U.S. Firmy Coast. Eng. Res. MR 76-5 (AD-A0z3 682) Reflection and Transmission Characteristics of Porous Rubble-Mound Breakwaters by Ole Secher Madsen and Stanley M. White MISCELLANEOUS REPORT NO. 76-5 MARCH 1976 DOCUMENT COLLECTION Approved for public release; distribution unlimited. Prepared for U.S. ARMY, CORPS OF ENGINEERS COASTAL ENGINEERING ce RESEARCH CENTER 20% Kingman Building OES\ Fort Belvoir, Va. 22060 mre Ae-S Reprint or republication of any of this material siall 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 desigriated by other authorized documents. TT iin 0 030] NAN SECURITY CLASSIFICATION OF THIS PAGE (When Data Entered) REPORT DOCUMENTATION PAGE . REPORT NUMBER 2. GOVT ACCESSION NO.| 3. RECIPIENT'S CATALOG NUMBER MR 76-5 TITLE (and Subtitle) 5. TYPE OF REPORT & PERIOD COVERED REFLECTION AND TRANSMISSION CHARACTERISTICS OF POROUS RUBBLE-MOUND BREAKWATERS Miscellaneous Report \4. PERFORMING ORG. REPORT NUMBER AUTHOR(s) 8. CONTRACT OR GRANT NUMBER(S) Ole Secher Madsen DACW72-74-C-0001 Stanley M. White PERFORMING ORGANIZATION NAME AND ADDRESS 10. PROGRAM ELEMENT, PROJECT, TASK AREA & WORK UNIT NUMBERS Department of Civil Engineering Massachusetts Institute of Technology ESI230 Cambridge, Massachusetts 02139 CONTROLLING OFFICE NAME AND ADDRESS 12. REPORT DATE Department of the Army March 1976 Coastal Engineering Research Center (CERRE-SP) IS TINUMBERIORIP/AGES Kingman Building, Fort Belvoir, VA 22060 ig (1, 14. MONITORING AGENCY NAME & ADDRESS(/f different from Controlling Office) 15. SECURITY CLASS. (of this report) UNCLASSIFIED 15a. DECL ASSIFICATION/ DOWNGRADING SCHEDULE DISTRIBUTION STATEMENT (of this Report) Approved for public release; distribution limited. 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 Friction factor Reflection coefficient Energy dissipation Hydraulics Rubble-mound breakwater Empirical relationships Porous material Transmission coefficient Wave reflection Wave transmission ABSTRACT (Continue on reverse side If necesaary and identify by block number) This report presents the results of a study of the reflection and trans- mission characteristics of porous rubble-mound breakwaters. An attempt was made at making the procedures entirely self-contained by introducing empirical relationships for the hydraulic characteristics of the porous material and by establishing experimentally an empirical relationship for the friction factor that expresses energy dissipation on the seaward slope of a breakwater. DD , brig 1473. ~—s EDITION OF | NOV 65 1S OBSOLETE SECURITY CLASSIFICATION OF THIS PAGE (hon Data Entered) PREFACE [This report is published to provide coastal engineers with the res of research on the reflection and transmission characteristics of porous rubble-mound breakwaters. The work was carried out under the coastal processes program of the U.S. Army Coastal Engineering Research Center @GERG)). The report was prepared by Ole Secher Madsen, Associate Professor of Civil Engineering, and Stanley M. White, Graduate Research Assistant, Ralph M. Parsons Laboratory, Department of Engineering, Massachusetts Institute of Technology, under CERC Contract No. DACW72-74-C-0001. The research was conducted at the Ralph M. Parsons Laboratory from 1 December 1973 through 30 November 1975. The authors acknowledge the assistance of Mr. James W. Eckert, who participated in the development of the accurate method for determining experimental reflection coefficients. The advice and encouragement of Dr. Robert M. Sorensen, Chief, Special Projects Branch, CERC, are greatly appreciated. Dr. R.M. Sorensen 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, 88 Congress, approved 7 November 1963. th JAMES L. TRAYER Colonel, Corps Engineers Commander and Director Tah IEA IV APPENDIX A B € CONTENTS INTRODUCTION. TRANSMISSION AND REFLECTION CHARACTERISTICS OF RECTANGULAR CRIB-STYLE BREAKWATERS. 1. Preliminary Remarks. : 5 2. Analytical Solution for Teanemieston od Reflection Coefficients of Crib-Style Breakwaters 3. Comparison with Experimental Results . 4. Discussion and Application of Results. REFLECTION COEFFICIENT OF ROUGH IMPERMEABLE SLOPES. 1. Preliminary Remarks. Bois ‘ 2. Theoretical Solution for the Reaieorion Coefficient of Rough Impermeable Slopes . 3. Experimental Investigation . 4. Comparison of Predicted and Observed Reflection Coefficients of Rough Impermeable Slopes. 5. Discussion and Application of Results. AN APPROXIMATE METHOD FOR THE PREDICTION OF REFLECTION AND TRANSMISSION COEFFICIENTS OF TRAPEZOIDAL, MULTILAYERED BREAKWATERS . S 1. Description of the Approximate Approach. 2. Determination of the Equivalent Restsupuler Breakwater. - 5 3. Computation of the Transmission, ait Reflection Coefficients for Trapezoidal, papier Breakwaters . 4. Comparison Between pceeered and Observed Transmission and Reflection Coefficients of a Trapezoidal, Multilayered Breakwater. SUMMARY AND CONCLUSIONS . LITERATURE CITED. GOVERNING EQUATIONS AND THEIR SOLUTIONS . EXPERIMENTAL DATA . DETERMINATION OF REFLECTION COEFFICIENT . TABLES 1 Information used in numerical sample calculations . 2 Comparison of measured and predicted reflection coefficients . 90 42 7A 10 CONTENTS TABLES--Continued Comparison of measured and predicted reflection Oe tleslemes ia Bsa ol 6) 5: eich @ ti Oe obeoro Information used in numerical sample calculations Evaluation of equivalent rectangular breakwater . Summary of calculations of external energy dissipation. Summary of calculations of reflection and transmission coefficients of equivalent rectangular breakwater based on AH / AH, =! : Summary of Calculations of Reflection and Transmission Coefficients of Equivalent Rectanguzar Breakwater based on AH / AH, given by equation (161) Predicted reflection and transmission coefficients of trapezoidal, multilayered breakwater . FIGURES Definition sketch . Transmission coefficient for crib-style breakwaters Reflection coefficient for crib-style breakwaters Empirical formula for flow resistance in a porous medium (Sollitt and Cross, 1972) Empirical formula for flow resistance in a porous medium (Keulegan, 1973). Comparison between predicted and experimental tramsmission, T, and reflection coefficient, R. Comparison between predicted and experimental transmission, 1, and reflection cockfivenent, Ros os % 2. « Comparison between predicted and experimental transmission, T, and reflection coefficient, R. Comparsion between predicted and experimental transmission, T, and reflection coefficient, R. RY enon 5 oe Comparison between predicted and experimental transmission, Tj. and srefilecttom coctiicient; Ro. = .'s.-5 2... 4 : 4 Page 73 76 89 91 93 95 7 16 22 23 52 34 35 35 38 38 59 CONTENTS FIGURES--Continued Page Comparison between predicted and experimental transmission, Tavand LerhectiongcOcksrGient yin Rivaks eis! «sy eee) sie sensi ite tenes 39 Comparison between predicted and experimental transmission COCREICHENE Sie ntianrn: ub May tie, ote net sity heme” ce 6 Ae ee ee mma each. Comparison between predicted and experimental reflection COCLETCMEMES MMe meta ciate Lehutes eh id Meets is cua. Vek 35 "ok ia, Seale Menem Deke puieaop eNO saa, Guo. 010. lider Gl Gt Gero noe deco Gc 48 Reflection coefficient, R, of rough impermeabie slopes. .... 53 Runup, R, on Tough, impermeable: Slopes. 202.92 -. .0s 6 3s ee D2 Slope friction-factor,: Piy- vveti ce hits bernaksanat © 9 2 ee 58 Slope roughness boards used in the experimental eS Calo aie HOTIan ey taste Leto Bee st, cglteenen a Ske ote. a ce OE Exe rameme al ese cuins weld, a aia We wegen cies kn sete, oo ter oi oniete er eee Wave amplitude variation along constant depth part of the Edney Aaa MAk sR wee Muay se os He eh ee ee eee tere) et OO Wave record showing pronounced second harmonics at a TOC Chins (cat Se UE eee: Le MLC ce Ss eng, a God at gh eee ote cee RO ROO Empirical relationship for the wave friction factor, Eu ate otis 68 Definition sketch of trapezoidal, multilayered breakwater and its hydraulically equivalent rectangular breakwater. .. . 83 Horizontal slice of thickness, Ah., of multilayered DOA a Cor eae see eR oe. ad Ae onan otis wee HS, has irs 84 Breakwater configuration tested by Sollitt and Cross (1972) . . 88 Comparison of predicted and observed reflection and transmission coefficient of trapezoidal, multilayered mea Waite ramets, eer ecu cee ts oc Sefaroueies nak fo mol ost 7G Pieaiter wet ue Uiusy sl elates 1 98 SYMBOLS AND DEFINITIONS incident wave amplitude maximum wave amplitude minimum wave amplitude reflected wave amplitude (complex) transmitted wave amplitude (complex) complex wave amplitude complex wave amplitude wave amplitude of equivalent incident wave complex vertical amplitude of wave motion on slope at stillwater level wave excursion amplitude fr1etron! factor average stone diameter reference stone diameter stone size as defined in Section III Zew/il average rate of energy dissipation energy flux nondimensional friction factor linearized bottom friction factor wave friction factor volume force pressure force inertia force slope friction constant acceleration due to gravity water depth along sloping breakwater face constant water depth seaward of breakwater incident wave height maximum wave height minimum wave height horizontal slice thickness head difference for equivalent breakwater head loss in material n head difference for trapezoidal breakwater VET: wave number wave number corresponding to constant depth region imaginary part of wave number real part of wave number momentum coefficient length of crib-style breakwater equivalent length of crib-style breakwater length of porous material n submerged horizontal length of impermeable slope wavelength distance parameter used in permeameter tests momentum flux porosity iT] number of experiments performed pressure pressure of water surface discharge per unit length discharge associated with slice } reflection coefficient critical Reynolds number particle Reynolds number particle Reynolds number of model measured reflection coefficient predicted reflection coefficient predicted reflection coefficient (simplified formula) ratio of runup to incident wave height reflection coefficient determined in Section II reflection coefficient determined in Section III parameter defined by equation (5) parameter defined by equation (28) time transmission coefficient wave period transmission coefficient calculated in Section II complex horizontal velocity component horizontal velocity component horizontal velocity component at the bottom horizontal discharge velocity component through a slice horizontal seepage velocity average horizontal velocity component horizontal velocity component at the free surface complex vertical velocity component vertical velocity component vertical velocity component at the bottom vertical seepage velocity vertical velocity component at the free surface horizontal coordinate parameter defined by equation (103) vertical coordinate laminar resistance coefficient constant associated with empirical formula for a turbulent resistance coefficient constant associated with empirical formula for 8 hydrodynamic characteristic of reference material angle of impermeable slope arbitrary phase angle parameter as defined by equation (18) complex free surface elevation maximum free surface elevation in front of breakwater maximum free surface elevation behind breakwater time-dependent free surface elevation added mass coefficient parameter as defined by equation (34) kinematic viscosity density 5 TT radian frequency, 2%, only a ‘transmitted wave, a,» is expected to propagate in the positive x-direction. The general solution for the flow within the structure is found (App. A), by introducing equations (7) and (8) in equations (3) and (4). The solution, which consists of a wave propagating in the positive x-direction, of complex amplitude a,_, and a wave propagating in the negative x-direction, of complex amplitude a_, is given by -ikx ik(x-2) Goud 2 ne Oly <0 (li2)) ae [Es n ae ene *) O Sy S=it with the complex wave number, k, given by key /Saf . = k, YS-if . (13) Equation (13) shows the wave number to be complex, i.e., to have a real as well as an imaginary part. The solution of equation (13) should be chosen such that the imaginary part is negative since this will lead to a wave motion exhibiting an exponentially decreasing amplitude in the direction of propagation as discussed in Appendix A. The general solutions for the motions in the three regions given by equations (9), (10), and (12) show the problem to involve four unknown quantities. These unknowns are the complex wave amplitudes an; ees wanvemtanid. a_and they may be determined by matching surface elevations and veloci- ties at the common boundaries of the various solutions. Thus, we obtain at x=0 from equations (9) and (12): Liddy Nitta Mle ners (14) i r + a and ; a, -a, = La, fees (15) YVS-iff and at x=2 from equations (10) and (12) ey Ber 2 (16) and ea ee sai San cceh (17) te + = VS-if To solve this set of equations we introduce the shorthand notation eae (18) Multiplying equation (16) by © and adding and subtracting equation qi) result in lte ikgZ = 19 a4 2e€ ay (19) and l-e as ee 20 a_ mete 2 (20) which may be introduced in equation (12) to yield the velocity within the structure Bee) Bie dee = rk (X=2) Tee (ik(x-2) Wy= fhe a, { Bae Psa S ere (245) Adding equations (14) and (15) and introducing a, and a_ from equations (19) and (20) yield, after some simple algebraic manipulations, an expression for the complex amplitude of the transmitted wave 4e yt eee (22) Gece Tee a tkt a ae a ab Similarly an expression for the complex amplitude of the reflected wave is obtained by subtracting equation (15) from equation (16) and introducing equations (19) and (20) a CEE Nie ea By a. Geos. Sears 20 These expressions may easily be shown to be identical to those given by Kondo (1975) when it is realized that the factor y used by Kondo is related to e through y = l1/e. To investigate the general behavior of the solution for the trans- mission and the reflection coefficient as given by equations (22) and (23) it is seen from equation (18) that Aah n “ n/VS ; (24) AS=1£ (50 7154 (£/S) and that the wave number, k, given by equation (13) may be expressed as Re hessae ft eS ee (25) O € 16) n/VS Thus, it is seen that the general solutions for the transmission coefficient = (26) Re (27) may be regarded as functions of the variables n/VS)-£/S, and nk? i.e., the general solution for R and T may be presented as a series of graphs, each graph corresponding to a particular value of n//S and giving R or T as functions of nk_& and f£/S. An example of this solution is presented in Figures 2 and 3 which correspond to a value of n//S = 0.45. As previously mentioned, a series of graphs is needed for different values of n//S. In fact such a series of graphs was developed corre- sponding to values of n//YS = 0.35, 0.40, 0.45, and 0.50. If it is assumed that the values of n, nk_&, and f are known, the graph to be used would depend on the value cRosen for the coefficient S given by equation (5). As discussed in conjunction with the introduction of the parameter S, its actual value is poorly understood except that it is expected to take on values in the interval 1< S< 1.5. Now, if taking 2| nNSyz 0.45 O O1 .O02, 03. 04,05 O06. OF Oseee2 nk ot Figure 2. Transmission coefficient for crib-style breakwaters. S,. de- fined by equation (28). For nko& < 0.1 use equation (35). ec nWS,z 0.45 O OPPO? 10.5. OF tO Oia Oia Oh: OS8nan OS nkot Figure 3. Reflection coefficient for crib-style breakwaters. S, de- fined by equation (28). For nk \& <0.) use equation (36): (22) m= 0045, nk 2 = 0.2, 10.45.06, and 0.8,. and £ = 5, one possible choice of S is to take it equal to unity, i.e., using the graphs corresponding to n//S = 0.45 with nk_% and £/S =f to obtain values of R and T. An extreme alternate choice would be to assume S = 1.67, i.e., using the graphs prepared for n/YS = 0.45/v1.67 = 0.35 with the values of nk & and) £/S) = £/1,67°= 3.0... It was found this way that the estimates Sf R and T varied at most by 0.01 with the above choices of S. This may be taken as an indication of the insignificant importance of the value assigned to the coefficient S. Thus, it is concluded that the value assigned to the coefficient S is of little consequence and that we may safely take S = 1.0. However, this result may be utilized to simplify the presentation of results. Thus, rather than presenting a series of graphs for different values of n/VS, one set of graphs, for example corresponding to n/VS, = 0.45, suffices. The factor S, is without physical significance and is deter- mined by requiring that the value of n/VS, = 0.45 for a given structure for which n, the porosity, is known. Thus, if n is known the value of S, is obtained from 2 e n Sa (745) ; (28) and Figures 2 and 3 may be used with nko and f/S, to obtain estimates of R and T. a. Simplified Solution for Structures of Small Width. For many breakwaters the width, 2, is of the same order of magnitude as the depth of water, h,. Thus, for relatively long incident waves, kh, and consequently kj% may be assumed to be small. Thus, with the assumption of k,& << 1, the general formulas for the reflection and transmission coefficients given in the previous section may be simplified considerably. The nature of the simplification is expressed by expanding the exponentials in terms of their Taylor series, i.e., ee i thes ony (29) and adopting 0(k2)° as the degree of accuracy of the simplified expres- sions. Introducing the expansion given by equation (29) in equation (22) yields: 24 a += —____4 _______, Ni = Se Clee alike Cle) ol -ak2) fh eaebee 2ie 1 2 oan + 0(k2) (30) 1+i + (S-if One) in which equations (13) and (18) have been introduced. Similarly, equation (23) may be simplified to read ke ; BN d= -(c eta) reo — - er Orkey- = —— + Oidenjieay (31) 1c L+i> 1. Thus, the simplified solution for the reflection coefficient is obtained from equation (31) as a | Dee, ee oe) (36) i f which Shows sthat Rr= A\/ (EX) af neo). «Thus, for £ > i. whaeheis usually the case, the transmission and the reflection coefficient are independent of the value of the coefficient S. This supports the finding discussed in Section II.1 where it was concluded that the value assigned to S was of minor importance. For later use, the simplified expression for the horizontal velocity within the structure is found from equation (32) to be Aha we 0(k2)7 : (37) 1+) i.e, the velocity within the structure is identical to the velocity associated with the transmitted wave. The simplified formulas derived here are limited to small values of nk,& by virtue of the nature of the approximation. The equations for T and R (eqs. 35 and 36) may be shown to be in good agreement with the general solutions presented in Figures 2 and 3 for values of nk £ 1.6.0, -Ry < “10, the flow and the resistance are purely laminar;for large values of Rg, as will be the case for most prototype conditions, i.e. , sie 1,000, the flow will be turbulent in nature. Rather than using equation (50) directly with the empirical formulas suggested by equations (51) and (52) it is illustrative to take the relationship for f as given by equation (54) and treating Rg, depending upon the solution through its dependence on |u;, as a known quantity. Introducing |u| from equation (37) leads to an implicit expression for f. R See te cre a 1] (57) aa aie (6) which may also be interpreted as an implicit formula for the factor A = k 2f/ (2n). This formula clearly reveals the possible scale effects associated with hydraulic modeling of porous structures to be an increase in the yalue of f, since Rg would be lower in the model than in the prototype if a Froude model criterion is used. With the empirical formulas for the hydraulic properties of a porous medium given, a completely explicit procedure for determining the transmission and reflection characteristics of a rectangular crib-style breakwater has been developed. 30 Sy Comparison with Experimental Results. a. Empirical Formulas for Flow Resistance of a Porous Medium. The procedure developed in Section II.2 for the prediction of transmission and reflection coefficients of porous breakwaters involves the use of empirical relationships for the hydraulic properties of a porous medium. Thus, only if these empirical relationships may be applied with confid- ence can the procedure itself be regarded as accurate. For steady flow the Dupuit-Forchheimer resistance law reads il ee te + BU)U , (58) in which H is the piezometric head. In permeameter tests it is customary to measure the head loss, AH, over a distance, Ly, for various values of the discharge velocity U. Rearranging equation (58) in a manner similar to that introduced in Section II.2 this may be written AHA bee 2 aan SHUT ee Ere. het eg ; (59) jan) = in which Rq and R, are given by equations (55) and (56), respectively and 8 has been introduced according to equation (52). Realizing that the porosity, n, of the porous material tested may vary it is convenient to introduce a reference porosity, nis and to write equation (58) in the form 1l-n 1l-n ae l-n Woe Bi = ig ee “Sy (60) n. fe) “ Ss 31 Ry IG * _ gd OH n> oT | H (Se * From experiments the value of Ce may be evaluated and plotted against Rq = Ud/v. The results obtained by Sollitt and Cross (1972, Tables F-1 through F-6) are presented in this manner in Figure 4 with the grain diameter, d, being chosen as the median diameter of the gravel tested and taking n, = 0.46 as the reference porosity. The data exhibit a remarkably low degree of scatter and are well represented by the relationship suggested by equation (59) with B, = 2.7 and R, = 170. For comparison, the curve corresponding to 8) = 2.7 and R, = 70, which correspond to the mean values of the ranges suggested by Engelund's (1953) analysis, equation (53) is shown. Although inferior to the curve corresponding to R, = 170, this curve provides a fair representation of the data. 3| 2) fo) 2 OM fe) : 10 = ee a= ad) ras “OL ue ee = 9 : Aytsoitod aduer1ejzer yitM (09) uotzenbso ‘dtysuotqzezor yTeotatdwy ‘13 S7T'O =P: @ [33 $290°0 = P : ¥ +33 120°0 = P @ ‘(og ysnoryi Ty SeTqeL ‘7Z61) Sssorp pue VITITOS Aq veep TeJUowTI1edxg ‘wntpew snoitod e UT 9dUeISTSAeL MOTF LOF E[NWILOF yeotatdug ‘p oansty Ne) t+ fo) I Ss a pec Trea ©) PN 2 pole ag”

