WHO! DOCUMENT COLLECTION Wave Loading on Vertical Sheet-Pile Groins and Jetties by J. Richard Weggel COASTAL ENGINEERING TECHNICAL AID NO. 81-1 JANUARY 1981 A -ZoRPS OF oe LN Approved for public release; distribution unlimited. U.S. ARMY, CORPS OF ENGINEERS COASTAL ENGINEERING Be RESEARCH CENTER PIKE Kingman Building Fort Belvoir, Va. 22060 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: Nattonal 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. fi Wee Ou / \ DOCUME \, COLLECT! = — ee, UNCLASSIFIED SECURITY CLASSIFICATION OF THIS PAGE (When Data Entered) REPORT DOCUMENTATION PAGE 1. REPORT NUMBER 2. GOVT ACCESSION NO.) 3. RECIPIENT’S CATALOG NUMBER CETA 81-1 - TITLE (and Subtitle) 5. TYPE OF REPORT & PERIOD COVERED Coastal Engineering Technical Aid 6. PERFORMING ORG. REPORT NUMBER WAVE LOADING ON VERTICAL SHEET-PILE GROINS AND JETTIES 8. CONTRACT OR GRANT NUMBER(S) AUTHOR(s) J. Richard Weggel 10. PROGRAM ELEMENT, PROJECT, TASK AREA & WORK UNIT NUMBERS RILZIZ PERFORMING ORGANIZATION NAME AND ADDRESS Department of the Army Coastal Engineering Research Center (CEREN-EV) Kingman Building, Fort Belvoir,, Virginia 22060 12. REPORT DATE January 1981 pa aes acne 17 15. SECURITY CLASS. (of this report) UNCLASSIFIED 11. CONTROLLING OFFICE NAME AND ADDRESS Department of the Army Coastal Engineering Research Center Kingman Building, Fort Belvoir, Virginia 22060 14. MONITORING AGENCY NAME & ADDRESS(if different from Controlling Office) 15a. DECLASSI SCHEDUL FICATION/ DOWNGRADING E 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) Groins Jetties Mach-stem reflection Wave loading Waves ABSTRACT (Continue on reverse side if necessary and identify by block number) A method is presented for calculating the distribution of force and over- turning moment resulting from incident water waves moving along the axis of a groin or jetty with vertical sides. Wave height at the structure is determined from experimental data on Mach-stem reflection. The distribution of force is assumed to be in proportion to the nonlinear shallow-water wave profile given by either the cnoidal or stream-function wave theory. An example problem demonstrates how the cnoidal theory may be used to estimate the wave force and overturning moment distribution along a structure. FORM DD , fpevee, WIA! EDITION OF 1 NOV 65 1S OBSOLETE 4 UNCLASSIFIED SECURITY CLASSIFICATION OF THIS PAGE (When Data Entered) AAI ce 0) gic at et cao et y 6 SME AT ls Ale e _ ema Saint ene, aga ire Salt Sha TAS YD MSEE ee 1 a wr | SNe ree eh emt Dae semnet MTR Lie ah danhvon Go Tie THs, te way eo Genk Qaibiea geld Er ert y ie ic ee Bie 3 j eres AN we ~~ anistity sachin as teary temo 7 one dae vom dics rea, a A ory iy et act een at rrproatelat eis i Mn) eae he Wik aes i \! r fi esdnws si il. bel eile ag nisin Sp Nee eto pi eerie “a ’ aril j ; } : Vina = horn wr Prt vemiera Ay hee ome ‘ ! ‘ | ( i ae ' a - , 1 AQ Z i Loe i ES : my — is mi RuAbeL smoke he ee ee ee ee eh on tf ; tt " i ' , a piynlate ictal Vaan a — ‘ apy ne i ‘ Taw ata ' 4 Y sis f wh i = % ‘ “ bred 4 maa 4 Rate, , . L¢ SiG a Ns Sk bea I rm oti £o4 L223 Vonia. 