot per year as determined from analysis of the aerial photos. However, it should be emphasized that the recently observed erosion rates reflect two effects: (a) The length of shoreline near the harbor has been protected by rubble and groins, and (b) the unprotected shoreline farther south is backed by dunes with an average height of about 120 feet with peaks more than 200 feet above the lake level. The rate of accretion at times on the south side may seem somewhat surpris- ing, but it must be recalled that the littoral drift often reverses (see Fig. 22) and in some years may be completely reversed. The variable direction of littoral drift is also evidenced by the two surveys in Figure 12, which shows the harbor entrance shoal from the south side as being apparently larger than that from the north in 1973. Using the approximation of 1 cubic yard of sediment per square foot of beach the 1950-73 observed average erosion rate for about 9,000 feet south of Holland Harbor is 0.75 cubic yard per year per foot of shoreline, plus the bluff contri- bution of 0.75 x 120/27 = 3.33 cubic yards per year per foot for a total loss rate for unprotected shoreline of 4.08 cubic yards per year per foot of shore- line. When compared with the erosion rates observed for the shoreline about 1 mile or more north of Holland Harbor of 3.18 cubic yards per year per foot, the difference of 0.9 cubic yard per year per foot of 22 percent can be attributed to the navigation structure. The effect is very small and should not be ex- pected to be readily apparent. The rate of loss of 0.9 cubic yard per year per foot of bluff material would correspond to a shoreline erosion rate of less than 0.2 foot per year for a bluff height of 120 feet. 56 The actual loss of land to the south of Holland (attributed to the naviga- tion structure) is relatively small, at the most 22 percent. This figure has been derived as a 23-year average, but with current high lake levels the pres- ent natural erosion rate is certainly higher than the average. The evolution of the shoreline at Holland was studied using the present model. The relevant physical data and the estimates of offshore sediment losses were used in the analysis. Interpolation of values shown in Table 1 gives the shoreline every 100 feet along the base line. The channel at the harbor en- trance was collapsed to zero so that the south reach ended at O- and the north at 0+. These shorelines indicate that the shore is stable at the breakwater; therefore, when the direction of transport is toward the breakwater it is as- sumed that the sand transported to the breakwater is entirely lost offshore. Monthly lake levels were taken from Figure 21. The results of these computa- tions are not given since they are essentially previously described results. The average beach slope at the waterline was determined to be 1:10; the beach profiles in Figure 17 near the breakwater show that this is a reasonable esti- mate although the slopes are not constant from profile to profile. The height of the berm is assumed to be 10 feet. The depth to no sediment motion was esti- mated at 30 feet, based on visual consideration of the offshore bathymetry and the use of Weggel's method (J.R. Weggel, personal communication). An offshore loss of 3.2 cubic yards per year per foot of beach is also included (see Fig. 26). The transport equation for this situation can be written (see eq. 53) A een 3 2 eh Be Os (KpQ) m dt dt where Q | COS G, Sin hy and 2 — (K5Q) m = beach slope at waterline D = dimensionless lake level ea = dimensionless offshore line loss This equation will be applied on each side of the breakwater. When the in- coming wave direction gives drift toward the breakwater, the boundary condi- tions expressed in Q are Q | b = COS a sin ap xX = wo Conversely, when the incoming wave direction gives drift away from the break- water, the conditions become Le) i] [e) and cos a sin On, 57 Choice of wave climate is the remaining input parameter to be