= 0.5, are utilized. Wilson's experimental data for R and T are plotted in Figures 6,7, and 8 as functions of the incident wave steepness, Hj/L. The predicted variation of R and T with H;/L following the procedure developed in Section II.2 is shown based on the assumption of Bo = 2-7, R, = 170, and R, = 70. In view of the results, presented in Figure 4 it is hardly surprising that the experimental data are represented better by the curves corresponding to R, = 170 than by the choice R, = 70. The predicted values of the transmission coefficient, T, are seen to be in excellent agreement with experimental values whereas the agreement between reflection coefficients leaves something to be desired. 35 ) fe) 2 fo) a CNVh se St S122 eh 8 — «fo, = UY *Z*z = Yi——f9r'0 = u AqTSOZod adue ~1ayer yITM *(09) uoTzenbs ‘dtysuotzeTes Teotzrdug 33 ggl°O =P: V ‘IF STTO =P: B as 97020 = PO) iar 8cS0 0 = P > Vlts 9500 = Pp US (0c oTqel: “eZ6T) ueSo[ney Aq eqep Te}UoWL4Iedxg ‘wntpoew snotod e& UT 9dUeSTSOI MOTF OF eTNULOF Jeotatdug ° S oin3sTy pllemo = pis = 2 eOl ge "5° > Z 201g 9 6 Z jOl 34 4omD Figure 6: Comparison between Predicted and Experimental Transmission, T, and Reflection Coefficients, R. Wilson's (1973, Table 5) data with kh = 0.482, d = 0.031 ft, 2 =h_ = 0.432 ft; gm: Reflec- tion Coefficient; @® : Transmission Coefficient. Predicted values; Bo Ste Lg R, SPA 0) meth men By = FET, R. = 70. R & Te 4 6 8 10-3 2 4 6 8 10-2 Hy AL Figure 7: Comparison between Predicted and Experimental Transmission, T, and Reflection Coefficients, R. Wilson's (1973, Table 6) data with: 0.45 <.k oh 0.51L,,d-= 0.0625. £52) = h.= 1.0 ft; ms Reflec- tion Coefficient; © : Transmission Coefficient. Predicted values; is G0 = 2.7, R= 170; —_— —: Oi 27, Rens 70. 35 The experimental values of the reflection coefficient were obtained from Healy's formula (Eagleson and Dean, 1966) Bax) Hnin = 61 R H + H y ’ ( ) max min where H is the maximum wave height (measured at the antinode) and ae is the minimum wave height (measured at the node) of the wave envelope in the reflected wave region. Equation (61) shows thate Haan is considerably smaller than Hy, when the reflection coefficient approaches unity. If it is assumed that H,,, is correctly determined but the value obtained for the minimum wave height incorporates an error, A, equation (61) may be written ie A Ree 2 Baa Sey Bin acer ea (62) max min Wate i H max min in which Hya, and Hain are assumed to be the true values. The error, A, in the experimental determination of H,;, will generally be positive due to nonlinear effects. Equation (62) therefore shows that the experimentally determined reflection coefficient will be lower than the true reflection coefficient due to the measurement error, A. This problem is addressed in detail in Section III.3; here it is just pointed out to illustrate that one must pay special attention to minimizing the experimental error in the determination of Hji,- No particular attention was paid to this problem by Wilson (1971) who applied equation (61) directly. It is clear from equation (62) that with the error A increasing with increasing nonlinearity of the incident waves, i.e., with increasing H;/L, a trend of determining an experimental reflection coefficient which decreases with incident wave height results. This may partly explain the behavior of the experimentally determined reflection coefficients in Figures 6,7, and 8 as being nearly constant with H;/L whereas the predicted reflection coefficients show R to increase with increasing values of H;/L. Since Wilson's (1971) experiments essentially correspond to scale models of the same structure, performed for different length scales, these experiments give an excellent exposition of the scale effects associated with hydraulic-model tests of porous structures. It is seen from the generally good agreement between predicted and observed transmission coefficients that the present analytical procedure may be used with confidence in assessing the influence of scale effects on experiments of this type. The Froude model criterion applies only so long as the flow resistance is predominantly turbulent, i.e., f is 36 given by equation (57) with R./Rg << 1. The scale effect is accounted for in the present analysis by the inclusion of the effect of the ratio R./Rq which in a Froude model will be greater in the model than in the prototype. An additional set of experiments is reported by Keulegan (1973). These experiments were performed for rectangular breakwaters of different materials and widths 2 = 0.25, 0.5, and 1 foot. As an example the experimental data corresponding to relatively long waves, h)/L = 0.1, as reported by Keulegan (1973, Table 12) are plotted in Figures 9, 10, and 11 versus H;/L. For comparison the predictions afforded by the procedure developed in Section II.2 are also shown. The choice of parameters B, = 2.2, R, = 70 yields a slightly better representation of the experimental data as could be expected from the comparison made in Figure 5. However, the predictions obtained from 8, = 2.7, R, = 70 are fairly good. The discrepancy between observed and predicted reflection coefficients is of the type noted in conjunction with the comparison with Wilson's (1971) data and may again partially be attributed to experimental errors in the determination of R. Keulegan's (1973) and Wilson's data on the reflection coefficient show the tendency of decreasing slightly with increasing height of the incident waves. However, it is noted that the experimental reflection coefficient (Fig. 9) increases slightly with H;/L. Since the reflection coefficient for this set of experiments: is relatively small, R= 0.3, the error in the experimental determination of H,j;y, may be expected to be rather small, thus essentially substantiating the previous hypothesis for the nature of the discrepancy. As a final comparison between the experimental data presented by Keulegan (1973) and the analytical procedure developed in this study, Figures 12 and 13 show a comparison between observed and predicted transmission and reflection coefficients for all the experiments reported by Keulegan corresponding to h)/L = 0.1 and Hj/hg = 0.1. With the generally good agreement between the experimental and predicted transmission coefficients exhibited in Figures 9,10, and 1l, the comparison given in Figure 12 shows the general applicability of the present procedure to predict transmission coefficients. The comparison of reflection coefficients given in Figure 13 is quite encouraging. However, it should be recalled that the predicted trend of increasing R with H;/L was not observed in the experimental data. 4. Discussion and Application of Results. A theoretical solution for the transmission and reflection characteristics of a homogeneous breakwater of rectangular cross section was obtained. The main assumptions were that the incident waves should be normal to the breakwater and that the motion should be adequately described by linear long wave theory. The general solution for the transmission coefficient, T, and the reflection coefficient, R, is presented in graphical form in Figures 2 and 3. For small values of 37 Big=4 22 Age Ween] GaSe ba 2 4, ia OR G@ee Hj 7L Figure 8: Comparison between Predicted and Experimental Transmission, T, and Reflection Coefficients, R. Wilson's (1973, Table 7) data with) kh) =90.5035. d= 0.125 ft; 2 = hl-;= 1.8) ft3\m: Reflection Coeffitient; @ : Transmission Coefficfent. Predicted values; BuHa2in aR = ie 9) 1B Dl 7k w= ae Ole fe} c (e) c R & T Hi 7L Figure 9: Comparison between Predicted and Experimental Transmission, T, and Reflection Coefficients, R. Keulegan's (1973, Table 12) data for ho/L =).071,) d =, 0.078 fe chu = Toft, 42 = 0.25 ecg es Reflection Coefficient; @ : Transmission Coefficient. Predicted values; th PBZ ed, Re =I70 8 eS eee eRe eo) re) c ie) a ac ‘ 38 4+AQ0D Figure 10: Comparison between Predicted and Experimental Transmission, T, and Reflection Coefficients, R. Keulegan's (1973, Table 12) data with h_/L = 0.1, d = 0.078 ft., h_ = 1 ft, 2 = 0.5 ft; @ Reflection Coefficient; ® : Transmission Coefficient. Predicted Bo = 2.2, R, = 70;—-—: 8, = 2.7, Rx = 70, values; R & Af 2 4 6 8 10-2 2 4 Hi ZL Figure 11: Comparison between Predicted and Experimental Transmission, T, and Reflection Coefficients, R. Keulegan's (1973, Table 12) data with h /L = 0.1, d = 0.078 ft, h = 1 ft, 2 = 1.0 ft; m:. Reflec- tion Coefficient; @ : Transmission Coefficient. Predicted values; B = 2.2, R. = 70;—- —: 8 = 2.7, R = 70. fe} ie fe} {es 39 ° © o PREDICTED TRANSMISSION COEFFICIENT Figure COEFFICIENT 0.8 O oO Gl PERFECT oY AGREEMENT ~y 7 0.2 0.3 04 05 0.6 O7 08 0.9 1.0 EXPERIMENTAL TRANSMISSION COEFFICIENT ION © o PREDICTED RE Figure o ws 9 9° raat a he) O 12: Comparison between Predicted and Experimental Transmission Coefficients. Experiments for H./h_ = 0.1 by Keulegan (1973, Tables 4, 8, 12 and 16). Predictions based on 8 = 2.7 and R, = 70. iy fo as / B = We 2 es rae wy PERFECT / AGREEMENT © Ol! 0.2, 0:3: 0:4 0.5 06 07,018 09 1.0 EXPERIMENTAL REFLECTION COEFFICIENT 3% Comparison between Predicted and Experimental Reflection Coefficients. Experiments for H./h_ = 0.1 by Keulegan (1973, Tables 4, 8, 12 and 16). Predictions based on ® = 2.7 and R, = 70. .. 40 the width of the breakwater, %£, relative to the incident wavelength, L, a set of simple formulas was derived for T and R, equations (35) and (36). From equations (35) and (36) as well as from Figures 2 and 3 it is seen that the transmission coefficient increases and the reflection caefficient decreases with decreasing values of nko&. This is in agree- ment with expectations since low values of k)& indicate a long wave relative to the width of the structure thus essentially making the structure transparent to the incident waves. An increase in frictional effects, which are accounted for by the linearized friction factor, f, is seen to cause an increase in the reflection coefficient and a decrease in the transmission, coefficient. -In.this respect it is. seen from equation (57), which is the explicit solution for the linearized friction factor, f, that the frictional effects increase with increasing amplitude of the incident waves, thus reflecting the nonlinear nature af the flow resistance of the porous structure. The procedure developed is, through the adoption of empirical relationships for the hydraulic properties of the porous mediun, entirely explicit. The required information is the incident wave characteristics (a; and L), the breakwater geometry (2 and hg), and the characteristics of the porous material (stone size, d, and porosity, n). The ability of the procedure to predict experimentally observed trans- mission and reflection characteristics of crib-style breakwaters was demonstrated. It was found that the procedure yields excellent predic- tions of the transmission coefficient whereas some discrepancy between observed and predicted reflection coefficients was noted. This discrepancy may be partly attributed to experimental error in the determination of the reflection coefficient. Numerical Example. The following numerical example is included to illustrate the application of the procedure developed for the prediction of transmission and reflection coefficients of a porous rectangular breakwater. The information which is assumed available is listed in Table 1. To illustrate the assessment of scale effects the problem is considered both for a prototype and for a Froude model with length scale ll “tow 25. As discussed in Section I the procedure developed in this Section of the report accounts for the partition of incident wave energy among reflected, transmitted, and internally dissipated energy. Thus, the present Section forms part of the ultimate procedure for the prediction of reflection and transmission characteristics of trapezoidal, multi- layered breakwaters. The energy dissipation taking place on the seaward slope of a trapezoidal breakwater is discussed in Section III which also includes a numerical example. The incident wave characteristics listed in Table 1 correspond to the incident wave assumed in the numerical example presented in Section III, Table 4, after subtracting the amount of energy dissipated on the seaward slope of a trapezoidal 4 _ Table 1. Information used in numerical sample calculations. Froude Model Prototype length scale 1:25 Incident Wave Amplitude ay in feet Wave Period T in seconds Water Depth Ay insect Incident Wavelength? Lan feet Breakwater Width 2 in feet Stone Diameter d= 1/2(d_ oy Porosity n vil may be obtained from linear wave theory using hy and T. *The porosity is assumed. Sensitivity of results to this assumption breakwater. The present numerical example together with the numerical example presented in Section III therefore illustrate the detailed calculations involved in the procedure for the prediction of reflection and transmission coefficients of trapezoidal, multilayered breakwaters which is developed in Section IV. The model breakwater characteristics ’ listed in Table 1 correspond to the characteristics of the crib-style breakwater which is hydraulically equivalent to the breakwater config- uration tested by Sollitt and Cross (1972). The determination of the hydraulically equivalent breakwater is discussed in detail in Section IN se 42 Ta use the general solution presented graphically in Figures 2 and 3 the value of S, is obtained from equation (28) AD 2 aii ae n 0.435 2 Gee) = Ge = WEE (63) The value of the parameter nkj& may also be determined directly from the information contained in Table 1 Es pee OS one nk 2 = (0.435) (2m) - = (0.435) (2m) Zee = 0.47. , (64) which is valid for the prototype as well as for the Froude model. It is noticed that the value of nk,& is sufficiently large for Figures 2 and 3 to be used. If nkj& had been below 0.1 the simplified formulas, equations (35) and (36), should be used with S = 1.0. The remaining task is the determination of the friction factor, f from equation (57). For the prototype conditions it is expected that turbulent flow resistance dominates so that the factor R./Rq may be neglected in equation (57). Therefore the remaining expression becomes: n 16 L el) ae ee ©: o>) ie) (0) 3 In this expression the value of 8 is taken according to equation (52) with Bo = 2.7, a reasonable estimate as discussed in Section II.3. Thus, ee ea ack tony at etal = ees om oe hi O [e) 02435 16 0.565 LEAS 63 = Cd 1+ 5 2.7 eS Se a CU 27 63/366 3m (0.435)° ER Op ee OAT lee 65 Uy = FDI BV. (66) This value of f is obtained for the prototype conditions assuming Rq 7> R, where Rq is the particle Reynolds number defined by equation (55) with |u|] given by equation (37). To check this assumption the value of } is obtained from equations (34) and (66) as 43 Kee Aon 6S) 3601228" = = ‘ 67 s 2n 0.87 pee con and therefore from equation (37) ‘ /g ewes Die 1 a Ju| = ay nh, “TH? Seeed'S 59-3 Creme) OVS4"£t/ sect. (68) This gives a value of the particle Reynolds number Ree ullcMoeo SAeSe)e ey aiok (69) d Ny 10> where the kinematic viscosity has been assumed given by vy = 10, 2£ta/seer This value is clearly much greater than the value of the critical Reynolds number, R,, which is of the order 100. Thus, the value of f determined by equation (66) holds for the prototype condition and the necessary parameters for use in conjunction with Figures 2 and 3 may be determined for the prototype Nee = (0047 oO Se 010955 Prototype ; (70) £/S, = 2.8/0.935, = 3.0 and Figures 2 and 3 yield for the prototype: Transmission coefficient = T = 0.22 Reflection coefficient = R= 0.71 . (71) For the Froude model one may as a first approximation adopt the assumption that Rg >> Rg, in which case the estimate of f obtained for the prototype still holds, a.e., £ = .2.8 1s a first estimate. wo evaluate the value of the particle Reynolds number, Rqy, the procedure is as previously outlined and from the well-known scaling of Reynolds numbers in a Froude model, (Reynolds number scale) = (length seaeye ; (72) 44 it follows from the length scale of 1:25 and from equation (69) that 5/2 1 _ Ree Sie (10*) Ge = 425. (73) This is not a value much greater than R, and it is therefore necessary to incorporate the Reynolds number effect in equation (57) when evalu- ating f. For this purpose it is assumed that a simple test has shown that Be cpl Osayhatay cei te (74) for the material used in the model. Taking Ry as given by equation (73) the expression for f (eq. 57) reads; R 16 L Eipenel Aidt saitlot Be Aimy cal ea ie) fe) 170 : 0.4] fl + (1 + p63 -1]) = 3.4 , (75) in which the analogy with the manipulations performed in equation (66) has been utilized. From this result an updated value of 4 is obtained since 3} = f/0.8 = 4.25. This value of X is different from the value. A = 3.5, used in determining the particle Reynolds number Rq used in the evaluation of equation (75). With this new value of \, equations (37) and (55) may be used to obtain a new value of Rq- This in turn may be introduced in equation (75) to get a new value of f and the procedure may be continued until convergence is achieved. It may be shown that +h) (76) in which Ry 5 is the new estimate of Rg» whereas Ry , is the previous estimate and Ay and 5 are the old and new estimate$ of \, respectively. This procedure is generally rapidly converging. Thus, the next iteration outlined above yields f = 3.46 which is reasonably close to the initial estimate obtained in equation (75). For the Froude scale model the parameters therefore become 45 nk 2 = 0.47 fe) S = 10). 