2 Pecans wes “ . bi ah md 2 Sete Senin: wih ce ORE a Une s wen i ate, Naga rwok Rete eer Lagat are) te: bite i she MB CST A MAN. AR ey eh SA 08 at - ai geint Ce ed ib ead” tain ee Eg d they Jat ich aes eto na e > K a a id hPa Ss Mm tae a" ly AR orlabarclirtchbethhetrvunlnncriipclt ms fe 784i ty nf hater PREFACE This report is an outgrowth of consultations with personnel from the U.S. Army Engineer District, Savannah, concerning wave loading on a concrete sheet— pile groin proposed for Tybee Island, Georgia. The Shore Protection Manual (SPM) provides methods for calculating wave forces due to waves acting normal to vertical-sided structures. The SPM also gives a method for reducing the dynamic component of force when waves approach at an angle to the structure; however, no methodology is presented for estimating the distribution of force and overturning moment along a structure's axis, particularly for waves propa-— gating with crests nearly perpendicular to the structure. This report provides such a methodology. For sheet-pile groins and sheet-pile jetty sections, cost savings can sometimes be realized if wales are assumed to longitudinally distribute some of the wave force. The wave loading methodol- ogy presented will provide the information needed to calculate the forces transmitted by the wales. The work was carried out under the coastal structures research and development program of the U.S. Army Coastal Engineering Research Center (CERC). The report was prepared by Dre J- Richard Weggel, Chief, Evaluation Branch, under the general supervision of N. Parker, Chief, Engineering Development Division. Comments on this publication are invited. Approved for publication in accordance with Public Law 166, 7gth Congress, approved 31 July 1945, as supplemented by Public Law 172, ggth Congress, approved 7 November 1964. Colonel, Corps bf Engineers Commander and Director VI CONTENTS Page CONVERSION FACTORS, U.S. CUSTOMARY TO METRIC (SI) ecccccccccccccscccee 5 WNIPNOWDWKGIILO Noo Goboo ob 00000 DDO ODDD ODD DOD ODD DDO UO D000 O0O000000000000000 7 SPM METHODOLOGY FOR COMPUTING FORCES AND MOMENTS. cccccccccccccccescce 7/ WANG FKSvail IRNTIMLWIRCIILONG GG 00D CDODOD0DD0DOO000000000000000000000005000000500 8 CALCULATION OF WAVE FORCE AND OVERTURNING MOMENT.cccccccccceccesceeee 10 EXAMP IE Ha PROBGEMCryeielelel chee velenslshelleielels)elcleliole) sieleiolcloleleliolelelels)eleejicielelelejer clepelalerorenersmmemnll SMW DCS G5 0GOCOOOCOD00 000000000000 000000000000 0000000 00000GD000000000 «IG LITERATURE CIID 50600 0000000000000 00DODODOD DOO 000000000 0NDODDDOOO00000 N7/ TABLE Vaatatiion or St omce) along! s tisucitsuicele © lel) leche) sl clielelioeleralonel slelicheleleiefolellelelcleleleneloronmnn (im FIGURES NSIILOC ECM jUEEGCMS OF A SoOllalkAiAy TENEG500000000000000000000000000000000 8 Oblique reflection of a solitary wave, experimental results; water depth, d = 0.132 EO Ost cio-eietieltetletieterekevelishellererel oie tevecetalie eucieceteveteveveveraietenenotenenn 9 Oblique reflection of a solitary wave, experimental resultS..crcccccece 9 Xela Ebatel eyeonbin jpre@ieslllao ogaocobG000G 0H DD0 000000000000 000000000000000000 «(LD 107422 Dimensionless cnoidal wave profile, K2 = 1 - O6000000000000000000 IZ Welrestenesl@ynl Cpe sE@seea shovel fmownsaine euloynys eaeoilino doogag0abg0G000K000DDD000D0000. iI Loadine sions wandwardissiidel ota CrcOsereohefelerolelelalefenelolelerslojenelalelolelelenelovenclcienonicicrenommne >) 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 Zo Sy 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 LOL? x 10-2 kilograms per square centimeter ounces 28.35 grams pounds 453.6 grams 0.4536 kilograms ton, long 1.0160 metric tons ton, short Oo9O72 metric tons degrees (angel) 0.01745 radians Fahrenheit degrees 5/9 Celsius degrees or Kelvins! 