determined, and is the most controversial. The wave climate most desirable for the study of shoreline evolution is a time series giving wave height, H, period, T, and direction, D. Unfortunately, this is seldom available and hence monthly sta- tistical summaries must be used, such as those by the SSMO for southern Lake Michigan. One possible use of the summaries is to construct monthly times t for each possible (H,T,D) triple, i.e., t(H,T,D), and then calculate the evo- lution of the shoreline as the (H,T,D) triples are chosen in some determin- istic order or at random. This method would be computationally very expensive and is not used. The most simple approach is to assume that these are but two (H,D,T) triples representing the gross transport north and south, each occur- ring for some length of time per month. The entire shoreline is alternately calculated for an incremental time, assuming the direction of the incoming wave iS positive, then negative. The period, T, used is taken to be the average T; i.e., ei Osis) 1b 2p (H,T) where the p(H,T) are the (H,T) probabilities given in Table 19 of the SSMO (National Oceanic and Atmospheric Administration, 1963-73). The choice of (H,D) for north and south, denoted (Hjy,Djy) and (Ho,Dg), respectively, must now be made. This choice is subject to the condition that the actual northerly transport, as calculated from the statistics and given a straight shoreline for the reach of interest, be preserved; 1.e., QT oa : a = D2) . tH cos a, sin a, = t, » p(T) "p.(H, D) HE cos a, Sin o% Eel Dis gaivemnne: north transport where t; = number of hours in a given month p(T|H) = conditional probability T occurs given H p(1,D) = probability of (H,D) pair (using Table 18 in SSMO as a data base) A, = Dy - shoreline orientation Ci a(S) ty = number of hours the average wave condition exists Hy, = average wave height ae =) Dy = shorelanel orientation Dy = average direction and similarly for the directions giving south transport. The average directions of the shoreline at the breakwater are calculated from the historical records. The directions are chosen for the incoming wave angles since the complex geometry of the harbor breakwater shields the nearby shoreline from waves arriving from most directions. At present, the time dura- tion of waves arriving from the north is assumed to be the same as from the 58 south; therefore, ty = tg = 0.5 tz. The conservation of transport equation is then used to calculate the average wave heights Hy and Hs. The results of these calculations are given in Table 5. Table 5. SSMO averages for Holland Harbor. ? Month ANDMNNMUNNKHUNNUNAAD er Oa 6 noe Oitenieiie °° ° ° ° 68) -0 .5 6 oS) ol 8 -8 -0 -6 -0 of 3 NILWNWWDNODNDDNPNON BG 0. Wea oOo Of Gasol 0 Oa GS-6 OLD DADUNFPWAHEHLODANW CMIWVUOUADWWHEUWOFH Annual + ho “I ° tEor Da 885 Ohne The computer program described in the Appendix was used to calculate the evolution of the shoreline from September 1967 to May 1968. The historical 1967 and 1968 shorelines, as well as the computed 1968 shoreline, are shown in Figure 29. The calculation assumes that Ax = 100 feet with > = 1 which gives a value for At which varies from 8 to 20 hours depending on the monthly wave characteristics. The principal discrepancy between the predicted and actual 1968 shoreline occurs near the breakwater. Although the shapes agree, there is an erosion in the calculated shoreline which is probably due to the approximations used in the calculation of the diffraction coefficients, and incoming wave angles which are functions of x in the shadow region of the diffraction zone. The unaltered theory of Penny and Price (1944) was incor- porated into the numerical scheme since most breakwaters can be represented as line barriers, and hence is almost always useful. However, for the case of Holland Harbor a universally valid prediction of the shoreline would re- quire the detailed calculation of the diffraction effects due to the geometry of the breakwaters. Also, the convenient