935 Model £/So) 02055 46/ 009555 = 951.57 and Figures 2 and 3 yield: Transmission coefficient = T = 0.19 Reflection coefficient = R = 0.73 (77) (78) By comparing the predicted results for the model (eq. 78), and for the prototype (eq. 71), it is seen that the scale effect has increased the reflection coefficient, whereas the transmission coeffic- zent is décreaséd. For the present example the scale effects aregnoe pronounced, but in other situations it may be a very important factor to consider (Figs. 6, 7, and 8) 46 III. REFLECTION COEFFICIENTS OF ROUGH IMPERMEABLE SLOPES We Preliminary Remarks. In the previous section of this report an analytical solution for the idealized problem of wave transmission through and reflection from rectangular breakwaters was obtained. Since most breakwaters are of trapezoidal, rather than rectangular cross section, a considerable amount of energy may be dissipated on the seaward slope of the breakwater. This external dissipation of energy is not accounted for in the analysis of porous crib-style breakwaters. To account for the external dissipation of energy on the seaward slope of a trapezoidal breakwater the associated problem of energy dissipation on a rough impermeable slope is considered both theoretically and experimentally. A theoretical analysis of this problem is based on the following assumptions: (a) Relatively long normally incident waves which may be considered to be adequately described by linear long wave theory. (b) Energy dissipation on the rough impermeable slope may be represented as the energy dissipation due to bottom frictional CLLeCts The first assumption is identical to the assumption made in Section II of this report. The second assumption presumes that the effect of energy dissipation due to wave breaking is minor. This may seem to be a restrictive assumption. When realizing that the seaward slopes of breakwaters are generally steep, this assumption is quite reasonable. At any rate, the main purpose of the theoretical analysis is to produce a rational framework within which the experimental results for reflection coefficients of rough impermeable slopes may be analyzed. The essential features of the mathematical manipulations and the derivation of the governing equations are presented in Appendix A to enable the treatment to be relatively brief and to the point. The frictional effects on the rough slope are accounted for by introducing a term relating the bottom shear stress to the square of the horizontal orbital velocity through the use of a wave friction factor, re analogous to that introduced by Jonsson (1966). The bottom shear stress is linearized and a theoretical solution for the reflection coefficient of rough impermeable slopes is obtained in terms of a linearized slope friction factor. By using Lorentz' principle of equivalent work an implicit solution for the reflection coefficient of rough impermeable slopes is obtained in terms of incident wave characteristics, slope geometry, and the wave friction factor, f , which expresses the effect of slope roughness. e 47 An extensive experimental investigation is performed to establish an empirical relationship for the wave friction factor. The experimental investigation utilizes measured reflection coefficients of rough impermeable slopes in conjunction with the theoretical results to obtain values of the wave friction factor. The experimental investigation shows the need for a method for obtaining accurate estimates of reflection coefficients from experimental data. Such a method is developed and the end product of the experimental investigation is empirical relationships for the wave friction factor. The experiments are performed with slope roughnesses modeled by gravel and are therefore applicable only when the slope roughness elements consist of natural stones. Separate experimental investigations should be carried out to establish empirical relationships for f,, corresponding to other surface roughness elements, e.g., concrete armor units. The result of the combined use of the empirical relationship for f and the theoretical developments is a ''semiempirical' procedure for estimating the reflection coefficient and hence the energy dissipation of rough slopes. The procedure requires knowledge of the incident wave characteristics (amplitude and wavelength) and the slope characteristics (slope angle and stone size). The procedure was tested against a separate set of experiments and yielded quite accurate results. Des Theoretical Solution for the Reflection Coefficient of Rough Impermeable Slopes. The problem to be considered is illustrated in Figure 14. Figure 14. Definition sketch. With the assumption of relatively long incident waves the governing equations for arbitrary bottom topography are derived in Appendix A, equations (A-20) and (A-21). The linearized forms of these equations are given by equations (A-24) and (A-25) 48 an | P) ey meme vox Oa 12 (79) and aU an i. E. a ASE ox at £) wu =F Olee (80) in which n is the surface elevation relative to the stillwater level, U is the horizontal velocity component, h is the local depth, g is the acceleration due to gravity, w is the radian frequency, w = 27/T, of the incident waves and a is a linearized friction factor defined by equation (A-23) (81) in which f, is a wave friction factor relating bottom shear stress, Th fluid density po, and the velocity, i.e., 1 Ti oe | Ul Obe. (82) The linearized equations (eqs. 79 and 80), are solved by assuming a periodic solution of radian frequency, w, and introducing complex variables defined by ater Real{z(x)e (83) = iT] and Lut) (= iT} Real{ u(x)e < (84) in which the amplitude functions ¢ and u are functions of x only, i =v-1, and only the real part of the complex solution constitutes the physical solution. In the constant depth region, h = ho» in front of the slope, bottom friction is neglected (i-e.5 £,, = £,,1= O*% forex >'2.) andthe general solution reduces to the solution given in Section II.2 for x< 0. With the change of the orientation of the x-axis (positive away from the slope as seen from Figure 14) the general solution for x > Re 1s 49 ee) ; (85) a ik x ~ik)x u= - oe (a,e -ae ) where a; is the amplitude of the incident wave, which without loss of generality may be taken to be real, and ay is the complex amplitude of the reflected wave. The wave number, k, = 27/L, where L is the wave- length of the incident wave, is given by keene : (86) On the rough impermeable slope, 0< x< %,, the effect of bottom friction is retained and equation (80) may be written pg Roe IG ae a Ta(iaiee) Bx Sg ee (87) when equations (83) and (84) are introduced. Multiplying equation (87) by h to obtain an expression for 0(uh)/0x and introducing this in equation (79) yield the governing equation 2 w” (1-if,) ae aot + ———— 5 = 0 ; OK xK< Le 3 (88) g With the depth varying linearly on the slope, i.e., =X tans . 5 NO Spates be ; (89) equation (88) is seen to be a special form of the Bessel Equation (Hildebrand, 1965) with the solution /u* (1-if,)x CHA IS ge tana, ? ye ORS he (90) 5O in which J, is the Bessel function of the first kind of order zero. The general solution includes also the Bessel function of the second kind, Yo: However, this solution blows up at the origin, x = 0, and for the solution to remain finite at x = 0 this part of the general solution is omitted. For x + 0, J, approaches unity so that the arbitrary constant A in equation (90) is the complex vertical amplitude of the wave motion at the intersection of the stillwater level and the slope, i.e., |A may be interpreted as a measure of the runup on the slope. It should be realized that the linearized solution, as discussed in Appendix A, is based on the assumption that |n| << h. Thus, since h = 0 at x = 0 the solution given by equation (90) cannot be considered valid near x = 0. However, Meyer and Taylor (1972) have shown that a linear solution gives essentially the same value of the runup as does the more realistic solution based on the nonlinear shallow-water wave equations. Thus, some physical significance may be attached to the magnitude of A, |A|, as being an approximate value of the runup on the slope. With c given by equation (90) the horizontal velocity is evaluated from equation (87) . fuo* (1-48, )x ae, (I-f,)x tan’ Tews tan. pO SEES een) in which Jy is the Bessel function of the first kind of order one. With the general solution given by equations (85), (90), and (91) the complex amplitude of the reflected wave, a,, and the complex runup amplitude, A, are determined by matching the solutions for ¢ and u at their common boundary, x = he: Thus, at XxX = s ik jf, / «ik de ae ate SOPH AT kt WiETe) (92) 1 36 fo) oS b and ik o£, ik o£, 5 ‘au -ae =A —— J, (2k.2 V1-if, ) (93) 1 G MR 1 Oris b b in which ho = Le tanB | and equation (86) have been introduced. These equations are readily solved to give the complex amplitude of the reflected wave. S| : : i anes J (2k ok, v¥1-if, J- ——_J, (2k, 2. v1 if)) a lees i2k 2. == e (94) a. F fered (2 Oy Vier Rie =— J, (2k,%, /T=if,) Suan a STEER and the complex runup amplitude ikj%, Nee e . (95) a i WAOket li One aes J Cho Vi=ie) See vI-if, It can be seen from equation (94) that baat which is the physical amplitude of the reflected wave, is equal. to a; for f, = 0. Since f, = 0 expresses the condition that the energy dissipation on the slope is zero, this is to be expected. Equations (94) and (95) show that the important parameters in determining the reflected wave amplitude and the runup amplitude are the length of the slope relative to the length of the incident waves in front of the slope, 2./L, and the friction factor, f,, arising, from the linearization of the bottom friction term. Since the linearized friction factor appears in’ the form vyl-if), it is expedient to introduce the friction angle $ defined by tan29 pees : 0< 42< 5 5 (96) since VI-if, = (1 + econ, ee ee (97) In terms of the relative slope length, 2./L, and the friction angle, o, the reflection coefficient, R = la Wee may be determined from equation (94). This solution is presented in graphical form in Baguxre 15). Similarly the nondimensional runup amplitude, 7 2a ’ (98) 52 IN DEGREES O VALUE OF $) |a,|/a; REFEECHION -COER. R tee Figure 15. Reflection coefficient, R, of rough impermeable slopes as a function of ¢ and &/L. 93 is obtained from equation (95) and is presented in graphical form in Figure 16. The solutions for R and R, were obtained through the use of complex computer programs for Bessel functions with complex arguments which are part of the Massachusetts Institute of Technology (MIT) Information Processing Center's IBM System 370 computer library routines. With the solution for R as presented graphically in Figure 15 it is seen that knowledge of, 2 ,/L and $ enables one to determine the value of R or conversely, if R 4nd 2<,/L were known Figure 15 may be used to obtain the corresponding value of 9. a. Determination of the Friction Angle, >. The value of the linearized friction factor, f,, or the friction angle, ¢, was considered constant (i.e., independent of x and t) in the analysis presented in the preceding section. This friction factor was introduced through equation (81) and corresponds to a linearized bottom shear stress as given by equation (82) tn p£, wh = owhU tan2¢ : (99) With the rate of energy dissipation per unit area of the slope given by Ep = tpU, as discussed in Appendix A, the average rate of energy dissipation per unit area of the slope is given by cosB ae S h dx We cosB 0 Ss T i | te dels (100) s 0 Ey = pwtan2 > in which U is the real part of the solution given by equation (84) with u given by equation (91). Since U is necessarily periodic the time averaging in equation (100) is readily performed so that ED OD tang QR J pw tan2¢ 7 = | ox ul? s (101) 0) From equation (91) it is seen that J, (2k,2. /I=i tand9 ye iy fee OX u = -14 ’ (102) Vi=a, tan2o | — $ g 54 VALUE OF $) IN DEGREES 4.0 O (OVS) a 3:0 oS N ~ sh iT) r=} o as =e Zz =) ac 1.0 COMO MNOS OD WMO4 TOS cOGurOn Oe Po/L Figure 16. Runup, R., on rough impermeable slopes as a function of ¢ and L/L. 55 in which yee 103 y Q 2 ( ) Introducing equation (86) this may be written fa ety = SS 104 Jul = [Al ue, = os (104) fe) UY, in which Ys kot Y¥l-1 tan2o : (105) Inserting equation (104) in equation (101) the average rate of energy dissipation per unit surface area of the rough slope may be written: 2 M2 Me 2 ‘1 ae (CNS, ) Ea {= plALE 3, : tan8 y A dy }stan2¢. - 3) C06) D 2 h 2 Ss S Jo vyil 2 ie) If the average rate of energy dissipation is evaluated using the nonlinear expression for the bottom shear stress (eq. 82) one obtains cos W lj at al Shs dx iL Ze ED = > ef res | cos. {rs J. |U|U dtd a (107) where U again is periodic and given by equation (84) with u given by equation (91). Performing the time averaging and introducing |u| as given by equation (104) the average rate of energy dissipation per unit area of the slope becomes 3 ll J, (avy!) yl? 1 iT} 3 1 2 A SS) a ef zw he | dy : (108) h ) y oO Using Lorentz' principle of equivalent work by equating equations (106) and (108) results in the following expression for the friction angle ¢: 56 A it tan26 = fT h tan ne , (109) in which Be is a slope friction constant given by (110) With ¥ given by equation (105) it is seen that the slope friction constant, as given by equation (110) is a function of the relative slope length, L/L, and of the friction angle, >. The evaluation of the integrals and hence F, = F.(%, /L, ¢) is performed numerically using the IBM System 370 computer li rary routines previously mentioned and He is presented in graphical form in Figure 17. b. Methodology for the Determination of the Reflection Coefficient of Rough Slopes. It-is' clearstrom Figures l¢ and 17 that equation (109) is an implicit relationship for the friction angle, $, since |A| as well as F, are functions of 9. If it is assumed that the wave friction facton;” £,,, in-equatwon (109) is known for given incident wave (aj, L, hg) and slope (2,) characteristics, equation (109) may be solved in an iterative manner. For an assumed value of ¢ and knowing £,/L, Figures 16 and 17 may be used to obtain values of |A| = Rees and ere With these values introduced in equation (109) a new value moe > is obtained and the procedure is continued until convergence is achieved. Once the value of the friction angle is determined, the reflection coefficient, R, is readily obtained from Figure 15. The preceding methodology for obtaining the reflection coefficient of rough slopes is straightforward. However, it does rest on one very important assumption--that the value of the wave friction factor, f, is known. Although similar to Jonsson's (1966) wave friction factor his expressions for fy, are not expected to hold in the present context which justifies asking: What has been gained by the theoretical development presented in the previous sections? To answer this question imagine that the problem had been approached on a purely empirical basis. Then, the effects of slope geometry and incident wave characterisitcs in addition to the slope roughness would have had to have been considered. The present theoretical development has circumvented such an extensive experimental investigation by establishing an analytical model, which essentially accounts for the oY IN DEGREES O VALUE OF ‘a SLOPE FRICTION CONSTANT, F, O OO uiOl, WHO2.) 103.7 046, 105 106) NO7msOe g/L Figure<17.4; Slope’ friction constant, Boo in equation (109) as a function of ¢ and &/L. 58 effects of slope geometry and incident wave characteristics leaving the burden of expressing the influence of slope roughness on the wave friction factor, fy. This considerable reduction in experimental effort is the reason for the theoretical development. Thus, the theoretical analysis has identified the fundamental unknown parameter as the wave friction factor, fy. For the method to be applicable an empirical relationship for f£, must be established and here the physical interpre- tation of f. as a wave friction factor may be used as a guide. For fully rough turbulent flow conditions, Jonsson (1966) found for waves over a rough boundary that his wave friction factor is a function of the boundary roughness, d, relative to the excursion amplitude, Ay, of the orbital particle motions above the bed, i.e., er Tee (111) W Ww Ay For fully developed steady flow over a rough boundary, the characteristic length scale is the water depth and the friction factor corresponding to fully rough turbulent flow conditions is a function of the boundary roughness, d, relative to the depth of flow, RSE eS) ; (112) In the present context it is expected that the flow is a mixture of a boundary layer-type flow (eq. 111) and a fully developed flow (eq. 112),and it may therefore be expected that the empirical formula for the wave friction factor, fe in equation (110) is of the form: d an og As a representative value of the excursion amplitude, A,, the value obtained from the theoretical solution (eq. 91) evaluated at x = 0 is taken, i.e., a, = tule = Ab aia) a tanB . d f apne S (113) and h = hy is taken as a representative value of the depth. Thus, it is anticipated that an empirical relationship d tans . 5 age Sona , fe , (115) for the wave friction factor exists. To determine this empirical rela- tionship is the purpose of the experimental investigation described in section III.3. 59 Se Experimental Investigation. The theoretical analysis of the reflection coefficient of rough impermeable slopes described in Section III.2 suggests a rather simple experimental procedure for the determination of the value of fy. Imagine that an experiment is performed in which the reflection coefficient, R, is determined for given slope (d and 2.) and incident wave characteristics (a; and L). With 2,/L and R known, Figure 15 may be used to obtain the the corresponding value of $¢. WAS Ry, 2a; and Fz may then be obtained from Figures 16 and 17 and equation (109) written in the form: — os h zi Oi «bes fs tanB . TAT F tan2o , (116) may be used to obtain the value of fi Performing a series of experiments for various slope and incident wave characteristics and analyzing the results as outlined above will produce a number of values of f,, from which an empirical relationship of the type suggested by equation (115) may be established. It should be pointed out that this procedure for the analysis of experimental data relies heavily on the theoretical development presented in Section III.2. The resulting empirical relationship for f,, therefore incorporates not only the true physical dependency of fy on the relative roughness, but reflects also inadequacies of the theoretical development. This is important to keep in mind, since it means that the resulting relationship for fy, becomes an integral part of the entire procedure for the determination of reflection coefficients of rough impermeable slopes. From the preceding the aim of the experimental investigation is to determine accurately the reflection coefficient of rough impermeable slopes for a variety of slope and incident wave characteristics. a. Experimental Setup and Procedures. The experiments were performed in a wave flume at the Ralph M. Parsons Laboratory at MIT. This flume is glass walled and is 80 feet (24.4 meters) long, 15 inches (0.38 meter) wide, and the constant water depth in front of the slope was for the major part of the experiments kept at h_ = 1 foot (0.305 meter). A piston-type wavemaker capable of producing periodic waves of periods within the range of 0.6 second < T < 2.2 seconds is located at one end of the flume. Experiments were performed for three wave periods T = 2.0, 1.8, and 1.6 seconds which with h_ = 1 foot correspond to depth to length ratios of the incident waves h_/L=0.092, 0.105, and 0.12, respectively. 5 A variable slope of rigid construction was installed approximately 60 feet (18 meters) from the generator. Care was taken to completely 60 seal the gaps between the variable slope, the glass-side walls, and bottom for each slope angle tested to eliminate the effect of leakage around the slope and ensure that the slope was truly impermeable. To develop various slope roughnesses plywood boards with glued-on roughness elements, gravel of diameter d = 0.5, 1, 1.5, and 2 inches (1.25, 2.5, 3.8, and 5 centimeters) were attached to the slope. In this manner experiments for various slope roughnesses were readily performed for a given value of the slope angle, tan8,. Photos of the various roughness boards are shown in Figure 18. Each experiment for a given value of T, d, and tan8, was performed by running the wavemaker continuously. After approximately 2 minutes, a quasi-steady condition was established in which the wave motion at any point along the flume was periodic with period T equal to that of the wavemaker. When this quasi-steady state was established the free surface variation with time was recorded at 4-inch (10 centimeters) intervals along the flume over a distance of approximately 10 feet (3 meters) of the constant depth region of the flume. The free surface variation was measured by a parallel wire-resistance wave gage and was recorded on a two-channel recorder (Sanborn). The slope and the instrumentation are shown in Figure 19. From the measurements the incident wave height and the reflection coefficient are determined as discussed in Section 3.b. This procedure was repeated for four values of the incident wave height by changing the wavemaker stroke with everything else being unchanged. It was found that a quasi-steady state could be achieved only for wavemaker strokes below a certain value. Therefore, experiments are limited to values of the incident wave heights below approximately 2 inches (5 centimeters). This, in turn, means that the incident waves do not break on the slopes tested, thus corresponding to the assumption of nonbreaking waves made in the theoretical analysis. From the preceding discussion of the experimental setup and testing procedures it is seen that a total of 48 experimental runs were performed for each value of the slope angle, (four different wave heights times three different wave periods times four different slope roughnesses). An additional 12 experiments were performed for a smooth slope for each value of the slope angle. For a smooth slope, which corresponds to d v 0, the relationship suggested by equation (115) is unrealistic. For a smooth slope a dependency of the wave friction factor on a Reynolds number can be expected. The experimental results for smooth slopes were analyzed without resulting in a useable relationship for f,. All the data, including the data obtained for smooth slopes, collected in the experimental investigation are presented in Appendix B. b. Accurate Determination of Experimental Reflection Coefficients. Since the reflection coefficient obtained from each experimental run is used directly in conjunction with Figure 15 to obtain a corresponding 6| "UOTIESTYSOAUT [eJUSWTIedxe oY UT pesn spzeoq ssouysno1 adots (wo g'¢) seyout ¢ 8 T T 9inst 4 62 Iapiooe1 uloques pue od8es oAeM (q) ‘dnjzos Te .uowtTredxg "61 oansty aoetd ut sdojts ysnoy (ze) 63 value of the friction angle,¢, it is of extreme importance that the reflection coefficient be accurately determined from the experimental data. According to linear wave theory the wave motion in the constant depth region in front of the slope is given by equation (85). Introducing the expression: (117) for the reflected wave amplitude in equation (85), where 6 is an arbitrary phase angle, the resulting wave amplitude, [zc » may be expressed as ag = [elena = ja [+ !2a:fa (|) cos (2k x 0 (118) which shows the wave amplitude to vary with distance along the constant depth part of the flume in a periodic manner. For values of kx + 6 = 0. +20, ete. (i.6.,,at the antrnodes) the resulting amplitude is a maximum, 1: a = al + la, = a; (1 Rona (119) max 1 and for values of 2k,x + 6 = + 7, etc. (i.e., at the nodes) the resulting amplitude is a minimun, Se kee la. =a Ryo: (120) Since the wave height, H, according to linear wave theory is twice the amplitude the preceding formulas show it, in principle, to be possible to determine the reflection coefficient, R, and the incident wave height, Hj = 2a;, by merely seeking out a node and an antinode along the flume. Thus, Aax ~ “min rea rae as a fee ane Hee: (121) max min max min and = ee Sg eee esse : Hin? ; (122) As discussed by Ursell, et al., (1960) the above formulas are valid also when the wave motion is weakly nonlinear, i.e., consists of a small second harmonic motion in addition to the primary first harmonic motion 64 of period equal to that of the wavemaker. The simple method for obtaining the reflection coefficient from an experiment, i.e., simply seeking out a node and an antinode, and using equations (121) and (122) appears to be the method used by Wilson (1971) and Keulegan (1973) as discussed in Section II.3.b. However, this may be a dangerous procedure to use when the reflection coefficient is large and nonlinear effects are pronounced as is often the case in experiments involving relatively long waves. To illustrate this, the theoretical variation of the wave amplitude relative to the maximum wave amplitude is found from equation (118) to be CS a (yt a (cos(2k_x + 6) -1))'/? : (123) A nax max (1+R) : . When the raw data for the wave height variation along the flume is plotted in this fashion versus x/L, the experimentally observed variation (open circles) in Figure 20 is seen to be somewhat erratic and not resembling the variation predicted by an equation such as equation (123). If one, in spite of this discrepancy between theory and observations, evaluates the reflection coefficient directly from the raw data shown in Figure 20 one finds 0.59 < R < 0.65 with the estimate depending on which node and antinode are chosen. This may not seem to be an alarming variation, but a critical inspection of the surface profile recorded near a node (Fig. 21), reveals that the wave height observed at a node is practically entirely due to a second harmonic motion whose presence manifests itself clearly because of the near vanishing to the first harmonic motion at the nodes. Since the theoretically predicted wave amplitude variation along the flume is based on linear theory, it applies only to the fundamental motion which has a period equal to that of the wavemaker. At each station along the flume where the free surface variation with time was recorded, the amplitude of the motion with a period equal to that of the wavemaker was extracted from the wave record by means of a Fourier series analysis. The Fourier series analysis is performed on the Ralph M. Parsons Laboratory Hewlett-Packard computer; the program is presented in Appendix C. When plotting the variation of the amplitude of the first harmonic motion with distance along the constant depth part of the flume (full circles in Figure 20), apparent disorder becomes extremely organized and the observed variation of the amplitude of the first harmonic motion is in excellent agreement with the theoretical prediction afforded by equation (123) with R = 0.88. The surprising thing to note from the data presented in Figure 20 is the drastically different reflection coefficient obtained from the 65 LN elidel. Nid cepa edna eZ LINEAR WAVE THEORY II o RAW DATA @® CORRECTED DATA 8 1.0 ® e . LQ e ® 6 ° 0.9 of sl Wows 0.8 ° {o) 2 ° ° @ fe) ° a Orr, Guo ans Gmax 0.6 ‘a oy Ae ° Coy ry 1OXS Q 0.4 6 . ° 0.3 s ~ $ 0.2 % e 0.1 g 0.0 fl A et a ray |e eh ia tle 0.0 0.1 0.2 OS Of! PiOl Se O26 Oka 0:8: O19 iO) RS IE Figure 20. Wave amplitude variation along constant depth part of the flume. T = 2.0 sec., Tan 8, = 1/1.5.,. Curve. corresponds, Co theoretical variation, equation (123), with R = 0.88. Figure 21. Wave record showing pronounced second harmonics at a node for experiments presented in Figure 20. 