TP 76-17 Floating Breakwater Field Assessment Program, Friday Harbor, Washington by B.H. Adee, E.P. Richey, and D.R. Christensen ~ TECHNICAL PAPER NO. 76-17 OCTOBER 1976 Approved for public release; distribution unlimited. Prepared for U.S. ARMY, CORPS OF ENGINEERS COASTAL ENGINEERING RESEARCH CENTER . 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: National Technical Information Service ATTN: Operations Division 5285 Port Royal Road Springfield, Virginia 22151 Contents of this report are not to be used for advertising, publication, or promotional purposes. Citation of trade names does not constitute an official endorsement or approval of the use of such commercial products. 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. UNCLASSIFIED SECURITY CLASSIFICATION OF THIS PAGE (When Data Entered) READ INSTRUCTIONS REPORT DOCUMENTATION PAGE 1. REPORT NUMBER 2. GOVT ACCESSION NOJ| 3. RECIPIENT'S CATALOG NUMBER We \YOoNy 4. TITLE (and Subtitle) 5. TYPE OF REPORT & PERIOD COVERED Technical Paper 6. PERFORMING ORG. REPORT NUMBER 8. CONTRACT OR GRANT NUMBER(s FLOATING BREAKWATER FIELD ASSESSMENT PROGRAM, FRIDAY HARBOR, WASHINGTON 7. AUTHOR(s) B.H. Adee E.P. Richey D.R hristensen 9. PERFORMING ORGANIZATION NAME AND ADDRESS Ocean Engineering Research Laboratory University of Washington Seattle, Washington 98105 11. CONTROLLING OFFICE NAME AND ADDRESS Department of the Army Coastal Engineering Research Center (CERRE-OC) Kingman Building, Fort Belvoir, Virginia 22060 14. MONITORING AGENCY NAME & ADDRESS(if different from Controlling Office) DACW72-74-C-0012 10. PROGRAM ELEMENT, PROJECT, TASK AREA & WORK UNIT NUMBERS F31538 12. REPORT DATE : September 1976 13. NUMBER OF PAGES 224 15. SECURITY CLASS. (of this report) UNCLASSIFIED 15a. DECL ASSIFICATION/ 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) Breakwaters Wave reflection Floating breakwaters Wave attenuation Waves Friday Harbor, Washington Wave transmission 20. ABSTRACT (Continue on reverse side if necesaary and identify by block number) A theoretical model for predicting the dynamic behavior of a floating break water is presented along with a report on a field experiment designed to pro- vide basic data for verifying the model. Additional data were taken from the literature and from auxiliary laboratory experiments. 0 mu =a mu m ins oO =) a =) m oO oO ll The dynamic behavior characteristics investigated were: (a) Total trans- DD , ros 1473 = EDITION OF Tt NOV 65 IS OBSOLETE UNCLASSIFIED eee eee SECURITY CLASSIFICATION OF THIS PAGE (When Data Entered) UNCLASSIFIED SECURITY CLASSIFICATION OF THIS PAGE(When Data Entered) prediction model was developed from two-dimensional, linearized solutions of the hydrodynamical equations formulated in terms of a boundary value problem for the velocity potential. Some nonlinear effects are considered. Results for the predicted transmission coefficients were in good agreement with laboratory and field data, and they showed how the influence of fixed-body transmission, and of sway, heave, and roll motions on the transmission coefficient changed with increasing values of the parameter, beam (width) to wavelength ratio. The shape of the curves predicting the mooring line forces as a function of the beam (width) to wavelength ratio (or of wave frequency) followed those for the measured responses, but predicted magnitudes did not agree closely with meas- ured values. The floating breakwater at Friday Harbor, Washington, was used as the field experimental platform; it was instrumented to record the incident and trans- | mitted waves, mooring line forces, and the acceleration components of sway, heave, and roll. Ninety-five 17-minute records were obtained during the period 30 December 1974 to 5 May 1975. Statistical summaries of all data are presented with analyses of selected transmitted waves, transmission coefficients, and acceleration components. The summaries and analyses constitute a performance report of a particular floating breakwater as well as an input to the develop- ment of the theoretical model. 2 UNCLASSIFIED Sib ele EN SS Eee ; SECURITY CLASSIFICATION OF THIS PAGE(When Data Entered) PREFACE This report is published to provide coastal engineers with a basic analytical procedure in the evaluation of certain floating breakwater types as structures for protecting particular sites against wind waves. The work was carried out under the coastal construction program of the U.S. Army Coastal Engineering Research Center (CERC). This report was prepared by Dr. Bruce H. Adee, Assistant Professor of Mechanical Engineering, Mr. Derald R. Christensen, Research Engineer, and Dr. Eugene P. Richey, Professor of Civil Engineering, of the Ocean Engineering Research Laboratory, University of Washington, Seattle, Washington, under CERC Contract No. DACW72-74-C-0012. Special appreciation is extended to the port of Friday Harbor, Washington, for the use of the floating breakwater for the field assess- ment part of the study. Mr. Robert Hovey, Port Engineer, and Mr. Jack Fairweather, Port Superintendent, provided generous assistance with the numerous logistics problems in the installation and maintenance of the measuring equipment. The sensor monitoring and recording package was adapted from a design developed in a contemporary project sponsored by the University of Washington Sea Grant Program for monitoring two other floating breakwaters of a different type. Data from these two sites were used for comparative purposes in the analyses of the Friday Harbor breakwater. Dr. D. Lee Harris, Chief, Oceanography Branch, was the CERC contract monitor for the report under the general supervision of Mr. R.P. Savage, Chief, Research 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. OHN H. COUSINS Colonel, Corps of Engineers Commander and Director IV APPENDIX CONTENTS CONVERSION FACTORS, U.S. CUSTOMARY TO METRIC (SI) SYMBOLS AND DEFINITIONS . INTRODUCTION. THEORETICAL ANALYSIS. 1. Linear Theoretical Model. 2. Nonlinear Theoretical Model 3. Results . FIELD DATA. Layout. ans Instrumentation . Wind Data . Waves . Cable ROReeS. Motion Package. ‘ Data Acquisition Syston é . Data Processing and Analysis. ONIADMHHPWNE COMPARISON OF THEORY WITH FIELD DATA FOR FRIDAY HARBOR BREAKWATER. CONCLUSIONS . LITERATURE CITED. HYDROSTATIC RESTORING FORCES AND SPRING CONSTANTS . MOORING ANALYSIS. LINEAR HYDRODYNAMIC COEFFICIENTS. FLOATING BREAKWATER ANALYSIS. DERIVATION OF PRESSURE TO SECOND ORDER FOR TWO PROGRESSIVE WAVES AT DIFFERENT FREQUENCIES. PHYSICAL PROPERTIES OF SEVERAL FLOATING BREAKWATERS . DATA SUMMARY SHEETS FOR FRIDAY HARBOR FLOATING BREAKWATER. .. . INCIDENT AND TRANSMITTED WAVE SPECTRAL PLOTS. Page 71 WZ 74 79 104 107 148 156 160 180 CONTENTS APPENDIX-Continued 11 12 13 I LOW-FREQUENCY SPECTRAL ANALYSIS OF FORCE DATA . J HIGH-FREQUENCY SPECTRAL ANALYSIS OF FORCE AND MOTION DATA . OG SKOOL OOO GL Toon DRS K WAVE MEASUREMENT . TABLE Summary of anchor cable force statistics. FIGURES Aerial view of Friday Harbor breakwater . A two-dimensional floating breakwater . Linear system representative of a floating breakwater . Filtered low-frequency seaward mooring line force, Tenakee, Alaska. Transmission coefficient for proposed Oak Harbor breakwater . Transmission coefficient for a rectangular breakwater . Transmission coefficient for a rectangular breakwater restricted to sway motion only . Transmission coefficient for a rectangular breakwater restricted to heave motion only. Transmission coefficient for a rectangular breakwater . Transmission coefficient for Alaska-type breakwater model Transmission coefficient for rigidly fixed Alaska-type breakwater model Transmission coefficient for Alaska-type breakwater, Tenakee, Alaska. Theoretically predicted transmission coefficient, Friday Harbor breakwater. : Page 192 200 218 61 11 14 20 21 24 26 27 29 30 Sil SS 34 36 14 15 16 IL 18 19 20 Zi 22 25 24 25 26 27 28 29 30 31 32 CONTENTS FIGURES-Continued Theoretically predicted sway motion response, Friday Harbor breakwater . Lh pce anna Theoretically predicted heave motion response, Friday Harbor breakwater . be : Theoretically predicted roll motion response, Friday Harbor brealkwalcerir wind mctaraNs a eaten is Seaward mooring line force for proposed Oak Harbor breakwater . Seaward mooring line mooring-force coefficient, Tenakee, Alaska . Recorded time series, Tenakee, Alaska .... <=.. Theoretically predicted long-period sway response of Alaska- type breakwater, Tenakee, Alaska . Theoretically predicted seaward mooring line mooring-force coefficient, Friday Harbor breakwater. Theoretically predicted long-period sway response, Friday Harbor breakwater. SME EONS PAs General location map. Field experiment site location map. Instrumentation location plan, Friday Harbor breakwater . Instrumentation and recording package layout. Average transmission curves for Friday Harbor breakwater. Transmission coefficient for Friday Harbor breakwater . Sway acceleration response for Friday Harbor breakwater . Heave acceleration response for Friday Harbor breakwater. Roll acceleration response for Friday Harbor breakwater . Seaward mooring line mooring-force coefficient, Friday Harbor breakwater . pane Page 38 39 40 42 44 45 46 47 48 50 51 52 55 59 65 66 67 68 70 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 acres 0.4047 hectares foot-pounds 1.3558 newton meters 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) 0.1745 radians Fahrenheit degrees 5/9 Celsius degrees or Kelvins' 1To obtain Celsius (C) temperature readings from Fahrenheit (F) readings, use formula: C = (5/9) (F — 32). To obtain Kelvin (K) readings, use forumla: K = (5/9) (F — 32) + 273.15. SYMBOLS AND DEFINITIONS Ay ,A2 Amplitudes of two incident waves a Amplitude of sway, heave, or roll motion for i = 1,2,3 B Characteristic beam of breakwater Co Body contour Cy Transmission coefficient Fj (t) Sway, heave, or roll exciting forces or moment for j = 1,2,3 KH; j Hydrostatic restoring-force coefficient for force in the jth direction due to motion in the ith direction KM j Similar to KHjj but due to the mooring system k,,ko Wave numbers of two incident waves L Incident wavelength mij Mass or moment of inertial when i = j, O when i # j n Unit interior normal to body surface P(x,y,t) Pressure Tr Vector from center of gravity to a point on the body surface CA WOH pfeis, Sway, heave, or roll motion; speed or acceleration 6 Phase angle 61552 Phase angles for two incident waves n(x,t) Free-surface elevation nz (x,t) Wave surface elevation for incident wave ny (x,t) Wave surface elevation for transmitted wave ij Damping coefficient for force in the jth direction related to velocity in the ith direction Added-mass or inertial-force coefficient for force in the "a jth direction related to acceleration in the ith direction ) Fluid density fC) Velocity potential w Frequency W1, W2 Frequencies for two incident waves FLOATING BREAKWATER FIELD ASSESSMENT PROGRAM, FRIDAY HARBOR, WASHINGTON by B.H. Adee, E.P. Richey, and D.R. Christensen I. INTRODUCTION Floating structures for use in the attenuation of water waves were introduced by Joly (1905). Little was done with the concept until the Bombardon floating breakwater was deployed to form a harbor during the Normandy invasion of World War II. The use of mobile harbors for po- tential military applications provided the incentive for extensive work during the postwar years. Representative articles from this period include those by Minikin (1948) who discussed floating breakwaters in general terms, Carr (1951) who used basic mechanics to predict trans- mission characteristics, and the review of the performance of the Bom- Bardon by Lochner, Faber, and Penny (1948). In 1957, the Naval Civil En- gineering Laboratory, Port Hueneme, California, began a concerted ex- ploration of the existing knowledge of transportable units that could serve as breakwaters or piers. Results of the study are summarized in Naval Civil Engineering Laboratory (1961), which was an invaluable state-of-the-art assessment with particular emphasis on military uses under the rather severe site criteria of an incident wave with a 15-foot height, 13-second period, minimum water depth of 40 feet, inshore trans- mitted wave height of 4 feet, and tidal range of 12 feet. A sequel to the earlier study (Naval Civil Engineering Laboratory, 1971) surveyed concepts for "transportable" breakwaters, including over 60 in the "floating'' category. Although no breakwater system was disclosed which would meet the stringent military site criteria and transportability requirement, these state-of-the-art reviews sparked renewed interest in the floating breakwater for nonmilitary applications. A review of de- velopments in floating breakwaters was summarized by Richey and Nece (1974); Seymour (1974) introduced a new and innovative concept for wave attenuation using a system of tethered floats which may have application over a wide range of wave conditions. Continually increasing pleasure boat ownership has nearly exhausted the available supply of moorage space in many areas. The need for addi- tional moorage space in conjunction with escalating construction costs and more stringent environmental restrictions require careful scrutiny of alternatives to the traditional fixed breakwater and excavation tech- niques employed in marina construction. Productive time in weather- dependent, waterborne activities such as construction, logging, and cargo handling could be increased if protective floating, transportable break- waters were used. Other uses in the control of shoreline erosion and in the emerging mariculture industry may also be found. The information on the performance of floating breakwaters, i.e., their wave attenuating characteristics, mooring line forces, and motions, is contained primarily in reports of laboratory scale. model tests with monochromatic incident waves; the few exceptions are the early analytical work by Carr (1951) and the occasional piece of information from a full- scale test like that performed by Harris (1974). There is a need for a fundamental analytical procedure to predict the performance characteris- tics of floating breakwaters with arbitrary cross section when exposed to a given incident wave. This procedure could be used to systematically compare performance information available in the literature, to examine new design proposals, and either eliminate or reduce and systematize auxiliary experimental studies. The development of the predictive procedure wasthe primary thrust of the project with the concommitant field assessment of a full-scale floating breakwater in operation at Friday Harbor, Washington (Fig. 1). The analytical model developed from the two-dimensional, linearized so- lutions of the hydrodynamical equations formulated in terms of a boun- dary value problem for the velocity potential. The model was refined progressively by comparisons with results already reported in the 1i- terature, by auxiliary laboratory tests, and by the results from the Fri- day Harbor field program, where measurements of incident and transmitted waves, mooring line forces, and acceleration in sway, heave, and roll were measured over a 6-month period. Ke) *Io.eMYeoIq TOqIeYH ABPTIy FO MOTA [eTIIV T oin3sty I] II. THEORETICAL ANALYSIS In the analysis of complex systems such as floating breakwaters, there is a great need for model-scale experiments to predict their per- formance and provide data for the application of rational engineering design principles. Full-scale measurements are also extremely valuable in verifying scaling relationships and in providing confidence that the data obtained from smaller scale experiments are reasonable. When one considers the myriad possible breakwater configurations which have been proposed to date and the different conditions which pre- vail at each potential breakwater site, the number of required model tests and the attendant expense are very large. To avoid this expense ‘and also to permit parametric studies aimed at obtaining optimum break- water configurations, a theoretical model was developed. The goal was to theoretically predict the performance which could be measured in la- boratory studies or at prototype installations. The initial restriction imposed on the theoretical model was to consider only two-dimensional conditions. Under this restriction the breakwater is assumed to be very long in one direction with long-crested waves approaching so that their crests are parallel to the long axis of the breakwater. At most breakwaters where the wave climate results from wind-generated waves, this condition would rarely be approached. How- ever, experiments performed using a boat wake to generate incident waves on the beam and at an angle to a breakwater indicate larger breakwater motions and larger transmitted waves when the incident wave crests approach parallel to the long axis of the breakwater (Stramandi, 1975). As a design tool, a two-dimensional theory provides information on the worst conditions which might be expected to occur. In addition, the extensive two-dimensional wave-channel experiments provide the data need- ed to test the theoretical model. Throughout the development of the theoretical model, every attempt was made to orient the model toward providing a useful tool applicable to realistic problems. To perform the calculations the user need only know the incident wave frequencies of interest, the contour of the breakwater cross section (catamaran- or trimaran-type cross sections are permitted), and the physical properties of the breakwater (these include mass, mass moment of inertia, and the static restoring-force coeffi- cients). The approach used here has been to employ the techniques which naval architects have developed to deal with ship motion problems. Mathema- tically, the hydrodynamic equations are formulated in terms of a- boun- dary value problem for the velocity potential. Solution of this complete problem is presently impossible because the free-surface boundary condi- tion is nonlinear. An approximate solution may be obtained if restric- tions are imposed on the boundary value problem, and the procedure of linearization is applied. The restrictions limit the applicability of 12 the solution to cases of small incident wave amplitude and small motion response of the breakwater. When using the linearized theory which is presented here, one must be well aware of the limits of applicability which are imposed on the results in order to permit the formulation of a tractable mathematical problem. Care must also be exercised because these restrictions may exclude phenomena which occur in nature from appearing in the mathemati- cal analysis. For instance, field observations clearly demonstrate the occurrence of mooring line force oscillations at periods greater than those which could be attributed directly to wind-generated wave exci- tation. Using a linearized approach, these long-period oscillations would not appear in the analysis. A theoretical model which includes nonlinear behavior of the system is required if these long-period os- cillations are to be included. A possible nonlinear mechanism for the transfer of wave energy to lower frequencies has been postulated and is presented to supplement the linear analysis. 1 Linear Theoretical Model. The problems involved in theoretically predicting the performance of a two-dimensional floating breakwater are illustrated in Figure 2. Here an incident wave approaches the breakwater on the beam. A part of the energy contained in the incident wave is reflected, part passes beneath the breakwater, and some is lost through dissipation. Another part of the incident wave energy excites the motions of the breakwater. These motions are restrained by the mooring system. The oscillating break- water in turn generates waves which travel away from the breakwater in the directions of the reflected and transmitted waves. The total trans- mitted wave is the sum of the component which passes beneath the break- water and the components generated by the breakwater motions. The total reflected wave is composed similarly. In completing the calculations, the information which is of most interest to the designer includes: (a) Total transmitted and reflected waves including their components. (b) Wave forces on the breakwater. (c) Motions of the breakwater. (d) Forces on the mooring lines. For the two-dimensional breakwater, definitions for the motions are shown in Figure 2. Sway is defined as the oscillation perpendicular to the long axis, or along the x-coordinate axis. Heave is the vertical 13 (ayia SAVM GSLLINSNVYL atx l *ZoVeEMYeoIg BUTeOTF TBUOTSUSWTPp-OM} VY NOISN3L 3NIT ONINOOW (a)eL ——_—_—_—_ > fn SJAVM G3L93 1334 ctx) ‘Z eIn3ty —<$}—______—_—- SAVM LN3SGIONI (iin 14 motion of the breakwater along the y-coordinate axis, and roll is the rotation about the long axis or the z-coordinate direction. As long as the problem is linear, computing the performance of a floating breakwater may be separated into three parts: (a) Formulate equations of motion, Calculate hydrostatic forces and moments. Evaluate hydrodynamic coefficients in equations of motion. Compute exciting forces on breakwater. Solve for the motions and motion-generated waves. Compute forces in the mooring lines. (b) Solve for the waves diffracted by a rigidly restrained breakwater. (c) Sum components to obtain total reflected and total trans- mitted waves. When combined, these parts of the calculation provide complete performance data for a two-dimensional breakwater. a. Breakwater Motions. In deriving the equations of motion, Newton's law is used. mj 2 = y forces; (1) here’: CHa motion of the breakwater in sway, heave, and roll for i = 1,2,3, respectively. The dot above indicates differen- tiation with respect to time. ta = mass or mass moment of inertia when i = j and zero when i # j. Expanding this equation to include the various forces in the summa- tion yields: ila 3 a, = Fj (inertial) + Ba (wave damping) + Fi (friction) + ie (hydrostatic) + . (mooring) + Bs (wave exciting) The inertial force (or addedamass force) arises when the breakwater acce- lerates, which also accelerates the fluid around it. The motion-gene- rated waves are moving away from the breakwater and result in the wave- damping term. A term representing the forces due to viscosity is in- cluded, but these forces are neglected in the analysis, Experience in ship motion analysis (Salvesen, 1970) has shown this to be acceptable for all motions but roll, where damping may make a more significant con- tribution than for sway and heave motions, At present, thé main reason for neglecting the frictional forces is that they lead to nonlinear terms in the equations of motion, which make their solution far more complex. Hydrostatic forces arise because of changes in the displaced volume of the breakwater when it moves. In this analysis the mooring forces are modeled as simple springs with their contribution to the damping and inertial forces considered small in comparison to similar terms resulting from the breakwater motion. The wave exciting force results from the incident waves striking the breakwater. If we neglect the nonlinear terms and assume that the fluid is in- viscid, then the equations of motion describing the coupled sway, heave, and roll motions of the breakwater are of the form: 3 : {(m.. + ¥45) a, + di a, + (KH, + KM; 5) a } = Ds (t) (2) LOD iie=) lor The symbols are defined as follows: Bae, OF added-mass coefficient with the Hij oH representing the J added-mass force or moment in the jth direction due to acceleration in the ith direction. = damping-force coefficient relating damping force or moment oe in the jth direction to velocity in the ith direction. Mile = = hydrostatic spring constant relating the restoring force or moment in the jth direction to displacement in the ith direction. Bole = similar to KH j but due to the mooring system. F, = exciting force or moment in the jth direction. In order to solve these equations, the physical mass and moment of iner- tia, added mass and damping coefficients, static spring constants, and the exciting forces must all be known. Mass and moment of inertia are computed directly from the specifications of the breakwater section. The KH jj are derived directly from hydrostatic considerations in Appendix A, while approximate values for KM;; are obtained by using a discretized approximation for the mooring line as described in Appendix B. Potential theory and the principle of linear superposition permit derivations for the hydrodynamic coefficients and forcing function Wije ij and F(t). Steady-state solutions of the form: a. (t) = a, sin (wt + 55) LO Vale =yeley Zi, S) (3) are assumed. Substitution of the assumed solution (eq, 3) into the equations of motion (eq. 2) yields a set of linear algebraic equations which may be solved for the unknown amplitudes and phase angles aj and 6,. Transfer functions, Hj, are then defined by the a; and 5; since the incident waves are assumed to be sinusoidal. b. Hydrodynamic Coefficients and Waves. Potential theory is em- ployed in computing the reflected and transmitted waves, hydro- dynamic coefficients and the