choice of incoming wave direction obscures the fundamental problem of how to properly use the statistical wave summaries. IV. CONCLUSIONS AND RECOMMENDATIONS The basic idea of Pelnard-Considere (1956), i.e., to investigate shoreline evolution by concentrating on conservation of mass as a special one-dimensional problem, has been generalized to essentially its limits of applicability. These physical processes of refraction and diffraction (where applicable) have been incorporated, including deterministic variations in lake level, bluff height, and beach slope. The inclusion of refraction makes possible the proper use of the known physical relationships between wave energy and littoral drift on a priori basis without necessarily determining these as results from the past recorded shorelines at a given location. Accurate determination of shore be- havior in the lee of a breakwater requires inclusion of diffraction in some 39 1967 SSS eS S> 1968 ACTUAL —— -—— 1968) /PREDICTED BREAKWATER PLANFORMS CALCULATED —40 —30 —20 -10 0 10 20 20 40 50 60 70 80 90 x100 FT Figure 29. Computed data versus actual data. form. This could be done in a heuristic manner, either in the global approxi- mation described previously or in the use of the constant depth theory of Penny and Price (1952). It could also be done in a more rigorous manner which would include the effects of a sloping beach. Thus, quantitative predictions of the shoreline can, in theory, be attempted in situations where onshore-offshore transport of sand is either negligible or is known from other sources. The resulting theory is presented in two equivalent forms, one in terms of the behavior of the shoreline y(x,t) alone, the other expressed explicitly in the longshore transport Q(x,t) and implicitly in y(x,t). The former has the advantage that numerical schemes, such as that of Crank-Nicolson can qualita- tively indicate the behavior of the shoreline in regions of rapid change. How- ever, the conservation of mass is difficult, if not initially impossible to achieve since any approximation of a transport-derived term (i.e., a term arising from 3Q/3x) will alter the transport balance. The later form allows the use of analytical or numerical approximations in the transport equation which will not disturb the total sand content of the system, but only its local distribution. The most severe and unavoidable limitations to the engineering application of these methods are the use of the statistical wave summaries. One possible use of these statistics was shown; however, many others are possible. Efficient and accurate use of the offshore wave statistics is endemic to the problem of large-scale shoreline prediction, and must be achieved before any theory (whether one line, multiple lines, or grid) can successfully produce accurate results. The problem of shoreline evolution sensitivity to time step in the input wave climatology would require further research. Despite this limitation, if the effects of wave refraction, wave diffraction, and change of lake level are taken into account (as in this report), and if the method is generalized, then a mathematical model with multiple bottom contour lines could be formulated which would (if the problem of wave statistics input is solved) permit a cal- culation of the evolution of the complete bottom topography. 60 It is important to point out that wave refraction effect on shoreline evo- lution has been found particularly important. It is particularly necessary in order to determine a planform stability criteria which can be established from the present formulation. The problem of shoreline stability needs to be investigated, both physi- cally and numerically, as for some deepwater wave angle, shoreline perturbance may increase by instability instead of being flattened out. Perhaps one of the most significant results of this mathematical approach to shoreline evolution is to point out the need for more research to quantify the phenomenology relevant to