66 raw data (R ~ 0.62) and from the corrected data (R = 0.88). By using this procedure it was found that reflection coefficients determined from the variation of the amplitude of the first harmonic motion generally were within a range of + 0.02 whereas a variation as large as 0.45 < R< 0.75 was found for a Single experimental run when the raw data were used directly. With the intended use of the data in mind, it is quite obyious that the accurate, although tedious, procedure of subjecting the wave records to a Fourier analysis had to be used throughout this study. The pronounced effect of second or higher harmonic motions on the accurate determination of the reflection coefficients from experimental data is closely related to high values of the reflection coefficient since this entails the near vanishing of the first harmonic motion near the nodes. However, it should be noted that the decision of whether or not to use the time-consuming Fourier series procedure cannot be based solely on the magnitude of the reflection coefficient. Test number 33 (App. B) showed pronounced higher harmonic effects with a reflection coefficient of R = 0.60 whereas test number 62 exhibited only insignifi- cant higher harmonics although the reflection coefficient for this test was 0.80. A visual inspection of the recorded wave profile at each station generally led to an accurate assessment of whether or not the Fourier series analysis was called for. C.1 Empirical Réelationshap )tor ithe Wave: Friction Factor, £%., To establish a sufficient data base from which an empirical relationship for the wave friction factor, fy, may be obtained, two series of experiments were performed for values of the slope (tan8,) = 1/2.0 and 1/3.0, respectively. For each slope a total of 48 experiments were performed as discussed in Section III.3.a. The data were analyzed in the manner described in Section III.3.b to yield values of the reflection coefficient, R. With the value of R determined for each experimental run and the value of & and the corresponding value of f was obtained from equation (116). Anticipating an empirical relationship for f, of the type suggested by equation (115) the semiempirical values of f, are plotted against the the value of |A|/(d tan6.) in Figure 22. Although exhibiting a consid- erable amount of scatter the data do form four reasonably well-defined bands depending on the relative roughness, d/h), in conformance with the anticipated behavior. The experimental data and the details of the analysis leading to Figure 22 are presented in Appendix B. Two families of straight-line approximations of the data are shown in Figure 22. One, the dashlines, has a 1 on 1 slope and leads to an extremely convenient empirical relationship for the wave friction factor -0.74 d tané . r d f= 10, 25 ine Saas ik (124) 67 M -"} $10]DVF UOTIOTAJF 9ACM 9YyZ BOF drysuotyejyet Teodtatdwyq “zz oainsty (GZ|) uolonby —— (p2l) uolonby ——— O2:| adoig SgQil=] -e- (2pO'O= °U/P) G00 (€800=°U/P) Ol e (S21 O= Cusp) Gl @ (291'0=°u/P) O'2 ® (SSHONI) SSSNHONOY S3ININ°U/P LNVLSNOO ZvOO €800 SZl'O ‘2910 °u/P 4O SSNIVA 4'YOLOV4S NO ~ M 68 The other, the full lines, has a 1 on 0.7 slope, (Fig. 22) and corresponds to an empirical relationship for the wave friction factor, q = O45 ka tang. On, f =O 29 7 Gan? . ¢1125) The second relationship, equation (125), is superior to equation (124) in representing the data from single subsets of the experiments in which only the amplitude of the incident waves varied. One such typical subset of data points, corresponding to d = 2 inches (Sveentimeters)), 1 — 1.8 seconds jand) tan 8.9= 1/2.0), as/indicated by full circles with arrows in Figure 22. The slope of a line connecting these data points is approximately 1 on 0.7; this slope reflects the experimental observation that the reflection coefficient generally decreased with increasing height of the incident waves. This is a consequence of the nonlinear nature of the energy dissipation on the rough slope and equation (125) is therefore to be considered superior to equation (124) which is included primarily because it possesses some convenient features. 4. Comparison of Predicted and Observed Reflection Coefficients of Rough Impermeable Slopes. With the empirical’ relationships for fy,,/" (eqs. 124: or’ 125)',"the semiempirical procedure discussed in Section III.2.b for the prediction of the reflection coefficient of rough impermeable slopes is now complete. Whereas the theoretical analysis identified the wave friction factor as the physically fundamental parameter, the friction angle, $¢, is the important parameter for the use of Figure 15. However, by merely introducing the empirical relationship for f, in equation (109) an implicit equation for > may be obtained. By introducing equation (124) in equation (109) the following equation for > is obtained: 0.26 d tan2¢ = 0.25(—) eee (126) ho s This equation is in principle implicit, since the slope friction constant Fo, as, seen from Ficure )l/iis.a.ftunction/ofso... However, ‘for)smald, values of 2./L (2, /L< 0.3), F, is only a weak function of > and equation (126) may therefore be regarded as an explicit equation for $ requiring knowledge of only the relative slope roughness, d/h). This may be a somewhat surprising result since it means that the value of the slope friction angle, 4, and hence the reflection coefficient obtained from Figure 15 is independent of the amplitude of the incident waves. As mentioned previously the main part of the experiments presented in 69 Figure 22 exhibited a slightly decreasing reflection coefficient with increasing incident wave amplitude which is not reproduced by this simple relationship for the slope friction angle. However, the feature of ¢ being independent of a;, when equation (126) is adopted, is extremely convenient for use in problems where the incident wave is given in terms of its amplitude spectrum rather than as a monochromatic wave. This is the reason for including equation (126) in the present report and it leads to reasonably accurate results. Upon substituting the relationship for f,, given by equation (125) in equation (109) a less convenient but more accurate implicit equation for ¢ is obtained. OZ hes A (F cane ae ¢ (tz) fo) 5 tan2o = 0.294.) fe) This equation may be solved iteratively by assuming a value of $¢ and evaluating |A| = R, 2a; and F, from Figures 16 and 17, respectively. With these values a new value of ¢ may be obtained from equation (127) and the iteration may be continued until convergence is achieved. Since |A| is a function of a;, the incident wave amplitude, ¢, is a function of a;; the use of equation (127) will therefore reflect the observed decrease in reflection coefficient with increasing incident wave amplitude. Although seemingly more cumbersome, it should be mentioned that equation (127) is solved after a limited number of iterations (two iterations generally suffice). For given incident wave, and slope. characteristics, ..a:,,.L, hos) es and d, either of the relationships for $ may be solved; the reflection coefficient is then obtained from Figure 15. To use this procedure to "predict" the reflection coefficients observed for the slope angles tanB, = 1/2.0 and 1/3.0 does not constitute a test of the procedure since these data were used in establishing the empirical relationships for fy, and hence the procedure. However, with the degree of scatter exhibited in Figure 22 this may be a meaningful comparison in that it will indicate the ability of the procedure to reproduce the experimentally observed reflection coefficients. To perform a more meaningful test of the procedure two separate sets of experiments were performed as previously described in Section III.3.b but for values of the slope angle tan§, = 1/1.5 and 1/2.5; each of these tests consisted of 48 individual experiments. From knowledge of the incident wave and slope characteristics the procedure was used to predict the reflection coefficient of the slope. For each experiment two predicted values of the reflection coefficient, Ros and R,, were obtained depending on whether the slope friction angle ¢ was obtained from equation (126) (Rs) or from equation (127) (Ry) - The predicted reflection coefficients were compared with the 70 measured reflection coefficient, R,. The comparison was performed by evaluating the mean value of the quantity, R Z(R_/R_) — = — ; (128) and its standard deviation, DN 2 R E( - R-/R_) oe) = ise Rb dll ee ene US i (129) p N-1 in which N is the number of experiments performed for a given value of the slope. The data used for this comparison are presented in Appendix B, and the results are presented in Table 2. Table 2. Comparison of measured and predicted reflection coefficients of rough slopes. As is evident from the comparison in Table 2, the procedure is quite accurate in reproducing the reflection coefficients obtained for tanB, = 1/2.0 and 1/3.0 in spite of the scatter exhibited in Figure 22. The procedure also predicts the reflection coefficients obtained for tanB, = 1/2.5 with a comparable degree of accuracy. Thus, for slopes 1/3.0 < tan8, < 1/2.0 the procedure is quite accurate in predicting the reflection coefficient of rough slopes. The more elaborate empirical formula (eq. 127) for the slope friction angle, 9, is superior to the simpler formula (eq. 126) as could be expected. However, it is noted that even the simple formula leads to reasonably accurate estimates of R. it For the steepest slope tested, tan8, = 1/1.5, the procedure leads to consistent estimates of the reflection coefficient as evidenced by the low value of the standard deviation. However, the mean value of R,/R,, , is somewhat different from unity. Basically the measured reflection coefficients are on the average only about 90 percent of the values predicted by the semiempirical procedure developed here. The reason for this discrepancy may be sought in the steep slope angle which violates the basic assumption made in the theoretical analysis that the slope be relatively gentle. This assumption, which is discussed in Appendix A, means that the horizontal velocity is taken to be representative of the velocity parallel to the bottom. For smaller slopes this is a fairly good assumption, but as the slope becomes steeper the velocity parallel to the bottom may approach U/cos8, where U is the horizontal velocity. This increase in near bottom velocity will increase the energy dissipation due to bottom friction and hence lead to smaller values of the reflection coefficient. Since the details of the wave motion on a steep slope are not known, the above discussion is only to be taken as a tentative explanation of the discrepancy between observed and predicted reflection coefficients for steep slopes. For steep slopes the reflection coefficient is generally close to unity so this discrepancy may not be of a severe nature. It is recommended that the procedure developed here be used also for steep slopes with 1/2.0< tan8, < 1/1.5 with a correction factor being applied to..the predicted. reflection. coefficient. -For-a-slope of -1- on 17S “this correction factor is taken as (Table 2);for slopes between 1 on 1.5 and 1 on 2 an appropriate correction factor, smaller than unity, may be chosen corresponding to a linear interpolation between the values of Sy All experiments discussed so far were performed for a water depth, ha = 1 foot (0.305 meter), im the constant depth part of the £flumeseA separate short series of tests was performed with h, = 16 inches (0.41 meter) for a value of tan8, =1/3.0 and a surface roughness d = 2 inches (5 centimeters). The results of these tests are shown in Table 3. The comparison between predicted and measured reflection coefficients presented in Table 3 shows an excellent agreement. It is of particular interest to,note that a reflection coefficient as -low ‘as ‘0.42. was observed and predicted. Since the average rate of energy dissipation on the slope is obtained from 2 Ey = (1-R’) Ey 5 (130) in which Ep is the energy flux asscoiated with the incident waves, this means that 80 percent of the incident wave energy is dissipated on the te Table 3. Comparison of measured and predicted reflection coefficients (hy = 16 inches (0.41 meter). Incident R R Wave Period Wave Height in R Predicted Predicted (in seconds) centimeters | Measured (eq. 127) (eq. 126) slope. This is a surprisingly large energy dissipation, particularly when it is realized that the incident waves showed no sign of breaking on the slope. This considerable energy dissipation is therefore due mainly to bottom friction. The procedure enables one to determine the wave runup in addition to the reflection coefficient. The determination of the wave runup, |A|, is an integral part of the procedure itself. Figure 16 is used both in establishing values of f from the experimental values of R, and in solving the empirical relationship (eq. 127) for the slope friction angle, ». Thus, although not explicitly appearing in the final result, the procedure for the prediction of reflection coefficients of rough slopes relies implicitly on the runup prediction afforded by Figure 16. Therefore, it would be somewhat disturbing if the runup predicted by Figure 16 was drastically different from the runup occurring in the experiments. For this reason a simple observation was made of the runup in the experiments used in establishing the empirical relationships for f,, and the observed runup was compared with the runup predicted from Figure 16. This comparison, involving the 96 experiments listed in Appendix B, shows a mean value of R, (predicted)/R, (observed) to be 1.15 with a standard deviation of 0.28. This agreement is not sufficient for the runup predictions afforded by Figure 16 to be used in actual design, but it does show that its use as part of the procedure 73 for predicting the reflection coefficient of rough impermeable slopes is warranted. This comparison of predicted and observed runup, which is independent of the determination of the reflection coefficient, may also be taken as an independent check of the soundness of the procedure and the theoretical analysis developed here. a. Limitations of the Procedure. The preceding comparison of measured and predicted values of the reflection coefficient of rough impermeable slopes has shown the semiempirical procedure to yield quite accurate results. However, it is important to realize that the significance of this favorable comparison is limited by the range of the independent variables tested here. Therefore, it should be used with caution whenever the values of d tanB,/|A| and d/hg are outside the range indicated by the experimental data in Figure 22. Furthermore, the procedure relies on an experimentally established empirical relationship for the wave friction factor, f,. In this investigation the empirical relationship was established from experiments in which the slope roughness elements were modeled by gravel. The procedure as it appears here is therefore applicable only for slopes whose roughness may be considered adequately modeled by gravel, i.e., natural stones, quarry stones, etc. To utilize the procedure for slopes protected by concrete armor units, an empirical relationship for f,, representative of these armor units, should be established in a manner similar to that presented in Section III.3. It is beyond the scope of the present research to establish more general relationships for fo Finally, the manner in which the theoretical results were used in the analysis of the experimental data to obtain values of f, makes the resulting empirical relationships for f, part of the procedure itself, i.e., the empirical relationships for f,, can be used with confidence only in conjunction with the present procedure for estimating the reflection coefficient of rough slopes. As mentioned in Section III.3.b, experiments were performed also for smooth slopes. For smooth slopes the roughness is negligible and the empirical relationships for ¢ (eqs. 126 and 127) are invalid. The type of empirical formula anticipated for f, was based on an assumption of fully rough turbulent flow conditions. To investigate if the experiments performed with rough slopes in this study do correspond to fully rough turbulent flow, the criterion established by Jonsson (1966) may be examined. For the present experiment the maximum value of |A|/(d tanB,) is seen from Figure 22 to be of the order 20. Since this value corresponds to the parameter aj,,/k used by Jonsson (1966, Fig. 6) it is seen that fully rough turbulent flow should exist for values of the Reynolds number 2 Re = HLA eats > 104 j (131) 74 with» = 100” ft-/sec. It is readily shown that the major part of the experimental data presented in Appendix B satisfy this criterion. The experiments and the empirical formulas developed for the wave friction actor, fs therefore correspond to fully rough turbulent flow conditions. When fully rough turbulent flow conditions exist in the model a Froude model will correctly reproduce the prototype conditions. A check of whether or not fully rough turbulent flow conditions may be expected in a particular model test may be performed in a manner similar to that described above, using Jonsson's (1966, Fig. 6) wave friction factor diagram. Be Discussion and Application of Results. A theoretical analysis of the reflection of water waves from rough impermeable slopes was performed based on the assumptions of relatively long, nonbreaking, and normally incident waves. The general solution for the reflection coefficient is presented in graphical form in Figure 15 which gives R as a function of the horizontal extent of the slope relative to the incident wavelength, &/L, and a slope friction angle, 9. The theoretical analysis accounts for the energy dissipation on the rough slope by including a term expressing the bottom shear stress. Therefore, the analysis introduces and identifies the physically fundamental parameter of the problem as a wave friction factor, fy. wave friction factor expresses the effect of slope roughness and is related to the slope friction angle, >, through the use of Lorentz! principle of equivalent work. A series of experiments was performed in which an accurate determination of the reflection coefficient of rough