1To obtain Celsius (C) temperature readings from Fahrenheit (F) readings, use tormulag € = (5/9) Cd? =32))o I Obtain Keiko (OX) teachings, wse cormillag, i = (5/9) (8 =32)) se 273050 % Tis susan au eeu tz te a © ~ . mas _a& < i oo Af yfagoe) fies ws Aeseaieng ALAA RED ANE . sande. A _ ; ae ‘ UA aes nh: rrr Pr) nh : Abagdo ‘Gt 5 ‘3 tae “OPER Hivpaeaae cs, rack : S shims OS », 3 SURPASS CMA TORS SSL dst: Sie EE eRe pee ane ane 1 : § os f : i if al + ie a4 3 : ‘ =i : or aes in ere ied Oia besa te Peet ee OD Ga pet. (Ge rile Sage tb es ae ZX EGE s RY =) ae y i Fe Been ke I | a a : 4 ere has thasd sv ere ASE WEEN = 3. eg .t5¢ 4 a) (er\E8 “NTE Met Seu asso thw TAY ill WAVE LOADING ON VERTICAL SHEET-PILE GROINS AND JETTIES by J. Richard Weggel I. INTRODUCTION The Shore Protection Manual (SPM) (U.S. Army, Corps of Engineers, Coastal Engineering Research Center, 1977) provides guidance for the design of verti- cal wall structures subjected to either breaking, nonbreaking, or broken waves; however, the design methods presented are for waves approaching perpen- dicular to a structure. The SPM also presents a rational, though untested, procedure for reducing forces when waves approach a structure at an angle. The SPM force reduction appears to be valid if the direction of wave approach is not too different from a perpendicular, iee, a > 70°, where a is the angle between a wave crest and a perpendicular to the structure. For structures essentially normal to shore such as groins and jetties, waves usually prop- agate along the axis of the structure rather than normal to it. For this sit- uation, the SPM force reduction factor, which varies as sin~ a, underpredicts the maximum wave force since a > 0° and sin® a > 0. Furthermore, the instan- taneous distribution of force along the structure is not uniform when a is small since wave crests act along some parts of the groin or jetty while wave troughs act along other parts. Consequently, the maximum wave force acts over only a small part of the structure at any one time and the point on the struc— ture where the maximum force acts moves landward along the structure as the wave propagates to shoree This report outlines a design procedure for esti- mating the force acting on a groin or jetty when waves propagate along the axis of the structure, iee.-, when a, the angle between the incoming wave crests and a perpendicular to the structure, is less than about 45°. II. SPM METHODOLOGY FOR COMPUTING FORCES AND MOMENTS The SPM presents three procedures for calculating wave forces on vertical or near-vertical walls. For nonbreaking waves, the Miche-Rundgren method is given. -The SPM design curves for this method were developed from Rundgren's (1958) equations and, for small values of the incident wave steepness H,/gT ; from Sainflou's (1928) equations where H; is the incident wave height, g the acceleration of gravity, and T the incident wave period. For breaking waves, the Minikin (1963) method is given in the SPM. When waves break against a structure they exert extremely high impact pressures of very short duration. The impact with the structure of the translatory water mass associ- ated with the moving wave crest causes the high pressures. The Minikin method attempts to describe these pressures but in applying the method the force is assumed to act statically against the structure. The problem is in reality a dynamic one. Wave pressures exceeding those predicted by the Minikin method have been measured; however, their duration is so short that the assumption that the force is a static one makes the method extremely conservative even though the force itself may be underpredicted. Since it is the impact of a translatory mass of water that results in the high pressure, it seems doubtful that waves approaching a vertical wall at an angle could cause them. The Minikin method is thus inappropriate for calculating forces by waves approach- ing a structure at an angle. For broken waves, the SPM