shoreline evolution. The insertion of empirical parameters has allowed the investigator to fit (to a large extent) a form of shoreline evolution; however, the processes involved in each of these param- eters are not well known, and, what fits at Holland, may not necessarily apply elsewhere. For example, the research topics which will improve the model are (a) onshore-offshore movement; (b) quantity of sand (silt) lost by rip currents or density currents; (c) percentage of sand in suspension; (d) distribution of sand discharge as a function of the distance from shore; and (e) more impor- tantly, how to treat the wave climatology in finite time intervals to obtain an equivalent result. The last topic may be the most difficult since the noise, (i.e., daily variation and effect of storm) may exceed the signal (i.e., the long-term trend). Shoreline evolution is due to a succession of extreme events separated by long-period time effects of equal importance. Therefore, the treatment of long-term evolution from an average wave climatology is question- able, since the complete time history may have to be considered on a daily basis. This topic can be investigated by a sensitivity analysis to wave definition; €.g., the present model could be used, as well as a research guideline for fill- ing many knowledge gaps in shoreline processes. 6| LITERATURE CITED COLE, A.L., and HILIKER, R.C., "Wave Statistics for Lake Michigan, Huron, and Superior," Department of Meteorology and Oceanology, University of Michigan, Ann Arbor, Mich., 1970. LE MEHAUTE, B., "A Theoretical Study of Waves Breaking at an Angle with a Shore- line,"' Journal of Geophystcal Research, Vol. 66, No. 2, Feb. 1961, pp. 495-499. LE MEHAUTE, B., and KOH, R.C., "On the Breaking of Waves Arriving at an Angle to the Shore," Journal of Hydraulte Research, Vol. 5, No. 1, 1967, pp. 67-88. LE MEHAUTE B., and SOLDATE, M., "Mathematical Modeling of Shoreline Evolution," MR 77-10, U.S. Army, Corps of Engineers, Coastal Engineering Research Center, Fort Belvoir, Va., Oct. 1977. MOBAREK, I.E., and WIEGEL, R.L:, "Diffraction of Wind Generated Waves,'' Proceed- tings of the 10th Coastal Engineering Conference, American Society of Civil Engineers, Ch. 13, Vol. I, 1966, pp. 185-206. NATIONAL OCEANIC AND ATMOSPHERIC ADMINISTRATION, ''Summary of Synoptic Meteoro- logical Observations (SSMO) for Great Lakes Areas," National Climatic Center, Asheville, North Carolina, 1963-73. PELNARD-CONSIDERE, R., ''ESsai de Theorie de 1'Evolution des Formes de Rivage en Plages de Sable et de Galets," Fourth Journees de l'Hydraultque, Les Energies de la Mer, Question III, Rapport No. 1, 1956. PENNY, W.G., and PRICE, A.T., "The Diffraction Theory of Sea Waves and the Shelter Afforded by Breakwaters," Phtlosophtcal Transactions, Royal Society of London, Serial A, Vol. 244, 1952, pp. 236-253. SAVILLE, T., Jr., "Wave and Lake Level Statistics for Lake Michigan," TM-36, U.S. Army, Corps of Engineers, Beach Erosion Board, Washington, D.C., Mar. 1953). US. ARMY, CORPS OF ENGINEERS, COASTAL ENGINEERING RESEARCH CENTER, Shore Protec- tion Manual, 3d ed., Vols. I, II, and III, Stock No. 008-022-00113-1, U.S. Government Printing Office, Washington, D.C., 1977, 1,262 pp. U.S. ARMY ENGINEER DISTRICT, DETROIT, "Detailed Project on Shore Damage for Holland Harbor, Michigan,"' Tetra Tech Report, Detroit, Mich., Apr. 1975. 62 APPENDIX COMPUTER PROGRAM This appendix presents the listing and brief explanation of the computer program written to investigate the behavior of a shoreline which contains a complete littoral barrier at x = 0. The numerical scheme is based on the finite-difference method of Crank-Nicolson, used to solve the cyclic nonlinear transport equations in Q and y. Although the program is written expressly for Holland Harbor, Michigan, hopefully, enough explanation is given to modify the program, if necessary, to suit a particular application. The program consists of a main program, which is simply a calling routine to the