impermeable slopes was used in conjunction with the theoretical analysis to evaluate the magnitude of the wave friction factor, f,. From these experimental values of f,, two empirical relationships for f, as a function of the relative slope roughness were obtained. One of these relationships (eq. 124) leads to a simple expression for the slope friction angle (eq. 126) which shows the value of ¢ to be independent of the incident wave amplitude. Therefore, equation (126) is particularly convenient for use when the incident wave is given in terms of its amplitude spectrum. The other empirical relationship for f, (eq. 125) leads to a more elaborate and accurate relationship for ¢(eq. 127). With this relationship the reflection coefficient of rough slopes decreases slightly with increasing incident wave amplitude, thus reflecting the nonlinear nature of the energy dissipation on the slope. This The resulting semiempirical procedure for the prediction of reflection coefficients of rough impermeable slopes was tested against a separate set of experiments and yielded excellent results for values of the slope angle 1/3.0 < tan, <_1/2.0. For slopes steeper than corresponding to tan$s = 1/2.0 the procedure overestimates the reflection coefficient and a correction factor varying from 1.0 for tanB . = hyo 30) iS) to approximately 0.9 for tanb . = 1/1.5 should be applied to the predicted Reswmlies. It is important to realize that the procedure is empirical and therefore is limited by the range of the independent variables used in the experiments establishing the procedure. These limitations of the procedure are discussed in Section III.4.a. Numerical Example. To illustrate the application of the procedure for prediction of reflection coefficients for rough impermeable slopes consider the following example specified in Table 4. Table 4. Information used in numerical sample calculations. Prototype Froude Model length scale 1:25 Incident wave amplitude a; in feet Wave period T in seconds Water depth hy in feet Incident wavelength?! L in feet Slope angle tans Civon) 1.25) Surface stone size d =1/2 idea) in feet Since a Froude model scales the slope frictional effects correctly for fully rough turbulent flow conditions, either the prototype or the model may be taken as the basis for the following numerical calculations. Choosing the model it is seen from the information presented in Table 4 that 76 Se ee Ot ye ee ee O12. (132) Taking first the simple expression for the slope friction angle (eq. 126) 0.26 tan2¢ = 0.25 es) eae (133) i in which Ee is obtained diréctly from Figure 17 Fe i=c0s83. , (134) since F, is not a function of $ for this low value of 2,/L. For higher values of £,/L a value of F, corresponding to an assumed value of $ is obtained from Figure 17 and substituted into the right-hand side of equation (133) to obtain a new value of ¢. With this new value of ¢ a better estimate of F, is obtained from Figure 17 and the procedure is continued until convergence is achieved. In the present case equation (133) may be evaluated directly to give 0.26 2) 0183 =" 0.25 (0.56) (0283) = O16 C1s5) tan2¢ = 0.25 Ce 167 and the value of » is obtained baste ten= Bisy a (136) With £./L given by equation (132) and 9 = BiSee Figure 15 gives the predicted value of the’ reflection coefficient Rog 7 0:94 (137) The present calculation corresponds to a steep slope, tanB. =)1/1.5, and as discussed in Section III1.4 the estimate given by equation (137) should be corrected by the factor given in Table 2 corresponding to $ obtained from equation (126) and tanB, = 1/1.5. The estimate of the reflection coefficient therefore becomes REE 0.94 = 0.92(0.94) = 0.86 . (138) 77 If choosing the more elaborate expression for » given by equation (127) the implicit relationship for > becomes x2 0.3 ASME sia tht (139) d tan2o = 02.29 (Cm) & ean é ) ) s The preceding calculation based on the simple formula for 9, may be used as a first guess for the value of 4. Therefore, with ¢ = 3.3° and &/L = 0.12, Figure 16 gives: cecal eae (140) u De\: , aL or VA 2.6a, = 2.6(0.069) = 0.138 foot , (141) and Figure 17 gives Pe = 0.83 as before. Equation (139) therefore becomes 0.2 OS) ne Om25 0.138 Fs Leas eye PolEEOt ye, Hc lelO7(E OF0R2: apne ecko 0.29 (0.64) (0.64) (0.83) = 0.099 (142) or eager ee . (143) One should now return to Figures 16 and 17 with this new value of 9 to reevaluate the values of R, and Fz. In the present example the values corresponding to this new estimate of ¢ (eq. 139) are practicaily identical to those obtained for $ = 3.3° so convergence is, in this example, rapidly achieved. With ¢ = 2.8° and &_/L = 0.12, Figure 15 yields a reflection coeEficient) of : Ruwi=, 10,395 c 144 (144) Due to the steep 1 on 1.5 slope this estimate should be corrected by the factor in Table 2, corresponding to ¢ obtained from equation (127), i.e., the best estimate of the reflection coefficient becomes 78 R LSVER /RS) 0895) =1102 89, (OLO5)1 = 0084 9. (145) P mp The two estimates for the reflection coefficient (eqs. 138 and 145) are in close agreement and they may be considered quite accurate since they correspond to values of |A|/(d tan6 .) =" Ionanded/h = OW07ewhach are within the range of the data presentéd in Figure 22.~ The value of the Reynolds number defined by equation (131) is 1.1 X 10%. This demonstrates that the flow in the model and therefore also in the prototype is fully rough turbulent and indicates that a Froude model will reproduce the energy dissipation on the rough impermeable slope correctly. As discussed in conjunction with the numerical example presented in Section II, the numerical example in this section accounts for the external energy dissipation whereas the numerical example in Section II accounts for the partition of the remaining energy among reflected, transmitted and internally dissipated energy. Subtracting the energy dissipated on the rough slope (the external energy dissipation) from that of the incident wave assumed in Table 4 shows that the remaining energy may be regarded as the energy associated with an equivalent incident wave of amplitude ay; = Raj. With a; = 0.069 foot, as specified in Table 4, and R = 0.84, from equation (145), the amplitude of the equivalent incident wave is ar = 0.84 (0.069) = 0.058 foot. This is seen to be the incident wave amplitude assumed in the numerical example in Section II, Table 1. The two numerical examples are therefore closely related and illustrate the details of the calculations involved in the procedure for the prediction of reflection and transmission coefficients of trapezoidal breakwaters, which is discussed in Section IV. “9 IV. AN APPROXIMATE METHOD FOR THE PREDICTION OF REFLECTION AND TRANSMISSION COEFFICIENTS OF TRAPEZOIDAL, MULTILAYERED BREAKWATERS di Description of the Approximate Approach. In Section II an explicit solution for the transmission and reflection coefficients of homogeneous rectangular crib-style breakwaters was developed. In Section III a semiempirical procedure for the prediction of reflection coefficients of rough impermeable plane sloped structures was developed. When viewing the interaction of incident waves with a trapezoidal, multilayered breakwater as a problem of energy dissipation the problem treated in Section II may be regarded as an idealized analysis accounting for the internal dissipation of energy within the structure, Ep int » Whereas Section III may be regarded as an idealized analysis of the energy dissipation on the seaward face of the breakwater, i.e., the external energy dissipation, Ep ext. This section presents a synthesis of the results obtained in Sections II and III into an approximate procedure for the prediction of wave reflection from and transmission through trapezoidal, multilayered breakwaters. The basic assumptions of this approximate procedure are those inherent in the analyses and procedures developed in Sections II and III: (a) Relatively long normally incident waves which may be considered adequately described by linear long wave theory. (b) Incident waves do not break on the seaward slope of the breakwater, so that the external energy dissipation may be considered mainly due to bottom frictional effects. (c) The cover layer on the seaward slope of the breakwater consists of natural stones, so that the empirical relationships for the wave friction factor developed in Section III.3.c may be considered valid. With these assumptions stated, the following procedure is suggested as being physically realistic although approximate in nature. For most trapezoidal, multilayered breakwaters, the stone size in the layer under the cover layer of the seaward slope is small relative to the stone size of the cover layer. As a first approximation the structure may therefore be regarded as resembling an impermeable rough slope. Thus, with the incident wave characteristics and the stone size, diy> of the cover layer as well as the seaward slope of the trapezoidal breakwater, tan8_, specified, the procedure developed in Section III may be used to approximately account for the energy dissipation on the seaward slope, i.e., the external energy dissipation may be estimated. This energy dissipation approximately accounts for 80 the dissipation of energy associated with the top layer of stones in the cover layer. The reamining wave energy may be expressed as the energy associated with a progressive wave of amplitude, ei GS Grau ayaa (146) in which a. is the amplitude of the actual incident wave, and Rrz is the reflection coefficient determined by the procedure developed in Section, FIL. With the energy dissipated on the seaward slope accounted for, the remaining energy is partitioned betweeen reflected, transmitted, and internally dissipated energy. This partition of energy is the problem dealt with in Section II of this report,.and it is evaluated by regarding the remaining energy as an equivalent wave of amplitude, a,, normally incident on an equivalent homogeneous rectangular breakwater. The role of this homogeneous rectangular breakwater is to reproduce the internal energy dissipation associated with the trapezoidal, multilayered breakwater, i.e., the two breakwaters should be hydraulically equivalent. A rational method for obtaining a homogeneous rectangular breakwater which is hydraulically equivalent to a trapezoidal, multilayered breakwater is developed in Section IV.2, based on steady flow consider- ations. By using the procedure developed in Section II, the partition of the remaining wave energy among reflected, transmitted, and internally dissipated energy is therefore approximately evaluated by determining the reflection coefficient, R;, and the transmission coefficient, T;, of the hydraulically equivalent homogeneous rectangular breakwater subject to an equivalent incident wave of amplitude, ar: Having now accounted for the external as well as the internal energy dissipation the amplitude of the reflected wave is found to be Ja] = Ryay = Ray > coal, and the transmitted wave amplitude is Ja, eles ol ee (148) The approximate values of the reflection and transmission coefficients, R and T, of a trapezoidal, multilayered breakwater are therefore R = = RR (149) 8| and a = TeR ; (150) The approximate procedure described above is used to predict the reflection and transmission coefficients corresponding to the laboratory experiments performed by Sollitt and Cross (1972). When considering that the predicted results are obtained without any attempt being made to fit the experimentally obtained reflection and transmission coefficients from Sollitt and Cross, the comparison between predictions and experiments is favorable. 2i Determination of the Equiyalent Rectangular Breakwater. From the description given of the approximate method for obtaining the reflection and transmission coefficients of a trapezoidal, multilayered breakwater, the missing link for carrying out this analysis is the determination of the hydraulically equivalent homogeneous rectangular breakwater. In Section II.2.a it was shown that a simple analysis, which essentially neglected unsteady effects, gave transmission and reflection coefficients (eqs. 45 and 46) equal to those obtained from the more complete analysis for structures of small width relative to the incident wavelength (eqs. 35 and 36). This observation suggests that a rational and reasonably simple determination of the hydraulically equivalent breakwater may be based on steady flow considerations. Therefore, a hydraulically equivalent breakwater is taken as the homogeneous rectangular breakwater which gives the same discharge, Q, as the discharge through the trapezoidal, multilayered breakwater. This definition will, according to the simple analysis presented in Section II.2.a, preserve the equality of transmission coefficients for the two structures and hence essentially give the same internal dissipation. This definition of the equivalent breakwater is illustrated schematically in Figure 23. Figure 23 shows schematically a trapezoidal, multilayered breakwater consisting of several different porous materials. These porous materials are identified by their stone size, d_, and their hydraulic character- istics, 8 , in the flow resistance formula (eq. 6}: To keep the following determination of the equivalent breakwater reasonably simple, the flow resistance is assumed to be purely turbulent although in principle it is possible to perform the determination of the equivalent breakwater based on the more general form of the Dupuit-Forchheimer resistance formula. Since the energy dissipation associated with the top layer of stones on the seaward slope has been accounted for, the rectangular homogeneous breakwater which accounts approximately for the internal dissipation should be hydraulically equivalent to the trapezoidal, multilayered breakwater with the top layer of cover stones removed. 82 EQUIVALENT RECTANGULAR BREAKWATER Figure 23. Definition sketch of trapezoidal, multilayered breakwater and its hydraulically equivalent rectangular breakwater. 83 The homogeneous rectangular breakwater consists of a reference material of stone size, d,, and hydraulic characteristics, 6). fale reference material should be taken to be representative of the porous materials of the multilayered breakwater. To find the discharge per unit length of the rectangular breakwater, the flow is assumed to be essentially horizontal and one obtains Oe armeeata eB. U i == a@ESsIh) in which 2, is the width of the equivalent breakwater and AH, is the head difference defined in Figure 23. The discharge per unit length is therefore obtained from equation (151) to be ie OLA = 2 (152) To evaluate the discharge per unit length of the trapezoidal, multilayered breakwater a horizontal slice of height, Ahj, is shown schematically in Figure 24. ‘| | | | P/pg | | | | Figure 24. Horizontal slice of thickness, Ah., of multilayered breakwater. J This horizontal slice consists of segments of the different porous materials of lengths 2,. From an assumption of purely horizontal flow it follows that the discharge velocity of the slice considered, U., must be the same in all segments and the total head loss across the breakwater must be equal to AH,» the head difference shown in Figure 23. From this 84 it is seen AH, = A : (153) in which AH. is the head loss associated with the segment of length, Le, and hydraulic characteristics, Bo. From equation (151) it is seen that ie eee a J J AH = 6 2 -——-= 8B — n g g non 2 Vet) Byers (154) in which 8, is the hydraulic characteristic of the reference material. Equation (153) may therefore be written: Gs oie Ten en ae (155) ag nN r in which the summation is carried out over the n different porous materials of the horizontal slice of thickness, Ee From equation (155) the discharge associated with the slice of thickness Nee is found to be gAH,, bie Ah. AQ. = aa =f B, } Geese spas 2 . (156) {2g Ly} Iga ae and by adding the contributions from all horizontal slices of the trapezoidal breakwater one obtains J x i nods ies ho aoa. OO ak ees Wie ete (37) Cea ay n at Thus, requiring that the discharges per unit length given by equations (152) and (157) be identical, the width, hes of the equivalent rectangular breakwater is 85 I es 1 ps eae (158) eam ( 8 1/2 hy AH, ; J (aE )} Ag This equation shows that the width of the equivalent breakwater may be determined from knowledge of the configuration of the trapezoidal, multilayered breakwater and the corresponding head differences, AH, and AH « As described in Section IV.1 the equivalent breakwater is subject to an equivalent incident wave of amplitude a; given by equation (146). A simplified analysis of the interaction of incident waves and a rectangular homogeneous breakwater of small width relative to the length of the incident waves was presented in Section II.2.a. This simplified analysis essentially neglected unsteady effects and any phase difference between the incident, reflected, and transmitted waves. The runup on the seaward slope is for this analysis given by equation (38) and this runup is taken as a representative value of the head difference, AH, across the equivalent breakwater. With this assumption, which neglécts the influence of a transmitted wave of small amplitude, one obtains AH, = (1 + Ry) ar = (1 + Ry) Rit as 5 (159) in which Ry is the reflection coefficient of the equivalent breakwater. The value of the head difference across the trapezoidal breakwater, AH7p, is in accordance with the argument presented for the equivalent rectangular breakwater taken as the runup on the seaward slope of the trapezoidal breakwater. This runup may in principle be determined by the procedure developed in Section III of this report. However, there is reason to believe that such an estimate, which would correspond to an impermeable slope, would be somewhat on the high side. In general one may, however, take ATM = ZR vdeo 5 (160) where R, is the best estimate available for the ratio of the runup to the incident wave height H; = 2a; for given slope characteristics. 0t R, is taken as determined from Figure 16 the estimate of AHy is expected to be conservative. Equations (159) and (160) show that the ratio AHS (ibe RRs = = SC——7" (161) AH, 2R 86 is a function of the reflection coefficient, R;, of the equivalent breakwater. Since this reflection coefficient cannot be determined until the width of the equivalent breakwater, 2 » is known one is faced with a tedious iterative procedure. However, in most cases a sufficiently accurate estimate of R; may be obtained by assuming initially that AH, / AHy is unity and use this estimate along with the best estimate of R, to obtain a new value of AH,/AHy; from equation (161). Numerical Example of the Determination of the Equivalent Rectangular Breakwater. To illustrate the procedure for the determination of the equivalent rectangular breakwater the trapezoidal, multilayered breakwater configuration tested experimentally by Sollitt and Cross (1972) and shown in Figure 25 is considered. The breakwater is divided into five horizontal slices, I-V (Fig. 25), and the reference material is chosen as the material with stone size dy = 0.75 inch (1.9 centimeters). Since the porosities of the three porous materials were reported to be essentially equal by Sollitt and Cross (1972), the porosity of all materials as well as the reference material is assumed to be 0.435. This in turn means that the ratio of the hydraulic characteristics of the various porous materials, Bn» according to equation (52) with B, = 2.7 reduces to the inverse of the ratios of the stone sizes, i.e., 6 4 d os AON ae (162) Bo From the geometry of the breakwater shown in Figure 25, with the top layer of cover stones removed, the equivalent breakwater width is readily calculated as shown in Table 5. Introducing the numerical result obtained in Table 5 in equation (158), the equivalent breakwater width, hes is obtained as 2 AH, AH, Le = (0.1819) nT, eee = e252 TH, feet . (163) Thus, the homogeneous rectangular breakwater which is hydraulically equivalent to the trapezoidal, multilayered breakwater (Fig. 25) consists of material of stone size d = d, = 0.0625 foot (1.9 centimeters) and porosity n = 0.435. The equivalent width, £., 1s given by equation (163) and a first approximation may be obtained by taking AH,/AHp = 1, Li sOiets Veua wee treet (0.77 centimeter). It should be noted that the homogeneous rectangular breakwater assumed in the numerical example presented in Section II, Table 1, has the characteristics of the hydraulically equivalent breakwater given 87 U es Ee (ysAvq “(ZZ61) Sssoij pue IITITOS Aq pojse} uoTJeINSTFuOD JojemyeotIg ‘SZ sANnBTYy OE 219 88 Table 5. Evaluation of equivalent rectangular breakwater. 8 { Gee: Ni ed. inches |} i (inches) | by equation (163) for the choice AH,/AHy = 1. Furthermore, the trapezoidal breakwater shown in Figure 25 has a seaward slope of 1 on 1.5 (tan8, = 0.667) which for the most part consists of stones of d = diz = 1.5 inches (3.8 centimeters). These slope characteristics correspond to those assumed in the numerical example presented in Section III, Table 4. The present numerical example of the determination of the hydraulically equivalent breakwater together with the numerical examples presented in Sections II and III therefore constitute an example of the computations involved in the procedure described in Section IV.1 for the determination of the reflection and transmission coefficients of the trapezoidal, multilayered breakwater shown in Figure 25. 