exciting forces. Under the assumptions of small incident waves, small breakwater motions and an inviscid fluid, the velocity potentials may be found and the problem subdivided using the principle of linear superposition. The total velocity potential: = (i) Nit Crozet ° Oneida °° Claas 7 8 Geeiten HE BS Hetoo OO is the sum of the incident wave potential, the diffracted wave poten- tial and the potential resulting from forced sway, heave, and roll mo- tions. The incident wave potential is well known and may be expressed directly. Obtaining the diffracted wave and breakwater motion poten- tials requires the solution of boundary value problems. These problems and their solutions are described in Appendix C. Appendix D provides the computer program used to calculate breakwater performance. When the velocity potentials have been obtained, the free-surface elevation at any position is found using the linearized free-surface boundary condition: n(x,t) = - = (x,0,t). (5) Here: free-surface elevation measured from stillwater level (y = 0), n(x, t) g acceleration of gravity, derivative of the velocity potential with respect to OX 0,t) time evaluated at y = 0. airente) Mee total Coe) ar g Lees incident (x,0,t) + >. diffracted (GeO), 10)) (1) * %+ motion (x,0,t)}. (6) The fluctuating component of pressure in the fluid and on the breakwater hull surface may be computed using Bernoulli's equation: P(x,y,t) =r p o, (x,y,t). (7) By computing pressures on the hull surface and integrating these around the contour, the forces on the breakwater may be computed. The force per unit length acting on the breakwater is then: F(t) = | Pn ds. (8) C Oo In this case, F(t) = force on the breakwater, aN n = unit interior normal vector on the hull surface, C contour of breakwater cross section. oO The rolling moment is: M(t) = | Prxnds, (9) C (0) where, a r = the vector from the center of gravity to a point on the surface. To compute the exciting forces on the breakwater in linear theory, the pressure due to the incident and diffracted waves is integrated over the hull surface. These forces and moments become: > > bos a eevee (Sola Os Hee rneua| Soe) m GS) oA, 0) { - a Lo O F(t) mdse t incident F(t) = { rxn ds} + k gle) =F &- @ ‘ We ameidene Gol) 7% qattecesaleeV lea ce (0) (10) Hydrodynamic coefficients are found using the potential resulting from forced oscillation of the breakwater. In this case the pressure integrated over the surface has a component in phase with acceleration and a component in phase with velocity, The component in phase with acceleration is normally referred to as the added mass, while the compo- nent in phase with velocity is the damping. The hydrodynamic coefficients shown in this section are derived in greater detail in Appendix C. Eo Mooring Forces. At the time the spring constants for the mooring lines are computed, mooring force coefficients are also calculated. These are: A : : c : : . aa = change in mooring line force per unit displacement in Qa: 1 sway, heave, or roll when i = 1, 2, or 3, respectively. The forces in the mooring lines may then be computed once the motions have been found. Mooring Force = The description of the linear system is now complete. The block diagram in Figure 3 shows the relationships among the calculations which are required. D Nonlinear Theoretical Model. Measurements taken at the Tenakee, Alaska, floating breakwater be- fore this research program was begun indicated the presence of a long- period oscillatory motion of the breakwater. These long-period motions were manifested most clearly in the measured mooring line forces. Look- ing at these, one can visually observe an oscillation with a period of about 60 seconds superimposed over the expected shorter period oscilla- tions. Figure 4 shows the results of a spectral analysis of the seaward mooring line data after a low-pass filter has been applied (the tech- nique for performing the spectral analysis is given in Section III of thaseneport)). The linear theoretical model permits the system to respond only at the frequency of the incident wave. In order to explain the presence of these long-period oscillations, nonlinearities must be included in the analysis. To perform a mathematically complete analysis including all nonlinear effects is beyond the present state of the art. However, in the case of the floating breakwater, one can show that if two incident waves are considered and second-order terms are retained, then an excit- ing force is present at the difference between the frequencies of the incident waves. The complete derivation in Appendix E shows that the nonlinear pressure may be expressed as: -rayemyeorq BUTAeOTF e& JO SATJEJUSSeIdeI WoISAS IBOUTT Toye=! BJAV3H™= | AVMS | =! AQ08 G3X!4 Aa G3SLLINSNVYL SAVM (BAVM LNAGIONI ONV AGO8 G3xI4) SNOILOW Y3S1VM W (SOINVNAG OILOW G AGOS aldIy) (SOINVNAG SNIYOOW ) °¢s ornsTy (4'x) Th SAVM LIN3QGIDNI 20 120000 100000 SMOGTHED AUTGSPECTRAL ESTIMATE (LBxx2-S) 80000 60000 40000 20000 °o 0.01 0.02 0.03 0.04 0.05 0.06 0.07 FREQUENCY (Hz) Figure 4. Filtered low-frequency seaward mooring line force, Tenakee, Alaska (record TK7-23). 2I Zkert 2k,y ki 0 2 2 1 2 2 2 P(t) = - = tw, Ae + W, A, e (k +k)y = 2u WA Are cos [(k, - k,)x (w) = wo)t + 6) - 6, ]}, (11) where, fe) = fluid density, W 1 2W5 = incident wave frequencies, Aj >A, = incident wave amplitudes, 2 2 OF O5 k_,k, = incident wave numbers = ——, ——, Lisi g g 8,555 = incident wave phase angles. Combining this pressure with the pressure obtained from the linear theory and integrating over the hull would provide additional exciting-force terms at zero frequency and at the difference frequency. Carrying the nonlinear exciting-force terms back through the linear response analysis should provide a quasi-linear approach. While there is no reason to expect this to provide exact correlation with measured data, the quasi- linear approach would at least permit the natural phenomena to enter into the mathematical analysis. One would expect terms to appear in the second-order pressure (eq. 11) at twice the incident wave frequency and at the sum of the inci- dent wave frequencies. Terms at twice each of the incident wave fre- quencies can be derived by applying the trigonometric relationships to the terms at zero frequency. While a term at the sum of the incident wave frequencies does not appear in the second-order incident wave po- tential, this term may result when the second-order potentials repre- senting diffraction or forced oscillation in calm water are included. A great deal more effort is required in this area to complete the analysis. There is also one other area where a nonlinear, or quasi- linear, analysis should be investigated. This is in the roll-damping- coefficient. Here, viscous effects seem to be important, and while the problem has not been dealt with within the present study, investigators have included a.term proportional to velocity squared in the equation for roll motion. Si Results. The computer program given in Appendix D has been developed to 22 calculate the values of hydrodynamic coefficients, breakwater motions, and the wave field. Input variables include: (a) The body contour, Co» represented by a series of points on the contour. (b) The physical properties of the body: mass, mass moment of inertia, and position of the center of gravity. (c) The mooring system spring constants. (d) The hydrostatic restoring spring constants. (e) The incident wave frequency, w. In this program the exciting forces and moments appearing in the equa- tions of motion and the fixed-body parts of the transmitted and reflect- ed waves are found by computing the forces, moments, and waves which result when a rigidly fixed body is struck by a sinusoidal incident wave of frequency ». Motions are found by computing the steady-state solu- tion to the three equations of motion. The hydrodynamic coefficients and the waves generated by the body motions are found by computing the forces, moments, and waves which result when the body is forced to os- cillate in stillwater in pure sway, pure heave, or pure roll. The physical properties used in the performance calculations for the various breakwaters are collected in Appendix F. a. Wave Transmission. To assess the performance of a floating breakwater, one quantity which is commonly used is the transmission coefficient. This is simply the transmitted wave amplitude divided by the incident wave amplitude, Inp(x,t) |/|nzx,t)| for monochromatic inci- dent waves. (1) Proposed Oak Harbor Breakwater. At one time the Corps of Engineers was considering a marina and floating breakwater at Oak Har- bor, Washington. Model experiments were carried out by Davidson (1971) ~ to determine transmission characteristics and mooring forces. The break water itself had a catamaran-type cross section. A comparison between the theoretically predicted and experimentally measured transmission coefficient is shown in Figure 5. This figure as well as the others plotted in this section and Section IV were drawn using a CALCOMP plot- ter. The plotting program uses a parabolic fit to determine additional points between the given data. Varying numbers of data points were used to describe each curve depending on its behavior. Data points were closely spaced in regions where the theoretical predictions indicated large changes in curvature. Wavelength is calculated in all the figures using the relationship between wavelength and period for waves in deep water. In this case, the results compare reasonably well except for the 23 *ZJoqemyeeiqg 1oqzey yeOQ pesodoazd 10F JUSTITFFOOS UOTSSTUSUeI] “Ss sinsTy HLONS TSABM/WUSd S°0 v°O €°0 ZO T°O 0 NOILOIGS Yd WOIILSIYOSHL Q (TL61) NOSGIAUG ‘SLNSWSYNSUSW TWWINSWTY3dX3 O) o°0 v°O S}O) 8°0 O°T IN3I3135309 NOISSIWSNUYL 24 predicted dip in transmission just above a B/L (beam/wavelength) of 0.2. There is also some difference at higher B/L ratios. The theory predicts that the part of the transmitted wave which would result where the body is rigidly fixed is almost 1 for a B/L less than 0.1 and drops rapidly at higher B/L ratios to the point where it is of little consequence beyond 0.2. Waves generated by the breakwater mo- tions play an increasing role for B/L ratios above 0.15. Heave motion is the major contributor to the transmitted wave in the very narrow band of B/L between 0.15 and 0.18 with a predicted heave resonance at a B/L of about 0.18. The dip occurs because the waves generated by heave and sway motions are almost 180° out of phase and cancel each other out. At B/L ratios above 0.25, sway motion assumes an increasingly dominant role. Roll motions are small throughout and generate only very small waves. (2) Rectangular Breakwater. A breakwater of rectangular cross section with the same beam and draft as the proposed Oak Harbor breakwater was tested at the University of Washington by Nece and Richey (1972). Results for the water depth of 29.5 feet are shown in Figure 6. Again the agreement is reasonable. Further experiments with this model have confirmed the existence of the trough at a B/L of 0.2. How- ever, this phenomenon can be observed only for very small wave heights. For practical purposes, the dip may be smoothed over considerably. The major discrepancy is at the high B/L ratios where the theory shows con- siderably greater transmission than is actually measured in the model tests. Since the transmitted wave is almost totally a result of sway motion, the problem must lie in the wave predicted by this motion. Over the entire range of wavelengths of interest, the predicted results follow the pattern previously discussed for the proposed Oak Harbor breakwater. The transmitted wave is almost completely a result of fixed-body transmission followed by regions of heave resonance, heave and sway cancellation, and finally, sway wave generation as the B/L increases. It is interesting to note that there is very little difference be- tween the open-well breakwater and the closed rectangle of the same overall dimensions. (3) Rectangular Breakwater Tested by Sutko and Haden, In some recent experiments Sutko and Haden (1974) have examined the effect that restricting breakwater motions has on the transmission coefficient. They used a rectangular breakwater model with a beam-to-draft ratio of 1.5. Plexiglas end assemblies were used to restrict the breakwater motions. Figure 7 shows the transmission coefficient when the breakwater is restricted to sway motions only. Here, the transmitted wave contains a component resulting from the fixed-body transmission and a component 25 G 0 “Zeyemyeorqg Ie[nsue,IeL e& OF JUSTITFF9OD UOTSSTUSUeI] *9 aIN3TY HLONATSAVM/WW39 v0 SO 20 T°0 0 NOTLI1GSud WOITLAYOSHL (ZL61) ASHIIY ONY 3ISN “13 S°6s = H1d3d *SLNSWSYNSUSW TWINAWIYSdX3 G1 20 a0) ou0 8°0 (1 INAI91444509 NOISSIWSNVUL 26 9°0 -ATuo uotzow ABMS 0} Po9zITIZSEL loyemyeorg Ie[nsuej,Ie1 e OF JUSTITJFJEO09 UOTSSTWSUPLI, *2 ean3ty HLONS TSAUM/WHa8 s*0 ¢°0 E°0 Z°0 1°0 0 NOILDIGSYd WOILASYOSHL (¥L61) N3QUH ONY OMLNS “SLNSWSYNSUAW TULNAWTYSdx3 3°0 v°O 8°0 ale | 4IN31I9145509 NOISSIWSNBYL eit resulting from the wave generated by the sway motion. At very low B/L ratios the fixed-body transmission is the more important component. At a B/L of about 0.1 both the fixed-body transmission and the sway-generat- ed wave are of equal importance. At B/L ratios higher than 0.215, the wave generated by sway motion dominates. The agreement between theory and experiment is quite reasonable for this case. A comparison when the breakwater is restricted to heave motion only is shown in Figure 8. ‘There is clearly a discrepancy in this case be- tween measured and theoretically predicted transmission coefficient near the Blip rat toot (Ol S) ay ASmasmatit cxmoruralct em che Eheomypnediicesmamneave resonance in this region which does not seem to be supported by the mea- sured data. In examining the mechanism used to restrain the breakwater motion, it seemed possible that this apparatus was introducing damping into the system. To test this supposition, transmission coefficients were com- puted with the calculated hydrodynamic damping increased by an arbitrary amount. The major effect of increasing the damping was to decrease the transmission near the heave resonance region. With damping at three times the hydrodynamic value, the results were quite close to the experi- mental measurements. Increasing the damping beyond this had very lit- tle additional effect on the predicted transmission coefficient. The scatter which appears in the experimental data in this region is a further indication that some nonrepeatable effect may be influencing the experimental results. So long as the additional damping is included in the theoretical calculations, the results compare well with experi- mental measurements. Figure 9 shows a comparison between model measurements when the model is unrestrained except by a horizontal mooring cable and the theoretically predicted results without mooring restraints. The theo- retical results are characteristic of the rectangular breakwater with the dip in transmission near a B/L equal to 0.2. The pattern of inter- actions between motion-generated waves and fixed-body transmission is similar to the previous description. The agreement between these re- sults indicates that the theoretical model may also yield the correct results when the model is free to heave only. At least further experi- mental investigation is warranted. (4) Alaska-Type Breakwater. The State of Alaska has embarked on an ambitious program for constructing moorages using floating break- waters. As part of a Sea Grant project the University of Washington has been studying the performance of this type breakwater. A theoretically predicted transmission coefficient and the transmission coefficient mea- sured in model tests are shown in Figure 10. The model tests were con- ducted using very small incident waves (wave heights on the order of 0.2 to 0.5 feet at prototype scale). Results for larger wave slopes were not included in the figure but do show the same trends with lower values of transmission coefficient. Theoretical predictions without added damping and with double the hydrodynamic damping are shown in Figure 10. 28 9°0 * ATuO uOTJOW BACSY OF P9ITISOL Jd}eMyYVoIQG Ie[Nsuej,IeI e IOF JYUSTOTFJOOO uUOTSStusueI] °g omn3Ty HLONS T3AYUM/WU3 S*O 7°0 €£°0 Z°0 1656) A 8 a in| NOILOIGSYd WOILSAYOSHL (ONIDWVG DINWVNAGOYGAH S3WIL 33YHL) NOILDIGSAYd WWIILIYOSHL (¥L61) N30BH ONY OMLNS *SINSWSYNSUSW TIWLNSWIYadxX3 °O v°O S20 8°0 O°T 4N319144303 NOISSIWSNBYL CAS) SIO) -JoIeMYeeIg Te[NBuejIeI e IOJ YUOTITFFOOD UOTSSTUSUeT]L °6 SANSTYA HLONS TSAUM/We38 S°0 v°O €°0 Z°O T°O 0 NOILDIGSYd WIILAYOSHL (7461) N3QBH ONY OWNS “SLINSWSAYNSUSW TWLNAWIY3dX3 Z°0 + (eo) 4IN313145303 NOISSIWSNUYL oe fo 2 oO Ort 30 Si i ‘To pow Lozemyesrq odkj-eyseTy Loy JUatoTFZe00 UOTSSTWSUBI], ‘OT emn3Ty HLONSTSAUM/WE3d Wels Ge Ih On 8°O 3°O v°O 3°O O (ONIdWWd JIWWNAGOUGAH JOIML) NOILOIGIUd a eee ‘WOILIYOSHL NOTLOIGAYd in) WOTLIYOSHL (43dO1S SAM TIBWS) SINSWSYNSUAW TWLINSWIYSdx3 {£N319144309 NOISSIWSNUYL 3] Clearly the increased damping makes a significant difference in the results. Some insight into the performance of this breakwater may be gained by following the theoretical results as a function of B/L. At very low B/L the fixed-body transmission dominates. The trough at a B/L of 0.4 comes mainly from the interaction of the roll-generated wave and the fixed-body transmission. For the next peak at B/L of 0.5 the roll- generated wave dominates as the roll resonant frequency is encountered. The next trough at B/L of 0.65 is a result of all three motion-generated waves interacting with the fixed-body transmission making only a rela- tively small contribution to the transmitted wave. The following peak at B/L of 0.7 results from interactions among the motion-generated waves which are of about equal magnitude. At a B/L of 0.86 the heave-generated wave dominates again, but as B/L increases beyond this the effect of heave and roll are rapidly decreasing while sway motion is becoming the dominant wave-generating mechanism. In the region of B/L between 0.4 and 1.0, changing the physical properties of the breakwater can have a marked effect in shifting the peaks and troughs by altering the heave and roll resonant frequencies. Experience with linear ship motion theory has shown that the worst agreement between predicted and measured motions occurs when rolling mo- tions are considered (Salvesen, 1970). This discrepancy is often over- come by arbitrarily increasing the computed roll damping to compensate for the viscous damping which is neglected. As indicated in Figure 10, when damping is added the theory gives a better prediction where roll motion plays a significant role. This places a significant restriction on the theory requiring careful monitoring of predicted roll motion. Where the theory predicts large roll motion, additional damping will be required to obtain results comparable to measurements. Figure 11 shows the predicted fixed-body transmission coefficient and the results of model tests. Agreement is quite close except at B/L of 0.78. The peak in predicted transmission may be due to a resonance of the waves within the well of the catamaran breakwater. There is another peak near B/L of 1.4 indicating the presence of higher harmonic resonances as well. Model tests show at least a slight hump in this region suggesting that the theoretical prediction clearly overestimates the effect of this phenomena, but that this probably is occurring in real life. For the data measured in the field, the transmission coefficient is defined as the square root of the transmitted wave spectral density divided by the incident wave spectral density. Figure 12 shows the transmission coefficient derived from the data obtained at the Tenakee,- Alaska breakwater. The theoretically predicted transmission coefficient with the computed hydrodynamic damping doubled is also shown for com- parison. Details of the technique used in the spectral analysis of the field data may be found in Section III. 32 *Tepou Tozemyeorq odkq-eysely poxty ATpT3tTI e LOZ JUSTITFJOON UOTSstTUSUeT]L ‘TT 9IN3TY HLON3 T3AUM/WY38 Onl 6°O 8°O L°O S}X0) S°O v°O £°O Z°O T°O 0 y SS fe) fo) 3 fo) IN3I91433509 NOISSIWSNEYL NOTLIIGSYd WOILIYOSHL “3 S Get (3d01S JAWM TIBWS) SINSWSYNSLAW TWWINSWTYSdX3 Ol (3d01S 3AUM YSOYNN1) SLNSWSYNSUAW TWLNAWIYadX3 © 33 “rezemyeerq odkj-eyse H1ONS TSAUM/WE38 SG SWE GS GG OG Be SP vk She Orbe GO. SO 70. |]O NG (ONIdWVG DIWYNAGOYGAH J9IML) NOILIIGIYd WITLIYOSHL (SZ-ZNL pLod0a) eysety “9ayeuo |, IV TOF JUSTITZZO909 uotsstusuezy INAWSYNSVAW ( ‘7T 9an3tTy qqsl4 © ‘ O e) oO INS31914453509 NOISSIWSNUYL a fo) Om 34 It should be noted that the model for the Alaska-type breakwater was not built to the correct scale to represent the prototype. Further investigation of the physical properties of the prototype after the model tests were complete revealed that it was heavier than originally predicted. The physical properties used in making the theoretical cal- culations are correct for all the comparisons made in this report. How- ever, care must be exercised in comparing the model test results and the field measurements directly. The physical properties for all the break- waters discussed in this section are in Appendix F. The first trough in the transmission coefficient curve results be- cause the wave generated by roll tends to cancel the fixed-body trans- mission. The sway-generated wave is small but cancels a little bit of the heave-generated wave. The total transmitted wave is then almost in phase with the heave-generated wave at a slightly reduced amplitude, Complex interactions among the components of the transmitted wave con- tinue to result in oscillations of the transmission coefficient up through a B/L of 0.9. At values of B/L above this, the transmitted wave is primarily a result of sway motion except for the peak at B/L equal to 1.4 which results from an increase in the fixed-body transmission. Considering the complexity of the breakwater response, the agreement should be considered to be reasonably good. (5) Friday Harbor Breakwater. The computed transmission coefficient for the Friday Harbor breakwater is shown in Figure 13. As in the case of the Alaska breakwater calculations, the computations of wave-damping coefficients have been arbitrarily doubled to reduce the excessive calculated motions in the region of resonant motions. In this figure the spacing of data points varies. More points are used to spe- cify the curve in regions of rapid change so that the plotted result accurately represents the theoretical prediction. In Figure 13, the first trough in transmission coefficient at about B/L = 0.5 results from heave- and roll-generated waves canceling the fixed-body wave transmission. This transmission coefficient is well be- low the transmission coefficient which would be obtained with the break- water rigidly restrained and only fixed-body transmission waves passing through. As B/L increases, there is a peak at about 0.7. At this point the heave-generated wave has almost vanished, and the fixed-body trans- mission is also small, The larger transmission coefficient is primarily the result of a roll-generated wave with a smaller component resulting from sway motion. The next trough at a B/L of 0.9 occurs as the heave motion-generated wave increases and cancels the roll and sway motion- generated components. The fixed-body transmission is very small at B/L of 0.9. As B/L increases beyond 0.9 the transmitted wave is almost totally the result of sway motion of the breakwater. At larger B/L ratios there are several oscillations in the 35 “rezemyesrIq Loqiey Aeptaiy ‘yuetotzFa09 uotsstusuer, (sutdwep otweuckpoxrpAy 991m2) perdtpead AT [@9TIO LOO YI, HLONZ TSAUM/WeS8 S v € Sj T "$T omn3Ty 1N3191443903 NOISSIWSNUYL 36 transmission coefficient curve. In this region one must be careful of the analysis because there are certain “irregular frequencies" or "John"! frequencies where the approach adopted here breaks down mathematically (John, 1950). These are described with reference to the integral equa- tion technique by Frank (1967). It is extremely difficult to predict where the first of these irregular frequencies will occur when the breakwater cross section is as complicated as the Friday Harbor break- water. If this cross section were rectangular with the same exterior dimensions as the Friday Harbor breakwater, then the first irregular frequency would occur at B/L = 1.7. In practice, one may watch for this mathematical phenomenon by checking the determinant of the matrix invert- ed to solve the system of equations. In fact, this does decrease in the region of B/L of 1.7 but does not indicate a singular matrix for the calculation in this region of B/L. Since this is beyond the frequency range of primary interest, it is best to simply view the results at B/L greater than 1.7 with extreme caution. The oscillations in the trans- mission coefficient in this region of B/L are probably the result of these irregular frequencies. b. Breakwater Motions. In the wave channel experiments perform- ed to date, there has been no attempt to compute the breakwater