controlling subroutine, and eight subroutines. The interrelationship of these programs is shown in the figure below; a brief discussion of each subroutine follows. INPUTQ | f : 3 ; Sica Hy f 5 a i : i OREGION { smmems:| CALCKDQ fem ; SREGION | be . H & ts 2 Venera f | e sea ee i hismesz es Figure. Program structure. MAINQ Calls subroutine EVOLVEQ EVOLVEQ The controlling subroutine which organizes the numerical method in a global sense and calls various other subroutines as required. The first call is to INPUTO, which reads the controlling parameters, his- torical shorelines and lake levels, and statistical wave information. For each month in the time period of interest, subroutine CALCKDQ is called. (It is assumed that during a month the diffraction coeffi- cients change negligibly.) For alternating equal incremental times the angle of the incoming waves is changed and subroutine SOLVEQ is called. This is repeated until a month's time has been completed. Then this is repeated until another historical shoreline is reached. 63 INPUTO This routine reads the information required to define a particular problem. Below are the cards required with an explanation of the variables required. Title cards are user information devices which are printed but never used. The card(s) are numbered by their order as defined by READ statements. CARD il: CARD 2: CARD 3: CARD 4: CARD 5: CARD 6: CARD 7: CARD 8: CARD 9: CARD 10: CARD 11: title card LAMBDA = A = At/Ax2 B = bluff height (feet) DO = depth to no sediment motion (feet) DX = Ax (feet) BERM = berm height (feet) BWL = length of breakwater (feet) BWD = depth of breakwater tip (feet) SLOPE = slope of shore profile at waterline XLINE = offshore line loss (square feet per hour) title card N1 = number of grid points to left of breakwater N2 = total number of grid points on both sides of the breakwater ND = number of historical shorelines NY = number of years lake levels are to read INYR = year of the first historical shoreline (must be year of first lake level as well) for all years with historical shorelines read year (I); I = 1,ND month (I); I = 1,ND hour (I); I = 1,ND where hour is the day x 24 for each grid point I read historical shorelines (feet) title card for each year read lake level for each month title card () for each month (including annual which is not at present used) input (in the notation used for Holland) Hg, Hy, Deep Dino Wp ie The input data above are printed without explanation along with several assays used internally. Note that in this routine i is redefined slightly so that a month's time is exactly 2nAt. 64 SOLVEQ CALCKDQ QREG ION SREG ION CS CETC Note I. Note II. This subroutine calculates the shoreline for one time step assuming a previous shoreline (and shoreline angle = tan7! (dy/ax)) and diffrac- tion coefficients as calculated in subroutine CALCKDQ. The scheme is essentially as is described in Section 2. The matrix inversion is achieved using a standard algorithm for tridiagonal matrices. This routine calculates the diffraction coefficient and incoming wave angle a = a(x) in the diffraction zone using the theory of Penny and Price (1952) for both incoming wave directions (the angle convention is shown in subroutine INPUTQ). Calculates diffraction in the ''Q'" region. Calculates diffraction in the "'S'" region. Calculates complex valued Fresnel integral. Calculates a given depth and period. In subroutine EVOLVEQ L1 = Ist year index L2 = last year index Calculation begins at date: year (Ll), month (L1), hour (L1) ends at date: year (L2+1), month (L2+1), hour (L2+1) In a general program L1 = L2 = ND-1 where ND is defined in subroutine INPUTQ ND must be passed to subroutine EVOLVEQ. Program as given in this report is set up to allow complete bypassing (lost offshore) when waves are not diffracted, i.e., To change program for no bypassing, four statements must be changed in subroutine SOLVEQ. 