presents a rational method for calculating forces based on computing the stagnation pressure. Again, this method is for normal wave approach. A modification of the Miche-Rundgren method as presented in the SPM appears to be applicable to breaking, nonbreaking, and broken waves for situa- tions when waves approach at an oblique angle. III. MACH-STEM REFLECTION Waves in the vicinity of a vertical wall are comprised of incident waves and superimposed reflected waves. For a perfectly reflecting vertical wall with waves of normal incidence, the reflected wave height equals the incident wave height resulting in wave heights in front of the wall equal to twice the incident height. When waves approach a structure at an angle, reflection is not complete resulting in a wave height less than the sum of the incident and reflected waves. For angles of incidence, a > 45°, a phenomenon termed Mach-— stem reflection may occure Instead of reflecting at the angle of incidence, near the structure the incident wave crest turns so that it intersects the structure at a right angle. This gives rise to three separate parts of the wave--an incident wave, a reflected wave, and a Mach-stem wave propagating along the axis of the structure with its crest perpendicular to the structure face (see Fig. 1). The amplitude of the Mach-stem wave, which is the wave causing the force on the wall, was investigated by Perroud (1957) and Chen (1961). The relative amplitude of the Mach-stem wave as a function of angle of incidence and H,/2d is given in Figure 2 (Hy is the incident wave height and d the water depth). A more general relationship tor average reflection data, independent of H,/2d, is given in Figure 3. For a > 45° the relative amplitude of the wave is about 2.0, iee., the wave height along the wall is the sum of an incident and reflected wave which has a height equal to twice the incident wave height; for a < 45°, the relative amplitude of the Mach-stem wave is less than 2.0. Hy = Height of Incident Wave Hr = Height of Reflected Wave Hm = Height of Mach-Stem Wave a = Angle of Incidence (angle between wave crest and perpendicular to structure ) r = Angle of Reflection Wave Crest Wave Crests Wave Crest coc “~SsS—— __ -Vertical Structure Vertical Structure Vertical Structure a2 202 20°45° Figure l. Reflection patterns of a solitary wave. Mach Reflection a Regular Reflection no reflected wave reflected wave | (no Mach-stem wave) appears appears Dimensionless Height of Wave Running Along Wall O 15 30 45 60 75 Angle of Incidence (a) Figure 2. Oblique reflection of a solitary wave, experimental results; water depth, d = 0.132 foot (from Perroud, 1957). B85) Mach Reflection Regular Reflection no reflected wave reflected wave appears appears ww) © eA Reiative Height of Mach-Stem ew Wave Running Along Wall, Hms /H; oO Average Curve o oO Relative Wave Heights (Hms /Hj and H-r/H; ) Relative Height of Reflected Wave, Hr/H; O ley? 802 45° 60° ROS Angle of Incidence (a) Figure 3. Oblique reflection of a solitary wave, experimental results (from Perroud, 1957). IV. CALCULATION OF WAVE FORCE AND OVERTURNING MOMENT The Miche-Rundgren method in SPM Section 7.32 provides a method for deter- mining wave force and overturning moment when either a wave crest or trough acts against a wall. For oblique incidence some parts of the wall are acted on by wave crests while others are acted on by wave troughs. Thus, there is a variation in force along the wall that can be assumed in proportion to the wave profile along the wall; iee., the maximum wave force corresponds to the wave crest, the minimum wave force corresponds with the wave