motions. While the transmission coefficient is the primary measure of breakwater performance, the motions may be very important to the designer, parti- cularly if boats are to be tied to the breakwater. For the theoretical analysis, this is a critical intermediate step where extensive experi- mental measurements used for comparison would be invaluable. Friday Harbor Breakwater. The theoretically predicted motions of the Friday Harbor breakwater are shown in Figures 14, 15, and 16. The motion response is almost the same as one would expect from an uncoupled spring, mass, dashpot linear system. The only unusual behavior is the null response in heave at B/L of about 0.75. This null occurs at a point where there is a phase shift in the "'added-mass" force, a phenomenon which has been observed in experiments with catamaran-type cross sections (Lee, Jones,and Bedel, 1971), and is a result of resonant wave conditions within the open well of the catamaran. Ce Mooring Line Forces. In recent years a great deal of effort has been expended in understanding and predicting mooring line perfor- mance, particularly for moored ships and drillingrigs (e.g., American Society of Civil Engineers, 1971). While many of these analysis techni- ques could be applied to the moorings of floating breakwaters, this has not been done to date. There are also very few model-scale experiments in which mooring forces have been measured and only a few cases where good field data are available. Two techniques for calculating the spring constants for mooring lines have been used. At first the catenary equations were applied to find the change in force per unit displacement. While this approach leads to a fairly simple algorithm for the calculation, there are a few problems. In several cases spring constants were needed when the mooring 37 “loJemyeorq Logiey Aeptiy ‘esuodsexz uotiow Aems (Sutdwep otweudpoipAy 991M2) po tperd AT [eoTIeLOSYL HLONJTSAUM/We38 v € Sj i "yl omnsty © ~ . (eo) (e) {o) SONLI dW SAGM LNSQTONI/SONLIIdWe ABMS ee fo) 2 —_ 38 *Zo1eMyeoIg Toqiey Aeptiy ‘asuodsar uotTiow saeoy (Sutdwep otweuxporpAy 99TM2) po,otpead ATT edTIJOeLOOYL HLONSTIAGM/We38 S v € 3} if "ST omn3ty © = ey oO oO oO SJONLITdWe SAGM INSOISNI/SCGNLI dW SANSH 2 fo S — 39 “LoJemyeotq Toqzey Aeptary ‘asuodsexz uotjow [Tor (Sutdwep stuweukpoxrpAy 991M) po,otpoad ATTeotTIeZOeUL “OT sAN3TY H1ON3TSAUM/We38 S v € Sj if (6) Bu fe) Q re 2 AN — — SONLIIdWY SAUM LNSOISNI/SONLI dW 1108*WbS8 w e N 2 ‘ap) 40 line was too taut to allow it to become tangent to the bottom at the anchor. If this condition occurs, or as it is approached, the catenary equations no longer apply. For many full-scale installations, a combi- nation chain and synthetic line anchor cable is used. This combination anchor cable presents problems in attempting to use the catenary equa- tions. Comparisons between the mooring line forces calculated using the catenary equations to predict spring constants showed poor agreement with measured results (Adee, 1975). While the general trends were re- produced, an increase in the predicted spring constants of about a factor of 4 would have been required to bring the theoretical prediction into agreement with the measured results. To overcome the problems encountered in using the catenary equa- tions, a system based on discretization of the mooring line and static equilibrium was developed. This method is described in Appendix B. (1) Proposed Oak Harbor Breakwater. One of the few model tests in which mooring line forces were measured was performed by David- son (1971) for the floating breakwater proposed for Oak Harbor, Washing- ton. The model configuration with properties scaled to the prototype is included in Appendix F. The shape of this breakwater is basically an inverted bathtub with foam flotation. Applying the theory to predict the mooring line force in the seaward anchor line at a water depth of 29.5 feet, one obtains the results shown in Figure 17. The mooring-force coefficient is defined as the amplitude of the force oscillation divided by incident wave amplitude times the weight per unit length of the breakwater. In this figure, the large range of the experimental results is directly related to incident wave amplitude. The smaller incident wave amplitudes generally produce lower measured mooring line forces per unit amplitude except at the beam to wavelength ratio of 0.49. Since the linear theory is mathematically cor- rect only in the limit as wave amplitude tends to zero, one would expect the best correlation between theoretically predicted and measured results for small amplitude incident waves. The results shown in Figure 17 are consistent with this expectation. However, the very large difference in mooring line forces as incident wave amplitude increases indicates a highly nonlinear response. A potential explanation for the nonlinear response observed in these experiments results from the condition of the mooring lines at the 29.5-foot water depth used for the model tests. Under these conditions, the mooring lines no longer maintain a catenary shape. When the initial tension in the mooring lines is increased to this level, they respond with very large changes in mooring line force for very small displacements of the breakwater. Consequently, small deviations in the planned posi- tioning of the anchors will lead to large changes in forces in the moor- ing line. This condition clearly should be avoided in prototype instal- lations where very large mooring line forces are to be avoided. 4 0 ‘tojeMyeorq Ioqi1eH yeQ pesodoad xox 9dz10F SUT] Butzoow pxemeos “/T oan3ty HLONATIAVM/WW3d v0 €°0 NOILOIG4ud WOIITLIYOSHL (TL6T} NOSOIAKG “1344 S°O (1Z61) NOSGIAUG “1334 € Z°0 T'0 @ LHOISH 3AUM ‘SINAWSYNSUAW TELNSWTy3dX30) LHOISH 3AUM ‘SINAWSYNSUAW TYLNSWTY3dX3O IN319144309 49Y04-ONIYOOW 42 A second possible explanation of the nonlinearity results when the "drift force" on the breakwater is considered. If one carries the hydro- dynamic analysis to second order, there are terms at zero frequency which yield a force on the breakwater in the direction in which the in- cident waves are traveling. This force has the same effect as increas- ing the initial tension in the mooring line and is proportional to wave amplitude squared. Increasing the initial tension tends to increase the spring constants of the mooring lines leading to larger oscillating forces as well. (2) Alaska-Type Breakwater. Mooring-force coefficients ‘theoretically predicted and measured for the Tenakee, Alaska, breakwater are shown in Figure 18. For the field data the mooring-force coefficient is obtained by taking the square root of the mooring-force spectral density divided by the incident wave spectral density and then dividing by the weight per unit length of the breakwater. Again,as with the Oak Harbor model experiments, there is good agreement, especially in predict- ing the peak in the curve near B/L of 0.65. One important aspect of the mooring line problem which should not be overlooked is a comparison between the model-scale results and the field measurements. For the Alaska-type breakwater, all the measured results indicate the amplitude of oscillation in mooring line force is in the order of hundreds of pounds, not thousands of pounds, as was pre- dicted for the Oak Harbor breakwater in the model-scale tests. When the mooring line tension data recorded at Tenakee are plotted as a function of time as in Figure 19, one observes that there clearly are oscillations associated with the incident waves. However, there are also low-frequency oscillations which are of greater magnitude. A com- plete explanation of the origin of these low-frequency forces has not been developed. However, one possible explanation is that these forces are a result of breakwater oscillation at the sway resonant frequency. Since the spring constant for sway motion is very small, one would ex- pect a long natural period. Theoretically predicted sway motion response for the breakwater is plotted in Figure 20. Predicted natural periods are 64, 37, and 29 seconds for tidal conditions of mean lower low water (MLLW), +10 and +20 feet, respectively. By applying a high-pass filter to the field data, one obtains the spectrum of force oscillation shown in Figure 4. Here, a peak is at a period of about 53 seconds (tide height = +7 feet). The predicted sway natural frequency is at 45 seconds when the tide height is +7 feet, which indicates that this explanation is plausible. (3) Friday Harbor Breakwater. The predicted performance of a seaward mooring line on the Friday Harbor breakwater is shown in Figure 21 for a tide height of +5.33 feet. The Friday Harbor mooring lines are different than those at the other breakwaters. They are composed of a section of chain attached to the breakwater, followed by a length of nylon rope and, finally, another section of chains at the bottom. This particular tidal condition was chosen because it is the condition during record FH 7-8 used later for comparison. 43 “(€Z-LNL Pt0sez) BYSeTy “seyeue] ‘USTITFJ9OD 9ILOF-BUTLOOW BUTT SUTIOOW pieMesS “gT oandTy HLON3 TSAUM/WY38 SiG O°% ST O°l S*O 0 a ee / ys Feces] NOILOIGSYd TWOILSYOFHL | SOO Oro Ww a) oO 020 IN3T91433509 39Y0I-ONTYOOW 44 BYseTy ‘seyeuay, “(€Z-LNL pr0902) “SOTIOS aT popzosey "61 ean3sty (S) SWIL 1139 GVO1 GYVM3YOHS OOl- 001 (91) JOYOS 45 ATIDE HEIGHT = +20.0 FEET @TIDE HEIGHT = +10.0 FEET MOTIDE HEIGHT = 0.0 FEET (MLLW) 16 14 — N —_— j=) oO) os SWAY AMPLITUDE/INCIDENT WAVE AMPLITUDE 0 0 85 1S 65 35) 45 35 25 PERIGD (S) Figure 20. Theoretically predicted long-period sway response of Alaska-type breakwater, Tenakee, Alaska. 46 @zS *Joqemyeoiq Ioqie}] Aepts4 ‘JUOTOTFFIOO 9ILOF-BUTLOOW OUT] BuUTIOOW paieMess po .dTpeid ATT eITeIOSUL HLONS TSAUM/We3S8 Say, O°v srs One S°3 0°23 SP Ib OP (! “IZ oansty 5}0) O O yoo 900 80°0 oro ol O vlO 910 IN3I9145509 3DYOS-ONINOOW 47 Low-frequency predicted sway motion and resonance are shown in Figure 22 for MLLW, +5.33 feet, +10 feet, and +15 feet tide heights. +TIDE HEIGHT = +15.0 FEET ATIDE HEIGHT = +10.0 FEET OTIDE HEIGHT +533 FEET MTIDE HEIGHT = 0.0 FEET (MLLW) 16 14 — N — (en) ep) SWAY AMPLITUDE/INCIDENT WAVE AMPLITUDE oo 35 30 25 20 1S 10 PERIOD (S) Figure 22. Theoretically predicted long-period sway response, Friday Harbor breakwater. 48 If1. FIELD DATA Ie Layout. The site of the floating breakwater instrumented in this study is located at Friday Harbor, Washington, on San Juan Island, just east of Victoria, British Columbia (Fig. 23). The breakwater is 25 feet wide, 904 feet long, anchored in approximately 40 feet of water, and was in- . Stalled in October 1972. The structure is made of Polyolefin flotation tanks linked together by a matrix of large wooden timbers. It is laid out in an expanded L-shape, the inside angle being 115°, with the short- er leg (227 ft.) directed toward shore and the longer leg (627 ft.) toward magnetic north. Themsmtewintesel f mS iprocected ony threemsides by San Juan and Brown Islands off the harbor entrance. This leaves an 0.25-mile-wide channel into the harbor with a northeasterly fetch of about 1.7 nautical miles. Southeasterly winds can also generate waves of importance parallel to the shorter leg where the fetch is about 1 nautical mile. De Instrumentation. The shorter leg was instrumented in this study for two reasons: (a) the most frequent winds are out of the southeast, and (b) barges were to be tied to the longer leg during the winter months for added protec- tion. However, the wave gages are positioned to give the proper inci- dent and transmitted wave data for all relative wind directions (Figs. 24 and 25). Four types of time-dependent data which are basic to describing the response of the breakwater were collected: (a) wind velocity and direction; (b) wave heights at key locations; (c) anchor cable forces; and (d) directional acceleration and angular motions of the breakwater. The locations of the measuring sensors are shown on Figure 25. Signals from the sensors were Carried by underwater cable to the recording system which was located in a small building mounted near the center of the short leg. So Wind Data. Windspeed and direction were measured by Weather Measure Corpora- tion's W121 sensor. Some additional circuitry was required to record the windspeed, and the sensor was recalibrated to this circuit. The sensor was mounted on the breakwater at the intersection of the two legs at 20 feet above the water surface. an Waves. 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Two spar buoys instrumented to measure wave elevation were located outboard of the breakwater and positioned so that one measured the incident wave field, and the other measured the incident plus reflected wave field. Two stationary gages were attached to pilings behind the breakwater to measure transmitted wave height. All four gages were of the resistance type. The spar buoys were used outside the breakwater to help reduce navigation hazards and because of the costs and logistics of placing stationary piling at these locations. The buoys were made of two sections of PVC pipe, the lower section being 6 inches in diameter and 15 feet long, and the upper section of 3-inch diameter and 12 feet in length with the upper 8 feet wound with a resistance wire. Four feet were exposed above the water surface, and a 2.5-foot-diameter disc was attached to the bottom to damp vertical motions. The natural periods in heave and roll, respectively, are 18 and 14 seconds, well above the anticipated maximum wave period of about 4 seconds. See Appendix J for a complete description of the wave staff and buoy designs. By Cable Forces. Anchor cable forces were measured using a bonded strain gage-type load cell that was placed in the anchor chains beneath the water surface. These cells and the associated electronics were designed and built for this project. They have an overall system accuracy of 0,75 percent of the designed or rated total load cell capacity over a temperature range of 10° Celsius (design load 12,500 pounds), These load cells employ a four- arm wheatstone bridge circuit which has two strain gages in each leg of the bridge and are self-temperature compensating. The units are O-ring sealed and wired directly to the bridge amplifier circuitry mounted in the recording package. Oi Motion Package. Breakwater accelerations were measured using three Kistler servo- accelerometers (Model 303T). One accelerometer, oriented horizontally, was mounted at the center of the breakwater to measure the sway accelera- tion. The other two were oriented vertically and mounted at opposite outboard edges of the breakwater to measure the vertical accelerations. The heave acceleration was obtained by taking the average of the signals from the two outboard accelerometers; the roll acceleration was obtained by taking the difference of these two signals and dividing by the distance between them. The accelerometer locations are indicated in Figure 25. 7. Data Acquisition System. The data recording and electronic package was built around the Sea 53 Data Corporation's Series 610 four-track incremental digital cassette tape recorder. The complete package, which included all the electronic circuitry for the individual transducers plus the tape recorder, was housed in a watertight, 6-inch-diameter PVC cylinder 5 feet in length. The system was designed to be operated manually or in a completely automated mode, thus requiring only periodic tape changes (Fig. 26). In its automatic mode, the system was activated when the windspeed reached or exceeded a preset value and stayed there for at least 1 min- ute. At this point, a single 17-minute sample of all the inputs was taken. Each 68 minutes following this, another 17-minute sample was recorded if the wind was still above its preset value; if not, the system was shutdown until the windspeed increased. Each 17-minute record con- sisted of 2,048 samples, taken at 0.5-second intervals, of all 13 chan- nels plus a clock channel. Twenty-five of these records could be record- ed on a single cassette tape. 8. Data Processing and Analysis. The initial step in the data handling was to transfer the data from the individual cassettes to seven-track magnetic tape by means of the Sea Data reader. These tapes were then converted to a computer compati- ble format on the University of Washington's CDC 6400 computer. The histograms for all records plus the basic statistics, i.e., the minimum, maximum, mean values and standard deviations as well as the transmission coefficients based on these standard deviations, were then computed and tabulated (App. G). A digital filter, with a cutoff frequency of 0.05 hertz (Gold and Radar, 1969) was applied to the transmitted wave data prior to these tabulations to remove tidal drift. The transmission coefficients given in these summary sheets are a ratio of the standard deviations for the transmitted and the incident wave gages. In the initial conversion, the data were checked for reader errors. These points were smoothed using a linear interpolation between the pre- ceding and the following good data points. Following this, the data were checked for extreme values. Data points departing from the mean by more than five standard deviations were smoothed in the same manner as were the reader errors. In no case did the number of errors warrant elimina- tion of a complete record (greater than six bad points). Record FH 11-1, however, had bad data for channels 3, 4, 5, 7, and 8. This record was run manually while calibrations were being made, and the affected chan- nels were not connected properly at this time. The final edited data were then stored on magnetic tape. The autospectra for all the wave data for all records were computed with a more complete analysis of the force and acceleration data applied to the more desirable events. Digital filtering techniques were used prior to spectral analysis on all the wave and force data. The procedures used follow those given 54 Initiate data record Data Stream Pf onl A777 Ch. 1 LIZLZZLN LLL, Wind- Amplifier and speed Bias Circuit IN Voltage to freq. converter 0-10 volts input serial to Che ; Wind Amplifier and Voltage to freq. » ae | | 4-bit par- converter 0-10 aH 4+ allel con- volts input ae version we | D Che 3 PA | y N.W. Bridge & Ampli Voltage to freq. o g Load Cell fier Circuit COMES 0-10 ; B Z volts input q Hy Ch. 4 7 Cassette S.W. " " @ Transport (5200 ea) Ch. 5 N.E. Pe > Load Cell Ch. 6 Load Cell Trans .Wave Staff #1- Staff #2 Ch. 7 Y Amplifier and Voltage to freq.| |4u.9 Bias Circuit converter 0-10 iment Ch. 8 / vo inp =o 2) o) hh = i‘) < o) Reset Counters Load Shift Registers oltage to freq. converter Q-10 Figure 26. Instrumentation and recording package layout. 55 by Gold and Radar (1969), The first step in the development of this filter function is to assume an ideal filter response function, 1, OS [eel se co = { ONE [sell so EE Cc = (12) D5 Os [sls x. n= { Link Sel se where f£, is the cutoff frequency, fy the Nyquist frequency, and F,(f) and F (4) are the ideal low-pass and high-pass filter response functions. The ideal filter response function is then Fourier-transformed to the time domain, giving the impulse response function, which is trun- cated by using an appropriate window function and transformed back to the frequency domain giving a complex frequency response function. The number of points used in representing the filter function is allowed to vary and the resulting convolution with the original time series is accomplished by using the overlap-add method of convolving smaller ser- ies with larger ones. This allows for more economical filtering proce- dures . This gives three variables to choose from in the final filter func- tion design: the length or number of points used in the filter, the type of window used to truncate the impulse response function,and the number of points to be truncated. This procedure is analogous to spectral estimation techniques except for the truncation of the impulse response function. The larger the number of points used in the filter function, the better the estimate. The smoother the window function, the broader the transition band. In addition, the ripple or Gibb's phenomena is reduced. Generally speaking, the more points that are truncated (set to zero) the better the result- ing approximation. In practice, the actual number is determined experi- mentally by comparing results for different truncation values. This results in setting approximately 20 percent of the impulse response func- tion to zero. 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LG6°E€ S5S°8T L16°T2 206° 92 002°%T MN SUOTIETADP pLepueIs “OT qe L ~€THS -E€THS -€THS -€TH4 -€THS -€TH4 -€THS -€THd @T-eTHs dTt-eTtHs 9T=2TH4 STt-2TH4 5T-2THS €t-eTHd 2T-2TH4 Tl=2eTHd OT-2THJ -eTH4 -eTHS -2eTH4 -2TH4 -eTH4 -2THd @cTHd ~-2THd =-2THd 9T=-TTH4 €T=-TTHI 2T-TTHd Tt-TTHd OT-TTHI -TTHS -ITHd -TTH4 -TTH4 -TTH4 -TTH4J -ITHd -ITHd -OTH4 -OTHI -0THS -OTHI Se@- 6Hd4 4e@= 6H4 €e- 6Hd @é- 6H4 ANMETNH OP © ANAT NOP OO ANMMITNMETHOKF OO 62 outside leading the inside gage by approximately 180° over the fre- quency range of 0.25 to 0.37 hertz. This indicates that the forces are relatively uninfluenced by waves above approximately 0.37 hertz. This frequency range is also where the transmission curves rise to near unity. This agrees with the low-frequency analysis and suggests that the response is similar over the complete frequency range below 0.37 hiewtz)- The acceleration force, autospectral and cross-spectral analysis results, are also given in Appendix J for the higher frequency range for record FH 7-8. No dominant features were observed in the motion spectra. Their peak values and spread of energy with frequency appear to follow the general character of the incident wave spectra in all records ana- lyzed. This implies that any natural frequencies in each of the motions is outside the range of significant incident wave energy. The cross- spectral analysis shows a high coherency and zero phase shift between the heave and roll accelerations. In both the sway and roll, and the sway and heave accelerations, the sway acceleration leads by approxima- tely 180° over the range of significant incident wave energy and then tapers to near-zero phase shift at higher frequencies. Also, the cohe- rency is high enough over the incident wave energy band to imply near linearity between all three motions. These conclusions are based on positive sway being outward from the short leg (south), heave positive up, and the positive roll to be clock- wise around a positive axis pointing westerly toward shore, 63 IV. COMPARISON OF THEORY WITH FIELD DATA FOR FRIDAY HARBOR BREAKWATER Although the Friday Harbor breakwater has a very complex geometry and does not respond as a rigid body to the incident wave excitation, it is important to draw some comparisons between the theoretical prediction of performance and the field measurements. In seeking a "'typical event"! from the enormous quantity of data gathered, the goal was to find a case where the wind was reasonably close to being on the beam of the short leg of the breakwater. The one striking item which emerges from the data is the similarity of all the transmission coefficients examined. These curves seem iden- tical no matter what the wind direction. This was not expected because there were barges tied to the breakwater along the entire long leg, while there were none along the shorter leg. A further investigation of the reasons for the similarity is certainly warranted. The record selected for comparison with the theory was FH 7-8. Figure G-3 in Appendix G shows the incident and transmitted wave spectra and transmission coefficient. This record is also listed in the statis- tical summaries of Appendix F. The spectral analysis using a high-pass filter was performed as described in Section III. A comparison of the theoretically predicted and measured trans- mission coefficient is shown in Figure 28. So long as the calculated hydrodynamic damping is doubled in the theoretical analysis, the results are quite good. As described in Section II, the peak in the transmission curve at a frequency of 0.95 hertz probably results from the "irregular frequency" phenomenon which occurs in this mathematical formulation. Comparisons of sway, heave, and roll acceleration predictions with measurements are shown in Figures 29, 30, and 31, respectively. Here, the acceleration response has been nondimensionalized by multiplying by the beam or beam squared, as appropriate, and dividing by the accelera- tion of gravity times the incident wave amplitude. In the case of sway acceleration, the theory overpredicts the values throughout the entire frequency range. The peak at 0.5 hertz appears in the correct location, but the measured values would need to be doubled to bring the curves into better agreement. For heave acceleration the curves appear to be in closer agreement, at least above the frequency of 0.4 hertz. Below 0.4 hertz there seems to be little correlation. Roll acceleration seems to show the worst agreement of all. Here again, the predicted accelerations are considerably higher than the mea- sured values. There are several possible explanations for the discrepancy between predicted and measured accelerations. In the field, even if the wind 64 “(8-ZHd P0901) ToemyYeerq Ioqzey Aeptay TOF JUETITZZ009 uOoTSSTWSUBI], ‘gz oanstTy (ZH) AINANDSYS 6°0 8°O LO 9°O SO v°O €°O 3°O LO) iy (Sutdwep ptueuXkporpAy 99TM2) @NOILOIGIYd IWOILAYOAHL LNAWSYNSVEW G1TSIa—> \ 6°O IN3I9144309 NOISSIWSNUYL 65 °(8-ZHA prosez) re1emyesriq Ioqzey Aeptiy oF osuodsexz uotieLoTeooe ABMS ‘67 9AN3TY (ZH) AINSNDSYS (OP Us 6°O 8°0 IO) 9°0 S°O v°O €°O GO) 70) O LNAWSYNSVAW Gals | SINVNAGONGAH 3OIML) NOILOIGAYd \ WOILSYOSHL a \ wo fo) NOILBYSTSIIE ABMS TBNOISNSWIONON (BLI*O/A*9 J | | \ | ca 66 *(8-ZHd Poder) Iajyemyeoriq Toqiey Aeptiy Loz osuodsexr uotieLeTeI0e sAVSH “OE SANITY (ZH} AQNSNDAYS On 6°O 8°O (50) (0) S)() v°O €°O 2°O 1°O ie eS LNAWSYNSVIAW vi \ (ONIDWVG SINVNAGOYGAH JSOIML) NOILOIGSYd IWOILAYOSHL NOILW¥Y31S998 SAUSH TIWNOISNSWICNON (U1L3*9/H*9 J 67 *(8-ZHA Ploser to9zemMyeetq Ioqiey Aeptiy IoF asuodsor Worscasceses ee (ZH) ASNENDSY4 6°0 8°0 LO) 3°0 76) ¥°O E°0 BO) IX LINSW3YNSV3W A71al4 (ONIGWVG DINVNAGOYGAH SOIML) NOILOIGSYd WOILSAYOSHL “Tg ean3sty ae) Y S ae S) ie om m NI NC =~ NOT LBY3TS93N 1104 TWNOISNSWIONON (ULA¥*D/ USER J 2 + 68 were blowing directly on the beam of the breakwater, one would not find the condition of long-crested waves impinging directly on the beam of the breakwater. As a result, the breakwater is not excited uniformly along its entire length. Therefore, the breakwater itself provides restraint against motions which are excited in a local area. The con- struction of this particular breakwater is also quite flexible, which allows for considerable internal damping of the wave-excited motions. The barges tied to the long leg also serve to restrain the motion and provide additional damping. There is a strong need, in this case, to provide laboratory data on the breakwater motions, which could be further correlated with the theory and the measured motions. If one looks at the measured accelerations by themselves, a consi- derable resemblance in all three degrees of freedom appears. Further, if these accelerations are viewed along with the incident wave spectrum, considerable similarity appears again, suggesting that further investi- gation of the measurement scheme would also be welcome. The final comparison to be made is between the theoretically pre- dicted and measured mooring-force coefficient. The theoretical predic- tion and measured data for the seaward mooring line is shown in Figure 32. The correlation appears to be quite good in this case. In looking at the time series of force on the mooring lines and the windspeed, one can observe a definite correlation between the wind gusts and increases in the mooring force. This is probably a result of the large barges tied to the structure which act almost as sails. If this is the case, the increase in tension caused by the mean wind on the barges needs to be accounted for. No attempt has been made to do this. The most common method of presenting the spectral data obtained in the field uses a frequency scale rather than the nondimensional beam/ wavelength scale used in Section II. In this section the comparisons are made using a frequency scale. For the Friday Harbor breakwater (beam = 25 feet) the conversion is: 2 Beret 2 La a7es L g assuming deepwater waves. 