2 statements after 1001 change QOLD (N1P1) to QOLD (N1PT) (1-ID) x QOLD (N1P2) 0 1 statement after 1001 change QOLD (N1) = (ID-1) x QOLD (N1M1) to QOLD (N1) = 0 1 statement after 3000 change P(N1) = 1. - (1D-1) x Q(N1M1) to P(N1) = 1 3 statements after 3000 change Q(N1P1) = 0 + (2-ID)/P(N1P1) to Q(N1P1) = 0 65 aaAgD nan aan 1001 1oo2 1100 diol 1104 PROGRAM MALTNQCINPUT® OUTPUT eo TAPES@YNPUT 9 TAPEGSOUTPUT) CALL EVULYEQ CALL EXIT EMD SUSRUUTINE EVOLVEW COMYONs BUATAZ LAMBUAV Re DQ eDX > HERMyNI pN@oBWLOBWOVDXHATsSLUPE OAL INE CUOMHONSOTFRES KKNeC!2]0115) 9 AAC 20115) COMMONS SSMUS DIR(130¢) oPERC 1302) OWH( 1502) oXB( S306) oAMPC I B02) COMMON SLAKELVLS KL( 24942) j COMMON /SFRLINEZ VATE (398) oSL(195,A) COMMUNSZ3S/23013,2) OIMENSTUN HOLES) oVCRISIOICCLIS) oVxCSL5) 0VV(4S5) oRGLC115) oMD(13) REAL LAMBDA DATA MO/31 6289315309319 3500310319 300310300319 305/ LIBIMNnEX UF FIRST BHOKELINE L136 Lato FREAD tPUT DATA CALL phPUTa NiMpsniel HTP LENle) NLPeauie?2 NaMy ence] TXde( fol) eOxneDK OU 1001 T3t0NJ Txz] aoa ICiT)sIx Tapely NIPLEN1 064 DUO {0ne LT3N{PyoNa Tut]la,0x IC(L) 1k DEFL%¢ ORIGINAL SHORELINE AND SHORELINE ANGLE Gai ,4(8+090) OU $1n0 1£4—N2 YCL)85bCT0L1) oF E970 e5/4( R00) ZE(yCadev(1))sOxHat vyy¥(y)aATaN(Z) OO inl Te2aeNtM) ZelCy(yeloev(Tel)y%e9 V¥( 1) 24TAN(Z) ZeCv(N LOT ONIME))/DAMAT VV ONL) FATANCZ) ZaCv(nNiPe)@eV(NIPy))/OKHMAT VY(NIPL)BATAN(Z) DO ine LENyPagnamt Zz(V( pol) oV¥( Tol), #69 VV¥(T)@ATAN(Z) Z3(Y(N2Z)°Y(N2M1) )/DKHAT VY¥(N@)2ATAN(Z) MaIn aNALYSIS LOOP IVE AR eh 50 9009 L®LicLa@ Toisk 0009 OD LTy1s0aTe (lol) IMieOsTE Ceol) Ly2eDaTE (lolol) T#7sNaTe (eet od) NMONaCTY2° 7 v1) *1arlh2ermMiol MUNaITMi ef QO 2379 LeLeteN¥4On MUNEMNG | IF (MON gNE, 13) GUTO 3 MOMNe] 1VEARSIYE ARS) TTLIMaXS (ONS) ZL3Z3( "UNG ) 2P205("UNO2) CALL CAL CKNUC Ye MON) DTHAT=xXS("0ONG3) DIRLEN Ik CHONG) OLR eanIk (MONG?) IF(LL o£%, 1) GOTO 11 IF(LL EG, NMON) GOTU 42 N1TaaS¢(*UNeu) O1HaT=*£S(M0N93) 66 oan aan 200 3000 too0 lool MLGEXS (HONG 2) GOTU 44 X¥EMQe“UN) S24 FaCTS(KMN@SLATE( SSL) ISXM GUTG 45 MPEMI(PON) O24 FACTSnATE(S9L)/x™ XLOSFACTEXS( HON) 2) CTBRACTEXS(MONGUY OTMATSFACTOXS(4ONG 3) DLLOTEXLCIVEAROMAN) sed, NOLLHATELLEOTeDTy(b+b0) PRINT Jug FURMAT(C IHL» /) PRINT 150eLeoTVYEAR MONO TILIMeD HAT ,OLLHAT ODT FURMAT(® FOR PARAMETEKS®o STO0® TTL IMPDTHATODLLHATeDIa¢eJO0 3EI1,3) PRINT 151 0MON¢ (25 (MUNOK) oKE1 96) FURMAT(/e® LISTING OF ARRAY KS POR MUNTH# oI 5S060F10,3) OxeeNULLHATsSL OPE OO 2991 TetelTlLI™ TALL SULVEQC VO VY oXoVXoZLoXLVeDOLLHaToOTHAT es te AMP( MONG 1) oDT) CALL SULYEQCMoVKeVoVVYoZ2oXLVoDLLHaToDTHATe 20 AMP( MONO?) 0 DT) CONTINUE PRINT SHOKELINE atl END UF MONTH JVEARSTYI®{2eLLeaeIMi JMONZ VE AR@L2¢(JyEAR/I2) 64 JVEARZJYEAR/I2 WRITE (60200) JYEARe JMON FURMAT(C//0% FOR YEARS eT90% MONTHS, FSe% LISTING UF FINAL SHORELINE 1%) DO 3090 I34oN2 RELCT)SVCI)/E OO 30ot 13198 JLS(Teol) 91561 J2zsiel4 IF(T (60, 8) JesqiS WRITE( 60201) (ICU) o JO eJ2) FORMAT(///98 X 991518) WRITE( 60202) (RSLC J) odadioJ2) FORMAT(s»® ¥ So15F B01) CONTINUE CONTINUE CONTINUE RETURN ENO SUBROUTINE SOLVEQ(VVeVeXXo SLANG o LyeXLGoOLLHAT oDT AT 2100 AMP_DT) SULVE FOR ARRAY yX USING IMPLICIT SCHEME FUR TRANSPORT QNEW CUNMONS BDATAS LAMBDAH DOD oRERMeNI oN2e BML eBWDODXHAT SLUPE OC ALINE CUKMONSUTERES KeMeCPo1IS) eAaCeoLI4) CIMENSTUN MXCLISDOVYCALD) oVC115) oSLANGCIIS) eVYOLUCI15) ovV6119) DIME SIUM POLLS) oWCLIS)oL (115) 910415) eNC113) oGNEW( 11S) pQOLN( 115) Re AL BGA SCOANG Eat} o/b 8 NIM] Bry leo] NLP Bylo. NIPQ2an1¢2 N24 sr eol D) $000 T8yeN2 VILOCPIBYYCT) CaLCULATE VLD & (VOLO) BASED UN OLD SHORELINE AND FRESENT DIRECTION DU 1001 ITFt—eN2 V¥CTIsVCI) AUGaaAal(IOcl)ev( Ty BaAnGeahGeZy QILOCTISAMPETOS( ANG) SSINC BANG) XK H2CT Del) GILDONIDECTOSL) #QULDONI MY) CILUCNIP1) 3 £2810) *8OLD(NIP2) auGebal lot) BANGTanGeZy QLT4isa“P COS (ang) *SIN(C BANG) #XKDA(TD01) 67 amor ana aon aan 2000 2001 2002 eos 3000 5001 40090 4001 4902 4003 5000 5901 5002 AIGZAQ(TDON2) BaAnGesNnGely Q.1M4 «PF ANP3COS (ANG) ® SINC BANG) ®&XKD2(IDONG@) TTLasy33 9) 9999 ITsyolTLaST CALCULATE akRAYS LoD D) 2090 TeaeniMj MMI OCT) SQOLO(CT) LOCI) ® (CULD( 11 )e2,%00LU( 