trough, and the variation in force between is assumed proportional to the ordinate of the wave profile. It remains to select an appropriate description of the wave profile. For waves in shallow water the cnoidal theory outlined in Section 2.26 of the SPM provides a satisfactory description. Stream-function wave theory (Dean, 1974) will also provide a satisfactory description of the profile; however, for pur- poses of illustration and convenience the cnoidal theory will be used here. V. EXAMPLE PROBLEM The computation of forces and moments is best illustrated by an example problem. GIVEN: A concrete sheet-pile groin perpendicular to shore is subjected to waves 6 feet (1.83 meters) high with a period of 8.0 seconds. Waves approach the groin so that the angle between the wave crest and a perpen- dicular to the groin is 30°. The water depth at the end of the groin is 10 feet (3.05 meters). The beach profile along the windward side of the groin is given in Figure 4. Concrete Sheet-Pile Groin Elevation +70 ft -15 -300 -200 -100 (0) 100 200 300 400 500 600 700 Distance from Shoreline (ft) Elevation (ft) ro) Figure 4. Beach and groin profile. FIND: Ignoring changes in wave direction as the wave propagates toward shore (refraction) and changes in depth near shore (shoaling), determine the maxi- mum wave force and overturning moment acting on the groin when the wave crest passes a point 400 feet (121.9 meters) from shore (100 feet or 30.5 meters from the seaward end of the yroin) and estimate the distribution of force and moment along the groin at that instant. 10 SOLUTION: An estimate of the reflection coefficient can be obtained from Figure 2 or 3. Calculate = 0.30 2d 2(10) From Figure 2, with H,;/2d = 0.3 and a = 30°, read BRA 2c = Oo47o Wnereiore,, at = 0.47(2d) = 9.4 feet (2.87 meters) Alternatively, from Figure 3 for a = 30°, Led lle = 1.61 or H_. = 1.61(6) = 9.7 feet (2.96 meters) This estimated Mach-stem wave height is subject to limitations imposed by the maximum breaker height that can exist in the given water depth. This breaking wave height is given approximately by Hp/d = 0378 or, for the example Hh = Osta = OSC) = Jo8 wees (2538 mekeies)) The wave at the structure is thus limited to a height of 7.8 feet. The remainder of the calculations are based on this maximum wave height. The maximum wave force (crest at the wall) can be estimated using SPM Figure /7-/0 by assuming that the maximum wave height of 7.8 feet is the sum of an incident wave and a reflected wave, each 3.9 feet (1.19 meters) high. Calculating and el —— = 0.00189 eT (3202) (8) ~ From SPM Figure 7-/0, read from the top part of the figure, F eS _ Wale = 1.14 and from the bottom part of the figure, ae es 0.38 wd 11 where F, and F, are the wave force when the crest and trough are at the structure respectively, and Ww is the specific weight of the water (64 pounds per cubic foot (10,000 newtons per cubic meter) for seawater). Therefore, F, = 1.14(wd?) = 1.14(64)(10)? = 7,300 pounds per foot (106,500 newtons per meter) and F, = 0.38(wd*) = 0.38(64)(10)? 2,400 pounds per foot (35,000 newtons per meter) Similarly, from SPM Figure 7-71, the maximum and minimum overturning moments are M, = 0.64(wd3) = 0.64(64)(10)? = 41,000 foot-pounds per foot {182,400 newton meters per meter) M, = 0.11(wd3) = 0.11(64)(10)3 = 7,000 foot-pounds per foot (31,100 newton meters per meter) where M and M, are the overturning moments caused by the waves when the crest and trough are at the structure. To determine the distribution of force along the groin, the appropriate cnoidal wave profile must be found. Calculate g SoD 7/2 B/S = 14.36 and A.