69 ’ (g-ZHa pi09ez) IajemMyeerq 1oqzey Aepl4iy ‘USTITFJ9OD 9OLOF-BuUTLOOW SUTT BuTLOOW preMeeS “7g 9IN3TY (ZH) AINANDSAYS Ont 8°0 9°0 v°O ZAn 0) (e) INSNSaUNSVAN aldid fa 900 800 Olo re) LIN3IdI1343509 3DYOI- ONIYOOW vlO 91°0 TO V. CONCLUSIONS Results for the predicted transmission coefficients were in good agreement with laboratory and field data, and they showed how the in- fluence of fixed-body transmission, and of sway, heave, and roll motions on the transmission coefficient changed with increasing values of the beam to wavelength ratio. The curves predicting the mooring line forces as a function of the beam to wavelength ratio (or of incident wave frequency) followed those for the measured responses. Care must be exercised in the analysis of mooring line forces because there is strong evidence of nonlinear be- havior. An extreme storm event did not occur during the sampling season at Friday Harbor, nor during two winter sampling periods on the Alaskan breakwaters; however, the anchor forces measured were about an order of Magnitude less than anticipated. The barges tied to the long leg of the breakwater did not noticeably affect the transmission coefficients above a frequency of about 0.3 hertz, since the curves for all incident directions were approximately coincident above that mean frequency. Below the frequency of 0.3 hertz, it appears that the barges may have reduced the transmitted energy somewhat. The extension of the theoretical model to include second-order terms showed the presence of additional exciting-force terms at zero frequency and at the difference frequency of the incident waves. Additional work on the basic theoretical model is needed to incorporate these terms into the calculations for mooring forces. The most appropriate means of veri- fying the role of the second-order terms may be in a model basin, where breakwaters of simple cross section and incident wave spectra having only two or three components could be employed under controlled condi- tions. va} LITERATURE CITED ADEE, B.H., "Analysis of Floating Breakwater Mooring Forces," Ocean Engtneertng Mechanics, American Society of Mechanical Engineers, New York, 1975. AMERICAN SOCIETY OF CIVIL ENGINEERS, "Berthing and Mooring Ships," Pro- ceedings of a Nato Advanced Study Institute, Lisbon, Portugal, July 1965. CARR, J.H., "Mobile Breakwaters,'' Proceedings of the Second Conference on Coastal Engineering, Nov. 1951, pp. 281-295. CHRISTENSEN, D.R., and RICHEY, E.P., "Prototype Performance Characteris- tics of a Floating Breakwater," Marine Technical Report Series Number 24, 1974 Floating Breakwater Conference Papers, University of Rhode Island, Kingston, R.I., Apr. 1974. DAVIDSON, D.D., ''Wave Transmission and Mooring Force Tests of Floating Breakwater, Oak Harbor, Washington,'' Technical Report H-71-3, U.S. Army Engineer Waterways Experiment Station, Vicksburg, Miss., Apr. 1971. FRANK, W., "Oscillations of Cylinders in or Below the Free Surface of Deep Fluids,'' Report 2375, Naval Ship Research and Development Center, Bethesda, Md., Oct. 1967. GOLD, B., and RADAR, C.M., Digital Processing of Stgnals, McGraw-Hill, New York, 1969. HARRIS, A.J., 'The Harris Floating Breakwater,'' Marine Technical Report Series Number 24, 1974 Floating Breakwater Conference Papers, University of Rhode Island, Kingston, R.I., Apr. 1974. JOHN, F., "On the Motion of Floating Bodies," Commmtcattons on Pure and Applted Mathematics, Vol. 3, 1950. LEE, C.M., JONES, H., and BEDEL, J.W., "Added Mass and Damping Coeffi- cients of Heaving Twin Cylinders in a Free Surface,'' Report 3695, Naval Ship Research and Development Center, Bethesda, Md., Aug. 1971. LOCHNER, R., FABER, O., and PENNY, W., "The Bombardon Floating Break-ater water ,'' The Ctvtl Engineer in War, The Institution of Civil Engineers, Vol. 2, London, 1948, p. 256. MINIKIN, R.R., "Floating and Foundationless Breakwaters,'' Engineering, Dec., 1948, pp. 557-579. NECE, R.E., and RICHEY, E.P., "Wave Transmission Tests on Floating Break- water for Oak Harbor, Washington,"' Technical Report No. 32, University of Washington, Charles W. Harris Hydraulics Laboratory, Seattle, Wash., Apr. 1972. v2 SALVESEN, N., TUCK, E., and FALTINSEN, O., ''Ship Motions and Sea Loads," Transactions of the Soectety of Naval Architects and Marine Engineers, Vol. 78, 1970, pp. 250-257. STRAMANDI, N., ''Transmission Response of Floating Breakwaters to Ship Waves,'' Masters Thesis, University of Washington, Seattle, Wash., 1975S. SUTKO, A.A., and HADEN, E.L., "The Effect of Surge, Heave and Pitch on the Performance of a Floating Breakwater,'' Marine Technical Report Series Number 24, 1974 Floating Breakwater Conference Papers, University of Rhode Island, Kwngston, R.I., Apr. 1974. NAVAL CIVIL ENGINEERING LABORATORY, ''Mobile Piers and Breakwaters - An Exploratory Study of Existing Concepts,'' Technical Report 127, Port Hueneme, Calif., Apr. 1961. NAVAL CIVIL ENGINEERING LABORATORY, ''Transportable Breakwaters - A Survey of Concepts,'' Technical Report R-727, Port Hueneme, Calif., May 1971. 73 APPENDIX A HYDROSTATIC RESTORING FORCES AND SPRING CONSTANTS Hydrostatic restoring forces and spring constants are computed for the two-dimensional analysis under the following assumptions: (a) The body rotates about the origin of the coordinate system and all forces and moments are computed about that point. (b) The body has vertical sides in the region of its waterplane. (c) All motions are small. le Sway Motion. In the horizontal plane the body is in neutral equilibrium. There- fore, there are no hydrostatic restoring forces and KH = KH = KH =e (A-1) 2, Heave Motion. Vertical displacement of the body results in a change in the buoy- ant volume of the body-and consequently a change in the buoyant force on the body. Since this force must be perpendicular to the waterline, there is no change in the horizontal force as a result of vertical dis- placement and KH,, = 0. (A-2) If one considers a small vertical displacement, éy, there is a resulting change in volume: 6éV = - éyA, (for dy + upwards). Here, A is the waterplane area. The vertical force then is: F = KH, 5 Sy = - pga oy, or KH, 5 = pga = eg lx, - x]. (A-3) In this equation xg and x, denote the sides of the body as shown in the Figure in this appendix. Since the vertical force may be regarded as act- ing at the centroid of the waterplane area, x,, the moment may be expressed. 74 ‘gueTdx0jeM OY) IveU ToIeMYeoIG BUTIeOTF FO DTIeWeYyoS T[-y °omNsTY SJNITYSLUM TVNIJ ¢ JNM JWMOA / G3SY3WWI A 7s AJWNIOA G39YN3WS NOILWLOY Boe Tes) M = K,,6y = - pga x. oY: Substituting for Xo and A, yields; KH 1 2 = = = - _ = — - A-4 93 ogA Xx ogA 2 [x + x | 2 pg [x 2 ] . ( ) So Roll Motion. The analysis of roll motion-induced forces and moments is compli- cated by the fact that the body is assumed to rotate about the origin of the coordinate system and not the centroid of the waterplane. The problem is illustrated in the figure. Here, line 2, the water- line after rotation through and angle 60 must pass through the inter- section of the y' coordinate axis and the initial waterline. Equations for lines 1 and 2 may then be obtained. bine Ibe sy Se Co Line 2: y = mx + Db. The slope of line 2 is: m = Be 2 = tan 60. Ax Line 2 must also pass through the point P so that: X = +c tan 66 and lel Oe These equations yield the relationship: b = c(1 + earn 88). To find the force acting on the body as a result of the rotation, the net lost or gained volume is needed. % 6V = | (mx + b - c) dx Xx a Xp D | [(- x tan 66 + c(l + tan™66) - c] dx 5K a 76 Thine tx, 2 By applying the "small angle" approximation and neglecting terms of the order of 602. Then, 2 2 aS |] tan 60 + cx, - x] tan’ 66. ine 2 2 Vv * 5 [x, =e ] 66 and the force is: sol 2 2 F 5 eglx, =r ] 60. The x and y components of the force are: it 2 = 2 =F COSIIO8 1% > eglx, = 2s ]5@ cos 68 ie 9] iH] and : Fe | 2 2 : By = F sin 60 5 pg[x, cn ]66 sin 66. Again applying the small angle approximation one finds: zed: 2 D FE 5 pglx, UX, 160 and ig 8 O, The hydrostatic spring constants coupling roll to sway and heave are then: sO (A-5) and iA 2 2 gn = SES TS aE oo To obtain the moment induced by roll motion compute: Tiler? 2 Sleeve: - (Ss x, tan 59) (= x,) eo 66, Moment of Gained Volume Moment of Lost Volume * Xb (Ne) wire and Moment of Original Volume = WY, 68. In this formula, W = weight per unit length (AG and ny = distance the center of buoyancy is below the center of gravity. The total moment then is: SAO nS, dh eags M 3 (x) Xo )66 + WY, 66, and the spring constant becomes: ees S 3 cs KH, me 04, Xa ) + WY: (A-7) | Expressed in traditional naval architecture terminology, this reduces to: = -8 KH, WGM, (A-8) where GM = metacentric height. 4. Collected Results. Shi 2 hia S hig @ Man 7 Mon = © bg pelx, is sa Ming > Llgp = pglx,” ; yd a! KH. = oa ie,” = xX ay + WY, 78 APPENDIX B MOORING ANALYSIS I Purpose of the Program. Computer program BRKMOOR computes the forces and moments imparted by a pair of mooring cables on a floating breakwater section. BRKMOOR also computes the changes in the mooring cable tensions and the spring- constant values for the moorings as the breakwater moves in sway, heave, or roll. Z. Program Description. Prograin BRKMOOR is written primarily in FORTRAN IV although FORTRAN II print statements are used. The program consists of the main program BRKMOOR and the subrou- tines LINE2, CHAIN, NYLON, EQULIB, SPRING, and LTERPS. BRKMOOR calculates the forces in a mooring cable by using a discre- tized approximation to the cable. The cable is divided into the number of segments specified in the input data. Each segment may be of a diff- erent material or size. Each segment is in turn divided into a specified number of sections. The cable is considered to be made of these sections with the weight of each section concentrated at the node at the bottom of the section. Connecting each node is a straight but elastic section. The main part of the program specifies 15 different angles at the attachment, ranging from nearly vertical to nearly straight to the farthest reasonable anchor position. A first guess at a top tension is made. LINE2 then sums down the cable computing forces and coordinates of each node starting with the initial angle and initial tension. The po- sition of the end of the cable is compared with the specified water depth at the anchor. The initial tension is adjusted and the summation repeated until the cable ends at the proper depth. Control then returns to the main program. LINE2 calls the subroutines NYLON or CHAIN to compute the strain of the cable section of the appropriate material. If other materials are used new subroutines should be written for strain computation, along with the appropriate calling expression in LINE2. At each angle the cable forces at the attachment and the anchor position are stored in arrays. EQULIB then computes the breakwater equilibrium position for the specified conditions. SPRING is called by EQULIB. SPRING computes the change in mooring 79 cable tensions with breakwater displacement in sway, heave, and roll and the spring constants of the moorings on the breakwater. LTERPS is a linear interpolation subroutine which computes the slop, AY | and the interpolated value of Y for a given X and an array of X vs Y values. LTERPS is called by EQULIB and SPRING. 3. Type of Computer and Peripherals. BRKMOOR was written for use on the CDC 6400 computer. It uses about 40,000g words of memory. No peripherals other than the card reader and line printer are required. 4. Input Data. The input to BRKMOOR is as follows: Card #1 - Title card, Format (8A10). 80 alphanumeric characters max. Card #2 - Breakwater geometry card, Format (5F10.0). YCG = Vertical location of breakwater CG relative to water surface. XCAB(1) = x coordinate of cable #1 attachment to break- water (the CG is at X = 0 and cable #1 is de- fined as the cable with its anchor in the +x direction). y coordinate of cable #1 attachment to breakwater. x coordinate of cable #2 attachment to breakwater. YCAB(2) = y coordinate of cable #1 attachment to breakwater. Card #3 - Number of desired conditions Format (12). (Also number of condition cards to follow) Card #4 - Condition cards, Format (4F10.0). (One card for each condition) FEXT = Force applied to the breakwater not due to moor- ings in x direction (could be due to wave action, tide, wind, etc. force in pounds). SEP = Anchor separation in horizontal direction (feet). TENS1 = Nominal tension in cable #1 (lb.). TENS2 = Nominal tension in cable #2 (1b.). It should be noted that only the following condition combinations are possible: SEP SEP+FEXT TENS1 TENS1+FEXT TENS2 TENS 2+FEXT TENS1+TENS 2 YCAB (1) XCAB (2) 80 Card #5 - Tide Card,Format (11,9X,5F10,0). NTIDE = Number of tide values to follow (max = 5), TIDE = Tide position in feet relative to that at which the anchor depths are given. Card #6 - Cable #1 Parameters, Format (12,8X,2F10.0). NSEG = Number of different segments (types of cable ma- terials) from which the cable is constructed. DEPTH = Depth of water at the anchor (feet). BSLOPE = Slope of bottom in region of anchor (feet/feet). Card #7 - Cable segment properties Format (15,5X,2F10.0,A10,F10.0), One card for each of the number of segments listed in card 6 parameter NSEG. NSECT = Number of sections into which it is desired to divide the cable segment. ALSEG = The length of this cable segment. WPF = Weight per foot in water of the cable material in this segment. MATL = Material name (as the program now stands this must be CHAIN or NYLON (Name must begin in column SIL) 3 DIAM = Diameter of the nylon rope or of the chain link in inches. Card #8 and #9 - Same as cards #6 and #7 only as applies to cable, #2. Table B-1 illustrates the input cards for a test case. All the read statements for the program are in the main program along with com- ments and explanations of input requirements. 5. Mathematical Procedures and Program Limitations. The basic cable computations which take place in LINE2 require some explanation. As was stated previously, the weight of each cable section is cozsidered to be concentrated at the bottom of the section. In order to find the shape of the cable, summations of forces are computed for Static equilibrium at each node. At each node we know the tension in the cable section above the node as well as the angle of that section with the horizontal. Figure B-1 illustrates the cable about the ith node. If the angle $; is taken to be the angle from the horizontal, then the angle $;,, can be computed as follows: Ty singy + Wy tan) . = tan SQLS - skesyl Tj COS $4 > (B-1) where Tj = tension in section i, W, = weight of section i concentrated at node i. 8| TEST CASE -= MEASURED CHAIN TEST 3/11/76 Co C6 Oo le 58.02 F8e2l Table B-1l. 0. =-l. Oo 360 4&2 360 3006 022 CHAIN 025 efe2 CHAIN 025 Example input for program BRKMOOR. 82 Ly: Cable Section i ren Node i+l Weight eigh Tea] Figure B-1, Cable sections about node i and free body diagram of node i. This new angle is then used to compute the tension in the next section: Tj cos Fy ; Sy eee ee B-2 Wan cos 5 il ( ) LINE2 computes the angle and tension of each section starting from the top. At each section the angle is compared with the slope of the bottom. When the angle ¢ is parallel or more positive than the bottom then > is set to the slope of the bottom. The X and y coordinates of each node are computed. = =% Seam Re Garey OMA (ee >) 1+] = ¥, ind. B-4 Mean > Ma Mee, SB Gan 2 Coe i+l where Xs = X coordinate of node i Ve, = y coordinate of node i Ley = length of section when under tension. 83 At the last node the y-coordinate is compared with the depth of the anchor. If theme is) ay datterence) the ani tial tensaion values) ad= justed. Guesses at the first and second tensions are made. From then on a secant (discrete form of Newton Raphson) iteration method is used to compute the subsequent initial tension values. An error of 0.0001*depth is allowed. In most cases 4 or 5 iterations yield the desired accuracy. Some important values are printed for each iteration to aid in troubleshooting. Within EQULIB and SPRING interpolation is required to find the va- lues of tension forces and X coordinates which are between the points computed by BRKMOOR and LINE2. The linear interpolation routing LTERPS was chosen over higher-order interpolation schemes because of the asymp- totic nature of the tension versus X values. If values are requested be- yond the ends of the computer arrays, they can be extrapolated, but a warning message will be printed by EQUILIB. An iterative procedure is required within EQULIB if the anchor separation condition is selected. Again the secant iteration method is used. EQULIB prints out values at each interation which can aid in troubleshooting but which can normally be ignored. Subroutine CHAIN computes the Strain in a chain using the basic elastic properties of a steel bar with a total area equal to the area of both parts of the links, and a factor of 6 to allow for the deformation characteristics of the links. This factor of 6 came from a finite ele- ment computation. Subroutine NYLON computes the strain in a nylon rope using a power- LUNG Eon) fist) Ot Ehetoxm: e = AX , where eS Swieealn, Ks @.02052 , 8 6 O.2257, —— D, T = Tension (pound) , D = Diameter of rope (inches) - This function was determined using a least-squares power-function fit of experimental data provided by Sampson Cordage Works for their 2-in-1 nylon braided rope. An experimental verification test was conducted as a check of the program. A chain was suspended from a spring scale. Measurements were made of the length of the chain, its weight and the tension in two geometrical configurations. The program gave computed values of the tension very close to those measured. 84 6. Flow Chart. Figure B-2 illustrates the flow chart of BRKMOOR and its subrou- tines. To Program Comments and Glossary of Terms. The program listing contains many comments which aid in following the logic of the program. The important variable names are explained as well as the input requirements. 8. Run Time and Memory Size. BRKMOOR requires about 40 seconds on the CDC 6400 to compile and compute results for one value of the tide parameter. Each additional tide value requires about 30 seconds additional time. These values are for cables divided into 50 sections each. Time should be somewhat pro- portional to the total number of cable sections. The number of test conditions has much less effect on time than does the tide. As stated previously, a central memory of about 40,000 octal is required. oF Run and Card Deck Setup Procedures and Special Operation Instructions. In order to run the FORTRAN source program deck on the University of Washington CDC 6400, the following deck is required: 10. BMOOR, T40. Job card ACCOUNT (Account no., password) FORTRAN. LGO (LC=6000) LC = line count value; depends on how many tides and conditions are run 7/8/9 FORTRAN DECK 7/8/9 DATA DECK 6/7/8/9 Sample Output Data. Example output from program BRKMOOR is shown in Table B-2; a listing of program BRKMOOR is shown in Table B-3. 85 BRKMOOR READ DATA | ECHO DATA FIND NOMINAL LENGTH OF MORE CABLE DEPTHS LOAD ARRAYS WITH FORCES AT ATTACHMENT VS X FIND LENGTH AND WEIGHT OF EACH MORE CABLE SEGMENT INITIAL ANGLES LOOP THROUGH EACH TIDE POSITION LOOP THROUGH MORE EACH OF THE CABLES TWO CABLES NO COMPUTE INITIAL CABLE ANGLE AT ATTACHMENT PRINT ARRAYS FORCES VS X MAKE FIRST GUESS AT CABLE TENSION COMPUTE NOMINAL AND PERTURBED WATER DEPTH LOOP THROUGH INITIAL ANGLES LOOP THROUGH NOMINAL AND PERTURBED DEPTHS COMPUTE STRAIN GIVEN TENSION & CHAIN SIZE RETURN COMPUTE STRAIN USING e = A x xB GIVEN ROPE SIZE AND TENSION RETURN Figure B-2. Flow chart for program BRKMOOR. 86 LINE 2 LOOP THROUGH EACH CABLE SEGMENT IF MATERIAL NYLON IF MATERIAL CHAIN COMPUTE LENGTH OF SEGMENT UNDER TENSION COMPUTE X & Y COORDINATES OF EACH NODE COMPUTE X & Y DIRECTION FORCES IF LAST ITERATION, (NOTE=1) YES PRINT FORCES & COORDINATES ALONG THE CABLE COMPUTE SLOPE OF CABLE IF YES CABLE PARALLEL Boron TO BOTTOM & SLOPE > CABLE 0 nd ASSUMED TO SHORE REST ON BOTTOM LAST YES () CABLE SEGMENT ? Figure B-2. 87 PRINT X,Y, TENSION IF NOTE=1 LAST ITERATION YES RETURN IF END OF CABLE AT BOTTOM DEPTH NO SET NOTE=1 COMPUTE NEXT GUESS AT INITIAL TENSION Continued EQULIB LOOP THROUGH TEST CONDITIONS IF IS no} PRINT TENSION ANCHOR IN CABLE SEPARATION SORE IS GIVEN GIVEN? YES ¢ MAKE A GUESS AT Fy IN CABLE #1 COMPUTE Fx IN CABLE #2 COMPUTE X ff CALL FOR EACH CABLE LTERPS FIND DIFFERENCE BETWEEN COMPUTED AND GIVEN SEPARATION IS GIVEN TENSION TOO LOW? PRINT WARNING MESSAGE IS GIVEN TENSION TOO HIGH? YES COMPUTE x CALL Ee UT ERES THIS THE FIRST TIME THROUGH? MAKE NEW GUESS AT Fx IS TENSION IN SECOND CABLE GIVEN? COMPUTE A Fy OWEN | aes IN OTHER CABLE CALL NO SPRING (6) RETURN MORE YES CONDITIONS ge Figure B-2. Continued 88 SPRING GIVEN X AND Fy COMPUTE DFXDX COMPUTE T, DTDX Fy DFYDX PRINT THE ABOVE CALCULATE FORCES & MOMENTS ON BRKWTR CALL LTERPS CALCULATE SPRING CONSTANTS DUE TO MOORINGS PRINT FORCES, MOMENTS, SPRING CONSTANTS RETURN Figure B-2. Continued 89 CALL LTERPS LTERPS LOOP THROUGH ARRAY OF | COMPUTE SLOPE BETWEEN ADJACENT POINTS INTERPOLATE TO FIND VAY RETURN IN THE FOLLOWING TABLES = X RELe TO CG» Y RELe TO WATER SURFACE TEST CASE -- MEASURED CHAIN TEST 3/11/76 MOORING€ LINE NUMBERe 1 TIDE® -.000 Y x TOP TENSION FORCEX FORCEY -7e1é7 240605 50202 0 430 50184 70167 2502294 50802 0956 -5e?22 -70167 252893 604C7? 