3) *UOLD(Jol)) O01) 80LI41 OCLC (NIP1) 20, O02) sGLIKNP CALCULATE aRRAYS Pol P(1) 840 Q(1)8ne OU $059 IBaeNy™1 PCT) aL (CII (e( Tey) ee2odol, MC1)e_Cl)sO¢(7) PCy )el pe(1Dey)FQ( NIM) PCy Py )Z16 GC 41P 1) B00e(P@ID)/P(NIPI) DU 39g ITENLPQ29NDM] FCT)TL C1) 8(eS( lay )teedoly 91) FL CLIP (7) Pl tedsle SO-vé FUR THANSPORY GNEW UCL) 3nNC1)7P 04) DO 49nO TB ayNI™] UCT) S(OCTI oC (CL enCTel))7PC1) Onna (NL) EUCNI)EDEINI SPOON) DO dnl Tet oNtM] Jevyjeory AN aly) BOC S)FQNEwW(JO1) 6UCJ) UCYIPy)E0CNIPL) s/PCNIPI) DO 49n2 T=N{P20N—MI UCT ECOCTIoL C1) enClel)) PCT) Q\2a(s2)2U0N2)3D(N2) /P(N2) DU GOn3d TEnNyPyenant Jzv2en te] GNiw(JIEQCJ)OONEW( IOI I OUCJ) IPUINTEQ TE CIPRINT FQ, 0) GUTU 5O PRINT 400 FO«mar(/o*® LISTING UF QNEwS) PRINT GWOLe(UNEN(T) 0181 9N2) FOIMATC {Xe t OF 10,4) CALCULATE New SHORELINE M009) 20,98(QNEN( 2) PUNERW(1)¢QOLD(2)eGUL0(1)) DU $970 T82N1 4} e072 29 254 (QNEW( LOL CQNENW( Tel )OQHLOCIT¢1)eQVULO(I@1)) MMONL)F0,S8(QNEw( NI @UNEW(NI@] CQQLD( NI) PGOLD(NI@1)) AKENEP1) 20,59 (GONE MC NIO2) CQNEW(NI 61) *UOLDINIO2) CUOLDINIOS)) 00 5091 13NIP20ND™1 MKCL BO PS*(QVEMC TOL SQNen(CTel)¢QQLO0( 191) eGULD(I"4)) KH0N2) 50,599 (QNEW(N2) PUNEM(N204)¢QOLD(N2) eQ0LD( Nem) ) FaytaTs/Oxmay FL2C0LLMAT/S5LOPE F2AALTNESLT/(HODG) BF2 DU 5092 1419Ne MRC LT) 2F xn LyoVv¢( look leh ZECKA(2)OKK CL) sPAHAT 68 oan aan SLANG( 1) S47 AN(Z) 00 $03 1z2N1™1 Ze( UK Lop youn( ley) )9e9 S003 SLANG(I)FaTaN(Z) ZECKAOND oxn(Niey)) /DAHAT SLANG(NL)SATAN(Z) ZECHACNLPSVAKKCNG OLD) SDKHAT SLANG(N1¢})FaTANCL) DU 5094 TEN{P20NQM1 Use( anc Topeoxn( Loy) )Fb9 $008 SLAnG(J)saTAN(Z) Za(xtcN2)enx(n20y))/0RHAT SLANG (N2)BATAN(Z) TFCIT 4&9, ITLAST) GUTO 9999 00 6000 134 on2 VV (1) sSLANG(1) 6000 VOL y)akacT) 9999 CGT ue ME TURD END SUBRUWTINE CALCKHUCY Ye MUN) COMMUNS BOATA/S LAMBDA eRe DQ eDK yHERM NI pN@o BML eo BWDeDXHAT » SLOPE OAL INE COMMONSUTFROEs KKN2L20115) 9 AAC 20115) COMHURS SSMUZ DIR( 1502) PERC 1302) NPH( 1302) oXSC1S,6) pAMPC1 302) OIMENSIUN VY(S15) REAL «3 OxK230 S/UKMAT NYP Landed CALC OJFFRACTION FOR LH® A=DIR¢MON4) CALL GETC( RAD pPER( MUNG 1) 943) Ageo)k DU 3090 JE1oN]h Tanyeyed WekedDy Vatal.eVY¥(J) @(8+n0) Rypzytyovey R1sSUPTCRL) say Zeasy Oysatan(2) TFC AL olLT,. A) GOTO 4 CALL QHEGION( RS pao ALOR) AaCyoy) sa GOTU 2a 1 CALL SREGION(R 1p Ao AL OBS) AAC Joy) 344 2 xKD2CyoT) aK sony $000 CONTINUE OO 1091 JENIPLeND mKO2C e121, JOOt AAChoy) BA CalC mIFFRACTION FOR RRB AceDIR(MCN, 2) CALL GETC(AwDyPER( MUNG 2) on3) Aszeo/e DU 2000 JENIPLOND uaneDy veRoleoYY(1)*(B+00) Aysrty+yey RLSSURTCRL)/H3 Zzasy ayzatTan(lZ) TF(A1 eLT. A) GOTU 31 CALL nREGION( Rig AVAL ORY) Ad(Qoy)BeA GuUTO y2 11 CALL SREGION( RI odo AL oKS) AAC2e7) 2044 12 XKD2(aeT) BK 5eKS 2000 CONTINUE NC PO91 TB 0eN4 me O2C201) 51, 2001 Abl(2ey)zoa ae TUN b.0 SueBROLTINE JINPUTQ CLMMONS BUATAS LaMSL4e Re DOsDK eo BERMoNS gN2oBWL eo BwDoDXHAT eo SLOPE SALINE COMMU SSARLINES VATE ($08) oSL(115,A) 69 nan aon onan NMNAMNRANDQAANAAGAAIE CiMMON ZLAKELVLZ KL (C24—9}2) COMMUN SSHUO/ DIR( 1592) ePER( 13902) DWH( 1502) 0X90 1506) oAMP(I S02) COMMON/Z3S/7230130e) DIMENSION 72LCA4e12) 9 4D(E3) eo ALIST(40) REAL LAMBUAGLAMBNAM MATA MD/3L Abe Se S00SL odo SL odio SH yp S10 3003105057 PRINT 999 999 FORMAT(IH19% DATA A5 READ BY SUBROUTINE INPUT%9/) READ SPACITAL PARAMETERS READ(G¥L01) CALISTCI) eo f31010) JO1 FLRMNA? (1008) READ( 5190) LAMBNACBeDOODKe BERMe Bwl eo BWDe SLOPE $90 FORMAY (8510.1) READ( 5100) MLINE PEINT 1OLeCALIST¢1) 0181910) PFINT 100°LAMHDAs beDOODXeHERMO AML »ANDe SLOPE o XL INE READ HISTURICAL SHUORFLINE DATA READ(Ge101) CALISTCI) 0181510) READ(50102) NtoNaeNDoNVoINYR 1o2 FORMAT(5I10) READ( 50100) C(DATECJ oT) e131 9ND) oJato3) DL 1050 TBq_—he BEAD (50100) (SLC Ted) odsteND) $000 CONTINUE PFINT LOL CALIST(I) 9181010) PRINT 1000((DATE( Je) 0181910); Jai,3) OU 2000 1349N2 2000 PRINT 1000(SL( Ted) eJ840ND) READ HISTORICAL LAKE LEVELS AND COMPUTE DLLJOT Fet,/(8%00) READ(C S101) CALISTCI) 0191010) OU 1001 IateNy READ( 50100) C2LC Ted) odatosa) $001 CONTINUE PRINT JOLOCALIST (I) 0381910) Du 20m! 1349NV