| mm il (=) e —~N [oe) which is the limiting value of H/d. From SPM Figure 2.11 for the calcu- lated values of T vg/d and H/d, find the value of k2 = 1 - 1074*9. From SPM Figure 2-12 with k2 = 1 - 10-4*°, find 2 a = 940 d The wavelength, L, is therefore 3 3 Da al QAOCLO) L = 240 rae 7.8 = 30,770 & i] 175 feet (53.3 meters) WZ This wavelength is measured perpendicular to the wave crests. To obtain the wavelength along the structure, this value must be divided by cos ae Thus, pee L eli) é L Oana Gan aoe 203 feet (61.9 meters) where L' is the wavelength along the structure. The appropriate wave profile is obtained from SPM Figure 2.9 by interpolating between the pro- filles for k2 = 1 = 10p¢) and) k“ = 1 — 1072) (This! profile wa plotted in Figure 5 in dimensionless form. CAM ES a ee REE (0) 0.1 0.2 0.3 0.4 0.5 x/L Figure 5. Dimensionless cnoidal wave profile, k* = 1 - 10749 The distribution of force along the structure is then obtained by letting the maximum correspond to 7,300 pounds per foot and the minimum to 2,400 pounds per foot. Then ie ee 7,300 - 2,400 = 4,900 pounds per foot or 71,500 newtons per meter and F(x) = 2,400 + 4,900 n(x) where F(x) is the variation of force with distance along the structure, and n(x) is the variation of the dimensionless surface profile with x. Similarly, the distribution of the overturning moment along the structure can be determined. M, - Nhe 41,000 - 7,000 = 34,000 foot-pounds per foot or c 151,200 newton meters per eee and Tietotone, M(x) = 7,000 + 34,000 n(x) 13 Values of F(x) M(x) are tabulated for the example in the Table and plotted in Figure 6. The force distribution along the groin is shown in Figure 7. Table. Variation of force along structure. F(x) 2 M(x) 3 (lb/ft) (ft-lb/ft) 1 Computed from x = 203 x/L' Computed from F(x) = 2,400 + 4,900 n(x) Computed from M(x) = 7,909 + 34,000 n(x) fo?) o DS ° Moment (thousands of ft-!b/ft) on io) Force (Ibs/ft ) (eo) (oe) ie) (e) (0) 10 20 30 40 50 60 70 80 90 100 Distance Along Sroin (ft) Figure 6. Variation of force and moment along pzroin. Force (lb/ft) Elevation ( ft) -300 -200 -100 (0) 100 200 300 400 500 600 £700 Distance from Shoreline ( fi ) Figure 7. Loading on windward side of groin. As a check on the wave force when the trough is at the structure, the hydrostatic force can be computed using the minimum water surface eleva- tion. From SPM Figure 2-13, the value of (y, - d)/H + 1 is found to be 0.850 where y, is the height of the wave trough above the bottom. Therefore, rearranging and solving for y,, y, = (0.850 - 1.0) H+ 4d or Vie -0.150 (7.8) + 10 = 8.83 feet (2.69 meters) Computing the hydrostatic pressure 2 De -—= = Sa(8:B DEE 2,495 pounds per foot or 36,400 newtons per meter compared with F, = 2,400 pounds per foot for the value computed from the Miche-Rundgren figures in the SPM. When actually computing forces on sheet-pile groins and jetties, the re- storing force caused by water and wave action in the structure's leeward side must also be considered. The critical design situation occurs when the water surface on the leeward side is a minimum; iee.-, when a wave trough acts there. For the example, the worst case for overturning moment exists when the water level on the leeward side is equal to y, = 8.83 feet. This corresponds to a minimum restoring force of 2,400 pounds per foot. A critical factor not included in the example that must be considered in any real design problem is the force arising because of the differential sand elevation on each side of the structure. These forces must be based on estimates of the maximum deposition and scour that will be experienced dur- ing the lifetime of the structure. It is also possible that this critical condition will occur during construction unless a scour blanket is placed adjacent to the structure. VI.