1.574 -6.211 -7.167 26.387 72158 2032€ -6.776 -7.167 260804 8.101 3.257 =-70417 -70167 270174 9.278 40419 8.158 -70167 272507 10.776 52899 -9.019 —7ol67 270814 12.704 7.809 -10.021 -70167 28.099 15.254 10.338 -11.216 -7.167 28.366 18.714 13.777 -12.664 —70l67 282619 2320557 18.601 140454 -70167 28.859 30.675 252698 16.755 -70167 292089 41.694 360687 -19.809 -70167 292290 61.608 560444 -24.690 -7.167 292410 121.346 114.814 -39.276 70239 24.544 50251 2 434 50233 -70239 250258 5.848 2963 52768 -7.239 252839 62494 1.595 =60295 =-70239 260346 70240 20353 60847 -70239 26.769 8.190 3.292 72499 -70239 272142 9.380 4.468 80248 =70239 276477 10.900 5.967 -9.122 -7.0239 27.790 12.833 7.888 -10.122 70239 : 20.077 152417 102449 -11.336 =70239 26.347 18.906 13.919 -12.795 -7.239 28.602 : 23.805 18.797 -14.606 7.239 28.844 31.065 252971 -16.935 70239 292677 420115 372058 -20-010 =-7.0239 290277 622505 570266 -252050 =70239 290395 125.036 118.305 =40.470 72095 24.658 52169 0426 50142 -7.095 250328 5oT5T 0948 520678 -72095 250945 60327 1.555 -60133 -7.095 260427 72080 22301 62696 -70095 262839 8.016 3.223 -72339 -70095 272205 9.179 40372 -8.071 -7.095 27.536 10.657 5-833 -8.918 -7.095 27.837 12.578 70732 92922 72095 282122 15.085 10.224 -11.092 -72095 282386 18.508 13.626 12.525 -70095 282636 230314 18.410 -14.305 -7.095 280874 30.353 252425 -16.579 -7.095 292102 41.256 36-302 -19.601 -7.095 29.303 60.738 5520647 =240341 =-72095 292426 117.875 111.530 -38.152 Table B-2. Example output from program BRKMOOR. 90 TEST CASE -= MEASURED CHAIN TEST 3/11/76 MOORING LINE NUMBER= 2 TIDE -.000 Y x TOP TENSTON FORCEYX FORCEY -7.1¢7 —-24.605 5.202 =2 430 520184 =-70167 250294 5-802 =0956& -5e722 -70167 -25. 893 60407 -1.574 -60211 70167 -26.387 70158 20326 =60770 -7.167 =26.804 8.161 -3.257 -7.417 -7.167 -27.174 9.278 -4.419 -8.158 =7.167 -272507 10.776 -5.899 -9.019 -70167 -27.81% 12.704 -7.809 -10.021 -72167 -28.099 150254 5 -10.338 -11.216 -70167 =280 366 18.714 13.777 -12.664 -7.167 =-28.619 232557 -18.601 14.454 -7.167 -28.859 30.675 =25.695 =16.755 =7e167 =292089 41.694 =-36.687 -19.809 =7.167 =29.290 61.608 560444 =24.690 -70167 =292 410 121.346 1140814 39.276 72239 =24.544 50251 0434 -5 0233 72239 -25.258 5-848 =. 963 5.768 -72239 25.839 62494 =-1.595 60295 =7.239 -26034%6 70240 =20353 60847 -70239 -26. 769 8.190 =-3.292 70499 70239 =270142 9.3€0 4.468 -8 6248 72239 —270477 10.900 5.967 9.122 7.239 =-27.790 12.833 7.888 -10.122 70239 -28.077 15.417 -10.4%49 -11.336 72239 28.347 18.906 -13.919 -12.795 70239 =28.602 23.805 -18.797 -14.606 =-72239 —-28. 844 31.005 -25.971 -16.935 70239 -29.077 42.115 -37.058 =20.010 =70239 =290277 62.505 57.266 -25.050 -720239 -292395 125.036 -118.305 -402470 =7.095 =24.658 52160 =0426 50142 =72095 252328 5 e757 2948 5.678 -7.095 252945 60327 -1.555 -6.133 -7.095 260427 7-080 22301 60696 72095 260839 8.016 -32223 -72339 -72095 27.205 9.179 =4.372 -8.071 -72095 -27.536 10.657 =-5 833 -8.918 -7.095 -27.837 12.578 7.732 9.922 -7.095 -28.122 15.085 -10.224 -11.092 =-7.095 =-28.386 18.508 -13.626 -12.525 72095 -28.636 232314 -18.410 -14.305 -72095 —-28.874% 30.353 250425 -16.579 -72095 292102 41.256 -36.302 -19.601 -7.095 -29.303 60.738 55 0647 -24.341 -7.095 292426 117.875 -111.530 =38.152 Table B-2. Continued 9) TO#3€62°C TO+3E6L°€C NOISN31L OOF34ET°SG— O0+34ET°S= AQAsQG TO+49¥%E° T= TO+39¥E°T XGA30 TO+482E°TH TO*38Z2E°T AQx sd penutquop *z-g oT qe]. TO*#392920°T = EEN o° = 2tdn T0#30ST69°2— = TEWH NOTLIZAYIG 11OY SLNVISNOD ONTYdS O° = €2ny TO+39LZ920°T = 224n 0° = Ted» NOTLIZYTG JAV3H SLNVLSNOD ONT ads TO+IZEIS9°Z== ELAy 0° = é2TAN T0+#3900T9S°6 = TTIW» NOILIZYIA AVMS SLNVLISNOD ONTUdSs 0000° a4W E2ES°LE= sH4 0000° =S4 SINT7 ONTYOOW OL 3NG WATYGIINGA LV AILVMNV3Yd NO SLNJWOW ONY S394D4 TO+ICOd>°*T= TO+IE024°T TO+s6TOL*> 2 TO*+IEOCH°T TOFFEO2S?%°T T0+36Tb6L°¥= T ydia AQLQ XG1d °ON 378V9 TO+3T8L°%— LOT°L= TTO0°62= 99L°8T= £E6°CE= 2 TOt3TSL°¥= L9T°%L= TTO°62 99L°8T= EE6°?E T XOxdd A x Ad XJ °ON 3718V9 °a7 900° s 2 318V) NI NOISN3L IVNIWON °g1 000°= s T 318V) NI NOISN3L IVNIWON 4334020°8S sd3S ‘NOLLVd3d3S YOHINVY TVLNOZIaOH °d1000° 21X34 639903 IWINOZIYOH CII IddvV ATIVNYALXS 00°= = 40TL e- SNOILIQVOD 3HL YODA 9L4/TT/E LS4L NIVHD G3dNSVJW -- 3SVI 1831 92 BMOORUD VOLO OTLOTOTA LOTTO TOOT LETT TELL TTT TATOO AA RUNT VERSICN FEB 74 B 13:04 04/09/76 PROGRAM PRKMOOR(INPUTsOQUTPUTs PUNCH» TAPES=INPUTs TAPEG2CUTPUT) Cc : C PROGRAM BRKMOOR COMPUTES THE FORCES AND SPRING CONSTANTS THAT A PAIR C OF MOORING CABLES IMPART ON A FLOATING BREAKWATER SECTION Cc CHRERERE EEE CHER ERE ERASE RARER EERE SEER ERO REE REAR RAH ESS EKER EERE REAR HE EES C INPUT C FIRST CARD-=TITLE = 80 ALPHANUMERIC CHARACTERS C BREAKWATER GEOMETRY=— C NUMBER CF TEST CONDITIONS C TEST CONDITIONS=-ONE CARD FOR EACH SET C TIDE CARD--NUMBER OF TIDE CONDITIONS AND THE CONDITIONS C FOR FIRST CABLE--NUMBER OF SEGMENTS ANCHOR DEOTH AND BOTTOM SLOPE C FOR EACH OF ABOVE CABLE SEGMENTS ==CARD WITH SEGMENT PROPERTIES C REPEAT --NUMBER OF SEGMENTS AND THEIR FROFERTYES FOR SECOND CABLE CREPES SERRATE ERE ERE EER REREKEAEKCEA EES HRR AD SRS SAA AKRK EKER EERE EEE RESEEEEHH C 3 COMMON/ONE/NSEG(2) »NSECT(295)9WSECT(295)5 MATLI(295) DI AM( 295)» 2 ALSECT(295) 3 COMMON/TWO/WY( 253) 9 EX (253020) 9 FX( 292520) 0 FY (293920) » TENS (293920) » 2 FEXT(9)sSEP(9)5 TENS1(9)5 TENS2(9) »NANGLE» NCOND» TITLE(8) 3 COMMON/FOUR/YCGoXCAB(2)5 YCAB(2)5 TINE(E) os ITIDE 3 DIMENSION WPF( 255) sALSEG(295)5 NEPTH(2)9BSLOPE(2)sALKNOM( 2) 3 PI 2301415926535 C#**READ A TITLE CARD -- 80 CHARACTERS MAX 5 READ 3sTITLE 12 3 FORMAT(8A10) 12 PRINT 16sTITLE 20 16 FORMAT(CIH195X»8A10///) C##*READ AND ECHO THE BREAKWATER GEOMETRY 20 2 READ 59 YCG»sXCAB(1) 9 YCAB(1) »XCAB(2) 9 YCAB( 2) C YCGsy COORDINATE OF CG RELATIVE TO WATER SURFACE 42 5 FORMAT(5F10.0) 42 OUTPUTs YCGsXCAB(1) »YCAB(1)5XCAB(2) 9 YCAR(2) C XCAB(T)=X COORD OF CABLE I ATTACHMENT RELATIVE TO CG C YCAB(I)*sY COORD OF CABLE Tf ATTACHMERT RELATIVE TO WATER SURFACE C NOTE--CABLE NUMBER 1 IS THE CABLE WITH ITS ANCHOR IN THE +X DIRECTION C*#**IJNPUT THE NUMBER OF DESIRED CONDITION CARDS MAX NUMBER®9 72 READ 10»NCOND 100 10 FORMAT(I2) C%**READ AND ECHO DESIRED CONDITIONS 100 00 17 ICOND=1,NCOND 102 READ 159 FEXT(ICOND)» SEP(ICCND)» TENSI(T COND) » TENS2¢1ICOND) 121 15 FORMAT(4F1020) 121 17 OUTPUT» FEXT(ICOND)» SEP(ICOND) » TENSI(TCOND) » TENS2( ICOND) FEXT=EXTERNALLY APPLIED FORCE (HORIZONTAL DIRECTION) LBe SEP =sANCHOR SEPERATION IN THE X DIRECTION FT. TENSL= TENSION IN CABLE 1 LBeo TENS2*TENSION IN CABLE 2 LBo INPUT SEP» OR TENS1 OR TENS2 CR BOTH TENS] AND TENS2 C#*#*READ AND ECHO TIDE CONDITIONS C NTIOESNUMBER OF TIDE CONDITIONS MAX 25 C TIDESTIOE POSITION RELATIVE TC NOMINAL DEPTH MEASUREMENTS FT 151 READ 205 NTIDEs(TIDE(I)» I=1sNTIDE) 166 20 FORMAT(T199X55F1020) aAagaan Table B-3. Listing of program BRKMOOR. 93 RUNT VERSICN FEB 74 B 13:04 04/09/76 166 201 203 217 217 223 223 226 230 264 264 326 326 331 332 343 357 375 377 403 405 406 411 411 421 423 431 443 444 445 446 456 461 463 470 OUTPUT» NTIDE» TIDE C€ LOOP THROUGH THE TWO CABLES OO 65 L#ls2 C#** INPUT THE CABLE PROPERTIES AND BOTTOM DEPTH AND SLOPE READ 229NSEG(I) »DEPTH(T) sBSLOPE(T) 22 FORMAT(I298X»2F10.0) C NSEGe NUMBER OF CABLE SEGMENTS OR MATERIALS C DEPTH= NEPTH OF THE WATER AT THE ANCHOR FT. C S8SLOPE2 SLOPE OF THE BOTTOM (FT RISE/FT) PRINT 25 25 FORMAT(///5Xy*%I NUMBER SECTIONS SEGMENT LFNGTH WT PER FOOT* 2 4X* MATERIAL DIAMETER*/) NS=NSEG(T) 00 30 JelsNS C#¥#*FOR EACH CABLE SEGMENT YNPUT C NSECT(I)=® NUMBER OF SECTIONS INTO WHICH CABLE SEGMENT I IS DIVIDED C ALSEG(T)=LENGTH OF CABLE SEGMENT J FTe C WPF(I)® WEIGHT PER FOOT IN WATER QF CABLE SEGMENT J LB/FT C MATL=MATERTAL OF CABLE SEGMENT EITHER NYLON CR CHAIN Cc MUST RE LEFT JUSTIFIED IN CATA FIFLD C ODYAM=DIAMETER OF ROPE OR CHAIN LINK INCHES READ GSOgNSECT( Io J) sALSEG( Io J) oWPF(C Tod) MATL (Is J) oDIAM(I5 J) 40 FORMAT(I555X» 2F10.09A105F10.0) 30 PRINT 505 JoNSECT(T os J) oALSEC(I oJ) oWPF( To J)» MATLI(Is J) sDIAM(Is J) £0 FORMAT (Xo I59 OX T5 9 B8XoFl1Oe24XnFlOe2s9XsAl10s 5X9 F603) C#*#FIND THE NOMINAL LENGTH OF THE CABLEs LENGTH AND WEIGHT OF SECTIONS ALNOM(T) #0. NG 60 J#l»NS ALNOM(I)=ALNOM(T)#ALSEG( Is J) ALSECT(Ta J) #=ALSEG(Is J) /NSECT(I 5 J) €0 WSECT( Is J) 2WwPF (Io J)®ALSECT(I9 J) €5 CONTINUE BSLOPE(2) e=BSLOPE(2) C#**_LOOP THROUGH THE TIDE POSITICNS 0G 400 ITIDE=lyNTIDE C##*LOOP THROUGH THE CABLES 00 150 I=ls2 PRINT 70 70 FORMAT(1H155X*NOTES-IN THE FOLLOWING TABLES ¥ AND Y ARE MEASURED 2RELATIVE TO THE CABLE ATTACHMENT#//) C D#Y DIRECTION SEPFRATION BETWEEN ANCHOR AND ATTACHMENT D=DEPTH(I)+TIDECITIDE) +YCAR(I) C#**COMPUTE INITIAL ANGLES TO BE USED C NANGLE*NUMBER OF ANGLES USED MAX#2N NANGLE#®#15 PHIMINSASIN(D/ALNOM(T)) DELPHI #(PI/2e-PHIMIN) /FLOAT(NANGLE 41) PHTONE==PT/2. C#**COMPUTE A FIRST GUESS FOR THE INITIAL TENSION FOR STEEPEST ANGLE ALSUM=0. TZERG2#0. DAF2D+( ALNOM(T )-D) *BSLOPE(T) NS*NSEG(T) O00 90 J#l»yNS NSS#NSECT (Is J) 00 90 KelyNSS Table B-3. Continued 94 -RUNT VERSTICN FEB 74 B 13304 04/09/76 471 ALSUM®ALSUM4ALSECT(I 5 J) 477 IFCALSUM .GT. DAF) GO TO 95 502 90 TZERO=TZERO+WSECT( Is J) 515 G5 CONTINUE C CCMPUTE THE NOPINAL AND PERTURBED DEPTHS 515 DREFe=D 517 DELD=DREF/100. 521 DPLUS2DREF4¢DELD 522 DMINUS=DPFEF=DELD C#**LOOP THROUGH THE INITIAL ANGLES 524 DO 100 KelyNANGLE 525 PRINT 975 T9KsNANGLEs TIDE(ITIDE) 541 C7 FORMAT(/X¥CABLE NUMBER #118 INITIAL ANGLE NOo *22* OF *I2 ane TIDE = ¥*F5.2/) 541 PHIONE=PHIONE*DELPHT C***LOOP THROUGH THE NOMINAL DEPTH AND PEPTURBED DEPTHS 543 DO 100 J=153 545 IPRINT2O C®eeeeeee TO SKIP THE PRINTING JF EACH CATINAPY = INSERT A GO TO 1111 546 IF(JS oEFO. 1) ITPRINT#=1 551 1111 IF(JS e€Q5. 1) D=DREF 555 IF(J e€Q. 2) DeDPLUS 5€1 IF(J e€Q. 3) D=DMINUS 565 WY (Ia J)2YCAB(T)-D 575 OUTPUTs JoKoDo YCAB(I) sWY( Is Jd) 5 PHIONE 627 CALL LINE2(I»sPHIONEs TZERGsE oBSLOPE(T)» X59 Vo FORCEXsFORCEYs IPRINT) 642 EX( Is Jo K) =XCAB(T)—X* (=1) ¥¥7 660 FX (I JoK)®FORCEX*(=(—1) *¥*I) 674 FY(T»J9K)sFORCEY 703 TENS(IoJsK)=2TZERO C wY(TIsJ)*#Y COORD OF THE ANCHOR TO NOe 1 CABLE==WATER SURFACE=ORIGIN C EX(IsJ9K)=x% COCRD OF ANCHOR RELATIVE TC CG OF BREAKWATER C TENS*TENSION AT ATTACHMENT C FY¥=sFORCE AT ATTACHMENT IN X DIRECTION C FY=FORCE AT ATTACHMENT IN Y DIRECTION 712 100 CONTINUE G END OF CABLE LOOP 716 150 CONTINUE 720 PRINT 102 724 102 FORMAT(1H1s5X*IN THE FOLLOWING TABLES = X REL. TO CGs Y RELo TO WA 2TER SURFACE#//) 724 00 160 I#ls2 726 PRINT 1O3sTITLE 733 103. FORMAT( 5X»98A10 ) 733 PRINT 105919 TIDE(ITIDE) 744 105 FORMAT(///5X*MQORING LINE NUMBER= *Y15*% TINE=*F6.3// 2 10> *VS14XoLHXs8Xo¥* TOP TENSTON*7X¥*FORCEX*7X¥FGRCEY*/) 744 00 120 Je®ls3 746 00 120 K=1ly»NANGLE 747 PRINT L1OsWY (To J) oD EX( Io do K) 9 TENS( To do KI 9 FX( To do KI oFY( Io do K) 1013 110 FORMAT(5X55(F1103594xX)) 1013 120 CONTINUE 1020 PRINT 125 1023 125 FORMAT(1H1) 1023 160 CONTINUE 1025 CALL EQULIB Table B-3. Continued 95 RUNT VERSICN FEB 74 B 13804 04/09/76 Cc END OF TIDE LCoP 1026 400 CONTINUE 1031 sTcP 1033 END Table B-3. Continued 96 a RUNT VERSION FEB 74 B 13804 04/09/76 SUBROUTINE LINE2(K»PHIONEs TZEROs DEPTH» BSLOPE>» Xs Yo FORCEX»FORCEY»s IP) 15 COMMON/ONE/NSEG(2) oNSECT( 295) 9WSECT(255)5 MATL( 295) 9DIAM(255)5 2 ALSECT(295) C#*THE INPUT TO SUBROUTINE LINE C PHIONE=INITIAL ANGLE OF CABLE C TZERQ*INITIAL GUESS OF TENSION AT TOP OF CABLE C**SUBROUTINE LINE COMPUTES C TZERO= TENSION AT CABLE TOP C FORCEX=FORCE IN X OIRECTION AT CABLE TOP C FORCEY=FORCE IN Y DIRECTION AT CABLE TOP C XsHORIZONTAL SEPERATION BETWEEN TOP AND BCTTOM CF CABLE C Y#VERTICAL SEPERATION BETWEEN TOP AND BOTTOM OF CABLE C#**GO DOWN THE CABLE SECTION 8Y SECTION COMPUTE TENSION» ANGLE» C EXTENDED LENGTHsX AND Y COORDINATES 15 PT23014159 16 NITER=0 17 MNITER®#25 21 NOTE=0 22 TsTZERO 23 152 NITER#*NITER+1 25 IF(IP 2EQ. C) GO TO 153 27 IF(NOTE -EQ. 0) GO TO 153 31 PRINT 155 35 155 FORMAT(//5X#1 J X vf TENSION LSECT* 2 6Xs*LEXT PHI-DEGREES FCRCEY FORCEX*/) 35 153 YeO. 37 X20. 43 PHI=PHIONE 44 NSS=#NSEG(K) 47 158 00 200 I#lsNSS 51 NS=NSECT(KsT) 56 DO 200 Jel»sNS 57 PHID#PHI*180./PY 61 IFCMATL(KsI) o€Q. SHNYLON )GO TO 165 67 IFCMATL(Ks TI) e&FQe SHCHAIN )GO TO 160 75 160 CALL CHAIN(DIAM(Ks TI)» T»STRAIN) 105 6G TO 170 111 165 CALL NYLON(DIAM(KsI)5TsSTRAIN) 121 170) =ALEXT#ALSECT (Ko I)*(10+STRAIN) 134 KaX+ALEXT*COS(PHI) 145 YeY*ALEXT*SIN( PHT) 152 TCOS*T#COS( PHI) 156 TSIN® T#SIN( PHI) 162 IF(IP .EQ. O) GO TO 185 170 IF(NCTE -EQ. 0) GO TO 185 172 PRINT 1805 To Jo X9 Vo To ALSECT(KsI) sALEXT» PHINs TSINs TCOS 231 180 FORMAT(%s21558F10.3) 231 185 IF(L eEQ. NSS cANDe J o£O. NS) €O TO 200 247 SLOPE=(TSIN+WSECT(Ks I) )/TCOS 257 IF(SLOPE oGEo BSLOPE) SLOPE=BSLOPE 262 PHTI®ATAN( SLOPE) 266 T=TCQS/COS(PHI) 271 200 CONTINUE 302 FORCEXsTZERO*CGS(PHIONE) 311 FORCEY=TZERC¥#SIN(PHIONE) Table B-3. Continued 97 RUNT VERSION FEB 74 B 13104 04/09/76 320 OUTPUT» NITERs Ys Xs FZERO C#**THE SECOND GUESS OF INITIAL TENSION ITS COMPUTED 350 IF(NITER oGTe 1) GO TO 220 353 TZOLD*TZERO 354 TZERO=TZERO#ABS(DEPTH/Y) 365 YOLD2Y 367 T=TZERO 370 GO TO 152 C#** THE SUBSEQUENT INITIAL TENSIGNS ARE COMPUTED USING SECANT ITERATIC 370 220 RELER#ABS(1.+Y/DEPTH) 402 IF(NOTE -€0. 1) GO TO 309 405 TF(NITER oGEo MNITER oORe RELER ol fe o0001 ) NOTE#1 424 DEROLD=DEPTHt+YOLD 426 DERR=DEPTH+Y 430 T=TZOLD=DEROLD*(TZERO=TZOLC) /(DERR=—DFROLD) 437 IF(T oLEe Oo) T2TZERO/2. 443 YOLO®Y 445 TZOLD=TZERO 446 TZERO®T 447 6A TO 152 447 300 RETURN 450 END JNT VERSICN FEB 74 B 13804 04/09/76 SUBROUTINE CHAIN(DsTsSTRAIN) 6 P1=#=3.14159 7 E=30.E6 11 AREAGD¥*D#PI/20 Cc CZELONGATION FACTOR -=- C#6 FCR OVAL CHATN 14 C=606 15 STRAIN®C#T/( AREASE) 21 RETURN 22 END UNT VERSION FES 74 B 13204 04/09/76 SUBROUTINE NYLON(DsTsSTRAIN) 6 X=7/(0*D) 10 A=.02052 12 B=.2237 13 STRAINGAtX¥#B 20 RETURN 21 END Table B-3. Continued 98 RUNT VERSION FEB 74 B 13304 04/09/76 SUBROUTINE EQULIB 2 COMMON/TWO/BY( 293) 5EX( 293020) 9FX(20352CISFV( 293920) 9 TENS (293520) 0 2 FEXT(9) »SEP( 9) 9TENSI1(9) 9 TENS2(9) »NANGLEaNCOND»s TITLE(8) 2 CCMMON/THREE/X(2)5F(2) 2 DIMENSION SEPDIF(3)»FO(3) C¥***EQULIB FINDS THE BREAKWATER EQUILIBRIUM POSITYON C*#**LOOP THROUGT THE TEST CONDITIONS 2 DO 100 IC#lsNCOND 4 TF(SEP(IC) eNEe O-) GO TO 20 7 IF(TENSI(IC) oNE.0.)GO TO 10 13 IF(TENS2(1C) oNEeO2)GO TO 12 17 PRINT 155 23 155 FORMAT(//X#NO INITIAL CONDITIONS SPECIFIED*) 23 6a Te 1¢0 C#**FOR THE CASES WHERE INITIAL TENSION TS GIVEN THE FOLLOWING IS USED 24 10 T#TENSI(IC) 27 Tel 31 Je2 32 Gd TO 14 33 Ve Te TENS2(IC) 36 Is2 40 Jel 41 14 QUTPUTsT»NANGLEsT 57 IF(T oGEe TENS(I9151)) €0 TO 18 70 PRINT 16 74 GO TG 100 75 18 IF(T oGEe TENS(IslsNANGLE)) PRINT 17 111 16 FORMAT(//5X*GIVEN TENSION CLOSE TO OF LESS THAN WEIGHT OF VERTICAL 2 MOORING LINE*/5x*NO FURTHER EVALUATION ATTEMPTED *//) 111 17 FORMAT(//5X*GIVEN TENSION TOO GREAT FOR FVALUATION wWITHOUT* 2 *EXTRAPOLATION*/5X¥*USE RESULTS WITH CAUTION*/7/) 111 CALL LTERPS (Iolo NANGLEs TENS9 EX» To X(1) »DUMMY) 123 OUTPUTsX(T)oT 137 CALL LTERPS (Iolo NANGLEs TENS oFX»T»F (I) » DUMMY) 151 OQUTPUTsF(T) 162 IF(I e&Qe 1 cANDe TENS2(IC) eNEo 0) GO TO 12 177 IF(TENS1(IC) oNEo O eANM. TENS2(IC) o.NE~ 0) GO TO 40 214 F(J)2-F(I)-FEXTCIC) 224 QUTPUTsF(J) 234 CALL LTERPS (Jo lsNANGLEsFXsEXsF (J) 9X (J) DUMMY) 250 OUTPUTsxX(J) C NOTEs= F(I)*#X DIRECTION FORCE ON CABLE I » X(I)=*X COORC OF ANCHOR 261 GO TO 40 C#**FOR THE CASE WHERE ANCHOR SEPERATION IS GIVEN C MAKE A FIRST AND SECOND GUESS AT FORCE 262 20 TA®(NANGLE#1)/2 270 EPS=SEP(IC)*.0001 274 DO 30 Ilels2 275 XC L)FEX( 1915 TA) 305 FCLISEX( Isls IA) 315 OUTPUTsF(LIo IIs FEXT(IC)oSEPC IC) sIA 344 FOCII)#F(1) 351 F(2)2-F(])-FEXT(IC) 361 OUTPUTsF(2) ; 371 CALL LTERPS (29 1sNANGLEs FX9EXsF(2)9X(2) 9 DUMMY) Table B-3. Continued 99 RUNT VERSION FEB 74 B 13:04 04/09/76 405 ASEP®X(1)=-X(2) 413 SEPDIF(II)=SEP(IC)—ASEP 421 OUTPUT s IIs TAs X(1)9X(2)5F (1) F (2) sSEPCIC) sASEPsSEPDIF(II) 466 IF(ABS(SEPDIFCII)) oGT. EPS) GO TO 24 476 GO TO 40 477 24 IF(SEPDIF(II) eGEe O.) GN TO 26 503 TAs1 505 GQ TO 30 505 26 TA®NANGLE 507 30 CONTINUE C#*##USE SECANT INTERPOLATION FOR THE SUBSEQUENT FORCE TRIALS 511 MN220 513 DQ 34 K=1l54N 514 FO(3) SFO(LI“SEPDIF(1)*®(FO(2)-FO(1) /(SEPDIF(2)—-SEPDIF(1)) 540 IF(FO(3) oLEo Oo.) FOC(3)=FO(2)/2. 552 F(1)2F0(3) 557 F(2)2=-F(1)=FEXT(IC) 567 DO 32 {#152 570 32 CALL LTERPS (Iolo NANGLEsFXoEXoF (I) 5 X(I) o DUMMY) 605 ASEP=X(1)=x(2) 613 SEPDIF(3)*#SEP(IC)—ASEP 621 QUTPUTs Ks X(1)5 X02) 5F(1)5F(2) sASEPs SEPDIF(3) 657 IF(CABS(SEPDIF(3)) oLEe EPS) GO TO 36 667 IF(K oEQ. MN) GO TO 36 672 FO(1)2FO(2) 677 FO(2)2FO(3) 704 SEPDIF(1)=SEPDIF(2) 711 SEPDIF(2)=*SEPDIF(3) 716 34 CONTINUE 720 36 PRINT 37 724 37 FORMAT(/5X*®MAX NUMBER OF ITERATIONS REACHED*/) 724 38 DO 39 Te#ls2 : 726 IF(ABS(F(I)) oGTo ABS(FX(I»1l»NANGLE))) PRINT 42 750 29 IFCABS(F(I)) ol Te ABS(FX(I9191))) PRINT 43 775 42 FORMAT(//5X*ANCHOR SEPERATION TOG GREAT FOR EVALUATION WITHOUT EXT 2RAPOLATING=-USE RESULTS WITH CAUTION*//) 775 43 FORMAT(//5X*#ANCHOR SEPERATION TOO LITTLE FOR EVALUATION WITHOUT EX 2TRAPOLATION=-USE RESULTS WITH CAUTION*//) 775 40 CALL SPRING(IC) 777 100 CONTINUE 1002 RETURN 10C2 ENO Table B-3. Continued 100 RUNT VERSICN FEB 74 B 13:04 04/09/76 o ane o 107 133 140 147 174 221 223 230 230 261 261 265 265 270 337 337 345 352 360 365 2 2 SUBROUTINE SPRING(IC) COMMON/TWO/WY( 293) 9EX( 293920) 9 FX(293920)9FV (253520) » TENS(293920)>5 FEXT(9) » SEP(9) 9 TENS1(9) 9 TENS2(9) sNANGLE» NCOND»s TITLE(8) COMMON/THREE/X(2)5F(2) COMMON/FOUR/ YC Gs XCAB(2)9 YCAB(2)5TIDE(5)»TTIDE DIMENSION DFXDX(2) sDFYDX(2) sDFEXDY( 2) 9D FYDY(2) »D(3)5FXX(293)5 FYX(293)9FV(2) oDTDX(2) 9DTDY( 2) 9DTOR( 2) 9T(293) REAL KM1IsKM1L29KM139KM219KM225KN239KM319KM329KM33 C#*#**SUBROUTINE SPRING COMPUTES THE BREAKWATER SPRING CONSTANTS C#**#COMPUTE THE SPRING CONSTANTS FOR EACH CABLE C Cc 1 nye 14 € 15 18 20 25 HOR VER Welw Pe 2 2 IZONTAL FORCE AT EQUILIBRIUMSF(T) FOR CABLE I T FORCE AT EQUILIBRIUM*FV(T) FOR CABLE I DO 14 Iels2 00 12 J#153 CALL LTERPS (Is JoNANGLEDEXo FX» (I) oFXX( Io J) 5DF) IF(J e€&Q-. 1) DFXDX(I)2=DF CALL LTERPS (Is JoNANGLESEXoTENSo X(T) 9T( 19d) 9 OT) IF(J eFEQe 1) DTDX(I)2=DT CALL LTERPS (Io JsNANGLEs EXsFYoX(1) sFYX( Tod) oD(J)) DTIDVC LACT 5 3—TU19 2) CWY (Ts 2)-WV(T93)) DFYDX(I)==-0(1) FVC I) ®FYX(I91) DFYDV( ID EC FYX( Is 2)—-FYX(L93))/0WY(Lo 2) -WYV(T9 3) *(—1.) DEXDYC IIS (CEXX( Is 2)oFXX (Ts 3d ISCWY ITs 2)-WY( 153) )¥l-1 2) CONTINUE PRINT 16s9TITLE FORMAT(1H15s6A10) PRINT 1559 TIDECITIDE) sFEXT( IC)» SEP( IC)» TENSI( IC)» TENS2(IC) FORMAT(///X¥*¥FOR THE CONDITIONS -=*/ 5X*TIDE = *F5.2/ S5X*EXTERNALLY APPLIED HORIZONTAL FORCE» FEXT= *F10.3*LBo*/ SX*HORIZONTAL ANCHOR SEPERATION, SEP= *F10.3*FEET#/ SX*NOMINAL TENSION IN CABLE 1 =*F10.3% LBo*/ S5X*NOMINAL TENSION IN CABLE 2 =*F10.3* LBo*//) PRINT 18 FORMAT(/5X*CABLE NOe FX*¥1LOX#FY*LIX» LHX op LIXFY*OXeDEXDX*7 Xs *DFXDY*7X9 DE YOX* 7X*DFEYDY%s 5X9 *TENSTON*®//) DO 20 [#152 PRINT 255 To FCI) oFVCI)oX( To WV (1591) sDFXDX( IT) »DFYDX(1) »DFXDY(I)» DFYDY(I)»T(Iol) FORMAT(9X51194(2X%9F1003)595(XsE11.3)) C®**®NOW CALCULATE FORCES AND SPRING CORSTANTS FOR THE BREAKWATER Cc Cc C Cc Cc C Cc Ses H=H RR FSe FHe EMR CH WAY MOTION +X DIRECTION FEFT EAVE MOTION +Y DIRECTION OLL MOTION COUNTERCLOCKWISE RADTANS FORCES CAUSING SWAY DUE TO THE MONRING LINES FORCES CAUSING HEAVE DUF TO MOORING LINES sMOMENTS CAUSING ROLL OUE TO MOORING LINES ANGE YCAB TO BE DIST TQ CG IN Y DIRECTION YCAB(1)*YCAB(1)=YCG YCAB(2)2YCAB(2)-YCG FSsF(1)+F (2) FHeFV(1L)+FV(2) EMR2EV(1)#XCAB(1)+FV(2)*XCAB(2)-F( 3) *YCAB(1)—F(2)*YCAB(2) C¥**CALCULATE CHANGE IN TENSIONS WITH BREAKWATER MOTIONS Table B-3. Continued 10] RUNT VERSICN FEB 74 B 13304 04/09/76 413 00 26 [#152 414 26 DTORCI)=DTDY(I) *XCAB(TI-OTOX(L)* YC AB(T) 432 PRINT 27 435 27 FORMAT(//5X*CABLE NO OTDOX¥*8X*#*DTIDY*8X*DTDR*//) 435 00 2€ T2ls2 440 28 PRINT 299 TsDTOX(I) »DIDY(I) sDTOR(T) 461 29 FORMAT(OXsT153(XE1104)) C SPRING CONSTANTS SWAY DIRECTICN 461 KMIL=(DFXDX(1) +DFXDX(2)) #(-1.) 471 KM12=(DFYDX(1) #DFYDX(2))*(-1e) 500 KM132(DFYDX(1) *XCAB(1) 4DFYDX(2)*xCA8(2)—DFXNX (1) *YCAB(1)—DFXDX(2)* 2 YCAB(2))*(=-1.) C SPRING CONSTANTS HEAVE 530 KM212(DEXDY(1) +DFXDY(2))*(-1.) 537 KM22=(DFYDY(1) ¢DFYDY(2))*(-1e) 546 KM23=(DFYOY(1) ®XCAB(1)+DFYDV(2)*xCAB( 2) 2 -DEXDY(1)*YCAB(1)=DEXDY(2)¥YCAB(2))*(—1.0) C SPRING CONSTANTS ROLL DIRECTION 576 K432=(DFEXDY(1) *XCAB(1)4DFXDY(2)*xXCAB(2)—DEXDX(1)*YCAB(1) 2 =DFXDX(2)*YCAB(2))#(-1.0) 626 KM322(DFYDY(1) *XCAB(1)+DFYDY(2)*xCAR(2) 2 -DFYDX(1)*YCAB(1)=DFYDX(2) *YCAB(2))*(=1.0) 656 KM332(XCAB(1)**2*DFYDY(1)+XxXCAB(2)**2*DF YOY (2) 2 +YCAB(1)#*#2*DFXDX(1)FYCAR(2) #*#2*DFXDX(2) 3 =XCAB(1)*YCAB(1) #(DFYDX(1)4DFXOY(1)) & =XCAB(2)*YCAB(2) #(DFYDX(2) +D0FXDV(2)) )*#l—1.) 756 PRINT 309FS» FH» EMR 767 30 FORMAT(///5X*FORCES AND MOMENTS ON BREAKWATER AT EQULIBRIUM DUE * 2 *TO MOORING LINES*/1lOX*FS= *F12.4/10X¥*¥FH= *F12.4/10X#MR= *F12.4) 767 PRINT 329KM119KM129KM13 1001 32 FORMAT(//5X*SPRING CONSTANTS SWAY DIRECTION®/10X*KM11 = *E€1205/ 2 lOX#*KM12 = €€12.5/10X*KM13 =¥E12.5) 1001 PRINT 349 KMZ219KM229KM23 1913 34 FORMAT(/5X*SPRING CONSTANTS HEAVE DIRECTION*®/10X*KM21 = *E12.5/ 2 lOX#KM22 = *E12.5/10X*KM23 =*EF12.5) 1013 PRINT 369KM31,KM329KH33 1025 36 FORMAT(/5X*SPRING CONSTANTS ROLL DIRECTION*/10X*KM31 = *E1205/ 2 1OX*KM32 = ¥E12.5/10K*KM33 =2*E12.5//) 1025 PRINT 38 1031 38 FORMAT(1H1) 1031 RETURN 1032 END Table B-3. Continued 102 RUNT VERSICN FEB 74 B 13304 04/09/76 10 20 SURROUTINE LTERPS (To JoNoXsVoXKXo VY oDYOY) DIMENSION X(293920)9Y(293520) NFOesN-L 00 10 K=l»yNMN LekK¢] TFCXX cEQe. X(IoJot)) GO TO 30 IFCABS(XX) oLTo ABS(X(IoJoL))) GO TO 20 CONTINUE DYDXS(V( To Jo LIKVIT oS oKIIAUXC Ts Jo LI—KX(To Jo K)) YY®V(Io do KV4(XX—XC Io d0K)) *0YOX RETURN IF(L .£Q. N) GO TO 20 Mell] BYOX2(CV( Ts Jo MIKVI Io Jo KD ISUKMC Ts Jo MI —X IT 9 do K)) YY*V(Is Jol) RETURN END Table B-3. Continued 103 APPENDIX C LINEAR HYDRODYNAMIC COEFFICIENTS The linear theoretical model used in solving the floating break- water problem has been discussed extensively by Frank (1967). He de- veloped the approach to solving the boundary value problem which has come to be known as the "Frank close-fit method". The reader is re- ferred to the original reference for a complete presentation of the method. In this approach, the classical linear boundary value problem requires that Laplace's equation be satisfied throughout the fluid do- main: V-0(x,y,t) = 0 for y <0. (c=) The free-surface boundary condition is applied on the undisturbed free surface: o,(%,0,t) + go, = 0 for y= 0. (C-2) The body-surface boundary condition requires that no fluid flow through the body surface: > VICAR) 9 Bi) ES N(e)) © 16). (C-3) S The bottom boundary condition for infinite depth is of the form: lim , ® = (0), = yo ty (aya) = 0 (C-4) In addition there is a radiation condition specifying that the waves travel away from the body. Because the problem is assumed to be linear, the velocity potential may be decomposed and several boundary value problems considered. If this is done the total potential becomes: d= ay + 2, Tons On + o. (C-5) oO iH} l potential representing pure sway motion in calm water, oe iT] 9 potential representing pure heave motion in calm water, oO iT} 3 potential representing pure roll motion in calm water, 104 body, Oe Ee incident wave potential. Another velocity potential may be defined: Ons potential for total fixed-body problem, so that 7A = o) + ,. ®, = potential representing the waves diffracted by a fixed Using this decomposition of the velocity potential, the boundary value problems may be expressed as: V-o.(x,y,t) = 0 for y <0, Q, (x,0,t) + go, = 0 for y=0, tt y pe ®. (x,y,t) = 0 y> -© 1 2Y>o > y and > vo. -n| =V.(s) - n(s) for i = 1,2,3 1 1 C oO or vo. ‘nl =O for i= 4,6. = € Oo (C-6) These boundary value problems are solved directly using the Frank method which distributes singularities over the hull surface. These singulari- ties satisfy the radiation condition, Laplace's equation, the free- surface boundary condition and the bottom boundary condition. To satis- fy the body boundary condition requires the formulation of a set of linear equations whose solution reveals the strength of each singularity distributed on the body. Once the velocity potential is found the pressure may be found from Bernoulli's equation: RiGee) =r pe (x, y,t). The force on the body surface is: = FE = | P(s) n(s) ds, C and the moment is: 105 (C-7) (C-8) M = | P(s) [rxn] ds. (C-9) G Oo The added-mass and damping coefficients are found by considering the cases i = 1,2,3. The forces and moments computed using these po- tentials may be separated into components in phase with acceleration and velocity. The component in phase with acceleration yields the added-mass coefficients and the component in phase with velocity yields the damping coefficients. Exciting forces and moments are computed when the case i = 6 is considered. Special Symbols for Appendix C. n(s) = unit interior normal vector to the body surface Ss = indicates arc length along body contour C = body contour P(s) = pressure on body surface V(s) = velocity of body surface ) = total velocity potential 106 APPENDIX D FLOATING BREAKWATER ANALYSIS I Purpose of the Program. Computer program BRK2D performs a performance analysis for two- dimensional floating breakwaters of arbitrary cross section. This ana- lysis includes predictions of the hydrodynamic coefficients, the dyna- mics and mooring line forces. 2. Program Description. Program BRK2D is written using both FORTRAN II and FORTRAN IV statements. The program consists of the main program BRK2D and the subroutines COEFF, COMP, PHYSCL, POTOUT, DYNAMC, MORTEN, CPV, LNEQF. The subroutines COEFF and COMP calculate the quantities needed to formulate the linear equations for the velocity potential. COMP calls on LNEQF to solve these linear simultaneous equations. Subroutine PHYSCL calculates the physical quantities including ad- ded-mass and damping coefficients and surface elevations per unit ampli- tude of motion. CPV is a subroutine which evaluates the Cauchy principal value integral in the Green function. LNEQF is a packaged subroutine to solve simultaneous linear equa- tions using the Gaussian reduction method. 