89 Se Computation of Transmission and Reflection Coefficients for Trapezoidal, Multilayered Breakwaters. Sollitt and Cross (1972, App. G) presented the results of a laboratory investigation of reflection and transmission characteristics of their model breakwater (Fig. 25). For the present purpose of comparison with predicted reflection .and transmission coefficients only the tests performed by Sollitt and Cross (1972) with relatively long waves will be used. Thus, the wave conditions to be used in the following for the purpose of demonstrating the computational aspects of the approximate method described in Section IV.1 are hy = lou Geeta splh 2.5) Seconds. 31h) = 14.56 Reet iy (164) and the breakwater configuration is that shown in Figure 25. a. Determination of the External Energy Dissipation. As discussed in Section IV.1 the first step in the approximate procedure for evaluating the reflection and transmission coefficients of a trapezoidal, multilayered breakwater is to estimate the external energy dissipation on the seaward slope using the procedure developed in Section III. From the breakwater characteristics shown in Figure 25 it is seen that the seaward slope consists of various stone sizes. Since the main part of the front face consists of stones of diameter dry = dy = 0.125 foot (3.8 centimeters) it is reasonable to adopt this stone size as the roughness of the slope. This is further justified by the fact that this stone size is the size in the cover layer near the stillwater level, where one would expect the major part of the external energy dissipation to take place. Hence, for the purpose of estimating the external energy dissipation the slope characteristics are taken as: Roughness = diy = 0lt25ychOot yc tan6 =) = 402667 Sx (165) This information in addition to the incident wave characteristics specified by equation (164) is sufficient to use the procedure developed in Section III for the prediction of the reflection coefficient, R of Lgl rough slopes when the incident wave amplitude, aj, is specified. Comparison of the incident wave characteristics (eq. 164) and the Slope characteristics (eq. 165) with the corresponding values for the Froude model characteristics given in the numerical example presented in Table 4 show that the calculations presented there correspond to the conditions being considered here. The numerical example presented in 90 Section III may therefore be taken directly as an illustration of the computational aspects inyolved in the determination of the reflection coefficient of the seaward slope, Rit Thus, with the detailed computational aspects given in Section III the computations corresponding to the incident wave amplitudes reported by Sollitt and Cross (1972, App. G) are summarized in Table 6. Table 6. Summary of calculations of external energy dissipation on seaward slope for experiments by Sollitt and Cross (1972, App. G). $ Degrees (eq al 273) OO AO. O31 Or Oy Oe tO Oo (OO ee WWWMHWwponwnwpnwwn wa wae we wa (=e ke (Ss 2) oot ol or Sie ulo) qos “{e) 5 y 2 e 5 5 55 5 5 : 5 ° ie IL Ihe Dr Dre Zi: Zi De oe Se 3 3ie he hy diy rc: Ir tané* OS 205 ne OF L077 55 (166) and the best estimate of the reflection coefficient, Rip is obtained by multiplying the predicted value, R., by the correction factor 0.89 associated with the steep slope, tanB. = 1/1.5. 9| For comparison the numerical example carried out in Section III is seen to correspond to an incident wave amplitude between the incident wave amplitudes of Run Numbers 470 and 469 in Table 6. b. Determination of the Internal Energy Dissipation. Having determined the energy dissipation on the seaward slope, the next step is to determine the internal energy dissipation. To perform this analysis one must first determine the characteristics of the rectangular breakwater which is hydraulically equivalent to the trapezoidal, multilayered breakwater shown in Figure 25. As an example of the computations involved, the homogeneous rectangular breakwater which is hydraulically equivalent to the breakwater configuration shown in Figure 25 was determined in Section IV.2. Thus, the equivalent rectangular breakwater has the characteristics obtained in Section IV.2, LSSae Stone size. = id= 0.0625 foot 1; Porosuty =m = 0.455 3, (167) and a width Qe given by equation (163) AH. Uon= (252). feet; (168) e AH. in which AH, and AHyp are the runup on the equivalent rectangular and on the trapezoidal breakwater, given by equations (159) and (160), respectively. As discussed in Section IV.2, the determination of 2, from equation (168) involves a tedious iterative procedure. A fairly good ‘first guess for the value of the reflection coefficient, R;, involved in determining the ratio AH,/AHy; from equation (161) may, however, be obtained by taking AH,/AHp equal to unity. Hence, a preliminary solution is obtained by taking de = 2.52 feet ‘ (169) The equivalent rectangular breakwater characteristics are therefore given by equations (167) and (169). The incident wave characteristics are given by equation (164) with an equivalent incident wave amplitude, ay, given by equation (146). The computations involved in determining the partition of the energy associated with the equivalent incident wave among reflected, transmitted, and internally dissipated energy is therefore carried out according to the procedure developed in Section II of this report. In Section II, the details of this computation were presented in the form of a numerical example. By inspection of the incident wave and breakwater characteristics used in the numerical example in Section II, these characteristics for the Froude model listed 92 in Table 1 are identical to those given by equations (164), (167), and (169). The numericai example may therefore be consulted for the details of the computational aspects inyolyed in the determination of the transmission and reflection coefficient of the equivalent breakwater, Ty and Ry subject to incident waves of amplitude ay The important parameters involved in the use of Figures 2 and 3 for this purpose were obtained in Section II 20 Tes) 2-52 = 0.47, (170) n okt =" OMMS5¢ oe and the porosity correction factor (eq. 28), ee pif, 2) FOeA3S 2 S,= Gas) = Gar = 0-93 (171) The computations corresponding to the equivalent incident wave amplitudes, a,, determined from the results obtained in Table 6 are summarized in Table 7. Table 7. Summary of calculations of reflection and transmission coefficients of equivalent rectangular breakwater based on AH ./ 4H, = 1. ou piece END EC Ln a a OX. ON 30d = td Sn SOS BOF SS NS 8 Us) 0. 0. 0. 0. O. 0. O. QO. 0. 0. 0. Whereas the reflection coefficient is essentially insensitive to the value of £.~ it is seen from Tables 7 and 8 that the transmission coefficient shows a significant change with the value of the equivalent breakwater width. The transmission coefficients obtained in Table 8 are considerably larger than the transmission coefficients obtained in Table 7, thus reflecting the smaller equivalent breakwater width. Since the equivalent breakwater width was obtained from equation (172) using the slightly conservative value of R,, obtained in Table 6, it may be anticipated that the equivalent breakwater width listed in Table 8 is on the lower side. Consequently, one may anticipate that the estimates of the transmission coefficients obtained in Table 8 are slightly on the high side. The preceding discussion essentially shows that the reflection coefficient, R;, may be regarded as relatively accurately determined whereas the transmission coefficient, T,_, is bracketed by the results : : I listed in Tables 7 and 8. 95 c. Determination of the Transmission and Reflection Coefficient of Trapezoidal, Multilayered Breakwaters. From the results obtained in Sections 1V.3.a and IV.3.b it is now possible to estimate the transmission and reflection coefficient of the trapezoidal, multilayered breakwater (Fig. 25) since the external as well as the internal energy dissipation has been accounted for. From the description of the procedure given in Section IV.1 it follows that the transmitted and reflected wave amplitudes, la, | and lal: are given by cay bangs Has ete A 173 | ea PRA Cys) [ane Reaa = ORE 174 r iGwAl fenton C7) The transmission coefficient, T, and reflection coefficient, R, are therefore obtained from the results listed in Tables 6, 7, and 8 since eI 175 EEL ea (7s) i! and EA hes ale RRiy : (176) i The resulting estimates are shown in Table 9. 4, Comparison Between Predicted and Observed Transmission and Reflection Coefficients of a Trapezoidal, Multilayered Breakwater. The preceding section has illustrated the use of the approximate method for the determination of transmission and reflection coefficients of trapezoidal, multilayered breakwaters described in Section IV.1. The breakwater characteristics as well as the characteristics of the incident waves used in Section IV.3 were chosen to correspond to the experiments performed by Sollitt and Cross (1972), and the predictions given in Table 9 may therefore be compared directly with the experimentally observed values of the transmission and reflection coefficients given by Sollitt and Cross (1972, App. G). This comparison between predicted and observed transmission and reflection coefficients is shown in Figure 26, where the values of T and R are plotted against the incident wave steepness, H,/L. From the comparison presented in Figure 26 the predicted reflection coefficients are in excellent agreement with the observed reflection coefficients for lower values of the incident wave steepness. For larger values of the incident wave steepness the predicted reflection 96 9Tqe 9Tqe eTge oTqe aTqe x ca T)Ily_|(2 eran) |(Z etgeL) | ITI, | TTI, | (9 etqet| (9 etqeL) | (9 eTqeL) | .oT x I MY al Nl fis lle as II T (tot *ba) wosz “yy/HV "(9 xtpueddy “7/61T) ssotg pue 123TTTOS Aq poqseq Tajemyeorq poroAkeT tz [nu “Teprozeder, Fo sJUsTITFJ909 UOTSSTUSULI} PUP UOTIIETFOL poJotpotg u T/°H “6 STIeL of, "(19T) worqenbe al, ) € Al; S) Aq uoat3 “HV/ HV uo paseq uotyItperd ——— ‘T = “HV/ HV uo paseq uot}otperd —— — ‘quoTOTFFOOD uoTsstusued} Te uoUTIadxo9 @ “USTITFFIOI UOT IT FJeL yequowtiedxy wp “(s'o = ux °9 ‘ddy ‘Z7Z61) Sssoatj pue 31TIIOS Aq pe ysey Laqyemyeoig parade[t yqnu ‘Teproz -adei} JO JUDTITJJOOD UOTSSTWSUBI} PUB UOT}IITJAL PaALasqo pue peqotpaad fo uostseduog 2167214 ‘97 oan3T4 98 coefficient is seen to increase slightly whereas the obseryed reflection coefficients exhibit a decreasing trend with increasing wave steepness. As discussed preyiously this trend of the experimental reflection coefficients is generally obseryed and may be partly due to experimental errors in the determination of the reflection coefficient. This was discussed briefly in Section II.3.b and in detail in Section iT Sib The transmission coefficients predicted based on the assumption AH,/AHp = 1 are seen to be lower than the experimentally obtained values. This, of course, is the expected type of discrepancy since the runup on the seaward slope of the trapezoidal breakwater is almost certain to exceed the runup on the equivalent rectangular breakwater. Adopting the theoretical value of the runup, R,, on the trapezoidal breakwater predicted by the procedure developed in Section III of this report is expected to give transmission coefficients slightly on the high side as discussed in Section IV.3.c. This anticipated behavior is not exhibited by the predicted transmission coefficients plotted in Figure 26. In fact, the agreement between observed and predicted transmission coefficients is excellent. A slightly different estimate of the runup on the seaward slope of a trapezoidal breakwater may be obtained by adopting, for example, the results obtained by Jackson (1968), who reported values of R approximately equal to unity for test conditions similar to those of Sollitt and Cross (1972). In the present case this value of R, would result in a slightly lower prediction of the transmission coefficient than the prediction indicated by the full line in Figure 26. The procedure developed here for the prediction of transmission and reflection coefficients of a trapezoidal, multilayered breakwater did not rely on the experimental data shown in Figure 26 to obtain a ''good fit''. The overall comparison between predicted and observed transmission and reflection coefficients, which is analogous to the comparison given by Sollitt and Cross (1972, Fig. 4-14), must therefore be considered very good. 32 V. SUMMARY AND CONCLUSIONS This report presents the results of an analytical study of the reflection and transmission characteristics of porous rubble-mound breakwaters. An attempt was made at making the procedures entirely self-contained by introducing empirical relationships for the hydraulic characteristics of the porous material and by establishing experimentally an empirical relationship for the friction factor that expresses energy dissipation on the seaward slope of a breakwater. The results are presented in graphical form and require no use of computers, although the entire approach could be programmed. The procedures were developed in such a manner that the information required to carry out the computations can be expected to be available. Thus, for a trapezoidal, multilayered breakwater subject to normally incident, relatively long waves the information required is: (a) Breakwater configuration: breakwater geometry and stone size and porosity of the breakwater materials (b) Incident wave characteristics: wave amplitude, period, and water depth. Only the porosity of the breakwater materials may be hard to come by. It is recommended that the sensitivity of the results to the estimate of the porosity, n, be investigated. The hydraulic flow resistance in the porous medium is expressed by a Dupuit-Forchheimer relationship and empirical formulas are adopted. The investigation shows that reasonably accurate results are obtained by taking Be 2.7 (177) 1150 R a oO in equations (51) and (52). To estimate reflection and transmission characteristics of a prototype structure only the value of 8, needs to be known. For laboratory experiments the value of the ratio,o,/8o is important in assessing the influence of scale effects. In a lapvoratory setup it is possible to determine the best values of a, and 8, from the simple experimental procedure used by Keulegan (1973). Thus, it was found that the porous materials tested by Sollitt and Cross (1972) showed a value of a) = 2,700, a better value than that given by equation (177). However, the important thing to note is that the analysis carried out in Section II of this report presents a method for assessing the severity of scale effects in hydraulic models of porous structures. The empirical relationships for the flow resistance of porous materials have been demonstrated to be fairly good for porous materials consisting of gravel-size stones, diameter less than 2 inches (5 centimeters). 100 The energy dissipation on a rough, impermeable slope was investi- gated in Section III. The experimental investigation revealed the need for an accurate method for the determination of reflection coefficients from experimental data. The simple procedure of seeking out the locations where the wave amplitudes are maximum and minimun, respectively, may lead to reflection coefficients which are much too low, unless the recorded surface elevation is analyzed and only the amplitude of the first harmonic motion is used to determine the reflection coefficient. Accurately determined reflection coefficients for slopes with roughness elements consisting of gravel led to an empirical determination of the friction factor (eqs. 124 and 125), expressing the energy dissipation on a rough slope due to bottom friction. Adopting this empirical relationship a procedure for estimating the reflection coefficient of rough impermeable slopes was developed. This procedure was quite accurate in reproducing the experimentally obtained reflection coefficients in a separate set of experiments. The procedure for the determination of the reflection coefficient of rough impermeable slopes is limited to slopes having roughness elements consisting of natural stones. To make the procedure generally applicable, empirical relationships for the friction factor should be determined for slopes whose roughness elements consist of models of concrete armor units. The synthesis of the investigation is the development of an approximate procedure for the prediction of the reflection and transmission characteristics of trapezoidal, multilayered breakwaters. This procedure is entirely self-contained and yields excellent results when compared with the model scale experimental results obtained by Sollitt and Cross (i972). It is emphasized that the analytical model for the reflection and transmission characteristics of trapezoidal, multilayered breakwaters developed here needs further verification before it can be used with complete confidence. However, the good agreement between predictions and observations exhibited in Figure 26 is encouraging and does indicate that a simple analytical model which may be used for preliminary design of rubble-mound breakwaters has been developed. 10l LITERATURE CITED BEAR, J., et al., Physical Princtples of Water Percolation and Seepage, United Nations Educational, Scientific and Cultural Organization, 1968. EAGLESON, P.S. and DEAN, R.G., ''Small Amplitude Wave Theory," Estuary and Coastline Hydrodynamics, McGraw-Hill, New York, 1966, pp. 1-92. ENGELUND, F., "On the Laminar and Turbulent Flows of Ground Water through Homogeneous Sand," Transactions of the Danish Academy of Techntcal Setences, Volts Noa. 42 19552 HILDEBRAND, F.B., Advanced Caleulus for Appltcations, Prentice-Hall, Englewood Cliffs, N.J., 1965. IPPEN, A.T., Estuary and Coastline Hydrodynamics, McGraw-Hill, New York, 1966. JACKSON, R.A., 'Design of Cover Layers for Rubble-Mound Breakwaters Subjected to Nonbreaking Waves,'' Research Report No. 2-11, U.S. Army Engineer Waterways Experiment Station, Vicksburg, Miss., 1968. JONSSON, I.G., "Wave Boundary Layers and Friction Factors," Proceedings lOth Conference on Coastal Engtneering, American Society of Civil Engineers, 1966, pp. 127-148. KAJIURA, K., "On the Bottom Friction in an Oscillatory Current,'' Bulletin of the Earthquake Research Institute, Vol. 42, 1964, pp. 147-174. KEULEGAN, G.H., 'Wave Transmission through Rock Structures,'' Research Report No. H-73-1, U.S. Army Engineer Waterways Experiment Station, Vicksburg, Miss., 1973. KONDO, H., 'Wave Transmission through Porous Structures," Journal of the Waterways, Harbors and Coastal Engineering, Vol. 101, WW3, 1975. pp. 300-302. MADSEN, O.S., "Wave Transmission through Porous Structures,'' Journal of the Waterways, Harbors and Coastal Engineering, Vol. 100, WW3, 1974, pp. 169-188. MEYER, R.E. and TAYLOR, A.D., "'Run-up on Beaches,'' Waves on Beaches, Academic Press, New York, 1972, pp. 357-412. MICHE, M., '"Pouvoir Reflechissant des Ouvrage Maritimes Exposes a L'action de Houle,"' Annalesdes Ponts et Chaussees, 1951, pp. 295-319. PEREGRINE, D.H., "Equations for Water Waves and the Approximations Behind Them," Waves on Beaches, Academic Press, New York, 1972, pp. 95-123. 102 POLUBARINOVA-KOCHINA, P.Y., Theory of Ground Water Movement, Princeton University Press apranceton. NiJs,,, 1962. SCHLICHTING, H., Boundary Layer Theory, 4th ed., McGraw-Hill, New York, 1960. SOLED, .CK. “and GROSS, Ree. “Wave, Reflection, and) Lransmisstonuat Permeable Breakwaters,"’ Technical Report No. 147, R. M. Parsons Laboratory, Department of Civil Engineering, Massachusetts Institute of Technology (MIT), Cambridge, Mass., 1972. URSELL, F. et al., "Forced Small-Amplitude Water Waves," Journal of Flutd Mechanics, Vol. 7, Pt. 3, 1960, pp. 33-52. WILSON, K.W., "Scale Effect in Rubble-Mound Breakwaters," unpublished Masters Thesis, Department of Civil Engineering, MIT, Cambridge, Mass., June V97 1% 103 APPENDIX A GOVERNING EQUATIONS AND THEIR SOLUTION Ik Long Waves over a Rough Bottom. To derive the approximate equations governing the propagation of long waves over a rough bottom we take the basic equations expressing conservation of mass and momentum for an incompressible fluid. In two dimensions these equations read (Schlichting, 1960) au; OW © ao a (A-1) DU _ 9p | ar ES Duar os: Woe Oe DW _ eps 4 i Cie Ne pg (A-3) in which U and W are the horizontal and vertical velocity components, respectively; p is the fluid density, p is the pressure, and g is the acceleration due to gravity. The coordinate system is defined in Figure A-1 and only the horizontal shear stress, t, is retained in the horizontal momentum equation. Figure A-1. Definition sketch. 