IFBIOTNYR@{ 2003 PRINT 99B01P o(ZLCled) pJeiol2) 998 FURMAT(IL 7oSu012F10—e2) K20 OC 10n2 13,eNV IvelNyReley LyYslvo48(lysu) Kr ko] ME CLoy I BCLLCT 2) edb (101) /MD(1) Mzts(y*Ly) MP seOre)em MLCLersCLLCTod elk (102). Km 99 10ne J3$e11 MLCLeseCdeCrodey edb Cred) )/MDCJ) IFC] ,tU,. NY) GOTU 1002 MOCTeyeeCZUCLerodrece (tote sMb(42) 1002 CONTINUE READ HIRECTIONSy PSXIUD8, WAVE HETGHTS FUR EACH MONTH DIRECTION CONVENTIONS CA & 1 6 2 OLEp WATER ANGLE LESS THAN 0 . x + 1 DEEP WATER ANGLE GREATER THAN O ° M 4 e a + % x o oh + one MOCO ye OOK OK €9=364415926/180, RrAD(5e101) CALISTCL) 0181510) 0) 1993 “B1913 R-61(60100) DeH( Mol) oDWHE Moe) oDIR( MOL) eDIR(Me2) oPER(My1) OPER (Moe) DIF(M,1)Z01TR(Me 1) *C9 70 1003 DiP(M,2)3UTR(Me2)9C9 PAINT LO1eCALISTCL) o 184010) D2 2003 “81013 DIRyerIR(h.1)/C9 DLAeIHITR(Me2)/C9 2003 PRINT 1L00eDRH(My 1) pOMH( Me 2) gDIR1 DIRS sPER( M91) PER( M2) aoaonan annan CALCULATE PARAMETEHS FON SCALING PINT 200 200 FIRMAT(/0% LISTING OF SCOLE PARAMETERS AS CALCULATED JN SUBROUTINE 1 [NPUTUe) OY LOn4 M3193 2180 025¢5eo5#DMH (Mel) / (Sel QUBBPER( My) 92) 225002585 eS8DWH (He 2)/ (Sef 2UBSPER(My2) 942) DLS Ko BSINKH( Mey) P2HPER( MGS) AATUT HE oBSENRH( MD) OR2EPER(MQ) ArmA, IrF( Op ,GT, As) azde K3(M04) B0XHAT BOK (8400) OTHATSLAMBDASDXHAT#E2 OTSDIHATS (RDO) Se37A W)TsMn(*)F12 NaxDT,/OTe] 13N Drandy/Nn 13 (Moa) SL AMBNAMa¢OT/0K 092) 8A/(BO09) W3(“oz)BOTHATSLAMBDAMPNKHATSOD x5(Meu)207 X5(™065)3N 15(M0q)3(B900)F(RODGISA MOBS 12UBRPER(M,1)PPER( M1) Z3CM9 4 E0255 eS MVAM (MeL ISNLO Z$0% 02) 800255 eS @VNK (Moe) SKLO A*PCM,L)BALZA wePOm PVEAIS“A POINT 2010Me(XS(MOK) oKSh 0G) oZ3( Moy) 02S( Moe) oAMP( M91) po AMP( M2) ZOL FORMAT (® MONTHB®,T30% AHRAY SAFO ZFOoSePOgL OFS ,O0F Oo 30% ARRAY 250% Le2booye® ARKAY AMPS eek O,3) 1004 CuNTINve RETURN END SJBNUNTINE SREGION( Ro Ag Al KS) CALCULATES KD IN S REGION (ASSUMES 1 8 DISTANCE/WAVE LENGTH) REAL wh P133,4415920535 C$BA2,n2R4271 25 STESING(CALSA) S24) SazSIu( (Alea) se.) C2zcIs(aiea) tCrzcus(alea) RZSSUPT(H) ULEC Sah S457 U2zoC yeh 305g MTP TEULFUL 2, CALL cS(%o360C6) ViF(1,755°C6) 72,5 w1z(3,°CO) 72, MEPL HV 2e2z2, Call CS(XS6sC6) V23(1,"S6°Co)/2,5 W2=(S,°CO) s2, XuBQesPlOKeC] MFZEQoePLORSCD CssCUs(x4) C52CU8(K5) SusSIn (ug) S325], 045) DasvielCUsveeCSowysSuUonoess Dozen} eC UenreCSayy*SUevo9ss KsSD4eDUeO5eN5 K52SGQT(KS) RETURW ED SIBROYTINE QREGION( Re Ao Alo K3) CALCULATES KM IN @ HKEGTON (ASSUMES R @ DISTANCE/WAVE LENGTH) REAL «3 7| mae: 10090 100 PT2304 415920535 CBBPeRAKILTIAS SIZGBIN (CALA) S24) SBPSIN((ALOA) S25) C1eCOs( Alea) CetCUsl Aiea) K32Supl(R) ULsC bak3es7 U2zelyek3esy yePTS 18UlLs2, CALL CS(1X9S6eC6) YLECL*S6°CO)/2, b1z(SeeCo) 2, VEIT) 2%UCs2, CALL C30 X9S6eC6) v2z(t ,°Sb°Co)s2, #22(94°CO) s2, wusesplexecy rS=etpl]eReco CuzCus(xu) C5=C IS (x5) Su=SIN(x4) SS=STN(KS) CUTC4eV IP CumHt FS ys V28C5on2esgy NSFeSurvitSuew{@sueVve®SSow2ecs HE2DUSUUCUSEOS KY=SSaRTCKS) RETURR END SUBROUTINE GETC (DoT ow 55 GIVEN OEPTH(O)e PERIODCT) CALCULATE WAVE LENGTH(W3) G3232,2 P12 301415926535 Pleas? 9P] W3EG3eT/F 12 ZyzPLoayst U$sutoI2¢0/(G3FTeT) VY$3E 4003) TFCKS ok Te 16,0) GOTO 3 Ceeb3eT/Ple Gulu 2a V3tEXP (KS) T3ECVyPL)/(V3e1) Ch2zGSeT@BUQT( TS) sPl2 0) 1000 124050 M$32%73/C5 YS3ZEXP( KS) TSE(VEPLI/(Y3O1) OssndeT3 M12489(C%9N3) M250 0en001%AK8(03) I-( xl okbe K2) GOTO 2 C3e(CyOOS) 2, CONTINUE w$eC3eT MITE (S0100) wdeC30D0T FuRMAT(/)® EAROR IN COMPUTATION OF WAVE LENGTHS04F10930% CALL EXIT 1%) CALl EXIT asectely RETURN (aD) SIHROYUTINE C8(KeG60C6) Z5R4AHS(K) Tt (24 oG1, 4,0) GUTO 4 Corsuet(Z3) So32Z3eCo Z$=(4 eZ3)%(4,¢7%) CAFC ((Se1IOTASE M1 182305 e2UU297ER9) O¢ 345045114267) 82343027 3300E 05) CorCom((COCS#L3¢4 D204 KE aS) #Z564 1 CASU4Em2)9Z23)¢1 6U0905E@1) S5B(CeoOTTHBIESI NALS 9S HASISKEOH)eZSOS COLIUI ESO) #23 S5ES6a0(0(5442,441616E 4) ¥7366,121 32EC3) 92548, 02049Fe2) GOTO a C5zCUS(23) $$291N0Z3) Thue/ly ASSCC (RB, THAC5HERUPZ50U,169289E 93) 82547, 97094 SE OS) *2106,792R01LeS) ASC CC A342 = 3, COC SUEM4) 92365, 972151 E93) 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