3. Type of Computer and Peripherals. BRK2D was written for use on the CDC 6400 computer. It uses about 25000g words of memory. No peripherals other than the card reader, line printer and card punch are required. 4. Input Data. The first cards in the data deck are label cards for the output. These are shown in the example input in Table D-1 for the example and are not included here. Following these cards, the input for BRK2D is: Card #1 - Title card, Format (8A10). 80 alphanumeric characters. Card #2 - Logical control card, Format (5110,6I5). N = Number of straight line segments used to fit the hull. NW = Number of points on the free surface where wave height is to be computed. 107 Card #3 - Card #4 - Candei5 = Card #6 - Card #7 - NWAVEL = Number of wavelengths at which computations are to be performed. = 1 for symmetric section. = Anything else for non-symmetric section. ISKIP = 1 Do not solve equations of motion, 2 Do not solve potential problem (read in coefficients from data), = Anything else solve potential problem and equa- tions of motion. Number of body segments which represent spaces be- tween multiple hull configurations (1 to 5). JC = Designates the segment number for segments repre- senting spaces between multiple hulls. Parameter card, Format (5F10.3,3A10). AREA = Crossectional area of immersed body. B = Characteristic beam as specified by BTITLE. D = Distance below free surface to origin of users coordinate system (all motions are referred to that point). ROE = Fluid density. GEE = Acceleration of gravity. BTITLE = Specifies B. Beam/wavelength specification, Format (10F8.5). BOL = Beam/wavelength ratios for computation (up to 10 different ratios may be used). Offset cards, Format (2F10.3). There must be N+l cards giving the offset points. In the version of the program used here, N must be less than or equal to 23 because of dimension statements. R(1,1I) = X-coordinate of offset point. R(2,1) = Y-coordinate of offset point. Hydrostatic spring constants, Format (9F8.3). This is read in subroutine DYNAMC. ISYM LC RKHYD (1,1) = KHyj. RKHYD (1,2) = KHj RKHYD (1,3) = KH]3 RKHYD (2,1) = KH RKHYD (2,2) 5 KH99 RKHYD (2,3) = KH93 RKHYD (3,2) = KH3> RKHYD (3,3) = KHz Physical properties, Format (6F10.3,3F5.2,15). This is read in subroutine DYNAMC. AREA = Crossectional area. B = Characteristic beam. XG = X-coordinate of the center of gravity. YG = Y-coordinate of the center of gravity. RMASS = Mass per unit length of breakwater. RINERT = Mass moment of inertia per unit length of breakwater. 108 Card #38 = Card #9 - Note: DAMP(1) = Added damping in sway. In the equations of motion sway damping will be 1+DAMP(1) times the computed hydrodynamic damping. DAMP(2) = Added damping in heave. DAMP(3) = Added damping in roll. NPUNCH = 0, punch data cards containing computed trans- mission coefficient, motion response and mooring- FORCE COekrTcIeMnit. Anything else, do not punch data cards. Mooring spring constants, Format (9F8.3). This is read in subroutine DYNAMC. RKMOR(1,1) = KM)q RKMOR(1,2) = KM}? RKMOR(1,3) = KM]3 RKMOR(2,1) = KM>1 RKMOR(2,2) = KMy> RKMOR (2, 3) = KM93, RKMOR(3,1) = KMz) RKMOR (3, 2) = KMz3 RKMOR(3,3) = KMz3 Mooring-line response parameters, Format (6F10.2). This card is read in subroutine MORTEN. DET Gis) a AF/Aa, for shoreward mooring line. This is the Change in mooring line force per unit displacement in sway. DELT(1,2) = AF/Aaz for shoreware mooring line. DELT(1,3) = AF/Aaz for seaward mooring line. DELT(2,1) = AF/Aa, for seaward mooring line DELT(2,2) = AF/Aaz for seaward mooring line. DELT(2,3) = AF/Aaz for seaward mooring line. The last 3 cards (#7, #8, and #9) provide the information needed for the dynamic analysis. If it is desirable to perform calculations varying the data, these cards may be repeated with different input data. There is a limit of 25 different sets of data. In the example data shown in Table D-1, there are 3 different conditions used, Mathematical Procedures and Program Limitations. The mathematics has been described in the report and Appendix C. The main limitations are that at most 23 offset points may be used A listing to describe the shape. This has been found to be very adequate for the configurations when more than the square of the number of points. considered thus far. Little change in the results occurs 15 points are used. Computer time increases about as of the program is given in Table D-2. Flow Chart. A flow chart is given in a figure of this appendix. 109 MU11/QM MU13/(04*8) MU22/0M MU31/(04*B) MU33/(QM*B*B) LAMBDAI1L/QD LAM30A13/(Q0*B) LAMBDA22/00 LAPBDA3L/(QD*B) LAMB0A33/(Q0*B*B) FX/QF MZ/( QF*B) GEN BY SWAY/SWAY GEN BY ROLL/ROLL(RAD) *8 INCIDENT/ETA TRANS BY FXD 3DY/ETA BEAM/WAVCLENGTH ADDED MASS QM = AREA*ROE MU12/0M MU21/Q6 MU23/(QM*B) MU32/(QM*B) LAMBDA12/QD LAMBDA21/0D LAMBDA23/(00*B) LAMBDA32/(QD*B) FY /QF GEN BY HEAVE/HEAVE REFLECTED BY FXD BDY/°TA REFLECTED + INCIDENT/ETA DIMENSIONAL FREQUENCY - H7 DAMPING QD 2 AREA*R JE ew WAVE FOZCES QF£AREA*ROE*ETA*W2PHASE REL TO ETA AT X29 = NEG PHASE REL TO BODY MOTIGN — DEGWAVE FIELD - AMPLITUDE 2ATTAPS POSITION = X/WAVELENGTH SWAY AMPLITUDE/ETA ROLL AMPLITUDE(RAD)*B/ETA GEN BY RESULTANT SWAY/ETA GEN BY RESULTANT ROLL/ETA TOTAL REFLECTED/ETA MOTION RESPONSE DIMENSTINAL POSITION = Xx HEAVE AMPLITUDE/ETA GEN 3Y RESULTANT HFAVE/STA TOTAL TRANSMITTED/ETA OAK HARBOR BREAKWATER = CORPS JF ENGINEERS TESTS 23 0 10 0 ty) l ue 1226 10.G 6.0 1.9905 32o2FULL BEAM el 0159290 0180 216311 °250 0280 0312298 e371 520 Le0 50 -1.25 5.0 2250 -5.0 -3.75 520 =F 0 4.583 -5.200 4.583 3.75 3.223 -3.75 -3.223 -2.50 -3.223 1.25 4.583 -1.25 4.533 0.00 42583 0.0v0 40593 “1.25 32223 -1.25 30223 -2.50 3.223 -3.75 4.383 -3.75 42583 5200 520 -5.00 5.0 -3.75 300 -2.50 5.0 =-1.25 300 0.0 900° Ve Ov 0.0 6405 020 320 0.0 12.6 10.0 0.0 2.34 2501 621. te) Oe Ve QO. 0. % 06 Ve Oo 12.6 ic. 300 —2.3% 2501 671.6 9 116.8 5224 166.2 52732 139.21 -3.372 159.9 2.963 -1376. 410.6 -1607. 1172. 28009 1713. 1206 10.G 0.0 2234 2501 621. 1 118.8 5024 166.2 -52732 10.21 32372 15909 2.953 -1376. 41006 -1607. 1172. 280.9 1713. Table D-1. breakwater) . 110 17 MAY 1975 0429 o4R7B25 1165. ° D. 0. 1 D6 ° do iol) 281.98 ° le 1. 26108 Example input for program BRK2D (Oak Harbor BRK2DHU FLUO TALETOA TAA TTT ALATA ETT RUNT VERSION FEB 74 B 17:12 04/23/76 PROGRAM BRK2D (INPUT »QUTPUTs PUNCH»s TAPESZINP'T, TAPFEZQUTPUT ) Ce**LATEST REVISION ***** 27 AUGUST 1975 C***PROGRAM BRK2D COMPUTES THE FIRST-ORDER RESPONSE OF AN OSCILLATING c CYLINDER ON CR NEAR THE FREE SURFACE OF AN TVEAL FLUID OF Cc INFINITE DEPTH 3 COMMON RI12625925)92K56(25925) 5 POUT( 25» 25)5 YOW(25s25) oF E( 2504)» RU(25925)9 2K(2594)5 2L(2594)>5 FB(3910)9 YELFIA( 37010)» HWR( 2596510)» LFI(2596)5 RI(25925)5 2RMU(393910)5 RLAM( 3935910). 3 DELW(255s6010) » XOL(25510U) 3 COMMON/ONE/X(25)9Y(25)5%8(25) 9 V¥8(25) 9 ANG( 25) 0DEL (25) 9 VV(25) LoFEIN(25)5 ETIN(25)59 RNORM(2593)9 JEC5) 3 COMMON/GNE2/CC3(29)59SS83(25) 3 COMMON /TWO/ NoNNWo NWAVEL» ISYMe TSKXT2o NO&o PLE x GAMMA oMs TK 9 TO 3 COMMON/THREE/ WAVEL( 10)» WN(10)5 RIL(LIISTIL 3 COMMON/SIX/XN(5)9°N(5) 3 COMMON /SEVEN/ AREA» Bs Ds ROE» GEEs ATTTLE( 3)» TITLES) 3 COMMON / ELGHT/LBLMU(35393)5 LSLAM(393049)5 LILFB( 353) oLBLYW2( 753) 5 ALBL(1093)0 DEG( 3010) 3 COMMON /NINE/ LBLRAQR(393)0 LELHWRISe3)0 LALRO 553) 3} 9 CGMMON/TEN/DELT( 293) 9FIR( 2910) o PHAS( 2019) oF IRND( 2010) » PHASI(2010) 3 REAL K 3 DATA KN/026356031L97TiS 5 bo GL3GI3SU591F 7s Fo EGSG 257 TING) » L 70603581C005659012 26406008 4427A/ 3 DATA CN/.5217556105535 0396666811983 .075942449E6K1 75 1 0003611758679 925 6000023369977239/ 3 PIEZATAN2Z (929-20) 7 TP=2.¥*PIE 49 GAMMA20.57721565 C*¥**BEGIN READING INPUT DATA AND PRINTING EC4) CHECK 12 3000 FORMAT (6419) C¥e*ekEAD LABLES FOR PRINT OuT 12 PEAD 30005 ((C(LBLMU(IoJoL)» L = Lo3)5 J = Lo4)o LY = 103) 36 READ 30005 (((LBLAM(IoJoL)» L = 193)9 J = 193) YT = 13) 63 READ 30905 (CCLBLFBC Job)» L = 1lo3d0 J = 193) 103 READ 3000» (CLBLHWB( Jobo L = 193)9 J = Ts?) 123 READ 3000» (( LBL Jol)» L = 1le3)d50 J 2 10910) 143 READ 30009 ((LELRAR(JoL)s L = 193)5 J = 153) 163 READ 3000» CCLALHAR( Job)» L 2 193)0 J = 155) 203 READ 30005 (LBLR( 1st)» L = Lod) 229 READ 205 TITLE 226 20 FORMAT (8Al10) 226 PRINT 300 TITLE 234 30 FORMAT (1H1» SALN///) 234 FREAD 505 Ns NWy NWAVEL» TSYM» I[SKTPs LCs JC 256 59 FORMAT (51105 615) C N = NUMBER QF STRAIGHT LINE SEGMENTS TO SE USED TO FIT C THE HULLeceoeoNJTE. THERE MIIST 3E N+1 OFFSET POINTS Cc NW ® NUMSER OF POINTS JN FREE SURFACE WHERE WAVE HEIGHT VS Cc TO 38 COMPUTED. THIS IS IN ADDITION TO THE COMPUTATION NF C WAVE HEIGHT 400 WAVELENGTHS ON FITHER SIDE GF THE RONDY Cc WHICH 1S PERFO2MED AUTOMATIZALLY C NWAVEL = NUM3ER OF WAVELENGTHS 47 WHICH COMPUTATIONS ARE TN Cc BE PERFORMED C ISYM = 1 FOR SYMMETRIC SECTION Cc = ANYTHING FLSE FOR NON=SYMMFTRTC SECTION Table D-2. Listing of program BRK2D RUNT VERSIUN FEB 74 B 17212 04/23/76 254 20u 262 264 267 273 315 315 342 342 367 367 372 406 406 41) 417 433 433 ISKIe 100 NOT SOLVE EGUATTINS JF “MOTTON 2 0U NOT SULVE POTENTIAL PROBLEM (READ IN COFFS) ANYTHING ELSE SOLVE FON® CNEFFYCTENTS AND DYNAMICA NUMBER OF 359DY SEGMENTS WHICH 2EPRCSENT FREE SURFACE RETWEE! CATAMAARAN HULS. SEGMENT NIMRERS SPECTFIED RY JC(5 NAW = Nw + 2 NC = N = LC NW1 = 25 - N — 2 IF (NW elLTe O) NW = O IF (Nw eGT.e NWl) NW = NwL PRINT 605 No Nwo NWAVEL® ISYMs ISK¥%, L%s» JC 60 FORMAT (1JX*NUMBER OF SEGMENTS =#5 T4// 1 LUX*NUMBER OF FREE=SURFACE STATTONS =*» Y4s/ 2 LOX*NUMBER OF WAVELENGTHS 2¥*5 I[4// 3 4 a uo of aNgNaNANN LOX¥*ISYM =¥*5914// LOX¥*ISKIP =*5 T4// LOX¥*NC 2 *159 *JC 2* 515 /) READ 70» ARFEAs Bs Ds ROE» GEE» (BTITLE(T)s» F = 1y3) 79 FORMAT (5F10.3»9 3A10) AREA = CRUSSECTIANAL AREA QF YMMERSEN B7DY B = CHARACTERISTIC LENGTH AS SPECTFIFD 3Y BTITLE OD = DOTSTANCE BENEATH SURFACE DF O29GTN VE USFRS COORNYNATE SYSTEM (+). ALL MUTIJNS REFE2ED TO THAT POINT AND BODY SHAPE SPECIFIED IN THAT SYSTEM ROE = FLUID DENSITY GFE = ACCELERATION OF GRAVITY PRINT 605 AREAs80 (3TITLEC(I)» I = 293)0 Mo RVEs GEE 80 FORMAT(1OX #AREA = *9 FLO03 //5 19K *B = €F1903 0 5K» 34106 2 //10XK% *D0 = * FLI063 // LOX FFLUID DENSTTY =* F10.5// 3 LUX*FACCELERATION JF GRAVITY =*» F12.3/) LFCISKIP 2&2. 2) GO TO 403 READ 1CUe (80L(1)»5 £ = le NWAVEL) 100 FORMAT (10F8.5) C SOL = BEAM/WAVELENGTH RAT( FOR SCIMPUTATTANS 00 é63¢ I = LyNWAVEL 600 WAVEL(I) = R/B0L(1) C WAVEL( I) = DIMENSTONAL WAVELENGTY DF TNCIDENT WAVES PRINT LlG» (89L0T)s IT = Jo NwAVEL) LLOQ FORMAT (1LOX*PEAM/WAVELENGTH RATIOS DF YNCTOFNTWAVES®//7( 25K 19°F. 1)) C#¥eeee INITIALIZE DUTPUT VARIABLES DO 113 IL = 1eldv WNCIL) = 9.9 BOL(IL) = vev OC 114 I = 193 FB(IsIt) = O.0 OELFB(IsIL) = ved DO 114 J = 153 RMU( 1» Jo IL) = 020 RLAM(I» J» IL) = GeO 114 CONTINUE OG 112 I = ls 25 XOL(IsIL) = GG OC 112 J = 196 HWB(IoJsIL) = 00D DEL W(IsJsIL) 20.9 112 CONTINUE ANMAAANANHA Table pD-2. Continued Ne RUNT VERSTON Feb 74 8 17:12 04/23/76 531 533 534 541 906 631 633 642 647 547 651 559 671 672 674 705 717 725 732 749 747 756 UUe 1007 1015 1023 1031 1034 1641 10%5 1045 1113 1113 113 CUNTINUE CeREeeCOMPUTE (B/WAVEL) AND NONDIMENSTINAL WAVE NO. DO 115 IL = ls» NWAVEL BUL(IL) = B/WAVELCTIL) 115 WN(IL)= TP * B/ wAVEL(IL) C#*#READ IN OFESETS OF CYLINDER NUP = N + 1 NUPD = NUP + 2 NToe = NUPP + NW - 1 Nl =N #1 # NW READ 130» (RI(LoI)oX£( 201) 9123 9NUP) 130 FORMAT (2F10.5) C RI (lel) oR (291) = DIMENSIONAL X» Y COORDINATE OF OFFSET C POINTS» RESPECTIVELY TF (NW eLT. 1) G3 TO 185 C***RZAD IN ADDITIONAL POINTS ON THE FRFE SYRFACF WHERE WAVE HFTGHTS (S ARE TO BE COMPUTEN. Tawi) STIRAGE LOATATIINS MUST BE LEFT BLANK Cc FOR THE POSITIGY 4 wAVELENGTHS FROM THE AONY READ 13Uy (RIC LoL) oRI (291) oT =NUPPONTOP) C RI (loldsRI(291) = COORDINATES TE ONINTS ON FREE SURFACE WHERE Cc WAVE HEIGHT IS TO BE COMPUTED. THIS [© TRUE FOR I eGTe N + 32 C#**NON-DIMENSICNALIZE YFFSETS DC 16u I = NUPPsNTOP M(I) = RICLsI)/B 482 YCI) &® -levwt-68 185 CONTINUE OC i90 I = ls»NUP YB(I) = RIC1ls1)/8 19u YB(I) = RI(291)/5 Cooo COMPUTE MIDPOINT» ANGLE AND LENGTH IF ST2ATGHT-LINE SEGHENTS. C#eeee AND COMPONENTS OF NORAAL TO BUDY JF =O OG 200 J2l0N X( J) Oo S*( MBI ISI+XB(IJt1)) YOS) =O. 5*(VACSI+VI(S+1)) Ti=YVR(J+1)-Y8( 4) T2=XB(J+1)-X8( 5) ANG (J) =ATAN2(T19T2) CC3(J) FCIOSCANG(J)) SS3(J)=SINCANG(J)) VVC SDSX(CSDECC ICID FV (J) *SS3 05) CEL(J ) = SORT(T2**2 + T1**2) RNORM(J ol) = -SS3(J) RNGRM(J 9?) = CC3(J) RNORM(J 93) = VV(J) 2)0 CONTINUE PRINT 305 TITLE PRINT 250 259 FORMAT (eex*®CYLINDER GENMETRYP///TOVRNTMENSTINAL OFFSETS #9 i LI X#NON-DIMENSLONAL OFFSETS%» SX*MTNPNINTS OF SEGMENTS¥*// 2 BKETE, LOXEKHs GDXRVR> LINK, WHVHs LIVEXH, GVH 3 18X¥*SLIPE*s SX*LENGTH*/) PRINT 2709 (IToRIC1lo Ll) WT (25T) oXB(T)roVYAC TIO KI TID VISTI oANG(T) © 1 DEL( LT)» IT2=1l5™) 270 FURMAT (X» [69 4(10X2F10.3)) NL =N #1 Table D-2. Continued WS QUNT VERSMON FEB 74 8 17:12 J4/23/76 4115 1143 1147 1147 1147 1152 120% 1204 1210 1213 421% 1222 1233 1242 1243 1245 1247 1252 1253 1254 1261 1265 1271 1300 1307 1312 1316 1317 1320 1326 1339 1344 1353 1356 2370 1403 1414 1424 1427 1431 1436 1443 1441 1469 PRINT 2700 NLo RIC LoNL)» RIC 29NL)9 KR(NLY® YACNL) PRINT 281 281 FORMAT (/7/19X» *PQOSITIONS FORK WAVE HETGYT CALCULATIONS */) 280 FORMAT (//) IF (NW ol Te 1) GI TN 299 PRINT 27lo (TeRI( LoL oRIC20T do X(T o VC TI 9 TENUOOONTIP) 271 FORMAT(X915s 2(10X%»5 2F1003)) PRINT 289 290 M=z=N + Nw + 2 C*¥**F*E TRANSFER TO CUORDINATE SYSTEM IN FREE SIIRTACE 00 285 I 2 lon YR( 1) = YB(T) = 078 235 YI) = Y(t) = D/B YACNUP) = Y3(NU9) = O/B C COMPUTE FACTORS GF I AND K INDEPENDANT SF FREQ'NENCY CALL COEFF YSURF = =-1.0£=-08 * 8 Ceooe START FREQUENCY ITERATIONS DO 301 IL 2 ls NWAVEL K = wN(IL) C ALL POTENTIALS INITIALIZED TO ZER. CeeeeeFE (Ig J) = NONDIMENSTONAL AMPLITUDE DE POTENTIAL AT POINT T NUE TO C#*eeEMNDE Jo (ASSOSTATED wITY CUS(WT) VY. FTC ToS) [5S SIMILAR TO FECT» J) C#**eeBUT ASSOSTATED WITA SINCWT) © J = 19203940555 IMPLY RESPECTIVELY C¥RRESHAYs HEAVEs ROLL» DLFRACTEDs INCTOENT AND OVFRACTED + INC TOFNT OO 1 T#ls25 DO 1 J = 1s$ FE( Is J)2#6. FI(IsJ)=0. 1 CGNTINUE C#¥*ea0N POINTS TO THE OFFSET ARRAY FOUR WAVELENGTHS FROM THE ORTGIN Cc ON THE FREE SURFACE K(N@2) = 4.0 * WAVEL(IL)/8B M(N#2) = -400 * WAVEL(IL)/8 Y(N+1) = YSURF Y(N+2) 2 YSURF C¥*eCIMPUTE INCTOENT WAVE POTENTIALS AND NIQMAL VELOCITIFS DO 4€2 1 = 1sM DOSU4 IC = 195 404 IFC(I eEQ. JC(IC)) GI TO 402 Ey = EXP(KeY(T)) CKX = COS(K*X(T)) SKX = SIN(K*X(T)) IF(I eGTe N) GO T) 443 FEIN(L) = EY#(SS3CT)*SKX + CC3(T)*TKM) FIIN(T) s=-EV#(SS3CLIECKX = CC3(TI*SKK) 403 FE(195) = EV*CKX*(1Lo/K) FI(I95) = EY¥SKX#(1./K) 462 CONTINUE AK = K/B WRF = SQRT(GEE*AK) WE = WRE/TP wT 2 1.0/WF PRINT 3000 Ky WRF» wFs WTs wWAVEL(TL) 300 FORMAT (//5 * WAVE NUMBER = K 2%5 FQ,5e S¥*CTRCULAR FREQUENCY =%» 1 F905 SK*FFREQUENCY =*5 FGe55 SX¥*¥PFRT JD =#5 Flicds Table D-2. Continued 114 RUNT VERSION FEB 74 B 17212 04/23/76 2 SX*WAVELENGTH 2%, FIL4) 1460 TK=2.0/K Cc FIRST-ORDER POTENTIALS ON CYLINDE® AQE FIRST CALCULATED. 1462 CALL COMP(K) Coos FIRST-ORDER PHYSICAL QUANTITIES ARF CALZULATCD. 146% CALL PHYSCL(K) 1464 301 CONTINUE 1471 1009 FCRMAT (10F8.5) 1471 ISKIPL = 2 C¥eeEEPUNCH RESULTS JF FOTENTIAL SOLUTION ON TARDS 1472 PUNCH 2€5 TITLE 15900 PUNCH 505 No NWo NWAVEL» TSYM» ISKTPl» LC» JE 1522 PUNCH Tus AREAs Bo' De REE» GEEo BTTTLE 1542 PUNCH 10095 (WNCIL)o IL = lolvds (BILCTLY» TL = lo lu) 1564 OG 310 I = 1s3 1566 PUNCH LUbd » (FSC LoL)» IL = Lodb)e COFLFAC To TIL)» IL = 14918) 1613 OO 310 J = 193 1615 310 PUNCH 1uG%> (RMUCTsJoIL)o IL 1919)5 (RLAM(ToJoIL)>o IL = 1510) 1652 DO 320 I = 1lsNNwW 1654 PUNCH 10099 (XOL(LoTILIoIL = 15910) 1673 CO 32u J = ls6 1972 320 PUNCH 10099 (HWO([LoJoTL)» IL = 10619) s(MELW( Todo IL)» IL = 1519) 1739 303. CONTINUE 1736 116 CONTINUE 1730 CALL DYNAMC 1731 302 CONTINUF 1731 SToP 1733 END Table D-2. Continued ks) RUNT VERSION FEB 74 8B 17:12 94/23/76 272 312 312 316 317 SUBROUTINE DYNAMC COMMON RI1L2(25525) »RKE6(25925)5 POT(25075)9 HOW( 25925) o FF (2506) 9 LFI(2596)5 RY(25925)5 RII25925)5 RKIP54)0 RL(2594)5 2RMU(303919)0 RLAM( 393910) 5 FB(3910)5 DELFP( 3910)» HWR( 25965910)» 3 DELW( 2596010) » XOL(25510) COMMON/ONE/K(25)0Y(25)5 XB (25) 5 VB(25)5 ANG (29)5DEL (25) 5 VV(25) LoFEIN(25)9 FIIN(25)9 RNURM(25593)9 JO(5) COMMON /TWO/ No NNWo NWAVELs ISYM>» TSKIP» NCs PIEsGAMMAs Ms TKs TP COMMON/THREE/ WAVEL(10)5 WN(10)»5 BOL(, ~_TL COMMON/FOUR/RAR (3910) »DELR(3910)9 HWR(2593519)5 DELWR( 2593910)» 2 HWT(25910) sDELWT (25910) »DRKHYD(3932)9 RKANR(393)0 RKT83(393)5 3 XGe YGs RMASS» RINERTs DAMP(3) COMMON /SEVEN/ AREA Be Do ROE» GEE» ATITLE(3)5 TITLE(B) COMMON/TEN/DELT (253) 0 FOR( 2910) » PHAS( 2519) 5 FE NRND( 2510) »PHASN(2510) OLMENSTON Al6s6)5 Cl6)59 ERASE(6) IF ( TSKIP o.NE- 2) GO TO 100 C¥*¥***eREAD POTENTIAL CUOEFS IF ISKIP = 2 READ 10095 (WN(IL)oIL = lsl9)5 (BSOL( TL)» TL = 1s 16) 1009 FORMAT(10F8.3) DO 110 I = 1953 READ 10095 (FA(ITsIL)» IL = 1910)5 (DFLFR(ToTL)» IL = 1510) DO 110 J = 193 110 READ 2U09» (RMUCTo Jo IL)» IL islO)o (RLAM(ToJoIL)» IL = 1519) 00 120 I = 1 » NNW READ 10099(XOL(Is IL)» IL = 1910) 00 120 J 2 16 120 READ Lud9o (HWE( To JoIL)» TL = lo tM) o(DFLW( I odo IL)» IL = 1919) 100 CONTINUE C*#**¥eeQUTPUT POTENTIAL COEFS CALL POTOUT IF(ISKIP et9O5. L) GO TO 14% CeeeeeREAD DIMENSITONAL JYOROSTATIC SPRING CONSTANTS READ LUGS, ((RKHYD( Io J) 9 J=193) »L=1e03) C#¥****START LOOPING THROUGH DIFFERENT DYNAMIC CONFIGURATIONS DO 140 K1 = ls5vu READ 1G10» AREA» Bo XG» YGs RMASSs PTNER TSN AMP(1) »DAMP( 2) sPAMP( 9) 5 1 NPUNCH aOLU FORMAT (6F10.35 3F5e2n 15) IF (EOQF95) 1l2ls 122 121 STOP 122 CONTINUE DU 5 K2 = 153 5 ITF(DAMP(K2) eEQe —God) DAMP(K2) = 9.2% C XGoYG = COORDINATES OF THE CENTER DF GRAVITY OF THE BODY C ecooeNOTEe MOMENTS AND MOMENTS OF INERTIA ARE COMPUTED C A8QUT THE CENTER OF GRAVITY Cc DAMP ADDS CORRECTION FOR VISCOUS 17° NON CUNE RR DAMPING C*¥**eeREAQ DIMENSTONAL MOGRING SPRING CONSTANTS READ 1LOG8s((RKMOR(I9J)9J=193) »T=193) 1003 FORMAT (9F6.3) C#eeeeNONDIMENTIONALIZE SPRING CONSTANTS »44S3S9 MOMNNT OF INERTIA C*#*eeeAND CG COORDINATES Q = AREA* ROE*® GEE/A 00 130 [ = 1593 DO 130 J = 193 Table D-2. Continued H6 RUNT VERSION FEB: 74 B 17812 04/23/76 320 331 346 372 402 406 412 413 415 416 421 445 471 531 552 576 623 633 643 652 654 655 666 705 723 736 753 771 773 774 1005 1006 1032 1047 1050 1051 1052 1054 1072 1111 1132 1155 1163 1204 1225 IF(RKMOR(IsJ) c€Q0 ded) RKMOR(IsJ) = 900 RKTB(I9J)2 (RKHYD(I9 J) + RKMOR(T0J))/0 130 IF(TcEQ. 3 oORe J cEQe 3IRKTB( To J) SRKTR(Ts J) /B RKTB(393) = RKTB(393)/8 RMASSB = RMASS/(AREA*ROF) RINERB = RINERT/(AREA*ROE*B*B) XGB = XG/3 YGB = YG/B C#eeeeSTART WAVELENGTH LOOP 00 150 IL = 1910 IF (IL »GT. NWAVEL) GO TO 1450 C*#*eekSET VALUES IN NONDIMENTIINALIZED ALGERRATC FQUATIONS OF MOTTON ACly1) = RKTB(isi) — WN(IL) * (RMASSBR + RMU(CLs>1sTIL)) A(252)= RKTB(292) = WN(IL) * (RMASSR + 2MU( 2s 29IL)) A( 353)" RKTB( 393) -YGB*RMASSB —WN(TL) * (RYINERB + RMUC3935TL) + 1 (XGB*¥*2 +YGB**2) * RMASSB) A(1l92) = RKTB(1ls2) — WN(IL) * RMU(T»25TL) Al(ls3) = RKTB(193) — WNC(IL) * (RMUC1>e35TL) -YGR*¥RMASSB) A(293) = RKTB(253) = wN(IL) * (RMU(2939TL) +XGB*RMASS3) A(291) = A(1s2) A(391) = A(1»3) Al3e2) = A(293) DO 2G I = 13 DC 10 J = 1593 A(I4#3»5 J*#3) 2 ACTo J) ACIs J+3) =RLAM( Is JoIL)*SQRT(WNC(TIL)) C ADO CORRECTION FOR VISCOUS DAMPING : IF (I c€Qo J) ACIoJ+3) = (1.0 + DAMPC(T)) #A( Ts J+3) 10 A(I+39J) = = A(IaJ+3) C(I) = FB(LoIL)*® SIN(DELFB(IsIL)) 20 C(I+3) = FB(IsIL) * COS(DELFS(IeIL)) SCALE = le NN = 6 C*¥#ee*SOL VE ALGESRAIC EQS OF MOTION. 8(1)5 Bl)» B(3) = AMPLITUDES C¥PeeEOF COSC WT)» 6(4)5 3(5)9 3(6) = AMPLITUDES OF STN(WT) FOR SWAY» HEA C#**#*AND ROLL AT CENTER FO USERS COORDINATE SYSTEM. LL = LNEQF(6oNNolaAsCoSCALE ERASE) DO 30 I = 153 C#**** AMPLITUDE AND PHASE OF RESPONSE RAR(IsIL) = SORT(C(T)*#2 + C(I+3)¥**2) 30 DELR( Is IL) = ATAN2(C(I)sC(I+3)) OG 46 I = ls NNW Aw = 0. BW 2& Oc 90 DO 50 J # 103 C¥*eexeRESULTANT WAVE AMPLITUDE AND PHASF FOR SWAYs HEAVE ROLLe HWR(IoJdoIL) = HWB(IoJ9IL) * RAR(JoTL) OELWR(LsJoIL) = DELW(1sJoIL) + DELR(JoTL) AW = AW + HWR(IoJ» IL) * SIN(DELWR( Ts Jo TL)? 50 BW = BW + HWR(IsJoIL) * COS(DELWR(UsJsIL) ) TFC(XOLCIsIL)eLTe O-) GO TO 70 C¥*e*e* TOTAL REFLECTED WAVE (VECTOR ADDITION) Aw = AW + HWB(Ts6sIL) * SIN(DELW(Ts%eTL)) BW = BW + HWB(Is6nI1L) * COS(DELW(T 65 7L)) GO TO 45 C¥**ee* TOTAL TRANSMITTED WAVE(VECTOR ADDITION) Table D-2. Continued 117 RUNT VERSION FEB 74 B 17212 64/23/74 1225 7 AW = AW + HWB(Is4sIL) * SIN(DELW(T»4s TIL) ) 1246 Ba = Bh + AWB(To4oIL) * COS(DELW(Ts 4s TL)) 1267 45 HwWE(Io TL) *SORTCAW**2 + BW**?) 1307 DELWT( Is IL) 2ATAN2 (AW, Rw) 1317 40 CONTINUE 1321 GO TO 1590 Ceeeee SET QUTPUTS FOR IL eGTe NWAVEL 1322 160 DO 170 I = 153 1324 RAR(IsIL) = GeO 1330 170 DELR(isIL) = Dod 1337 0O 180 [ = 15925 1349 HWT(TsTL) = Ged 1344 CELWT(1sTIL) = 00” 1351 00 180 J = Lea 1352 HWR(TeJoIL) = 0.0 1361 180 DELWR(IsJoiL) = 020 1374 15G CONTINUE C#eeee OUT PUT OYNAMIC RESULTS 1376 CALL I YNOUT 1377 NMOR = O 1400 DO 139 IP 2 1s3 1402 DO 139 LQ = 193 1403 IF (RKMOR(IPsTO) oEQ. eC) GO TO 139 1410 NMOR = NADR + 1 1412 139 CONTINUE 1416 1F (NMOR eNEo O) CALL MORTEN 1421 TF (NPUNCH eNEe 0) GO TO 140 1423 PUNCH 2000 1427 2060 FORMAT (*1112i11111%) 1427 PUNCH 20055 (SOLCTO)SHWT( Ls TO) s RAR C15 TO) »pPAQ( ArT Qs RAR( 35 TO) 0 1 TO = 1910) 1466 2005 FORMAT (5F1004%) 1466 PUNCH 2000 1472 PUNCH 20195 (80LCTQ)5FIRND(15TQ) oF NQNN(25TO)e 1021910) 1517 2010 FORMAT (F1904 2620.4) 1517 2140 CONTINUE 1521 RETURN 1522 END Table D-2. Continued 118 RUNT VERSION FEB 74% 8 17812 04/23/76 SUBROUTINE MORTEN C*#**eSUBROUTINE MORTEN COMPUTES FORCES IN THE MOORING LINES aAaaAD COMMON/FOUR/RAR( 3910) 9DELR(3591U) 0 HWRI 255 Ar 1V)5 DELWR( 255 3519)0 2 HWT( 25510) oDELWT (25910) sRKAYD(3932)5 PEMNE(353)5 PKTB(353)5 3 XGs YGs RMASS» RINERT» DAMP(3) COMMON /SFVEN/ AREAs 39 Ds ROE GFE» BTITLS(3)» TITLE(8) COMMON/TEN/DELT( 253) »FOR( 2919) » PHAS( 201 DV) oFIRND( 2910) » PHASD (2910) READ 10 ((DELT( Is J) 932103) 512152) 10 FORMAT (6F10.2) DELT(LoT)oI2193 = CHANGE IN FORCE IN SYORTWARD MOORING LINE PER UNIT DISPLACEMENT YN SWAY HEAVE AND ROLL DELT( 251) 912193 = CHANGE IN FORCE IN SEAWARD MOORING LINE PER UNIT DISPLACFMFNT IN SWAYs HEAVE AND ROLL CAB = 1.O/(ROE*GEE*ARE A) CONS = 180.0/ACOS(-1.0) DO 100 J = Is2 PRINT 20 20 FORMAT (////20X*MOORING LINE MODEL RESULTS*7/) IF (J oEQ- 1) PRINT 18 IF (J eEQe 2) PRINT 19 18 FORMAT (30X*SHOREWARD MOORING LINE*/) 19 FORMAT (30X*SEAWARD MOORING LINE*/) PRINT 305 (DELT(J9K) 9K=153) 3U FORMAT (* CHANGe IN FORCE PER UNIT DISPLACEMENT IN SWAY HEAVE* 1 * AND ROLL» RESPECTIVELY =*5 3F19.4//) C*#**COMPUTE FORCES IN MOORING LINES AND PYASE 00 50 I = lelu AA = RAR( is T)*DELT( Us) AB = RAR(2sT)*OELT( Js 2) AC = RAR(39T)*DELT(J593)/8 TS = AA*SINCDELR(151)) + AB¥SIN(DELR(20T)) * AC*SIN(DELR(30T)) TC = AA*COS(DELR(L5T)) + ABSCOS(DELR(20T)) + ACHCOS(DELR(30T)) FOR( JT) = SQRTCTS*TS + TC*TC) PHAS(J91I) = ATAN2(TCsTS) FORND( JT) = CAB¥FOR( Jo T) PHASD(J9T) = CONS*PHAS( Jo) 50 CONTINUE C#**PRINT RESULTS PRINT 805 (FOR(JoT)oL=L91O)5(PHASD( JoT)oT=1910)» LL. (FORND( Jo To T21510) 80 FORMAT (3X*MQORING LINE RESPONSE®/5X®FORCE AMPLITUDE/ETA*. 11%. i LWELO.3/EX*#PHASE REL TD ETA AT X¥8d = NEG %» ICF10.4// 2 5X30HFORCE AMPLITJDE/RIE*GFAREA*ETA, 10F10.3) 1v0 CONTINUE RETURN END Table D-2. Continued W9 RUNT VERSION FEB 74 8 17:12 04/23/76 SUBROUTINE COEFF Cc THIS SUBROUTINE CALCULATES THE PARTS OF T(ToJ) AND K( Ts J) Cc WHICH ARE INDEPENDENT OF FREQUENCY NUMBER Keo 2 COMMON RI12(255925)5RKE6(25925)5 POT(2%9?5)9 HOW( 25925) 9FE( 2596) 9 LFI(2506)9 RI(25925)5 RU(25925)5 RKI2A594)5 PL (259 4)» 2RMU(393919)9 RLAM( 393910)» FB( 3010)» DELFB(3510)5 HwR(2596910)» 3 DELW(2595910) » XOL(25910) 2 COMMON/ONE/X(25)9 ¥(25)5XB(25) 5YB(25)9 ANG (75)eDEL (25) LoFEIN(25)9 FIIN(25)9 RNORM(2593)»9 JOC5) 2 COMMON /TwO/ NoNNWs NWAVEL» ISYMs TSKIPs NC» PIEsGAMMAaMs TKo TP 2 COMMON/ONE2/CC3(25)5SS3(25) 2 N2 = N/2 6 0O 4 IT = lsM 10 IF(I .GT. N) GO TO 7 13 TFCISYM e£Qe 1 eANDe I oGTe N2) GO TN 7 26 X11 = X(T) - XB(1) 34 Yll*V(1)-Y8(1) 41 x21 2 X11 + XB(1) 45 Y212Y(1)+Y3 (1) 53 PPL2ALOG( X11 ¥*2+Y11**2) 66 PQLEALOG(X11¥**2+Y214*2) 101 TPL=ATAN2(Y115X11) 105 TOLZATAN2(Y215X11) 111 00 1 J = loN 112 X122X(TI—xXB(J4+1) 120 Yl2"YV(T)-Yb(J+1) 124 Y22sV(I)+Y8(J+1) 131 PP22ALOG(X¥12**2+Y12**2) 144 PQZBALOG(X12*¥*2+Y22**2) 157 TP2=ATAN2Z(Y12s9Xl2) 163 TQZ22ATAN2Z(Y229X12) Cc CORRECTION FOR DISCONTINUITY IN ATAN2 AT PTE 167 IF(X11 eGTe Oo oORe X12 oGTe O-) GI TO 4 201 IF(TP2 oGTe Oe eANDe TPL oh To Oo) TOL = TPL + TP 214 IF(TP2 eh Te Ge cANDe TPL oGTe Ue) TPL = TPL = TP 227 re) C3 = CC3(J) 232 $3=SS83(J) 235 A1L=PIE 237 IF(I=-J) 29352 241 2 A1l=TP1-TP2 243 3, A2=T02-TC1 245 A5=C3¥(—XBC JFL) 4XB( JI—K1L2*0. 5*PP24+K11 *0. 5# PPL 1+V12*TP2-YLI¥*TPLIFS3¥(YB(JI-VB( U4) -K12"TP2 1=Y12*0.5*PP2+X1LI*TPL+V114*G.5*PPL) 30? A6=C3#(=XB( SFL) XR (SI—X12%0. 