105 The boundary conditions to be satisfied by these equations are that the pressure be zero at the free surface, z = n(x,t), p=p = (0) fie A Sins (A-4) and the kinematic boundary conditions an an % - eRe ena UF a Bue =a Onmat sz n (A-5) and oh Wb + Us, ae OMiat) z= he (A-6) where subscripts n and b refer to the conditions at the free surface and at the bottom, respectively. To derive the continuity equation in the form normally encountered in work involving long waves, equation (A-1) is integrated over the depth of water n n n 3U QW U 3 | age -{ waz = | ax Cea y as. =; 2 3 (A-7) -h -h -h and the remaining integral is evaluated using Leibnitz' rule (Hildebrand, 1965) B(o) B | af az = =| £ dz -£(B) 2+ ea) A, (A-8) A(o) A in order to obtain one i an dh =| U dz + ue Sa ae =H =U BES = 0 ° (A-9) -h Introducing the concept of a mean velocity, U, defined by — if i) Wee ee | Wd zs 3 (A-10) -h 106 and realizing that the boundary conditions (eqs. A-5 and A-6) may be used to simplify equation (A-9), the continuity equation becomes an < (UGhenieeon ane (A-11) From the continuity equation in its basic form (eq. A-1), introduction of the typical length scales, the wavelength, L, in the horizontal direction and the water depth, h, in the vertical direction, shows that the order of magnitude of the vertical velocity component is given by W = 0 Gu) (A-12) Thus, for long waves, h/L << 1, we have that W<< U. This observation suggests that the vertical fluid accelerations, DW/Dt, in equation (A-3) may be neglected. For long waves equation (A-3) therefore simplifies to a statement of hydrostatic pressure distribution, Pi= PEM awN Ss (A-13) where the boundary condition (eq. A-4) has been invoked. Introducing p as given by equation (A-13) and making use of equation (A-1) the horizontal momentum equation may be written: Z DU _ aU , ou oUW an dT/P ; (A-14) nm Getartat ee Dt." ‘oe «19x oz This equation may be integrated over the depth to yield: n n aes ide ree Hoods ya ae Sueno u ae ot by Ox ay dhy _ aig ee s “Up {HW 2 Uae = -(h+n) g — + 5 a (A-15) By virtue of the boundary conditions (eqs. A-5 and A-6), the bracketed terms vanish, and by introducing the concept of the momentum coefficient, 107 kee (A-16) Oe Gen us the integrated momentum equation may be written: = { (htn) U} + = { (h+n) Ko) = -(htn) g one 2 ‘ (A-17) in which the shear stress on the free surface has been set equal to ZEGO Using the results from linear wave theory (Eagleson and Dean, 1966), it may be shown that the value of Km as given by equation (A-16) is 1.01 corresponding to a wave having h/L = 0.1. Thus, it is a good approximation to take K, in equation (A-17) equal to unity as is normally done in open channel flow calculations. With equal to unity and incorporating the continuity equation, equation (A-11), the momentum equation becomes: aU on is t x & 9x p(hn) pean The appropriate equations governing the propagation of long waves over a rough bottom are therefore equations (A-11) and (A-18). However, these equations cannot be solved until the shear stress in equation (A-18) has been related to the kinematics of the problem. To this end we may introduce the concept of the wave friction factor, f , as defined by Jonsson (1966), Ls — Mia f |U|U ; (A-19) where |U| is the absolute value of the velocity. When this expression is introduced in equation (A-18) the governing equations become: an Pe) z ge ax, tr) U)oae ae and dU dU on 5 é,lulu Btu oie eeexe (eae a) where the overbar notation has been dropped. 108 a. Linearization and Solution Technique. To obtain closed form analytical solutions to the equations derived in the previous section it is necessary to linearize these equations. To justify a linearization of equations (A-20) and (A-21) we must impose the condition that [neha (A-22) in which case h+n may be replaced by h. This condition implies that the term U3U/3x also may be omitted. Hence, the final task is to linearize the bottom friction term. Since we will be looking for periodic solutions to the linearized equations this is conveniently done by taking i il = 2 = 2o0W ‘ oe i cour Ska in which fp is treated formally as a constant and w is the radian frequency of the periodic motion. Performing the above linearizations of the governing equations, we obtain an 3 _ as BEL win” (A-24) and aU an a ae te ant £ wu =O0e (A-25) To illustrate the solution of this set of equations we consider the simple case of periodic waves of radian frequency, w, propagating over a horizontal bottom, i.e., h = hy = constant. To facilitate the solution complex variables are introduced by defining U = Real fu et *} iwt) n = Real {ZT e (A-26) where i = Y-1 and the amplitude functions, u and c, are complex functions of x. The physical solution is given by the real part of the solution as indicated by the notation Real { }. Introducing equation (A-26) in equations (A-24) and (A-25), these may be written in terms of the amplitude functions: joc + h. <= 10 (A-27) 109 and 7 1 as fs Ns iw (1-if,) US OURS (A-28) From equation (A-28) we obtain Se eee emer OSI (A-29) which may be substituted into equation (A-27) to yield the governing equation: y2r we + gay Ost) = Oe (A-30) eee which simplifies to the usual wave equation for fh equal ‘to! zero. The general solution of equation (A-30) is given by ge ger +a a ; (A-31) in which a, and a_ are complex amplitudes, whose magnitudes give the physical wave amplitude, and k is the complex wave number defined by 2 gp gee 7 ah. q-if) (A-32) which may be written in terms of the usual wave number, ioe (A-33) Yoh fe) and ne Popeye oT fee (A-34) fe) b fe) b zg where tan2>) = fh (A-35) Taking, for example, the wave solution of amplitude a in equation (A-3l1), this reads: : 110 4 ~ik) Vv il+f z (cosd) - sind, ) x ‘i -———S 4 -k eae 2 Sing, x -ik. Vv 1+f z cosd¢, x Leta age b b see b b ts > (A-36) where a, may be considered a real number. When this is introduced in equation (A.26) and only the real part is retained we obtain 4 -k_ v l+f 2 sing, x 4 Hoe ale b E cos (wt Ke v 1+e,* cosd, x) (A=37) which shows the solution to be that of a sinusoidal wave propagating in the positive x-direction with an exponentially decreasing amplitude. In the same manner the wave solution of amplitude a_ in equation (A-31) may be shown to represent a wave propagating in the negative x-direction with an exponentially decreasing amplitude in the direction of propagation. Thus, we have obtained a formal solution to the problem of a long wave propagating over a rough bed. However, the solution must be considered formal since it depends on the value of the linearization factor f, introduced through equation (A-23). To obtain the appropriate value of the constant f,, it must be related to the true friction factor, f,, and the wave characteristics. b. Application of Lorentz' Principle. To obtain an explicit solution for the linearized friction, factor, f,, we usé Lorentz’ principle of equivalent work. This principle is a useful tool for obtaining approximate results from a set of linearized governing equations in which the nonlinear term expressing the flow resistance has been linearized. The principle (Ippen, 1966) states that the average rate of energy dissipation calculated from the "true" nonlinear friction term and that calculated from its linearized equivalent should be the same. As shown by Kajiura (1964), the instantaneous rate of energy dissipation per unit bottom area may be approximated by Ee Ut : (A-38) For a given area, A, and assuming a periodic motion of period, T, the average rate of energy dissipation becomes: - 1 les Ey =X | dA {= (i Unde s (A-39) in which the double overbar signifies that spatial as well as temporal average is to be taken. In the present context use of equations (A-19) and (A-23) in conjunction with Lorentz' principle yields: AR 1 1 1 2 a a dA calece te luju" dt} = 1 mie 2 = dA {= i) ae, ov Wis Ghat (A-40) A T b A 0 which leads to a determination of fy in terms of known quantities. As an illustration we take a progressive wave in a constant depth of water, h = ho» and equation (A-40) may be written: f= a (A-41) in which U, is given by equations (A-26) and (A-29) corresponding to the surface profile given by equation (A-36), i.e., gk a Pe 2 Gernot eee len Maddy wvl-if, ~k5x Uy e cos (wt ~k x + ow) P (A-42) in which ob is given by equation (A-35) and re) 4 (A-43) wv 1l+f 4 112 and 4 AO WB wth Tuscan eee D ae k. -ik; = k= Ky 1+f) (cos, -i sing, ) ‘ (A-44) Introducing these expressions in equation (A-41) and performing the spatial average over an area of unit width and extending from x = 0 to x = &, the following equation is obtained: 4 ka Ne a ey Opes eee ee On De Ae eh 8 aye (A-45) W 2 3 -2k.2 (kh) a (ome) l-e Although not leading to an explicit equation for the friction factor, f,, this expression may be solved iteratively from knowledge of f, and the wave characteristics. The bracketed term in equation (A-45) arises from the spatial averaging process and becomes unity if ki << 1. Thus, in the immediate vicinity of x = 0 we have 4 ka Pay, ies oop Or, (A-46) W 2 (koh) which may be shown to lead to the same rate of amplitude attenuation as the formula suggested by Putnam and Johnson (1949). c. Limitation of the Solution. To discuss the limitations of the solutions obtained from the governing equations derived in the preceding sections we consider the solution obtained for a progressive wave in constant water depth, h,, without bottom friction, i.e., ans fy = d= Oz For this simple case we have from equations (A-37) and (A-42), Ns =) ay cos (k x -wt) a U, = im ete cos (kx -wt) : (A-47) The basic assumption made in the derivation of the equations leading to this solution was that the vertical accelerations were negligible. We may now reexamine this assumption by obtaining the expression for the vertical velocity, W,, from equation (A-1), W, = aw (1 + =) sin(k x -wt) , (A-48) oO and the leading term arising from the vertical fluid acceleration in equation (A-3) is 113 ow + 2 z se ea hy) cos(k x -wt) , (A-49) p which integrated over depth gives: fen a ee ere ot => cos (ka oe) (A-50) Re ee i aa ce Oo From the derivation of the pressure distribution (eq. A-13), the term expressing the presence of the wave is P, = © ga, cos(k,x -wt) , (A-51) and to justify the neglect of the term given by equation (A-50) we must have that et xP awh, << OUga 3 (A-52) which with the aid of equation (A-33) may be stated as 1 Zz, pC LD) cassis lan, (A-53) i.e., a requirement of long waves as previously stated. Now, in the process of linearizing the governing equations it was mentioned that this linearization was justified if a - =o << 1 ; (A-54) because the terms then omitted would be smaller than the terms retained by the factor a,/h. This in turn would be a consistent procedure only if the pressure “term given by equation (A-50) is greater than (a, /h) times the leading term given by equation (A-51), i.e., linearization of the governing equations and taking the pressure distribution to be hydrostatic is consistent only if a 2 + 2. p 2 ho >> 0 ga, Te ’ (A-55) et 114 which may be written as the requirement that the Stokes' parameter SS ae (A-56) Thus, the limitations on the solutions obtained from the linearized set of governing equations are expressed as the inequalities given by equations (A-53), (A-54), and (A-56). For a derivation of the appropriate governing equations when equation (A-56) is violated the reader is referred to Peregrine (1972). For a wave propagating over an uneven bottom a vertical velocity may be imposed by the bottom boundary condition (eq. A-6). So long as the velocity obtained from this boundary condition is smaller than that given by equation (A-48) the preceding limitations are applicable. This in turn may be stated as a requirement that the bottom slope, > : (A-57) satisfy the inequality, tan6 ae AG] : (A-58) 2. Long Waves in a Porous Mediun. To derive the equations governing the propagation of waves in a porous medium we consider an element as sketched in Figure A-2. In a porous medium of porosity, n, the discharge per unit area in say the x-direction is given U = nU, where U, is the seepage velocity, i.e., the actual mean velocity of the pore fluid, and U is termed the discharge velocity. With this definition it is seen that the discharge velocity, (U,W), for an incompressible fluid and medium must satisfy the continuity equation: Spy Ea aT (es (A-59) or for a homogeneous medium, aU OW. ax 27 = 0 . (A-60) 115 | | Us ,U | | Figure A-2. Definition sketch. To derive the horizontal momentum equation we consider the elementary control volume indicated in Figure A-2. The momentum equation states that the sum of the forces acting on the fluid within this element must equal the rate of increase in momentum plus the net momentum flux out of this control volume. Of forces acting on the fluid within the element we have the pressure force, gece SP gxe2 (A-61) Pp OX ; and a force resisting the fluid motion within the porous medium. In an unsteady motion this force will consist of a drag force, 5Fg, and an inertia force, 6Fy. For steady flow the drag force component is expressed as a volume force which may be taken as oF y = -o(a + BY u + W-)U OXOZ) © (A-62) where the hydraulic properties of the porous medium are given by the coefficients a and 8. The coefficient a expresses the laminar flow resistance and 8 is associated with the turbulent flow resistance. The inertial force, 6Fy;, is associated with the fluid acceleration. The fluid velocity, as seen by a solid particle in the porous mediun, is the seepage velocity and by analogy with inertia forces acting on a single particle we may take 116 DU DU S S oF, =.-p (1) + «) DET OS = -p(1 + kK) pe (-n) OxXOZ, (A-63) where 64M, is the volume of solids within the control volume and xk is the added mass coefficient. Little information is available on the magnitude of k for a closely packed ensemble of irregular grains. For isolated spheres k = 0.5 may Serve as an indication of the order of magnitude. - The rate of change of momentum within the fixed control volume is given by a aU (@¥U) =n a 6x6z , (A-64) and the momentum flux out of the control volume is given by bs 3 3 iF ME = p (5 tU nu.) + Zz tU nw} ) ox62 = ou. ou, en (U. ee aP We Sa 6xdz r) (A-65) where the continuity equation has been invoked. Now formulating the momentum equation, a + SF y + SFL ee My , (A-66) and introducing equations (A-61 through A-65) the horizontal momentum equation is obtained as DU pd +« (1 -n)) p= - OD otaee AVC Wee (A-67) aX which corresponds to the equation given by Polubarinova-Kochina (1962) for unsteady flow in a porous medium with the resistance term expressed by a Dupuit-Forchheimer type of formula. The coefficient attached to the acceleration term is somewhat different from the expression given by Sollitt and Cross (1972), but is believed to be correct in the present derivation. it A similar expression may be derived by considering the vertical momentum thus leading to the momentum equations for a homogeneous, imcompressible mediun, DU eS ROIS siege oD -oe{n(a + BY u- + W-)} U (A- 68) Dt OX Ss and DW 0S pee = - SB - vg -pfn(a + by Ue + W)} Woe (A-69) where Sade = om) a (A-70) is expected to take on values within the range 1< S< 1.5. The similarity of these governing equations and those given for waves over a rough bottom (eqs. A-1 through A-3) should be noted. Further development follows that given in Appendix A.1 and the boundary conditions to be satisfied are those previously given but now expressed in terms of the seepage velocity. Hence, integration of the continuity equation over depth gives: on n ots (Uth + n)} = 0 ; (az and with the assumption of long waves, equation (A-69) yields a hydrostatic pressure distribution, p= pein Zz) le (A-72) The horizontal momentum equation then becomes 1 DU n S pg Pe an (a + 8 |u| )U 5 (A-73) where U should be interpreted as the depth averaged discharge velocity. a. Linearization and Solution Technique. The linearized version of the governing equations may be taken as an P) Wee ta (HU), = 10 3 (A-74) 118 and S oy an mn) D Sit Ct Seas ag Oe gO (A-75) where the flow resistance term has been linearized by introducing the dimensionless factor, f, defined by f ~ = a el (A-76) in which w is the radian frequency of the wave motion, which is assumed periodic. The solution technique is that used in the solution of the linearized equations governing the propagation of long waves over a rough bottom, i.e., we take Real {z en vey =} iT} U = Real {u ey (A-77) For the case of constant depth h = ho introducing equation (A-77) in the governing equations leads to Bion) eae BONE he GICs E u io(S-if) ox ° eS) and the equation governing the amplitude function 9° ie = + (S-if)¢ = 0 (A-79) ene dX fe) Both of these equations are seen to be similar to those discussed in Appendix A.l.a and the results obtained there are readily generalized to give the solution for a progressive wave in a porous medium. The general solution consisting of waves propagating in the positive and negative x-direction is found to be given by tc=ae + ae ; (A-80) 119 ok. wlySaueeek Seer ee (A-81) and [Hr cane e = (A-82) As was the case in the solution for long waves over a rough bottom, the above solution is formal only, since the appropriate value to be assigned to the linearized resistance coefficient f has not been determined. To obtain the appropriate value for f, Lorentz' principle is used with the rate of dissipation per unit volume given by Eee OF (A-83) Thus, we obtain: = 0a Bia (A-84) aie n where the spatial average is to be taken over the volume occupied by the fluid. As a simple example we consider the velocity given by U = Uy cos wt , (A-85) for which equation (A-84) yields: eae 228 (A-86) w 3m fo) ; In the discussion of long waves propagating over a rough bottom the result corresponding to equation (A-86) was given by equation (A-46). However, although the friction factor, f,,, in equation (A-46) may be considered known by applying Jonsson's (1966) empirical results, we have not established a similar empirical relationship for the hydraulic properties, a and 8, of a porous medium. 120 Engelund (1953) reviewed a number of empirical formulas for the hydraulic properties of porous media consisting of sand. He recommended the following empirical relationships: 3 2 (1-n) v Ee a (A-87) n d and a 1l-n 1 B = By ei id > (A-88) in which d is the grain size of the porous material, v is the kinematic viscosity of the pore fluid, and dp) and 859 are constants whose values have been found to vary within the range, 780 < at Ss 1,500 or greater 158° < Bo < 3.6 or greater . (A-89) The coefficient a, which depends on the fluid viscosity, is expressing the Darcy-type resistance associated with laminar flow of the pore fluid. The flow resistance associated with the coefficient 6 is the velocity square-type normally associated with turbulent flow. From equation (A-86) it is seen that we may take n 37 a wD uae! + a BU, , (A-90) in anticipation of the domination of turbulent resistance in prototype flow of water through breakwaters. Written in this form the degree to which laminar resistance affects the results is given by Qa 31 a 31 2 50! v (A-91) where equations (A-87) and (A-88) have been introduced. This expression may be written as: (A-92) 12 in which R, is a particle Reynolds number, dU, Ry a ia 6 (A-93) and ‘a 4 Re ean ahs OTe (A-94) - 8 25 By is a critical Reynolds number whose value is of the order 70 if the mean values of the ranges indicated by equation (A-89) are taken. The fact that the. term (1-n)?n varies only slightly for. 0.4 < n< 0.5, hasvbeen used in establishing equation (A-94). Thus, for values of Rg >> R, the flow resistance is purely turbulent and with 8 related to the physical characteristics of the porous material through equation (A-88) with 8, taken as 2.7, the problem of determining f may be considered resolved. b. Limitation of the Solution. The basic assumption made in the derivation of the equations for the propagation of long waves in a porous medium was that the waves be long relative to the depth. Whereas this assumption in the context of long waves over a rough bottom was equivalent to the negligible effect of vertical fluid accelerations, the important term to consider in the context of waves propagating in a porous medium is the term expressing the resistance to vertical flow in equation (A-69). For the simple case of a progressive wave in a porous medium the solution for the horizontal velocity component given by equation (A-78) is Ge ieee vee eae fue ah ea ts One E EE (A-95) w, = -i——°_* (1+ ~ aoe (A-96) oO and the vertical resistance will contribute to the pressure distribution by an amount 122 Zi . Ww 5 dl Ziad 2 -ikx | of a te diz =a! x(1 + ae of g(k ho) ae 7 (A-97) ) 10) which is to be compared with the term retained, i.e., pg a. Thus, for the term given by equation (A-97) to be negligible we must require that 1 2 x £(k,h)) Sh Pes (A-98) which for values of f greater than unity is a more severe requirement than that given by equation (A-53). Thus, for values of kyh, ¥v 0.5 and f ~ 4, which are reasonable values, the inequality (eq. A-98) is only approximately satisfied. 123 APPENDIX B EXPERIMENTAL DATA This Appendix presents the experimental data obtained under the present research program. Tables B-1 and B-2 present the experimental data used in establishing the empirical relationship for the wave friction factor, Figure 22, and equations (124) and (125). Column 1 identifies the experimental run. Columns 2 and 3 give the slope roughness, d, and the period of the wavemaker, T, respectively. From the experimental data, when analyzed as described in Section III.3.a and Appendix C, the incident wave height, H;, and the reflection coefficient, R,, are obtained as listed in columns 4 and 7, respectively. The observed runup on the rough slope is listed in colum 6, and the value of the horizontal extent of the slope relative to the incident wavelength obtained from linear wave theory, £./L, is given in column 5. Thus, the first seven columns constitute experimental data. With the values of £./L and R, from columns 5 and 7 the corresponding value of ¢, the slope friction angle, is obtained from Figure 15. The value of ¢, listed in column 8, along with the value of 2) Experimental results for 1:3.0 slope, hy Nene ee ee oh ooo Neneh oho nono hohe on oon ono nono hon ooo oho kook ono—onolo) -28 ao - 36 - 36 736 Ske} -28 -28 ~28 -28 Ssh Gehl Bey Bene - 36 +36 - 36 - 36 +28 ay2;) +28 -28 -31 sep phil wictil - 36 - 36 - 36 . 