- SUMMARY The proposed method for computing the distribution of wave force and- overturning moment along a vertical sheet-pile groin or jetty is approxi- mate. It assumes that the force and moment are in proportion to the nonlinear wave profile as given by the cnoidal wave theory. Alternatively, any other appropriate wave theory to describe the profile could be used. The assumption that a wave crest or trough acts uniformly along structures oriented nearly perpendicular to shore grossly overpredicts the total force since only a small part of the structure is acted on by a wave crest at any instant. The use of wales on sheet-pile groins and jetties can distribute these forces longi- tudinally, allowing the safe use of smaller structural sections. 16 LITERATURE CITED CHEN, T.C., “Experimental Study on the Solitary Wave Reflection Along a Straight Sloped Wall at Oblique Angle of Incidence,” TM-124, Beach Erosion Board, Washington, D.C., Mare 1961. DEAN, R.G., “Evaluation and Development of Water Wave Theories for Engineering Application,” SR-l, U.S. Army, Corps of Engineers, Coastal Engineering Research Center, Fort Belvoir, Vae, Nove 1974. MINIKIN, R.R., Winds, Waves and Maritime Structures: Studies in Harbor Making and in the Protection of Coasts, 2d ed., Griffin, London, 1963. PERROUD, P.H., “The Solitary Wave Reflection Along a Vertical at Oblique Incidence," Ph.D. Thesis, University of California, Berkeley, Calif. (also Institute of Engineering Research, Report No. 99-3, unpublished, Sept. 1957). RUNDGREN, L., “Water Wave Forces,” Bulletin No. 54, Royal Institute of Tech- nology, Division of Hydraulics, Stockholm, Sweden, 1958. SANIFLOU, M., “Treatise on Vertical Breakwaters,” Annals des Ponts et Chaussees, Paris, France, 1928. U.S. ARMY, CORPS OF ENGINEERS, COASTAL ENGINEERING RESEARCH CENTER, Shore Protection Manual, 3d ed-, Volse I, II, and III, Stock No. 008-022-00113-1, U.S. Government Printing Office, Washington, D.C., 1977, 1,262 pp. 17 ORTIY AMUTARSTH B gooBtie onohd bud 1 a2 ee tein be om’ eh aah reeaae ap tegat nolaot?: lapsed of ksi tad am dson¥, 73 Heed weit NG a gulsstinhs a San teRe ~ hela “ aeiey Tapes: ’ %8 iy Oe trespdiuls 7 Site TOD EHR CE hort bainee wake apron PATE ye HE. wee hire sRiieeuns ieee ehh esate” ak igh . mer fags vad nuit prise *, eae a) Sli ie mae oie i gs 5 *: aint cee SY" oh ree Re ie r epi Wigs kasistor xapiorhh paren ee eet nay ,Gatog: OA et bint! ie ep iakce hie) \/edetres', ciactihaa te" gainer ‘ates 35 Lv “ ap nbd siqec | dacobiniagcar oe aR4Qe TVW ghee “Wash sHerewh eng ay 113% a6 4 i t 7 Us a€ i uw f te : « ¢ Ca | bail el i] ; " ; agfesl Yo ssustian! Inyok ,8¢ ool nPtellog! "Pesorrs aval pA7aW wl yee SCC) ~aehewt ,nlotilowsd . xo"! Herne do oo tety 0 eg ta vFOmMeS. 67 TT we ¥ Pw: ; kee ee d un ie al s di hecbictatal oe cabo uo Rew hee" Lad oe eeFahett” re ge Sib Sere. Joa : Att! ie q [ an FA P x] / , pee OURETMED MON ASU OE LANRN SORE SLAP eROO. ~ PeAe 40 wna? oy sie HHA aO- MOO sot SIR YET Rap? PD" 42 gia Hs Es) BE See i io al OS hs eee é HI sed PEP | ao Ae aD ‘tna t 1 anid = 7, F 7 129 1-18 *ou BITsSN* €02OL CYR. COW pre TeoTuyoe, *1eqUeD yoTeesey BSuTeouTSsuq TeISeOD *S°N :SeTIeS °*IT *OTITL °L *SOARPM °h *sa010J OAPM °C *soTtqqer °Z *suTOIy °T eAT0042 cAPM UOTIOUNFJ-MeVI1}Ss IO TepTousd ay azeyate Aq uoazts atTTyoid sAeM A9z,eM-MOTTeYS AeeUTTUOU ay OF uotj10doad ut 90q 03 pelinsse sf ad10J JO uoTINGTIISTp sy_ “UuoTIOeTJe1 wazs-yoey «uO §«6ejep)§6«6[equemtiedxe worlz poeutmisjep