5¥PO24+K1140.5*P 01 14Y22¥*TO2—Y21*TQ1)—-S3¥*(-YB( J) +YB(Jt1) -X12¥*T02 L=-¥22*0o5¥*PO2+KLI*TOL+Y21*0.5*PQ1) 351 4 k112*x1l2 353 Yil=yl2 354 Y212Y22 356 PP1=PP2 357 PQ1=PQ2 361 TPL2TP2 362 TQ12TQ2 364 RI12(IsJ) 2 Al = A2 Table D-2. Continued 120 COEFF RUNT WERSICN FEB 74 8 372 377 400 402 407 416 423 424 15219 03/13/76 RK56( 19d) = AS = AE 6d Te 1 00 209 L = lsN RT12(1lst) = eC RK56(isL) = 000 CONTINUE RETURN END Table D-2. Continued 12] RUNT VERSION FEB 74 B 17312 04/23/75 aon 13 SUBROUTINE COMP(K) THIS SUBROUTINE COMPUTES THE COEFFICIENTS DEPENDENT ON K AND CALLS ON LNEQF TO SOLVE THE SIMULTANENUS EQUATIONS FOR THE VELOCITY POTENTIALS FE(I90J) AND FI(ToJJ)sFOR FE2 COMMON R112(6250¢25) pRK56(25925) 0 PIT(25, 2F)o YNW(25525) 0 FE( 259 Ade LFI(2596)9 RI(25925)59 RJ(25925)5 2K(27504)5 BLI2594)5 2RMU(393510)5 RLAM(323910)5 FB(3910)5 DELFR( 3510)» HW8(2596910)o 3 DELW(2595910) » XOL(25910) COMAON/ONE/X 025) 9V(25)5XB(25) 9 YB(25) 0 ANG(25)9 DEL (25) ,VV(25) LsFEIN(25)5 FIIN(25)5 RNORM(25593)5 JC(5) COMMON/ONE2/€C3(25)59SS3(25) COMMON /TWO/ NoNNWs NWAVELs ISYMe YSKIP» NCo PIE» GAMMAa My TK» TP DIMENSION A(50050)98(5094)5ERASE(51) REAL K N2 = N/2 DO 1 IslsM™ DO 6 I2= 19% RK(InI2) = 000 RLCIst2) = 0.0 00 4 IC = 155 IF ( I eEQ. JCC(IC)) GO TO 1 IFCISYM .NE- 1) GO TO 8 IF(I oGT.o N2 oANDo TIT eLEo N) GU TO 9 XJ1 = X(T) = X8(1) X21 = X11 +XB(1) Y212V(1)+¢v3(1) PQLSALOG(KLL**2+V2Z1¥%2) TOL=ATAN2(Y215Xil) CALL CPV(XI19Y2159E21oCI1Ll»SI1»sAIIIs ALNIIoK) C2l1=Cll $212S811 A921=A911 A10212A1011 DO 7 J2l»N X12=X(1T)=XB(J4+1) Y22=V(I)+YB(J+1) PQ2B2ALOG(X1L2%*2+V22**2) TQZ2=ATAN2(Y225X12) $3=*SS3(J) C3=CC3( J) CALL CPV(X1L2 9 V220E22eC129S12sA9129 AL I12 6K) 00 13 IC = lod IF(J -EQ. JC(IC)) GO TO 41 A32A1011=-41012 AG=E21L*SLI-E224S12 A72S3% (0.5*(PQ1—POQ2) +A912—-A911) +C3*(TOQL=TQ2441011-A1012) AB=E2L¥*SIN(K¥*XLI-ANG(J) DeE22*SIN(K*XL2=ANG( SD) RI(IsJ) = 20 * AS + RIIZ(I9 J) IF (Y oNE. J) GO TO 3 RI(I9J) = RI(I9J) -— TP RJ( Tod) = -TP¥AS POT(IosJ) = TK*¥AT + RK5SS(Ip J) HOW( To J) = -TK¥PTE*AS 0O 10 L = 13 RK(IsL) = RK(IeL) + POT(Io J) * RNIRM( JoL) Table D-2. Continued 122 RUNT VERSION FEB 74 B 17:12 04/23/75 365 410 435 463 466 470 471 473 474% 476 477 501 502 504 507 507 512 513 527 545 546 547 555 560 570 601 627 655 66) 663 664% 666 667 675 677 Tol 762 712 725 726 730 731 737 741 743 754 765 776 1007 1014 1015 1017 1030 1035 1035 1035 10 12 16 31 15 41 2 22 27 RL(IsL) = RL¢I9L) + HOW( To J) * RNOQM( Jol) RK( 195) = RK(I94) — FEIN( J) *#POT( Is J) & FITN( I) #HOW( Ts J) RL(Is4) = RL¢I94) = FEIN( J) #HOW( To J) = FITN(S) *POT( TI» J) IFC URN) 29797 X112x12 Y212Y22 PQ1L=PQ2 TOL=TQ2 A9112#A4912 41011=A1012 C11=Cl2 S11=S12 E212E22 CONTINUE GO TO 1 us oC Mo iH we yh DO 12 L = 153 RK(JoL) = RKCISoL) * (—-1.0) #¥*L RLOIsL) = RLCISsL)¥*(=1.0)¥**L DO 11 J = lsN OO 16 IC = 15% IF (J eEO. JC(IC)) SN TO 11 JS 2 hl) oO di al RI(IsJ) RI(IS»s JS) RJ(IsJ) RI(ISsJ5) RK( 194) CONTINUE CONTINUE {2 = 0 CO 22 T=lsN DO 14 1C = 155 IF(I 2£EQ. JCCIC)) GO TO 22 I2*=12¢1 II = [2 + NC DO 31 L = 194% B(I2sL) = RK(TsL) B(IIsL) = RL( isl) J2 =u DO 22 J=lsN 00 15 IC = 195 IF (J -EQ. JCC(IC)) GO TO 22 J2 2 J2 +1 JN = J2 + NC A(I2,s J2) = RI(IsJ) A(I25JN) = = RJCIs J) ACII»sJ2) = RJ(Io J) ACIIs JN) = RI(Ts J) CONTINUE SCALE=le NN = 2¥*NC LLELNEQF(505NN»s 49 A995» SCALES ERASE) PRINT 279SCALE FORMAT(//95Xo *DETERMINANT= ¥*51PE1204) I2 = 0 00 26 I = 1oN Table D-2. Continued 123 RK(T94) — FEIN(J)*POT(ISs JS) + ETIN( J) FHOW( TSS JS) RL( 194) = RLG194) — FEIN( J) #HOW(IS9 JS) = FLING J) *PCT(ISs JS) RUNT VERSION FEB 74 8B 1041 1042 1050 1052 1054 1955 1065 1100 1103 1104 17 35 26 29 DO 17 IC IF(T e€Q. 12 3 I2 II = [2 OO 35 L FE(ToL) FICIsL) CONTINUE RETURN END + ou uw > 17:12 04/23/76 195 JC(IC)) GI TO 26 1 NC 194 B(I2sL) B(TIsL) Table D-2. Continued 124 RUNT VERSION FEB 74 83 17212 04/23/75 SUBROUTINE PHYSCL(K) 6 COMMON RI12(625925) sRK56(25025)9 POT(25925)5 HOW 25s 25) a FE( 25a Ede LFI(2596)5 RIM25925)5 WI25925)5 RKI 259 4)5 RL 250 4)5 2RMU(393919)59 RLAM( 3503910) FB(391%)0 DELF3(3910)5 HWB(2596910)» 3 DELW(2596510) » XOL(25519) 6 COMMON/ONE/X (25) 9Y(25)59XB(25) 9 ¥B(25) 9 ANG(25)0oDEL(25)sVV(25) LoFEIN(25)5 FIIN(25)5 RNURM(25593)5 JEU5) 6 COMMON /TWO/ NoNNWs NWAVEL» ISYMsy TSKI>» NC» PIEo GAMMAg Me TKoT®& 6 COMMON/THREE/ WAVEL(10)5 WN(10)5 BOL(10O)s1TL 6 COMMON /SEVEN/ AREA» 35 Ds ROE» GEEo BTTYLE(3)5 TITLE( 8) 6 REAL K 5 DO 3 [ = 1sN CeeeeeMODE 6 = INCIDENT + DIFRACTED POTENTYALS 7 FE(Is6) = FE(Ts4) + FE(Is5) 23 3 FI(Is6) 2 FIC I>4) + FI(Is95) 42 FACM = (B**2)/AREA 47 FACL 2 FACM*SORT(K) 53 FACF = FACM * K 55 DO lL = 153 56 DOl ML = 156 ot, RA = 0.0 60 RM = 020 61 IF(M1 .EO. 4) GU TO l 63 IF(M1 .EQ. 5) GO TO1l C##eeeINTIGRATE PRESSURE COMPONENTS OVER BONY 66 DO 5 I = IsN 70 RM = RM + FECT» M1) * RNORM( ToL) * DEL(T) 104% 5 RA = RA + FIC Io M1) * RNORF(IsL) * DEL(T) Ue) IF(M1 eGTe %) GO TO 8 C#*#*#ADDED MASS AND DAMPING IN DIRECTION L DUE TO MOTION M1 AT C¥*FeRWAVELENGTH ILeo 126 RMU(LoML» IL) = RM*FACH 136 RLAM(LoM1ls IL) 2 RA*FACL 145 co TO 1 C##eeeWAVE FORCE AMPLITUDE AND PHASE IN PTRESTIIN M1 DUE TA C¥x*eeINCIDENT wAVE AT WAVELENGTH IL 146 8 FE(L» IL) = SQRT(RM*¥*2 + RA**2) * FACE 166 DELFB(L»o IL) = ATAN2(=RAdRM) 209 1 CONTINUE 205 Iw =N +l 207 IMAX = N + NNW 211 DG 30 I = IW» IMAX 213 DO 6L = 1s% C*#*eeeCOMPUTE POTENTIAL AT FREE SURFACE POINTS USING GREENS THEQRUM 214 00 4 J = 1s N 215 O00 10 IC = 195 216 10 TF(J 2FEQ. JC(IC)) GO To 4 224 FE(IsL) = FECIoL) + FEC Jo LI*RICTo I) -FLCSoL) FRI To J) 255 FICIsL) = FIC IsL) + FEC JoLIFRIC Tod) +FT (Jol) *RI( Is J) 306 4 CONTINUE 311 FE(IoL) = (FEC ToL) - RK(Ts9L))/TP 326 6 FICIsL) = CFICLsL) = RLCI9L))/TP C*****MODE 6 = INCIDENT + DIFRACTED POTENTIALS 344 FE(I96) = FE(Is4) + FE(Is5) 361 FI(Is6) = FI(Is4) + FI(TIs5) Table D-2. Continued 125 RUNT VERSION FEB 74 B 17212 04/23/76 375 SOU 410 412 443 470 472 473 Te eN C*¥*#eeNON DIMENTIONALIZE FREE SURFACE POSTTLIN WELTY WAVELENGTH MOL(IIsIL) = X(I) * B/ WAVELC(IL) OO 2 M1 = 156 C*e#eeeWAVE AMPLITUDE AND PHASE AT POINT [If DUE TO MODE Ml AT WAVELENGTH 2 7 30 HWBCIIToM1sIL) = SQRT( FEC Is M1)¥*2 + FI(ToML)**2) * K DELW(IIsM1lsIL) = ATAN2(=FI(IsM1l)5 FE(To“1)) CONTINUE RETURN END Table D-2. Continued 126 RUNT VERSION FEB 74 B 17:12 04/23/76 ONNWNNhy NMNMNN = 10 12 32 Uv 150 221 223 224 241 324 326 327 334 337 353 355 436 449 441 460 534 542 544 561 632 633 1001 1002 1003 1004 20006 SUBROUTINE POTOUT COMMON RI12(25025) »RK50(25925)5 POT(25075)9 HOW(25925) 9 FE( 2594) » LEI(2596)5 RI(25925)5 RU(25925)5 RK(2594)5 PL(2594)5 2RMU(393910)9 RLAM(3593910)5 FB(3519)5 DELF9(3510)5 HWB(25s6910)5 3 DELW(25596510) » X0L(25910) COMMON /TWO/ NoNNWo NWAVEL» ISYMs TSKT% NO» PIE» GAMMAsMs TK oT? COMMON/THREE/ WAVEL(10)5 WN(10)5 BOL(1O)sIL COMMON /SEVEN/ AREA», Bs Do ROE GFEo B8TITLF(3)5o TITLE(S) COMMON / EIGHT/LBLMU( 35393) LBLAM(393032)9 LELFB( 393) oLRLHWR( 753) LLBL(1053)5 DEG(351G) 2 FORMAT(//3X9 3A109 / (5X9 3A1G5 19F19.4)) FORMAT (/73X%5 3A10»9 / (5X» 3A105 10F1004% /5X%» 3A1059 10F10.4/)) FORMAT( 5X5 3A100 LOF100% / 5X» 3419» 1LOFIO.4% /) FORMAT( //73X5 3AlLWs 2X» LOFLO64/ 3X» 3410» 2X5 10°10.4) PRINT 20665 8TITLE FORMAT (1H1s 20X%9 *NONOIMENSIONAL POTENTIAL COEFFICIENTS* /// 25% 1 * W 2 SORT(G/B)s wW2 = G/B*¥ / 25X*R = *» 3A10 / 1 25x*G 2 ACCELERATION OF GRAVITY*/ 1 25X*¥ROE = MASS DENSITY OF FLUYD*®/ 225X *ETA = INCTDENT WAVE AMPLITUDE */25X%o*WAVEL = INCIDENT OR GENE 3RATED WAVE LENGTH*//) 00 9 It = 1210 DEG(lsIL) = SQRT((GEE*BOL(IL)) /(TO*R)) PRINT 10040(LBL(19K)9K = 193)» (BOL( TL)» TL = 1910)>5 1 (LBL( 29K)» K = 193)5 (DEG(LoIL)» IL = 1910) PRINT 1LOQ1Ls (LBL(35K)9K= 1L93)5 (CC LELMUCTs IsK)oK= 1p 3) 0 LORMUCIo Jo IL)» IL = 1919)5 J = Lo 3)o TL = 193) PRINT 10601 s(LBL( 49K) 9K = Llo3)m9 (CKCCLSLAM(Ts JoK)oK = 193)5 LORLAM( Io JsIL)» LL= 1910)5 J = 1lo3do T 2 193) DO lI = 153 00 1 It = idslv DEG(IsIL) = 57.298 * DELFB(IsIL) PRINT 1002» (LBL(59K)5 K = 1lo3)5 ((LRLER( I 9K) oK= lo3)o LCFB( Ip IL) o IL = LolG)o (LBL(SeK)o K = lot)o(DEG(Io IL) sIL=15910) 25 I = 153) OG 2 f = IsNNW oo 8 IL = 110 DEG(JsIL) = OC TFCIL GT. NWAVEL ) GO T9 8 DEG(1sTL) =sXOL(foIL)*8/BOL( IL) CONTINUE PRINT 10025 (LBL(B9K)o K = 1Lo3)o (LEL(90KI® K 2 1la3)o 1 (CXOL(TsIL)» TL = Lo 10)5 (LALVICsK)» Kel53 Vo (DEGCL>IL)IoTL21510) 00 3 J = 153 CO 3 IL = 1910 DEG(JsIL) = 572-298 * DELW(IsJsIL) PRINT IGO3» ((LBLHWB(JoK)» K = 153)0 (YWB(ToSoIL)o IL & 19lC)o 1 (LBLG7sK)o K = 193)5 (DEG(JsTL)» IL = 2olD)o J = 153) IF (XOL(I51) oLTe--0-) GO TO 4 DO 5 IL = 1610 DEG(IsIL) = 57.298 * DELW(Is6sIL) PRINT 10035 (LBLHWA( 72K)» K=lo3)5 (HWA(IT»SoIL)» IL = LolG)s Ll (LBL( 69K) 9K2193)5 (DEG(loIL)» IL = 1919) GO TO 2 00 7 J = 193 Table D-2. Continued 127 RUNT VERSION FEB 74 B 17:12 04/23/76 635 00 7 IL = 1slv 636 7 DEG(JsIL) = 57.298 * DELW(IpJ+3» Tl) 655 PRINT 10035 ¢((LBLHWB(J9K)» K = 1Lyo3)0 CHWRUTs Jo IL)» IL = 1510)5 L(LBL(69K)5 K2lo3)5 (DEG(J—35IL)» TL = 1919)5 J= 406) 732 2 CONTINUE 735 RETURN 735 END Table D-2. Continued 128 RUNT VERSION FEB 74 B 17:12 04/23/76 FUNCTION LNEQF( Mo No N19 As Bo DTRMNTSZ) Ceo SOLVES SIMULTANEJUS LINEAR EQUATIONS 8Y GAUSSTAN REDUCTION. Ceo FORTRAN ITV EQUIVALENT OF LNEQS. 12 REAL AC M9M) 08 (Me) o ZIM) oDTRMNT» RMAX o> RNEXT» Ws DNV 12 NMLeN=-]) = 1¢ DO 46 Jel»NM1 15 JlsJ+l Coe FIND ELEMENT OF COL J» ROWS J—=N»o WHITH HAS MAX ASSOLUTE VALUE. 17 LMAXeJ 20 RMAX=A4BS( AC Je J)) 34 DO 6&6 KsJloN 35 RNEXT=ABS(A( Ks J)) 52 IF (RMAX oGEe RNEXT) GO TO 6 55 RMAXSRNEXT : 57 LPAXeEK 60 e@ CONTINUE 63 IF (LMAX NE. J) GO TO lo Coo MAX ELEMENT IN COLUMN IS ON DIAGONAL 65 IF (AC d9J)) 20994520 Ceo MAX ELEMENT IS NOT JN DIAGONAL. EXCHANGF 2OWS J AND LMAX. 73 10 DO 12 L#JoeN 73 WeA(JeL) 102 A(JoL)Z=ACLMAXsL) 113 12 AC(LMAXsL)2W 124 OO 14 L=loNl 125 W=B(JoL) 132 BlJoL)FB(LMAXosl ) 143 14 B(LMAXsL) =a 154 DTRMNT = -OTRMNT Coe ZERO COLUMN J BELOW THE DIAGONAL. 155 20 ZIJ)BL SAIS) 165 DO 30 K=Jl»sN 167 IF (ACKoJ)) 22930922 175 22 We=Z( J) *A(Ko J) 205 DO 24 L2JloN 207 24 AUKoL)2WAC JoL) +A(KoL) 230 DO 26 L2losN} 231 20 B(Ko Ll) =WeA( Jol) *B(KoL) 252 30 CONTINUE 255 40 CONTINUE 257 IF (ACN9N)) 4259749 42 265 42 Z(N)2Le/A(NoN) Ceo OBTAIN SOLUTION 4Y SACK SUBSTITUTION. 275 DO 50 L#®l»sN1 277 50 BI(NsL)=Z(N) *B(NoL) 315 DO 60 K2=l»sNM1 316 J=NoK 317 JlzJ+l 321 DO 58 L=eloNl 322 w20. 323 DO 56 T=Jl»N 325 56 WeAl(JoT)*B( ITs L) ew 342 58 BlJoL)I (Bl Jol d—w) *Z2 (0) 362 60 CONTINUE Table D-2. Continued LNEQF 129 RUNT VERSION FEB 74 B 17212 64/23/75 Ceo EVALUATE DETERMINANT. 364 IF (OTRMNT) 70974970 366 7G DO 72 J=l1eN 370 72 DTRMNT=DTRMNT*A( Jo J) 377 74 LNEQF=1 401 RETURN Coe SINGULAR MATRIX» SET ERROR FLAG. 401 94 LNEQF = 2 403 OTRMNT=0. 404 RETURN 405 END Table D-2. Continued 130 RUNT VERSION FEB 74 B 17:12 04/23/76 SUBROUTINE CPV(XsYsEoC1loS1sA9sA109K) C ceeeCAUCHY PRINCIPAL VALUE INTEGRAL. 13 COMMON /TWO/ NoNNWo NWAVEL> ISYMs TSKIPs NC» PIE» GAMMAa Ms TK9 TP 13 COMMON/SIX/XN(5)9CN(5) 13 REAL K 13 IF (Y eGEe 000) Y * -1-0E-08 15 TT=ATAN2Z( V9 X) 24 TH=PIE/2e+TT 27 TF CXeL Te.) THTH+TP 32 AABK*Y 34 E=EXP(AA) 43 BB2K*X 45 C1=COS(BB) 54 S12SIN(38R) 63 REK*SORT(X#*2+Y**2) 102 SUM1=0. 103 SUM2Z20¢ 104 IF(R.GEo10-IGO TA 13 107 SUM1120. 119 SUM22206 lis FAC=1.0 113 SUM1C=1. 114% SumM2C#1. 115 SDLTH=0. 116 CDOLTH=0. 117 ASSIGN 3 TO LOC 120 TF(X-FQe0.) ASSIGN 8 TO LDC 122 RFL21.0 124 OO 1 L=is109G 125 OL=L 126 FAC2FAC*DL 130 RL#R¥RL 132 OLFAC=FAC*DL 133 OLTH20L*TH 135 Al=RL/DLFAC 137 IFCABS(CDLTH) LE-L-E-G7)GO TO 2 151 SUM1C2ABS(Al/SUM1) 157 IF(SUMIC.LEeLeE“05)G0 TO 7 165 2 COLTH2=COS(OLTH) 171 SUM112A1*CDLTH 172 SUM1®SUM1+SUM11 174 7 GO TO LOCs (358) 203 8 SUM2C=06. 204 GO TO 5 205 3 IFCABS(SDLTH) eLEoleF-C7)GN TG 4% 217 SUM2C2ABS(AL/SUM2) 225 IFCSUM2C LEoleFr45)G0 TO 5 233 4 SOLTH=SIN(DLTH) 237 SUM222A1*SOLTH 24) SUM2=SUM2+SUM22 242 5 TFCSUMIC eLEoLo EIS eANDoSUM2CoLEoleE-05)GN TN E 261 1 CONTINUE 263 6 C=GAMMAtAL/IG(R)+SUML CooooeDISCONTINUITY OF 2PIE IF X NEGATIVE IN CT FUNCTION. Table D-2. Continued 131 RUNT VERSION FEB 74 8 17812 04/23/76 27. IF(XoLT eGo) TH=TH=TP 3u0 S=TH+SUM2 302 AG=E*(C1*C+S1¥*S) 306 ALUSE*(—C1*54+S1#C) 313 GO TO 9 Cooo LAGUERRE QUADRATURE-FIVE POINT. 314 13 DO 14 Y=ls5 316 A2XN(T)+AA 321 TERM=CN(I)/(A¥A+BB 488) 330 SUML=TERM#*A*SUM1L 333 14 SUM2=TERM+SUM2 337 F=le j 340 IF(XoLToCco) Fel. 343 AG=F*ePTE*S1T*¥E=SUM1 347 A1Q2-F¥*¥PIE*C1*E+BB*SUM2 353 9 RETURN 354 END Table D-2. Continued 132 RUNT VERSION FEB 74 8 17312 04/23/76 NM Mh SUBROUTINE DYNOUT COMMON RI12(025925) »RK56(25525)5 POT(25925)5 AOW( 25925), FE( 2556) 9 LFI(2506)5 RI(25025)5 RI(25925)5 2K(25,4)o RL( 25594) 5 ZRMU(393519)5 RLAM(393910)5 FB(3910)0 DELFR(3910)5 HW8(2596510)5 3 DELW(259691G) » XOL(25910) COMMON /TwWO/ NoNNao NWAVEL» ISYMs ISKYPs NCp PIEsGAMMAd Ms TKe TP COMMON/THREE/ WAVEL(10)5 WN(10)5 BSALCIO) «TL COMMON/FOUR/RAR(3¢1II5DELR(3s1U)5 HWRUA553e79)y DELWR( 25593910) 5 2 HWT(25510) pDELWT(25910) sRKHYD(393)9 2XMOR 1 9493)9 RKTB(393)0 3 XGs YGs RMASS» RINERT» DAMP(3) 3 COMMON /SEVEN/ AREAs Bo O» ROEs GEE» BTITLE(3)» TITLE(P) COMMON /NINE/ LBLRAR(353)5 LBLHWR(503)0 LALR( 553) ; COMMON / EIGHT/LBLMU(39353)5 LBLAM(393203)9 LALFB( 303) o LBLHWA( 793) 9 ILBL(1053)5 DFG(3510) LOOL FORMAT(4/3X5 3A100 / (5X5 3A1059 10F1004)) LVO2 FORMAT (/73X%5 34105 ¢ (5Xo 3A105 1OF19.4 /5X» 30105 10F10.4/)) L003 FORMAT( 5X5 3AL05 LOFIVe& / 5X9 34100 19F10.4% /) 1004 FORMAT( //3X%5 3AlUs 2X9 1LOFIO6e4/ 2X» 3ALO» 2¥%5 10F10.4) PRINT 200059 AREAs 39 XGe YGo RMASS» RINZERTODAMP(1) »DAMP(2) » DAMP (3) 2000 FORMAT( IHL» 20X*#DYNAMIC MODEL RESULTS#//* AREA#4*F10.325X#R22510, 3,5 L 5X EXG=RFL Oe 20 SKPYGE# FIO. 3» SKEMASS2HF 19,30 5K RINERTIA=*FIONW3// 3* ADDITIONAL DAMPING ADDED= IN SWAVY=*F6.2* LAMDALI IN HFAVF= 3*F6e2* LAMDA2Z2 IN ROLL=*F6e2® LAMDAI3*® //) PRINT 20019 ((RKHYD( To J) 5 J2193)0T2103)9 ((RKMIR( Ip J)o J2l53)5T2159) 2001 FORMAT(* SPRING CONSTANTS KUL K 1? K13 Kel 1 Keé2 K23 K31 K32 K33*/ 2* HYDROSTATIC*?7Xs9F1003 / * MOORING*¥LIXS9FIN.3///) 00 9 IL = 1010 9 DEG(1sIL) = SQRT((GEE*BOL(IL)) /(TP*8)) PRINT LVO4o(LBL(isK)oK = Le 3)5 CAILCTL)» TL = 15910)>5 1 (LBL(25K)5 K = 193)59 (DEG 1oIL)» TL 2 %519) DO 17 = 193 DO 2 Ik = 1910 1 DEG(ITsIL) = 57.298 * DELRI(IsIL) PRINT LOOD2s(LALR(1LsK)o K = Lo 3) o((LBLQRAR(ToK)» K = Lly3)o 1 CRARCTsILIo IL = Lold)o (LBL(69K) 9K = 153)0 (DEG(IsIL)oIL2=1010)5 TI 2 2 153) : NO 2 1 = LsNNw 0O 68 IL = 1510 8 DEG(1ysIL) =XOL(I5IL)*B/BOLCIL) PRINT Lud2o (LAL(39K)5 K 2 Lo3)o (LAL(99K)9 K = 193)5 1 (XOLCIs fl)» TL = Lo hG)»o (LALGLOsK)» Ke1l93 )o (DFG(1sIL)»IL=1529) OO 3 J = 153 DO 3 [IL = 1/10 3 DEG(JoIL) = 572298 *DELWR(Io Js IL) PRINT 10035 ((LOLHWROIJoK)o K = 293) (HWR( Todo IL)» IL = To1O)> L (LBL( 60K)» K = 193)5 (DEG(JoTids FL = 1010¥0 J = 1593) IF (XOL( 191) otT.e 0e) GO TO 4 bO 5 IL = 1910 DEG(2oIL) = 57.298 * DELWT( Is IL) 5 DEG(AsIL) = 570298 * NELW(Io6sIL) PRINT 10035 (LBELHWR(7oK)o K21l93)9 (HW9(TsAoTL)» IL = loll)» 1 (ULBL(69K)oK2=153)5 (DEG(LoIL)» IL = 1019) PRINT 10G35 (LBLHAR(4eK)oK2193)0 CHWT(To TL) oTL = 1912)> 1 (LBL(60K)» K = Ls 3)»y (DEG(2sIL)5 TL = WIN) Table D-2. Continued 133 DYNOUT RUNT VERSION FEB 74 B 17212 94/23/76 637 GU TO 2 640 4 OO 7 It = 151% 642 DEG(lo IL) = 57-6298 * DELW(ITs4%sIL) 655 7 DEG(291L) = 57¢298 * DELWT(IsIL) 670 PRINT 16uU35 (LBLHWB(4sK)5K = 153)5 (YWR( Te SoTL )oIL= 1lyelOdde L(LBL(6sK)o K = 1ls3)o (DEG(LoIL)s TL = Lo lh) o(LBLYWR( SK)» K=103)5 2CHWT(Lo tL)» IL = Lo lO)o(LBL( 69K) 9K = Lo 3)o(DFG(2sTL)o fL=1910) 1005 2 CONTINUE 1010 RETURN 1010 END Table D-2. Continued 134 BRK2D READ LABELS INPUT DATA COMPUTE BODY SEGMENTS CONTROL POINTS BEGIN B/L LOOP = COMPUTE INCIDENT WAVE POTENTIALS CALL COMP CALL PHYSCL PUNCH POTENTIAL COEFFICIENTS CALL DYNAMC Figure. Flow chart for program BRK2D. 135 7. Program Comments and Glossary of Terms. The program listing contains many comments which aid in following the logic of the program. Descriptions of variables also appear where they are read into the program. 8. Run Time and Memory Size. BRK2D requires about 70 seconds of central processor time on the CDC 6400 computer to compile and compute results for 10 different beam wavelength ratios. A central memory of about 55,000 octal is required. 9. Run and Card Deck Setup Procedures and Special Operation Instruc- tions. In order to run the FORTRAN source program deck on the University of Washington CDC 6400, the following deck is required: BRK2D,CM55000,T100. Job card ACCOUNT (Account No., password) FORTRAN : LGO(LC=6000) LC = line count value 7/8/9 FORTRAN DECK 7/8/9 DATA DECK 6/7/8/9 10. Sample Output Data. Table D-3 is the output for the Oak Harbor breakwater. 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The Laplace equation, Es TO SECOND ORDER DIFFERENT FREQUENCIES interactions of waves at two direction. The complete v9 = 0, (E-1) applies throughout the fluid below the free surface. The boundary condition, es ao ely Tay eee eM ‘are os dig .y (Vo>-Vo) = (E=2) at Y must be satisfied on the free surface, y = the bottom is: lim oo Vireo for an infinitely deep fluid. quiring the generated waves to travel away ensure uniqueness of the solution. = 0 n. The boundary condition on (E-3) In addition a radiation condition re- from the body is needed to In this formulation the x axis lies in the direction of incident wave propagation. The difficulty in solving this boundary value problem stems from the nonlinearity of the free-surface boundary condition. In order to "linearize" the free-surface boundary condition, expand the velocity potential, ®, in a Taylor series about the undisturbed free surface: o(x,n,t) = o(x,0,t) a. (ea yet) + $ ac pee »Y> 2t)) A OC). (E-4) y=0 y=0 Also expand n and » in power series: nxt) = en) (x,t) + e2n() (x,t) + o(e%), exy.t) = eo Vexy,t) + 276 cx y,t) + o(e%). (E-5) 148 athe Substituing the expansion for ¢ into the free-surface boundary con- dition: 1 € 9796 Guy, t) + ee ary ©) + ge ay + eee ag? ate at? Be of (1) Ces (2) (2 x + 2[e{ i i = ar e7{ ae i 5y jt]. (2) (2) qe.) Kv) Ze) [i = © Ia) ls = : _ (1) (1) (2) (2) 1 ao Ze oo > 2,0 > ao F144, . q Is ox ok ody j3 t ax dy 33] (1) (1) (2) (2) = @ r) f) $ re) Leh ees 4 one i + es jh + coe i + a jh: ag(l) » 96 (1) (2) (2) j ie (2 = co A i — 7H] + O(e°) = 0 (E-6) on y = Now use the Taylor expansion for 9$(x,n,t) and neglect terms of or- der e> in the boundary condition: 2, (1) 5, (a) 2, (2) aie Y) a a Smee) Gee) a ¢ 5 eee <2 da 9 3 at dyot at (1) 2,.Q) (2) + ge{——— 2g ame (8,8) ase ace nen oo oy ay? dy Cec) (OL) aaaQ) 2,9¢ a“ ag a) ae, j meilamax: vataxyj, say stay) ) 7 © ae Grouping terms by order: Rarsie Onder ie: 2,(1) (1) 2 - + g 2 =O ony = 0. (E-8) at Y Second Order eae AON Ne 4 eh LON a Cla Oe. 9t2 oy joy 92 oy ox dxdt CD) a) ad co) iy - E-9 = oe 0 Cmsy = Os (E-9) 149 Using the dynamic boundary condition on the free surface, one finds: Wiese) Ss 2S + = Vo: Vo} ony = (E-10) Substituting the expansions into this Tae yields: eq en) Gaye + ane” + 0(e>) = - 2 ees > Vo: Vo} i “oe y=0 Nisou boomer 2 z - Soy Oot” = Vo Vo} + O(n‘). (E-11) y=0 Substituting for >, the right-hand side becomes: go) (2) 2 @y) 2 a 2 -- 5 tet eS et A 20) 2,(2) - i C = a VE 4), ony = 0. First Order e: (1) (1) _ , 2 OO 0.8) : n (x,t) = - E yt ‘ (E-12) 2 Second Order e : (1) (1) (1) 2 (1) (2) _ 14 2 iL pei age 2 > Gat) & = g t dyat ot - 2g tC = CS oy ys on y = 0 or CRA ays Wee Srl Oe tee RO a , Ge me Oe Se Ae Ox 9g (1) , 2 (( mS CE FO (E-13) Using the first-order relationship above in the second-order boun- dary condition on the free surface (E-9), one finds: 929 (2) “94 (2) 1 9g (1) 9 2) 36 (1) er Mea oar bat aE > ae te ) at Y g at y L 9g (1) 97 2) i a (1) 9 2) ox dxdt oy dyot 150 (E-14) Ie First-Order Solution of the Boundary Value Problem. This solution results from the superposition of the velocity poten- tials for individual waves: gA CD Gear) = — e Loe cos (k}x - wit + 64) 1 gA DE AGE Y, tan © =) COS (kx - wot + 65). (E-15) Check the solution: Toe =o. F (1) Lim 0 + 0 because of exponential function. Wr dy 2a) (1) a> OUD ee ky wen goa tigi oie Shey ie ky - guA,e 27 1Cos (k,x - wot + 6.) kent + g {w Ale 1 cos(k,x - wt + 54) k.y ee + wA5e 2 cos (k,x - Wot + 6,)} = (0), 2 Therefore, this is a solution. Surface elevation then becomes: 1) (1) wy ae ag! (Cs Onie) ; n (x,t) = - 3 ee Ea & Ay sin(k,x 2 Oe & 61) - A, sin(k,x - Wt + 55). (E-16) To prepare for the second-order solution, construct the right-hand side of the free-surface boundary condition (E-14): vag a a2) ag ag (1) 94D) 2 ot dy t2 g oy OX dxdt (LE) oyu, (2) NA 87% ei os 2 “Or ene = @ {gA, Sin(k) x - wt + 54) + gh, sin(k,x - wot + 6,)}H0} - 2{- w A, sin(k,x - wt + 6,) (E-17) 1S] 2 2 - w A, sin(k,x - wot + 65) }xtw, A, cos (k,x - jt + oD 1 1 + 2 w oAy cos(k,x - Wt + 65) } - 2{w A, cos (kx - wit + 6) + 2 : WA, cos (k,x - Wot + 65) Ixfw, Ay sin(k,x - wt + 5) + Ww aN sin(k,x =| (i) 9 Ay t + 6,)} =) 0). 2 Since this condition is homogeneous, the first-order potential is the solution to the second-order problem. De Second-Order Results. The free-surface elevation will be modified when terms of second order are included: no?) (x,t) = 2 eae arou ea + ay) | ge (OE Byatt 2g ox oy y=0 man 4, (eA, sin(k,x -w t+ 54) + gA, sin(k x - Wot + 6,)} x Dies Bree {A, W, sin(k,x = wt + 54) + Ay W, sin(k,x - wot + 65) } 2 i : F - ag SG wi Ay sin(k,x 2 Orie 61) - WA, sin(k,x = Oye o 55) ] 2 + [w,A, cos (kx = Gye = 61) + WA, cos (kx - wot + 65) ] } or ane) Ge) = 0 Ay sin’ (kx = Wy 2 : ; + w) AJA, sin(k,x - Wt + 55) sin(k) x = W it + $) it + 51) 2 F ; + W, AJA, sin(k,x - wit + $1) sin(k,x - wot + 55) Cg) a + w, A, sin (k,x - Wot + 85) 1 2 ah RNS z i INS Satin (kx - wt + 5) + 2w,w,A,A sin(k,x - w Pah As t + 6,)sin(k,x - Wot + 5.) i 152 ZANZ GADD Ney Satin (k,x - wot + $5) + Ww 25 2 cos (kx, - wit + 54) + 2w,w,A,A cos(k) x - Ww e2hyAs t + 5) cos(k,x - wot + 65) 1 2 2 5 Nor COS (kx - wot + 55). Using the trigonometric relationships: 2 Nye ee? a 2 DN a ae? gin Gso8) F Wy A, sin (k,x - wt + 51) + Wo A, sin (kx = x6 ? 54) 1 2 zy AA, {cos[(k, = k,)x - (w) = W)t + 8) = 65] cos[(k, + k,)x - (w, + wo)t + 5) + 6,]} 2 W5 AA, {cos[ (k, - k,)x - (w, - wo)t + 5) - 55] = cos[(k, + k,)x - (w, + w)t + 5, + 5,]} - 5 tu A + wyA ey = w.w, A.A 141 2 22 AyA, cos[(k, - ky) x - (wo, - wo)t+ 8) - 65]. Combining further: (2) tel ele TP espie) = Fy Ay cos [2{k, x - wt + 6,3] 1 Or. 2 ae 2 -18 7 WA, cos[2{k,x - wot + 653] (E-18) - (w 2 + Ww AYA NN. COS [Ox & Tk Nie = "(@e = wae = Gs 6] 2 1 2 Ie 1 2 1 2 1 2 5 & (w ae 2w,w, + w 25 Ke COSIGR S Te xs S) (Gis 9S We OF = Se] 2 1 ile 2 ee? iL 2 1 2 1 Dn? which is the final form for the second-order term for free-surface ele- vation. 153 Now, turn to the equation for pressure which is necessary to com- pute the force on the body. Take the pressure to be zero at the free surface. Then Bernoulli's equation may be written: 3 1 P == p — - 5 p¥O-Vo = oBy- (E-19) Substituting the expansion for 9: (1) (2) 1) 2 2¢ 2 9 ae? Pe = Olé a v & “+ + 2 [e Cae (a) 2 ec C=) I) ey) = Oley. SuMEee 9 (2) = 0, we can drop this term and proceed to separate the equation by order: (1) mt) as — Bey, (E-20) and (ly) 2 (il) 2 p{2) pie ao) fs aS We (E-21) Substituting the velocity potential into the equation, one finds: per) sk pgtAes1” sin(k x - wt + 64) 1 k : + Ave 2 sin(k,x - wot + 65) + y} (E522) for the first order, and oe sin(k,x -w,t + 5) (4) . _ 6 P = - a Ue w A l 1 2 yee - wA,e 2 sin(k,x - wot + 65)] Kayan + [w,A,e 1 cos(k x = wt + 64) ky 2 + w Are 2 cos(k,x = w5t + 5,)] } for the second order. Note that this is identical to part of the 154 equation for surface elevation. The second-order pressure may be reduced to: (2) 2 e De PLAS? BR DA go P So tO, Ay Oe Op yes k_+k i 2u WA, Ane’ te 2 cost (k, - k,)x - (w, -w)t + 6, - 65]} (E-23) which indicates that the second-order pressure is composed of a component independent of time and at the "difference frequency". This is surprising since the equation for the free-surface elevation (eq. 18) includes terms at twice the incident wave frequencies and at the sum of these two frequencies. Using trigonometric relationships the first two terms in equation (E-23) could be expanded to yield terms at twice the incident wave frequency. A term at the sum of the two inci- dent wave frequencies may appear in the pressure computed using the velocity potentials representing wave diffraction or forced oscillation. It might also appear if the present analysis were carried to the third order. The derivation included here was intended to reveal the presence of a low-frequency component in the exciting force and has not been used to determine the other velocity potentials or carried beyond the second order. x Bo List of Special Symbols for Appendix E. Aj >A, = Wave amplitudes g = Acceleration of gravity ee b wy? w2" : 122 = Wave numbers, eb ee respectively X,Y = Cartesian coordinates (x-directed parallel to the direction of wave propagation, y-directed vertically upward) 84255 = Wave phase angles n(x, t) = Free-surface elevation o(x,y,t) = Velocity potential Wy »W5 = Wave circular frequencies 155 APPENDIX F PHYSICAL PROPERTIES OF SEVERAL FLOATING BREAKWATERS Li Proposed Oak Harbor Floating Breakwater (Davidson, 1971). a. Physical Properties. m = mass per unit length = 25.1 slug/ft I = mass moment of inertia = 621 slug-f£t7/ft X = x-coordinate of center of gravity = 0.0 ft. g (on centerline) ie y-coordinate of center of gravity = -2.34 ft (below WL) KH, 5 = 64.5 1b/ft/ft KHz, = 1,165 ft-1b/ft INLIb Oxelere Mel. 6 Ss O 1j b. Mooring Line Tension Response (change per unit displacement) . Ne ey iL L@ iey/sere AT _ 281 lb/ft Ay AT _ aa = boil Ue Go Computed Mooring Spring Constants (depth = 29.5 feet) KM, = 119 1b/ft/ft KM, > = -5.24 Ib/ft/£t KM, 5 = 166 1lb/£t KM, = -5.73 lb/ft/ft KM,, = 10.2 1b/ft/ft KM,, = -3.37 lb/ft KM,, = 160 lb/ft KM,, = 2.06 1b/ft - 156 = 2 = KM. 282. f£t-lb/ft Rectangular Breakwater Tested by Nece and Richey (1972). Physical Properties (at prototype scale). The cross section is a rectangle of beam 10 feet and draft 5 feet. m = 100 slugs/ft TS AW eioaeiees pee Ss = 0.0 ft (on centerline) i = -1.0 ft (below WL) KH,> = 640 lb/ft/ft Ming = So OM seat e All other KH.. = 0 13 All KM.. = 0. 