Sf os1njoni4s ay) 3e WSTeEYy aAeM *SepTS TeoTIAEA YyITM Aqqef AO uToAs e jo sTxe 94} SuoTe BZuTAOM SeAeM 19}3eM JUSPTOUT WorF BuT,[NseA Juowow Buyuinj1s9A0 pue a010J JO UOTINGTAISTp sy BuTIJeTNITed AOzZ pejueseid st poyjem y *sooueteayol TeoTydeisotTqtq sepnTouy (I-18 °oOU £ 1eqUeD YDIReSeY ZuTIseuTsUuq qeaseo) °s*n -- Pty Teoruyoez) -- ‘mo /z@ +: "TTF : ‘d [21] "1861 ‘e0TAIeg uoTIeWIOFJUT TeoTUYye], TeuOTIeN Worzy sTqeTTeae ‘ea ‘prety3utadg { aeqUeQ YoIPassy SuTlaeuTZuq TeqIseoD °S*m : “eA ‘SAFOATOG JIO¥ -- JesdaM pazeyoty °C kq / setaqef pue sutoi8 oeTTd-jeeys TeoTIIeA uo BUTpeOT sAeM °r paeyoty ‘Te33em L¢9 I-18 °ou eBIT8cn® €02OL CUS} COU Pte TeoTuydey, “*1equeQ yo1eesey BuTAseuTsug TeISeOD *S*p :SeTIIG “IT *OTITL °T *SOARM °h *so010J OAPM °C *soTqqeF °7% *suTOIy °T *AJOoy GAPM UOTIOUNF-Me|I3S IO TepToud aya azeyate Aq usatT3s eaTTyoid sAeM AezeM-MOTTeYS AesUTTUOU 924} OF uotj10doid ut 0q 07 polinsse sft 90103 JO UOTINGTAISTP syY_ “UuoTIIeTFJoeI weaqs-yoem uo ejep [ejJUemTiedxe woiz poeutwieqjep sft 91nj}oNI4s aya 3e 3y3TeY aAeM *SePTS TeOTIIeA YITM ARQef AO uToOAs e Jo sTxe 9y SuoTe BZuTAOW saAePM JejeM JUePTOUT WoAZ Buy [Nse’ Juewou Butuanz19A0 pue 90103 JO uoTINGTAASTp oy. BuTJeTNOTeD 10y pequeseid st poyjoW y *soouetojol TeoTyderZOTTQTq sepnTtouy (T-1g**:ou § daqUeD Yyo1eesey BuTiseuTsuq Teqseop *s*n -- Pty Teofuyoel) -- °wo 7/7 : “TIF +: «d [2T] “1861 ‘a0TAIeSg uoTJeMIOFUT TedTUYSE]L TeuotqIeN Wory osTQeTTeae ‘ea ‘pretzy8utidg { daaquej9 Yyoreessy SutiseuTsuq TeIseoD °S*p : °BA SATOATOG JIOq -- TessomM pazaeyoTY °¢ kq / setqqef pue sutois oTTd-jeeys TeoTIeA uo BuTpeOT asaAeM °f paeyoty ‘Tessom L¢9 eit} SOE eqTesn’ £072OL OUEST OO Pte TeoTuyoe, °ieqUeD YOIeesey BuTIeeuTsUuq TeISeOD *S* :SeTIeS *IT °OTITL °I *SOAREM °H *Sa0IOJ OAM °F *setiIef °Z *suTOIn °T *A1O9y} APM UOTIOUNZ-wWesiqs IO Teprouo ayq azeyate Aq useatTs aTtTyoid sAeM Ja}eM-—MOTTePYS APsUTTUOU sey, OF uotjiodoid ut aq 07 pownsse ST 9d1OF JO UOTINGTIASTP YL “UOTIIeTJer wajs-yoey uo ejep TequemTiedxe wory peuTmwiejep Sf 91n}9n13s ay2 3e WSTey eAeM *SeptTs TedTIAeA yITM Aqqef io uToAs e Jo sTxe 942 SuojTe SuTAOW seAeM 19qzeM JUEePTOUT Woiz Buy Z[Nse1 Juewom BZuyuInz1eAo pue ad10J FO voTINGTAISTp oy. BuTJeTNoTed AoF pequeseid st poyjem Vy *saouedtejel TedTydeiasoT[qtq soepnyoIuy (1-1g *Ou § Jaque9 yoIeesey BuTIe9uTsUy qTeqseop *s*n -- Pty Teotuysel) -- ‘mo /7Z : “TIF : ‘d [ZT] . "1861 ‘20TAIeg uoTJeWIOFUT TeITUYyIeT yTeuotieN Worzy eTqeTTeae ‘eq ‘pretg3utadg £ dJeqUueD YoIeeSsy Su~ieeuTZuq TeIseon °*S*n : “eA SATOATEG JIO_ — essay PpareyrTY °C &q / setaqef pue sutois eTTd-jJesys TeoTIeA uO SuUTpeOT sAeM *f paeyoty ‘T233em Le9 1 SOE eBITscn* €02OL OVISTRY POU Pre Teotuyoey, “*1eqUeD YyoIeesey BuTIseuTsug TeIseoD *s*f :SeTtes *IT *opan 1 *SOARM °h *so010J DAPM °F *sotiqer °Z *suToIg °*T eA10O9y SAPM UOTJOUNFJ-MesI4Ss IO TeptToud ay} reyate Aq uaat3 aTTyo1d saAeM AejeM-MOTTeYS AeeuUTTUOU sayz OF uotj10doid uf 0q 0} pollnsse sf |9d10F JO UOTINGTAISTp sy, *UOTIIETJeI wejs-yoeR uo ejep J[equeuTiedxe woiz peuTWiejJep St e1njon14s aya 3e WYSTeEY eAeM *SepTS [eOTIIeA YITM AQQZef AO ufors e Fo sTxe sy ZuoTe BSUTAOW SeAeM 19}eM JUEPTIUT Wory BuyA[Nse1 Jusuow SutTuinqisao pue a010f FO UOTINGTAISTP 9y} SuTJeTNOTeD AoFz pequeseid sp poyzeu Vy *saouedejyol TeoTyders0TTqTq SepnTIuy (I-18 *Ou £ taqUeD YTe8Sey SuTisouTsug qeaseop °s°q -- pty eoruyoer) -- ‘mo 77 : ‘TTF : ‘d [/T] *T86l ‘90TAIeg uoTIeWIOJUT TeOTUY Ie], yTeuotieN wWorzy osTqQeTTeae ‘ej ‘prety38uzadg £ aAequapQ Yo1eessy SutdseuTSuq Teqseop °s*n : “eA ‘SATOATOG Ja0q —- Essay pazeyoTY °C kq / setazqef pue suqToi3 oTTd-jo0oys TeoOTWIeA uo SBUTPpReOT 2ARM °c paeyoTy ‘Tes3emy