1J Rectangular Breakwater Tested by Sutko and Haden (1974). Physical Properties of Model. The cross section is a rectangle of beam 0.333 feet and draft 0.222 feet. m = 0.143 slug/ft I = 0.023 slug-ft“/ft x = Wo ft (on centerline) o Tg 8 -0.123 ft (below WL) KH, 5 = 200% Moy/see//stre KH, = 0.244 £t-1lb/ft All other KH.. = 0 1j AU KiNG 3 SO 1j Alaska-Type Breakwater. a. PhysicaL Properties. m =] 62.8 siya 157 I = 4,234 slug-ft/ft x 5S 0,0 3 g Ve = -1.3 ft (below WL) KH, 5 = 528 lb/ft/ft = 32,885 ft- KH, 32, ft-lb/ft All other KH.. = 0 1j b. Mooring Line Tension Response (change per unit displacement). Hl 07.0 Wyse Ax Nae iy > 90.5 lb/ft A Gea -572 1b Go Computed Mooring Spring Constants (tide = +7.0 feet). KM} = So(0) ioy/ree/see KM, = 0.245 1lb/ft/ft KM, 5 =-9.23 lb/ft KM, = 0.302 1b/ft/ft KM, 5 Ss Oi Moy/ierey/ste KM, =-2.68 lb/ft KM. =-9.52 lb/ft KM.5 =-2.82 lb/ft KM 3 = 88.9 ft-lb/ft Friday Harbor Breakwater. a. Physical Properties. m 61.02 slugs/ft i} I = 4,160 slugs-ft°/ft 158 x = O00 se (On Cemecieligne) y_=-0.49 ft (below WL) KH, 5 = 884 1lb/ft/ft KH. = 55,610 ft-l1b/£t All other kH.. = 0 1j Mooring Line Tension Response. AT Re 2 208 yee NT AR 2A ay 7 29-0 lb/ft AT ae = O87 Hb Computed Mooring Spring Constants (tide = +5.33 feet). KM) = 6.46 Ib/ft/£t KM, 5 = O50 Wo//ree/sre KM, 5 = 18,5 stello/see/ict KM, S 0,510) lores KM,5 = O. 590 Wo/ire/sce KM53 S Now sell /scie/scte KM, S 8,6 Mo/see KM,, = ov b/s KM, 3 = 64.6 ft-lb/ft 159 APPENDIX G DATA SUMMARY SHEETS FOR FRIDAY HARBOR FLOATING BREAKWATER (WINTER 1975) Appendix G contains a summary of all the data recorded at the Fri- day Harbor breakwater during the winter season of 1975. Seven tapes were recorded during this period, with a total of 95 records. The tapes are numbered in sequence from FH7-1 through FH13-8. The date of each tape is given along with the pertinent statistical data for each record in the tapes. The number of days and hours given for each record begins with the day and hour given for that particular tape. All minimum and maximum values are measured from zero mean. 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OT°%eT 99°9€ 9€ °682 92°62 Te9e 6 e°%6 XV 862T° 760° E60T° BcoT° B80T° 94€0° GEEC® 942°0E %66°OT %6T°EL 909°S2 0€°64 OT°Y AIGIS TOO°S 2£25°S 464%°G ELT°S eeT°o SG0°E %E4°2 06°S86 90°L2L eT°e88. 04°9L4 S°9ET 9°02 NVIW 9 909 S= 6dr = SEB — ec bcm OO om eerie) eileen O66. — 90°LE= 2T°9ET= 0%°9L= S°9ET=“0°6= °NIW SE° 92 é To9° €SE° 98€° TS9° 00S° 8Ttt° %eT° OT°2ET 96°52 88° LOE 09°LS ®°9L 8°2T *XVH 2L66T° TS2T° D5ET® E€OBT® 6SET° T6%0° 2690° 0€9°62 559°6 LTL°SL €22°82 69°92 9E°E AIGIS GTO°S O69°S 9TS°S E£52°S LLT°% 985° 6TS°E LE*A5TT 9T° 462 BE°TSTT 68°SOL O°T2ZT 8°02 NVIW €2 O6L°= LO0S°= 6L4°=— 99E°= eL2°— GS2°= LLEe= 44°99— 6T°BE= HO°EET— YECTOT— %°OET—¥°2T— °NIW LE° €2 T 662° eLc° 624° 2T8° 629° 602° OLT? 9G° EET Te°re 9E°OHE 09°L0T €°6S 24°6 SKVW (9398/93S/14) e414 *4s eld °13 sat sa7 say seq °930 HdW SUNOH °YJA°S °YOH H2ACN °43u °ONI 2 NVYL T NVYL 3S aN AS AN ould ods ONV $¥3L3WOYI1999V $39V9 3AVA $1739 avor GNIA QNIA SAVG °4309 NI °ON °SNVUL SWIL °9348 2902 5s SJIdWVS JO YIaWNW SW 00S = GOI¥3d SNIIdWVS (NV3W O83Z WOUS GIYNSVIW SINIVA °NIW ONV °XVH) (S2/02/€ - O€22 — ETHI) VILVANVIYG INTIVOIA YOeUVH AVOIYS BOI VIVO WOITLSTIVIS 40 AYVWWAS 178 0S92° 66562° S2l02° OTO°S ¢TS°S 984°S 26T° T= BO0E°T= 8Tb6°= O%T°T OLE°T E98° 13): So 4 Sh Sl 3 Oe 920°C 984°S 464°S C902 = Sa EG Chel 1sG° 944° 98E° (93$/93S/14) °Y3A°S ° YOK SUSAN SazLIWOU31359V 6902° %59T° %€20° L£92°S 042° 988€°9 GT¥°= 652°= 620°- T68° G9L° 280° BOZT° 969T° 2250° 222°S = TlL2°s 008°9 ISAS IGN 0 969° TT8° 812° °id 1d eld °34u “ONT c NV s39¥9 3AV (S2Z/02/€ - O0€22 1 A S€20° S96°S £10 °= 680° TSS0° sgEe°9 LoT°- 86T° *14 T NVaL €THd) 662°T9 29°89OTT £9°OET- €£°L02 81S°SsS QE Sey, CONSOLE »9°B86T say ERS YILVPHVIUYG ONTLVOIS YORUVH AVOTYS 6T2°9T €88°OTT %6°ToL BE °FOTT 96° E9— BE °EZZ= 90°24 29°09% 9T2°Lt 260° TIT Ts°292 2L°SETT 0S °6b4— 08° TEZ— 84°84 02°9%E $@1 $497 JN AS $1139 avant 69L°CE 97° 89S OT °STT—- 58°ER 262°04 60°96S 80°RTI— 26°T6 sey AN 90°84 00° AIGLS L£°G2T 6°42 NVIH L°S2T-T°OTt= °NIW 6°TTT ¥°OT %X¥W 49°69 %L°% AAOLS 9°€2T 8°22 NVIW 9°E2T-E°OT— °NIW G°S2Tt B°ET *XVH °930 HdW ula °dS QNIA ONIA 8T 5T° 92 8 LT 2e° %”e a SaunoH GNY SAVQ °3309 NI °ON °SNVUL JWIL °93¥8 2502 = S31dWVS JO YI8WNN SH 00S = GOIY3d INI IdWVS (NV3W OYSZ WOYS AIYNSVAW SINTIVA °NIW GNV °XVH) OS VIVG WOITLSTLVIS 40 AYVWHNS 179 APPENDIX H INCIDENT AND TRANSMITTED WAVE SPECTRAL PLOTS Appendix H contains the incident and transmitted wave spectral plots along with the corresponding transmission response curve for 11 representative records. The data for the first nine were recorded at Friday Harbor, Washington, during the winter of 1975. Figures H-11 and H-12 were computed from similar data collected in Alaska during the winters of 1974 and 1975. The original time series were high-pass filtered at a cutoff fre- quency of 0.05 hertz to remove tidal drift. Each series consisted of 2,048 data samples and were sampled at a period of 0.5 second for the Friday Harbor data and 0.44 second for the Alaska data. The standard deviations and corresponding overall transmission coefficients for each of the Friday Harbor plots are given in Appendix G iSO INSI9144309 NOISSIWSNWYL ae oo" t SSS —_ ee 2 > | a | (9-ZHA) e2ep Tearqoeds aAeM pojiTwsUeZy pue JUSpTOU.T “T-H ean3tTy (ZH) AONANDIYS 98-0 40 _79°0 6y°0 LE°0 2" 0 21°0 00° oo WHLOSdS SAVM GSLLINSNVYL 90° 0 €0°0 80°0 (Ss ~314) JLVWILSH TWWaldadSOLAVY GSHLOOWS 0 ot SANS VYULISdS SAVM LN3AGION! —= SSNOdS3Y NOISSINSNVYL—> Et°o 18] IN319144909 NOISSIWSNVYL 08°0 09°0 Ov"0 02 0 O0* T 00 (L-LHd) e2ep TerqOeds oAeM poi iTusUeI pue JUEPTOUT “7-H eansty (2H) AQNANDAYS 6v°0 LE°0 @2"0 z1°0 98° 0 4L°0 239°0 VuloadS JSAVM G3LLINSNVYL — SAUND =—- 3SNOdS3¥Y NOISSINSNVYL VYLo3sdS SAVM LNSGIONI— ‘gI*o S-,14) JLVWILS3 TWYL193dSOLNY G3HLOONS 8070 70°0 oo ot°0 ( 61°0 182 INS1I9144909 NOISSIWSNVYL 020 08*0 09°0 00° t 00: 98° 0 -(@-LHA) e2ep Teztqz0eds aaem poz jtTWsUeT] pue JUePTIUT (zH) AINanvaYs 6v°0 LE*O SAYND SSNOdS3y VaeLoasdS SAVM LN3SGIONI— NOISSINSNVYL "$-H ean3Ty S2°0 _~ 92°0 Lt°0 g0°0 00 14) JLVWILST IWuldIadSOLNVY GAIHLOOWS vS°0. (S “5 ovo 183 1N319144309 NOITSSIWSNVUL *(6-LHd) e2ep Tezq9eds sAEM pszaTUSUeT pue JUEepTOUT (ZH) AQNANDAYS 6r°0 Le*o ie §5°o 98° 0 41L°0 “63° 0 =, er Bees o V8LO3dS SAVM GSLLINSNVYL 09'0 plow (ereptaen =O Deuiae ‘y-H ean3ty 2° 0 Et°O0 SAYND Vu¥lo3sdS SAVM LNSGISNI—| JSNOdS38 NOISSINSNVUL —=! 00° 1 ood 600 43©=———S 0 id) FLVWILSA TWYLIadSOLNY G3HLOOWS SIO €1°0 eo (S 2 184 -(OI-ZHA) e2ep Tetq90ds oAeM poqitusUeIy pue JUSEPTOUT ‘*¢S-H euANndTY (2H) AINANDAYS 6y-0 Le-0 G2" 0 Z1°0 98° 0 4L°0 29° 0 00° r=) 02°0 SAYND =—-3SNOdS3¥Y NOISSINSNVYL OF°0 a9°0 —IN319144309 NOISSIWSNWYL 08°0 Vu¥LosdS SAVM LNSGIONI— 00° 1 cate (S ~ ia) ALWWILSS IWaLIFdSOLNVY GAHLOOWS vt°0 Tt°0 4070 vO°0 8t°0 185 IN3T9T4S4409 NOTSSIWSNVYL “(TT-LHA) e2ep Teaqoeds onem pojzjTUsUeI} pue JUepToOU, ‘9-H ean3Ty (2H) AINSNDSYS g5°0 98° 0 “vl 0 23° 0 6y'0 Vuelosds SAVM aalliwsnval OVO 02°0 09°0 SAXND a SSNOdS3uY NOISSINSNVYL 09*0 VYLO3dS SAVM LNSGIONI — Q0° tT LE*O S3°0 Zt°0 00° €0°0 or PP 90°0 €t°O. (S ~714) ALVWILSS WaloadSOLAVY GAHLOOWS _ 60°0 186 IN3I9144909 NOISSTIWSNVYL 08°0 0S°0 Ov°0 02°0 00° 1 “(ZI-LHA) e2ep Ter}yO0ds aAem poi Tusuet, pue JUSPTOUT */-H eins Tt. 98° 0 4L°0 VueloasdS JAVM LN3GISNI~™ (ZH) 6v°0 Leé°0 S2°0 “Et"°0 SAYND 3SNOdS3Y NOISSIWSNVYL oO o oo &P 10°0 €0°0 0 (S ~ 14) JLWWILSS WaldadSOLNWY GAHLOOWS vO" 90°0 (0) 187 LN319144509 NOISSIWSNVYL -({T-ITHd) e2ep Terq90ds sAem pojzTwsueI pue JUSpToUT (2H) AINANDSAYS 98° 0 vL"0 29°0 ev'0 Le°0 0a°0 eA , z ov"O 09°0 i 2 fas) ro) 00°t VuLOSdS JAVM LNAGIONI— SAYND SSNOdS3Y NOISSINSNVYL— "8-H einsty S2°0 $0°0 214) JLVWILSA WWulIdadSOLNVY GAHLOOWS *0 vO°0 LO°0 (S 9 60°0 188 LNATI1S4309 NOISSIWSNVYL 08°0 09°0 ov°0 02°0 o0°T 98°90 *(ZI-TTHd) e2ep Terj990ds aaem pajztusuety pue JUSpToUT “6-H sansTy (ZH) AQNSNDAYS L°0 29°0 8h°0 Le°0 So°0 VuLOadS SAVM G31 LINSNVYL—» 3ANND SSNOdS3Y NOISSINSNVYL a V¥lI3adS SAVM LNSIGISNI—- gt°0 19 Sa) so°0 (S-14) JLWWILSS WalIdadSOLAVY GAHLOOWS Te°0 Lo°0 189 INS319144509 NOTSSIWSNVYL ov°0 02°0 03°0 08°0 00° t “eysety ‘ssutaidg seyeuel ‘(1-LxL) eiep Ter.O.eds oAeM poejITWsUeIy pue jUEpToOUT “OT-H emn3sty (2H) AINANDSYS 86°0 ¥8°0 0L°0 9S°0 ov'0 82°0 vt°O 00°0 wo ay fo WsaLI3dS SAVM G3SLLINSNVYL SAYND JSNOdS3Y NOISSINSNVYL Vdlo3dS SAVM LN3AGISNI—= (=) €9°0 Lv°0 3&°O 9t°0 oo° Se alsi) ALWWILSS WWalisdSOLAVY GAHLOOWS 6L°0 ( 190 LN3I914433509 NOISSIWSNVYL 0s°0 Ov°0 02°0 08°0 00°t “eysety “exats ‘ (OT-pys) Biep Te1j.eds oAeM poz ITWSUeIY pue JUSpToUY ‘TT-H oaznsTy (ZH) AINSNDSAYS 86°0 v8°0 0L°0 SSvOm = ov°O 83° 0 vt°O 00°0 VueldsdS JAVM GSLLINSNVYL — VdlosdS SAVM LNSGISNI— y SAYND ASNOdS3Y NOISSINWSNVY L 6t°O €t°0 90°0 14) JLVWILS3 WWYLITdSOLNY GAHLOOWS Sé°0 Té°0 é (S- 19] APPENDIX I LOW-FREQUENCY SPECTRAL ANALYSIS OF FORCE DATA Appendix I contains the low-frequency autospectral and cross-spec- tral plots for record FH7-8. The data were recorded at Friday Harbor, Washington, on 6 January 1975 at 0030 hours. The original time series were low-pass filtered at a cutoff fre- quency of 0.2 hertz and every eighth data point used to generate a new time series. This gives 256 points with a sampling period of 4 seconds. 192 *(9-ZHd) earqoeds ao10z YSemYyINOS “T-T aan3stj (ZH) AINSANDSAYS 60°0 80°0 90°0 S0°0 €0°0 00°0 (24494 20 93) GQ3Y3L1I43 SSVd MOT 2 (S81) 0612 = 20 = JONVIYVA i) 12°0 vt°O Lo°0 00°0 ) JLVWILSS WULITdSOLAY GSHLOOWS 82° se°0 (20 40 % 193 "(8-ZHd) ertq0eds 9dZ0F AseoyIION *7-] oun3Ty (ZH) AINSNDAYS 60°0 80°0 90°0 S0°0 €0°0 20°0 00°0 (24494 2°0 = 93) G3431 113 SSVd MOT We) 891 ae = JONVIYVA 12°0 ¥t°O L0°0 00°0 (20 40 %) JLWWILSS IWULIFdSOLNY GIHLOOWS 82°0 Se°0 194 3t°0 *(8-ZHdA) erqzo0ds ad10F YsSeayynos ‘“¢-] sANnsTY (ZH) AINSANDSAYS 60°0 80°0 90°0 S0°0 €0°0 20°0 00°0 (24494 2°0 = 93) Q3Y311I4 SSVd MOT 2(*al) 0024! 50 = JONVIYVA 82°0 T2°0 vt°O L0°0 00°0 (20 30 %) JLWWILSA WWULIAdSOLNVY GAHLOOWS se°0 195 ‘sooqoy ySeayINos pue Isvey TOU UseEeMJ0q eSed “p-I SINSTY (ZH) 11°0 60°0 80°0 90°0 AINANDAYS S0°0 €0°0 30°0 00°0 €9°0- 68° T- ST°€- (SNVIGWY) JLVWWILS3 IWdlodds ASVHd GAHLOOWS €9°0 68° T 196 3t°0 “seoL0F ZSeeyInos pue Yseoey.ioU useMI9q ADUsLEeYOD *S-] saN8TY (ZH) AINANDAYS 1t°0 60°0 g0°0 90° 0 g0°0 €0"0 30°90 09°0 08°0 JLVWILSS AINIYIHOD G3YVNOS GIHLOOWS oo°t 197 “S8dLOF JSOMYINOS pue YSeeYyINOS ay. Uds9eMI0q aSeYg ‘9-] 9ANBTA (ZH) AINSNDAYS 1°0 60°0 80°0 90°0 g0°0 €0°0 i=) i=) e oO €9°0- 68° T- St°€- €9°0 *t (SNWIGVY) JLVWILSS TWuYldadS 3SWHd G3IHLOOWS St°e a 198 ot°0 Tt°0 “sa0r0F YSOMYINOS pue YSeoyyNOSs 9Yyi usemMzeq AoUaTEYO) (ZH) AQNANDAYS 60°0 g0°0 90°0 S0°0 "L-J eansty €0°0 20° 0 00°0 09°0 OF°0 02°0 00°0 JFLVWILS3S AINSYSHOD GaYWNOS GAHLOOWS 08°00 o0°t 199 APPENDIX J HIGH-FREQUENCY SPECTRAL ANALYSIS OF FORCE AND MOTION DATA Appendix J contains the incident wave spectral plot along with the autospectral and cross-spectral plots for the force and motion data for record FH7-8. The data was recorded at Friday Harbor, Washington, on 6 January 1975 at 0030 hours. The incident wave spectra was unfiltered. All the force and motion spectral data were digitally high-pass filtered at a cutoff frequency of 0.1 hertz. The autospectral data is plotted as a percent of the vari- ance, i.e.,the total area under the spectra. Wave heights, forces, and motions were measured in feet, pounds, and feet per second square, re- spectively. All spectra were computed from 2,048 data points sampled at 0.5- second intervals. 200 00°T L8°0 *(8-ZHd) eaqoeds onem YUepToUT (ZH) AINANDAYS SL°0 z9°0 0s°0 z(t) S¢90 ‘[-f ean3Ty Le°0 S2°0 ONIN 31113 ON (44) 101 = SH =50 = JONVIYMVA iz) 0 L0°0 (20 $0 %) ALWWILSA TWYLIIdSOLNY G3IHLOOWS 82°0 SE°0 20! 00° tT L8°0 *(8-ZHA) e23z9eds ad10F yseoyynos *7-f 9aINn3BTY (ZH) AINSNDAYS SL°0 z9°0 0s" 0 Le"0 S2"0 ot°0 00°0 (24484 |'O= 93) G3YN311I4 SSVd HOIH (Sqi)( Ol) 61:2 =_0 = JONVINVA ] €- 2] 82°0 12°0 vt°O Lo°0 “0 (20 $0 %) JLWWILSH TWHYLIFdSOLAY GIHLOOWS se°o 202 00°T L8°0 *(@-ZHd) exqoeds 9020F YseaYyIION “¢-L eANsTY (ZH) AINSNDAYS SL" 0 z9°0 0s"0 Le*0 S2°0 ot°0 00°0 (24484 1° = 24) G3N31I4 SSVd HOIH (, 01) 891=.0= JONVINVA 1Z°0 bro LO°0 00°0 (2-0 40 %) FJLVWILSA WWuldIadSOLNY GAHLOOWS 82°0 SE°0 203 00° Tt L8°0 (8-ZHA) eaq0eds adzZ0J YSeaYyIN0S “P-L 9AIN3TY (ZH) AINSNDAYS SL" 0 29°0 0s°0 LE°0 SZ" 0 (24484 |O= 94)G3Y31 113 SSVd HOIH (Ol) cv | rot) = JONVIYVA 00°0 82°0 T3°0 vt°O LO°0 00°0 Sé°0 (2-0 40 %) JLWWILSH TWYLIFdSOLNY GSHLOOWS 204 00° t L8°0 “S9DLOF YSeoYy INOS pue SeSsYyZIOU 9Y} US9M}9Eq 9SeUd (ZH) AQINANDAYS SL" 0 29°0 0s"0 LE"0 S2"0 "S-f oan3sty ot°o0 00°0 €9°0 €9°0- 68° T- ST°€- (SNVIGWY) JLVWILST TWYLISdS JSWHd GIHLOOWS 68°T ST°€ 205 00° T L8°0 “sadL0F YSeoyINOS pue seey.IOU sy} UseM}I0q ADUaTEYOD (2H) AINSNDIYS SL°0 29°0 0S" 0 LE°0 SZ" 0 °Q9-f 9aN3TA ot°O 00°90 09°0 0v°0 02°0 00°0 FLWWILS4S ADNAYSHOD G3INVNOS G3SHLOONWS 08°0 oo°t 206 00° Tt L8°0 “seoTOF ZSAMYNOS pue JSe9YAZNOS 9Y4} Us9Mz9q J9SeUd (2H) AQNANDAYS SL°O z9°0 0s" 0 Lg"0 SZ" 0 "L-¢ emn3ty ot°O g o aa" co0 = eSO- =—sTI- STS (SNVIGVY) JLVWILSS TWULITdS JSVHd GIHLOOWS Ste 207 L8°0 *sooLOF JSOeMY NOS puUe sesy Nos sy usemMzeq AduUsrTEYO) (2H) AQNANDSYS SL°0 29°0 0S°0 LEO S2"0 "S-f 9mn3TYy ot°0 00°0 09°0 ov0 02°0 JLWWILSS ADNAYFHOD G3AYVNOS GIHLOOWS 08°0 oot 208 oo°t £8°0 *(8-ZHA) e2390ds uotiOW aAeaHH *G-f eaNn3Ty (ZH) AINSNDAYS SL" 0 z9°0 0s°0 Le°o G2" 0 o1°0 00°09 (24494) °O= 93) GAYSLMS SSVd HOIH d “pbs = a a }) (, 91) evv of? 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L8°0 “T[OI pue aAeaYy usemjeq AoueTeYyoD “¢[-f oansty (ZH) SL°0 z9°0 0s°0 AINSNDIYS Le"0 S2°0 ot°0 00°0 00°0 o3°0 Ov°0 02°0 JLVWILS3 AINAYSHOD GIYWNOS GAHLOOWS o8°0 00° tT 213 0.87 1.00 0.75 0.62 0.37 0.50 FREQUENCY (Hz) 0.25 0.12 oO oO eo SI°e 69° I €9°0 €9°0- 68° I- SI'e-— (SNVIGVYU) JLVWILSS WuLlIadS ASWHd GAHLOOWS 214 Phase between sway and roll. Figure J-14. 00° T L£8°0 "TTO1 pue Aems usvamjoq ADuaTOYyOD “Ss T-f ean3Ty (ZH) SL°0 69°0 0S°0 XINANDAYS LE"O SZ°0 08°0 o9°0 Or°O 02°0 JLVWILSHT AINIYFIHOD GIYVNOS GIHLOOWS 00° t 215 “aAeoy pue Aems us9m}0q OSeYd “OT-f OaNsTY (ZH) AINANDSYS z9°0 0s°0 Le*0 G2" 0 00°0 68° €9°0 €9°0- 68° I- St°e ‘ST°E-_ (SNVIGVY) JLVWILSI WYLIIdS ISWHd GIHLOOWS 216 oot L8°0 ‘a ARay pue AeMS usEeM}0q ADUETAYOD “LI-f£ aan3ty (ZH) AINANDAYS SL" 0 29°0 os°0 LE°0 G2" 0 21°0 00°0 08°00 o9°0 Ov°0 0z°0 00°0 JLVWILSS AINIYSHOD GIYWNDS GIHLOOWS 00° T 217 APPENDIX K WAVE MEASUREMENT Ie Wave Staff Design. A block diagram of the wave staff and associated electronic cir- cuits is shown below: Precision Square Bilateral & AC Detector ey Wave Current Staff Buffer & Variable Oscillator Source Gain Amplifier The wave staff itself consists of a length of PVC tubing which is spirally wound with a resistance wire, such that when it is immersed in seawater, the electrical resistance varies in direct proportion to the length of the exposed staff. The electronic circuits driving the wave staff consist of a fixed frequency square wave oscillator (having a precisely controlled output amplitude) driving a precision bilateral current source with an output current directly proportional to the input voltage. Thus, the wave staff is driven by a current source of constant magnitude, but one which changes direction with each one-half cycle of the square wave oscilla- tor. The output of the wave staff then is a square wave voltage with a magnitude (peak to peak) that is directly proportional to the length of the exposed wave staff. This output is fed to a high input impedance voltage follower circuit which serves as a buffer between the wave staff and the ac detector circuit. The precision ac detector circuit uses two operational amplifiers in conjunction with two diodes to form a precision full-wave rectifier circuit that is capable of operating at very low input voltages. Ordinary diode detector circuits cannot operate on ac Signals of peak magnitude less than the forward voltage drop of the diodes and produce large conversion errors unless the sig- nal magnitude is large with respect to the diode voltage drop. A gain control has been incorporated in the detector circuit so that full-scale Output can be set at any positive value up to +10 volts with a wave staff resistance of 300 ohms up to 3,000 ohms. Alternating current is used to drive the wave staff to avoid both the corrosion effects that would occur if direct current were used and the de offset which occurs as a result of the use of dissimilar metals in a conducting solution. The latter is eliminated by use of ac coup- ling in the output from the wave staff. Bench tests of the wave staff electronic circuits were made using a1,000-ohm variable precision resistor in place of the wave staff. The 218 circuit was adjusted to produce an output range of 0 to 10 volts with the resistor varied from 0 to 1,000 ohms. Linearity was determined to be 0.1 percent of full scale over this range. Tests were also made to determine the effect of temperature on sensitivity and zero drift. A decrease in sensitivity was noted with decreasing temperature of about 0.03 percent of reading per °Celsius over the temperature range of 0 to 24°Celsius. A zero drift of 2 millivolts was also noted over the same temperature range. A +10 percent change in supply voltage from the nominal +15 volts produced no observable change in output. If we assume an operating temperature range of +5°Celsius, the maximum error in the wave staff electronics due to the combined ef- fects of nonlinearity and sensitivity variations with temperature is +0.2 percent of reading. Since the primary interest is in a dynamic mea- surement of waves, the zero drift noted will have negligible effect on the experiment since temperature variations of any appreciable magnitude will only occur over long periods of time compared to the wave periods. Further calibration tests were conducted using actual wave staffs of l-inch diameter and 20-foot lengths, and 3.5-inch diameter and 8-foot lengths at various depths of immersion in saltwater. These tests were conducted from a dock at Shilshole Bay on Puget Sound. Because of ripples and waves on the water of the order of 1 inch (peak-to-valley) it was difficult to obtain a highly precise measurement. The output was recorded on a strip chart recorder and it was therefore possible to average these variations to some degree. The readout resolution of the strip chart (and accuracy) is about +1/4 of a minor division. Full scale across the chart is 50 minor divisions and, thus, the resolution is about 0.5 percent of full scale. Some nonlinearity is noted near full immersion (see calibration curve). Some offset was expected because of the finite resistance of the saltwater path in the ground return which is not taken into account during initial calibration of the wave staff unit. The initial calibration is made with the wave staff on the dock where full scale and zero are set by making actual contact between the ground wire and the wave staff resistance element at the corresponding ends. However, measurements were made of the resistance of the salt- water path to ground in the same location where the wave staffs were immersed and the value of resistance measured (on the order of 10 ohms) does not account for the offset observed at full immersion. In addition, the offset should occur at all readings and it does not. Therefore, it is believed that the nonlinearity observed is a result of some other phenomenon as yet undetermined. Both units produced highest accuracy near center scale with decreasing accuracy toward either end. Overall accuracy including end points is about +3 percent. If the range of oper- ation is reduced so as not to use the last 1 foot on each end of the wave staff, the accuracy is improved to about +1 percent. The output from the wave staff electronic circuit is fed directly into a voltage to frequency converter; the frequency output is then counted and stored on separate storage registers, once every 50 milli- seconds. If an 8-bit register is used for the wave staff measurement, 219 the maximum count that can be stored is 255; therefore, the sample time must be on the order of 25.5 milliseconds (maximum count divided by maxi- mum frequency output from voltage to frequency converter) . The wave buoys use an §8-bit register with a 32.5-millisecond sample time while the wave staffs use a 16-bit register with a 250-millisecond sam- ple time. The error due to gain instability and nonlinearity of the voltage to frequency converter is of such low magnitude that it can be neglected and the overall accuracy of the recording is essentially the same as given for the wave staff unit by itself (i.e., between +1 and +3 percent depending on the range of operation on wave staff). a a De Spar Buoy Design. Spar buoys were used at two of the sites because of their advantage in handling and transport and because they minimized the placement diffi- culties due to navigational hazards, water depth, and tidal conditions. The spar buoys were made of two PVC pipes coupled together near the cen- ter of the buoy. The lower section is a 15 foot by 6 inch pipe filled with styrofoam. The top section is 12 feet by 3 inches wherein the upper 8 feet is wound with a resistance wire which measures wave elevation. The wave staff electronics are mounted inside the top section, above the waterline, with the remainder being filled with a wood core to add stiff- ness. The buoys also have a 2.5-foot diameter damping plate mounted on the bottom and are anchored using a dual point mooring system with the anchor lines attached at the center of drag on the buoy to prevent it from being pulled underwater in strong currents. One of these buoys was tested in the Puget Sound just north of Seattle. Its performance exceeded expectations both in terms of minimized response to the waves and accuracy of wave height measurement. Figure K-1 gives a sample of the output from the buoy's wave staff in saltwater for a plus and minus 1 foot excitation of the buoy in heave. This was accomplished by push- ing the buoy up and down by hand. Some distortion results from this approach which shows up in the output of the accelerometer mounted at the center of the response of the buoy in heave and roll in calm water. The natural periods for heave and roll taken from these plots are approxi- mately 18 and 14 seconds, respectively. These are well out of the range of the 3-to-4-second wave periods expected at the site. Visual obser- vations of the buoy in waves in excess of 1.5 feet indicated no ob- servable heave or roll motion, but some yaw about the anchor line caused by the current and wind. This motion resulted in less than a 1 foot variation from the buoy's horizontal position in calm water and appeared to have periods in excess of 30 to 60 seconds. For comparative mea- surements, the buoy was located about 30 feet from an existing four-gage array of l-inch diameter Oceanographic Services, Inc. resistance wire wave staffs. A comparison of simultaneous output from the two wave staffs (buoy mounted and stationary) is shown in Figure K-4. The auto- spectras computed from data obtained from one of the stationary wave staffs and from the spar buoy, in a 25-miles per hour storm with 220 1.6 £t/on* 1,25 £t/sec?/cn * ee + ZERO VALUE (KI _ SPAR BUOY WAVE STAFF t-—+ *0.4 cm/scale div. @— 125 mm/min Figure K-1. Wave and acceleration data for par _ uoy. HEAVE anil 4 Y | | Tf oT l L + : t --f---4-- H ; 1 | | | | | J is Sart 4-4 4 4 i Jk 4 Beate eae ‘1 1 1 1 SPAR BUOY WAVE STAFF 1 foot motion) | tbh ites { 1 | Hie | | ite can ont Hk kalo a hs [big A iis I Figure K-2, Spar buoy heave response. 22 [ POINT a ee (45) ROLL PERTOD earoris a i : HEEL iii Sanaa No} pap! : ae 4 ff dee SPAR BUOY WAVE STAFF tao 4 SEC. fap pf tga fad te tf fp eh Ref Sp Sp peep aff y= * nEee Caae o a) NT N\A LAN. aw 2 E : i : 2 ii 5 1 Tal fA J f a Has = SPAR WAVE HU Nab Se ae ann ef pe tems sig) thet este Set) pe ce sit et eee ore es Te rapes ape team ers hes (peat at *0.4 cm/scale div. <2—— 125 mm/min Figure K-3. Spar buoy roll response. Wes Breas Rts ie, EO Coan ie 204 BEC. _SPAR a WAVE STAFF, ., ee oe EAE SS sega aia eee ie aie WA AMT ik eae Lid fray a AGA in | a at | h /\ if wh vith NN i: NIA biti ip If Ten Ae ma ii vin nl ee es «| iM n a aau Pies 4 A inh if fA A n MA Ea Aly Kanha ara hs a FW | WY | TUM i < i} i | iy an yyy : Usofarioemla lag alee i ees r | foe at ee bs fe bene tea a te ae e tee ee ee ee a ee os T WAVE STAFF ae Oaas 0 *0.4 cm/scale div. 1S, saint Figure K-4. Wind wave data for spar buoy and stationary staff. 222 maximum wave heights in excess of 1.5 feet are shown in Figure K-5. These spectra were computed from simultaneous records of 20 minutes in length. 223 NORMALIZED SMOOTHED SPECTRAL ESTIMATE (Hz) OS! WAVE STAFF ite) © Ly) SPAR BUOY WAVE STAFF ¢ 2 | + Ca} @ ™ ° \ 9) \ \ (op) N WN \ es \ * \ NS \ fo) \ ~ (oe) eo} °09,00 0.20 0.40 0.60 0.80 1.00 Figure K-5. 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