36 R u 6 A H, i 1.70 1.85 2.08 2.06 2.21 2.32 2.50 alpsaks} 1.99 2.08 2.42 1.72 Ledi7, 1.83 15:93 1.51 1.68 uly /P4 72: 1.13 1729 1332 1.34 alee u 1.46 1.47 aor Ak 1.14 23) 125; 133) O72 0.90 0.99 0.06 1.03 1.34 1739) 1.54 0.95 ately) 1.22 1.34 0.98 1.06 aE s(0}2) Lel8 1.34 de D2 Ve75: 2.12 1.10 Ake 1.60 1.83 1.24 V2 37 1.60 Ne eee eee eee on onokon oho nono hohokolo ooo No loo nonololoMolelosololonelonolo komo) ooo ors NN NW MUA NOOO NWUWWDUDOOF UnNUnUNUKUUH NWODDE-E £& f+fu NNN uuu DBWEAO ofS woL fF&Hrekst rPaAMnwWwo fru Dwn FUuUnfS NNWN FworeuunFrfe OFUuUWDOWwWan Senou wWwuenr worst Wwrwoow 8 9 10 alay 12 13 14 A o Ro Be tan2o fe tanBed A (deg) 3 2.28 1.30 0.044 O10 ag) pile) L529) 0.067 0.14 2.0 2S) 1.30 0.070 0.14 ZS) 2.14 1.62 0.081 0.09 2.0 2.18 1.62 0.070 Ora Thy! 2.22 163 0.060 0.08 1.8 23:20 1.62 0.063 Oet2 1.8 2.46 1.88 0.062 0.05 20) 235, 1.94 0.069 0.06 Caryal 2.22 2.03 0.073 0.08 US9 2.40 sigeul 0.068 0.09 37 12933) 1.28 0.129 02/23) 105350 0.55 (0) 3:9 1.90 1828 0.135 0.29 8.89 0.56 0 4.1 1.87 1.24 0.145 0:26 10.69 0.54 10) 4.5 83) 125 0.158 Os22 | 14)522 0.51 0) 4.6 1.83 1.59 0.162 0.27 9.09 0.54 0 4.6 1.84 17359 0.161 0.21 11.44 0.52 (0) 4.2 1.89 1.58 0.147 0.34 5.60 0.57 0 4.7 1.82 T59 0.164 0.18 13.49 0.50 (0) Pins} 5 e9/ 1.86 0.186 0.29 tensa 0.44 0 Sie) 16 1.86 0.193 O35) Tigh 0.54 ie} 6.4 63) ei 0.228 0.38 95725 0.52 0 6.6 Fo?! 1 .5)7 0.233 0.30 11.74 0.48 10) 4.8 1.80 1.26 0.168 0.19 8223) 0.45 0 4.0 1.88 1.28 0.141 0.34 3.86 0.53 0 4.7 1.81 20 0.166 0.28 Sp54) 0.50 0 4.4 1.84 1.28 0.154 0.36 4.04 0.52 0 519) 1.69 dead (halal 0.37 4.49 0.50 0 6.8 1.59 153) 0.241 0329 6.75 0.46 0 hey) WW T2 1.55 0.200 0.25 6.19 0.47 0 5.8 Dey7a 1.55 0.204 0.23 6.75 0.47 0 6.1 147.0 1.79 0.215 0.41 3.54 0.48 0 6.8 1.58 1.81 0.243 O27) 5192 0.45 0 6.7 L59 1.79 O4239, 0.22 UiaiZak 0.44 0 7.9 1.47 73 0.282 0.27 7i(30) 0.43 0 4.6 1,.'82 12.8) 0.165 0739) 2.67 0.50 0 537 1.69 aaa 0.200 0225 5.03 0.43 0 Devil 1.76 1.29 OSLZ9 0.19 5.74 0.43 (0) 5.0 Ls 7 W27, 0.176 0.31 3.61 0.46 (0) Sal 1.78 Lod, 0.178 0.32 2.82 0.49 0 Did! LereyArp 157 0.182 On31 3,03) 0.48 0 (fess) 1.54, 1.52 0.259 0:33: 4.05 0.44 0 6.2 1.68 1.50 0.219 0. 37 35 0.47 10) (a¥/ 1.60 L748 0.239 0.33 3).130 0.45 0 5.4 nbew is) 1.89 0.189 O5.277, 3.02 0.46 10} 6.0 1269 63 0.218 0.48 1.94 0.49 0 6.2 1.66 1.83 0.219 0.45 2.15 0.48 0 SIHe) ao72 1.28 0.194 O27: 3.41 0.43 (0) S86) aay Ale DESY LIL 0.196 0.22 4.75 0.41 0 5.0 179 27 0.175 0.24 2.45 0.46 (0) 4.5 1.82 1.28 0.160 0.34 93 0.48 0 BE) L52 1.50 0.267 0.34 Z\ealal 0.43 (0) 625: 633) 13:50: 02231 0.41 Died 0.46 0 LAO} nay) LoS 0.249 0.34 2.93 0.44 (0) 559, 1.69 1.54 0.209 0.46 1.76 0.48 (0) 8.3 1.42 Ls\67; 0.297 0.41 2.61 0.42 0 8.9 1.36 1.65 0.320 0.35 31.28: 0.41 0 10.4 2.2. 1.48 Ow512 0.56 SCE) 0.40 9 730 1250 nEEAyAS} 0.271 0.45 2) 0.44 0 -57 D7, Bo -57 ~54 ~54 ~54 ~54 -58 -58 -58 9.8. -50 -50 - 50 - 50 -49 -49 -49 =49 -47 -47 +47 -47 -47 -47 47 -47 -47 +47 -47 -47 -44 -44 -44 -44 +45 -45 +45 +45 -45 «45 -45 -45 =43 =43 -43 -43 Table, B-3.. ‘Experimental; results, for. 1-:2..5) slope, hy = 1 foot. 1 2 3 4 5 6 7 8 d aL H R fal R R R a u lol m P ps Run : Number (in) (sec) (£t) 175 0.0 220 0.07 1.64 0.84 176 0.0 2.0 0.06 1.39 0.83 177 0.0 2.0 0.05 1.56 0.82 178 0.0 2.0 0.08 2.02 0.89 179 0.0 1.8 0.08 1. 82 0.85 180 0.0 ies} 0.07 1.64 0.82 181 0.0 1.8 0.06 1.48 0.85 182 0.0 1.8 0.05 1.46 0.86 183 0.0 156 0.09 ibs Sal 0.87 184 0.0 6 0.06 1.39 0.85 185 0.0 1.6 0.07 al sy4 0.86 186 0.5 2.0 0.09 1.86 0.65 0.65 0.66 187 0.5 2.0 0.07 2.03 0. 67 0.67 0.66 188 0.5 2.0 0.06 73. 0.70 0.68 0.66 189 0.5 220 0.04 oS On 71: (0)5 Zak 0.66 190 10 210 0.04 40 0.65 0.68 0.61 191 10 2.0 0.06 Les} 0.64 0.64 0.61 192 iO) 2.0 0.08 AL tes) 0.64 0.63 0.61 193 1.0 20) 0.07 W579 0.64 0.64 0.61 194 5 210) 0.05 1.66 0.63 0.64 0.57 195 Ale: 2.0 0.04 143 69 0.66 0.57 196 20 V(0) 0.06 lays} 0.59 0.60 0.54 197 2.0 2.0 0.04 ibs 7/0) 0.61 0.64 0.54 198 0.5 158 0.03 1.40 0.68 0.67 0.59 199 0.5 alate} 0.06 ays} 0.64 0.62 0.59 200 0.5 tg 0.07 1.79 0.63 0.61 0.59 201 0.5 1.8 0.08 1.96 0.63 0.59 0.59 202 1.0 abate} 0.08 1.95 0.58 0.54 0.52 203 SUS(0) 158 0307 1.79 0.59 0.56 0.52 204 1.0 18 0.05 1.88 0.60 0.59 0.52 205 alto) 1.8 0.03 Ql 0.64 0.62 OF S2 206 nls} 18 (@)ealal LZ: 0.56 0.48 0.47 207 nes) 1.8 0.08 15516 0.57 0.52 0.47 208 PS 18 0.06 a er hs} 0.59 0.55 0.47 209 ISS) 18: 0.10 iS y/7/ 0.58 0.48 0.47 210 2.0 a8 0.11 Hp? 0.52 0.45 0.44 21) 2.0 1.8 0.09 159) 0553) 0.48 0.44 212 2.0 Les 0.07 203) 0.54 0.50 0.44 23 2.0 158 0.03 1.39 O57 0.58 0.44 214 0.5 1.6 O17, 1.59 0.50 0.48 0.54 215 0.5 1.6 0.15 1.67 0.52 0.49 0.54 216 0.5 1.6 0.10 etey/ 0.57 O52 0.54 217 0.5 136 0.06 alee 0.60 0.56 0.54 218 1.0 16 Onl 7, 1.84 0.47 0.44 0.49 219 1.10) IE) 0.42 ales 0.53 0.47 0.49 220 10 16 0.09 1.62 0.56 0.49 0.49 221 1.0 16 0.06 Lew73 0.57 O52 0.49 222 5 16 0.15 1.46 0.48 0.43 0.45 223 1S) 1.6 0.12 1.65 0.50 0.44 0.45 224 15 1.6 0.09 ays} 0.52 0.47 0.45 22'5 1.5 06) 0.05 167; 0.54 0.53 0.45 226 2.0 1.6 0.10 1.46 0.48 0.45 0.43 227] 2.0 LAS 9.08 ee 0.49 0.47 0.43 228 2.0 6 0.06 165 Ou5a! 0.49 0.43 229 22.0) iba) 0.04 ao On52 0.52 0.43 27 Table B-4. Experimental results for 1:1.5 slope, hy = 1 foot. al 2 3 4 5 6 i 8 d T H, R = aoe R R R als u H, m Pp ps Run Number (in) (sec) (£t) 120 0.0 2.0 0.13 185, 0.82 Alyat 0.0 210) 0.09 1.29 0.91 122 0.0 20 0.12 1.35 0.87 DNS 0.0 8 0.05 1.74 0.87 124 0.0 1.8 0.06 1.74 0.85 125 0.0 alg) 0.06 201 0.85 126 0.0 alits} 0.06 2130 0.86 127 0.0 1.6 0.03 Dias) 0.87 128 0.0 AL) 0.05 2.08 0.81 129 0.0 IER) 0.06 2 Si: 0.89 130 O25 2.0 0.04 56) 0.87 0.95 0.93 ALS WE On 750) 1.06 1.39 0.86 0.95 0.93 132 OS 210 0.06 1339 0.86 0.95 0.93 133 OS 230 0.07 dG Bis 0.86 O!/95) 0.93 134 110 200 0.11 43 0.81 0.93 0.92 135 0 2.0 0.14 1.33 0. 80 0.92 0.92 136 10 240 0.10 2.46 0.81 0.93 0.92 137 Abe) 2)'0 0.05 Ae e1IG 0.84 0.94 0.87 138 a5 20 0.07 P32 0.79 0.93 0.87 139 aS 210 0.07 1.49 O81 0.93 0.87 140 alse) 2.0 0.09 29 0.81 0.93 0.87 141 2.0 230 Ok VL 2165) 0.83 0.92 0. 82 142 20) 2\.0 0.09 1.29 0.84 0.92 0.82 143 220 2.0 0.07 1.49 0.85 0.93 0. 82 144 20 2.0 0.05 0.43 On 85: 0.93 0.82 145 0.5 1S 0.04 1256 0.81 0.92 0.89 146 0.5 ies! 0.02 56 0.84 (02-93 0.89 R47 0.5 18 0.04 1.56 0.80 0.92 0.89 148 0.5 1 8 0.05 1.46 0.81 0.92 0.89 149 1.0 16 0.08 30) 0.80 0.89 0.91 150 1.0 18) 0.07 24 0.78 0.89 0.91 V5 185 (0) 1.8 0.07 it Dai 0.79 0.89 0.91 152 eS 1.8 0.05 OS 0.87 0.89 0.86 153 dae) alate! 0.08 0.97 0.78 0.89 0. 86 154 TRS) aS 0.10 0.87 OLA? 0.88 0. 86 155 ales) Pas 0.10 1.04 0.78 0.88 0. 86 156 220) 148 One 0.95 O272 0.87 0.79 R57 2.20 1.8 0.08 1.08 0. 74 0.88 0.79 158 20 1.8 O07 7h 0.74 0.89 0.79 159 2.0 a ts 0.06 4: 0.74 0.89 0.79 160 0.5 16 0.07 1.49 0.79 0.86 0.84 161 0.5 1e6 0.06 39 On77 0.87 0.84 162 0.5 36) 0.04 a eye) 0.79 0.88 0.84 163 0.5 1.6 0.03 TG 0.80 0.88 0.84 164 0.0 10.6 Oa 0.95 0.83 0.83 0.90 165 1.0 16 0.09 93) O73 0.84 0.90 166 10 16 0.10 153 0.73 0.83 0.90 167 a sO) ANE AS) Os2 15S 0.85 0.83 0.90 168 ni) 6 0.04 Sta 0.73 0. 86 0.85 169 1} 6 0.05 2.90 0.74 O85 0.85 170 Le 6 0.07 2.68 0.78 0.84 0.85 171 2.0) ILA (0) Ss} 2.30 0.69 0.79 0.78 ae 26.0) 16, 0.12 2.08 0.71 0.80 0.78 173 Zaid 1.6 0.08 2.08 On 71 0.80 0.78 E74 220 1.6 0.06 2.08 0.68 0.83 0.78 230 10; 72510) Oz 1.91 0.88 0.92 0.90 128 APPENDIX C DETERMINATION OF REFLECTION COEFFICIENTS During the preliminary experimental runs performed to determine the reflection coefficients of steep, rough slopes it was observed that an appreciable effect of second or higher harmonic motions was present in the wave flume. It was also found that if one sought out the locations along the constant depth part of the flume where the wave height, i.e., distance between crest and trough, was maximum and minimum, respectively, the resulting estimate of the reflection coefficient could vary as much as from 0.45 to 0.75 depending on the choice of maximum and minimum wave height. With the intended use of the experiments such a variation of the experimentally determined reflection coefficient is clearly undesirable. The theoretical foundation for using the formula for the experimental determination of the reflection coefficient, Gene . nn Sie Hie Seine poh re max min is based on an analysis assuming linear waves, i.e., the motion consists of purely sinusoidal waves of one frequency. Hence, the appearance of higher harmonics is an indication of the inapplicability of equation (C-1) for the prediction of reflection coefficients. Furthermore, the physical concept of a reflection coefficient really makes sense only if super- position, i.e., linear waves, may be assumed. If the motion in the wave flume was purely sinusoidal, the surface variation should at any point vary sinusoidally with a period, T, equal to that of the wavemaker. Since this was not the case it was decided to extract from the measured surface variation at each station the amplitude of the motion having a period equal to that of the wavemaker. This amplitude of the first harmonic is the one which, according to linear theory, should vary in such a manner that equation (C-1) provides a determination of the reflection coefficient, R. The experimental procedure used was the following. For a particular experimental run the wave generator was started from rest. The wave motion in the flume was allowed to reach a quasi-steady state in which the motion at any point along the flume was periodic, i.e., the motion, although not purely sinusoidal, repeated itself with a period equal to that of the wavemaker. It generally took 2 to 3 minutes for this quasi- Steady state to be reached in the present experimental setup. As mentioned in Section III of the report it was not possible to attain this quasi-steady state for large amplitude incident waves, which limited the test conditions for which reflection coefficients could be determined. ae) After reaching the quasi-steady state the wave gage was positioned at the first station. The motion was observed on the paper tape of the Sanborn recorder and was visually determined to be periodic. With the maximum paper speed, 100 millimeters per second, three to four wave periods were recorded. The gage was moved to the next location, 10 centimeters away, and the procedure was repeated. This was done over a distance of approximately one wavelength of the incident waves so that at least two maxima and minima were recorded. Of the three to four wave periods recorded at a given station one was chosen for analysis. This variation of the free surface during one wave period was digitized manually at intervals of 1/20 the wave period. Since the motion is assumed periodic the two end points of the digitized data should be identical and the mean value of the two end ’ points was chosen whenever they were not exactly the same. This way 20 equally spaced (in time) values of the surface elevation were obtained and these values were used as input to a simple computer program which performed a Fourier series analysis of the data and gave the amplitude of the first harmonic motion as output. When realizing that each experimental run required the measurement of the wave motion at some 30 stations it is quite obvious that the manual procedure of digitizing the wave records means that it is an extremely time-consuming effort to obtain the reflection coefficient. However, the end result (Fig. 20) rewards the effort in producing reflection coefficients which vary only within + 0.02 with choice of node and antinode. The experimental data listed in Appendix B (Tables B-2 and B-4) were all obtained and analyzed in this manner. Modifications of the Hewlett-Packard Computer System at the Ralph M. Parsons Laboratory made it possible to interface this computer with the wave tank experiments. This enabled the tedious manual reduction of the data to be circumvented and to feed the experimental data directly into the computer. A multifunction meter triggered the computer to start taking data by imposing a high voltage. After being triggered the computer starts taking data at the rate of 14.5 readings per second, i.e., a reading per At seconds, where: At = 0.069 seconds. (C=2)) The computer program was designed to take a total of 50 readings, i.e., cover a period of approximately 3.5 seconds, thus ensuring that an entire wave period is recorded. Of these 50 equally spaced values the first 15 are discarded to avoid transient effects and from the 16 reading on, the next D1l+1l values are adopted for computations where D1 is the integer most closely approximating (C-3) 130 These Dl+l1 values are treated as previously described for the manually digitized data for which Dl=20. All this is done internally in the computer and the output is the amplitude of the first harmonic motion at consecutive measurement stations. A search routine was also included in the computer program so that the maximum and minimum values of the wave amplitude were determined and the resulting estimates of the reflection coefficient were printed out for each experimental run. Although the computerized procedure was thoroughly checked against the manual procedure before the former was adopted for the experiments listed in Appendix B, Tables B-1 and B-3, the procedure of obtaining a paper tape wave record was continued to avoid possible loss of experimental data. An example of the added accuracy involved when a more exact method is inacted is seen by examining the example described in Section III.3.b, Figure 20. From Figure 20, as previously noted, the raw data from the experiment (the open circles) do not reproduce a well-behaved wave ampli- tude variation. Through the use of the Fourier series computer program, the raw data were corrected, i.e., only the first-order wave amplitude was retained, and the corrected data are seen in Figure 20 as the solid circles. It is observed from studying the corrected data that the minimum amplitude locations are not precise and one must fit a theoretical curve to the corrected data to determine the minimum wave amplitude and the resulting reflection coefficient. The theoretical curve with R=0.88 appears to fit the data presented in Figure 20 well, but as it will be demonstrated, the wave amplitudes immediately surrounding a minimum have to be determined at a much closer spacing than used in the experiments presented in Figure 20 if the curve-fitting procedure is to be eliminated. Figure C-l is a graphical representation of equation (123) close to a node location for various reflection coefficients. It is clearly seen that, as the spacing between measurements becomes larger, one is able to have relatively large errors in obtaining the minimum location especially for high reflection coefficients. For example, a 4-inch (10 centimeters) spacing corresponds to a measurement interval, Ax/L, of 0.03 for the 2- second wave in the present experiment. The maximum deviation would occur when the measurement locations were equally spaced around the minimun, i.e., Ax/L equal to 0.015 on either side of the minimum. If one had an actual reflection coefficient of R=0.88 one would obtain a value of a/anax equal to 0.115 for a maximum displacement from the actual minimum. Therefore,if one assumed that the reading of a/ap,, = 0.115 was the correct minimum value, then the reflection coefficient would be determined from Figure C-1 R=0.79 which is approximately a 10-percent error. Similarly, if one has an actual reflection coefficient of R=0.60 and collected data at the maximum deviation locations, the error incurred would only be of the order 3 percent. It is apparent that the errorsrors are most prevalent when one has high reflection coefficients. The computer program described previously allows one to collect a large number of data points and analyze them quickly so that the measurement interval can be reduced in the vicinity of nodes, thus resulting in smaller errors. [31 *SJUITOTFJFOOO UOTIDIOTFOI SNOTIVA LOF opou ke FO AJTUTITA oJeTpouWT oY? UT (¢FZT) UOTIeNbe wory ‘uoTIeTIeA opnyT{dwe oAeM TedTIELOOU] “1/X "T-) eainsty 32 After observing how important it is to have the minimum amplitude defined as accurately as possible, the experiment used in Section III.3.b was repeated with the measurement interyal surrounding the observed minimum reduced to Ax/L equal to 0.39 inch (1 centimeter). Figure C-2 is a plot of the raw and corrected data for measurements taken in the immediate vicinity of two node locations, L/2 apart. Two observations can be made. First, the raw data show a slight discrepancy between the minimum locations and, secondly, one cannot assume that the actual minimum, i.e., the first harmonic amplitude, will fall where the raw data minimum occurs. In order to ensure that the actual minimum loca- tion is found, one must use the smaller measurement intervals for a sufficient distance around the observed minimum to ensure the location of the actual minimum to be occupied. By substituting the results obtained from Figure C-2 into the calculations made in Section III.3.b, one will calculate an average reflection coefficient of R = 0.90 which is much larger than the reflection coefficient suggested by the raw data (R = 0.62) and slightly higher than obtained from the corrected data (R ~ 0.88) with a measurement interval of 4 inches (10 centimeters) when the best fit value of R is chosen. The accuracy, using the more refined acquisition system, does not produce, in this case, a far superior product. To avoid the somewhat subjective and tedious curve fitting procedure, the refined system of closely spaced measurements near nodes is recommended. Figure C-2 shows how the amplitude at the node as well as the location of the minimum amplitude itself is affected by higher order wave harmonics. Through the use of a high-speed computer, the data can be quickly examined and the resolution required may easily be obtained. The computer program which was used for the computerized procedure for determining the reflection coefficient is listed on the following pages. With the preceding description of the program and the extensive use of comments in the program this should be self-explanatory. 133 ‘OZ oan3ty ut paqueseid quowtzedxe ay 02 B8uTpuodseit105 apou e Fo AYIUTOTA O2ETPOWUWT 94} UT UOTIeTIeA opnzT{dwe oAem FO ejep pazd9IL09 puke PsATasqg “/X 2G OG: Go vo Co "7-9 oansty 34 REM **x* ANALYSIS OF A WAVE PROFILE *xx REM *** BY FOURIER SERIES APPROXIMATION *xx REM THIS PROGRAM MUST BE RUN FROM A POSITION A STATION BEFORE REM A MAXIMUM AMPLITUDE. CALCULATE THE NUMBER OF STATIONS TO BF REM MEASURED (S) AND THE NUMBER OF POINTS TO BF USED IN THE REM PROGRAM DEPENDING UPON FREQUENCY (D1). SI>D14+15 REM T=PERIOD,DI=NO. OF PTS. USED,KI=ORDER SOUGHT REM Q=MAX VOLTAGE TO.TRIGGER,S=NO. OF STATIONS EXPECTED REM SI=NO. OF DATA PTS.,T5=TEST NUMBER,CI=CONVERSIONCFT.-V) DIM AC161,B(181,C(50,10),F(581,F(3,501,G(5%4,5%),R(41,Z(3] READ. 1, DI,K1>.0,S,S1,, 15,6) GOSUB 1800 PRINT PRINT “ 2K TEST NUNBER, TS 9°22 2 a ck ak ak otek *” PRINT PRINT. | JCORREGILON HACTOR =" iC) PRINT FOR Nis! To Ss CAS Cle D SF Ree Des==*O THEN 22 GOTO 19 CALL Clg, F) Le De<= 1° THEN 25 GOTO 22 BOR MANZS le" LOR Sil GALEN CI, Dh) LET G{N1,N2)=N NEXT Ne HOR X=ileshO Dil-hd En LXE GONE X15) NEXT X GOSUB 75 NEXT NI GOSUB 380A GOSUB 7285 STOP REM SUBPROGRAM TO CALCULATE MAXIMUM AMPLITUDE USING FOURIER REM SERIES AT EACH STATION. FOR X=1 TO D1I+l LET RE Q= GUN xs NEXT X LET E=D1+1 LET G=K1l EEE BOSCH ten eID 72!] BET GE) = Fl) LET A4=1/2*F(1 J Leh Hea Abil FOR] J22770 (DA LET A4=A4+F(J] NEXT J LET A=1/0€2*3.1416)*H* (A441 /2*F( DI+1 1) i395 124 134 135 14a 152 160 165 170 189 190 195 200 205 218 230 240 2508 264 262 263 264 265 266 267 270 288 281 296 380 381 383 384 398 395 426 414 420 436 440 445 456 455 469 465 470 480 4908 495 500 5@5 510 FOR K=1 TO KI ah Ths LET AG= CRE) I*¥COSCT ID). 5 LET BO=CF(1 JK¥SINCT1I))*.5 FOR N=2 TO D1 LET X= (€(€2*3.1415)*(N=1)*H/T)*K LET A@= A@+F(N)*COSCX) LET B@=BO+F(NJ*SINCX) NEXT N LET T9=(€2*3.1416)*DI*H/T*K LET A@= AO+(F(D1+1 ]* COS(T9))*.5 LET B@=BO@+CF(D14+1 JX SINCTS))*.5 LET ACK )=2/D1*A@ LET BCX ]=2/D1* Ba FOR M=1 TOE LET T5=(3.1416*2)* (CC M-1)*H/T)*K LET C(UM,K J=At+AC(K ]*COSCTS)+BIK )XSINCTS) NEXT M LET Z(KJ=SQRC ACK J*¥ ACK IJ+B(K )*BCK ]) PRINT “STATION NUMBER:”, NI PRINT “H= ",H, SECONDS” LET ZtKI=Z0UKI7C! PRINT ™ MAXIMUM AMPLITUDE =",Z(K),"FT. RORe Ky =ci1K EET Stk, NP I=EZCCK) NEXT K PRINT PRINT RETURN REM SUBPROGRAM TO SEEK MAXIMUM AND MINIMUM LOCATIONS REM AND CALCULATES REFLECTION COEFFICIENTS AND INCIDENT REM WAVE HEIGHTS. CAN ONLY BE USED IF ATTENUATOR IS LEFT REM AT ONE SETTING THROUGHOUT THE EXPERIMENT. FOR K=1 TO K1 are (len LET AS=E(K,NI ] LET NI=N1+1 IF Nl>S THEN 599 IF E(K,NI)] >= AS THEN 49890 LET A6=A5 LET A5S=E(K,NI1 ) LET NI=NI+! IF NIl>S THEN 606 IF ECK,NI J <= AS THEN 445 LET R(1 J= CAG= A5) /CA6+A5 ) EET “Av=A5 LET AS=ELK,N! J LET NI=N141 IF NI>S THEN 614 IF E(K,N1)] >= AS THEN 482 LET R(2 J= (A5-AT7) /CA5+A7) LET A8&=A5 Peis) 520 538 555 548 545 5508 560 578 586 590 395 596 2199 600 662 693 695 648 618 611 612 613 614 615 616 617 618 620 621 622 6308 648 650 660 661 662 663 664 665 666 667 668 669 673 675 677 689 685 699 700 7102 LET A5=E(K,NI J LET NI=NI+1 IF Nl>S THEN 6298 IF E(K,NI1 ] <= AS THEN 520 LET R(3 = CA8- A5) /(A8+A5) LET A9=A5 LET A5=E[K,N1] LET NI=NI+1 IF Nl>S THEN 595 IF E(K,NI J >= A5 THEN 568 LET R(4)=CA5-A9)/(A5+A9) GOTO 640 LET A6=9 LET A7=@ LET R{1 1-0 LET R(21]=A LET A8=0@ ERRaR 1S 1228 LET A9=@ LET R[4]=94 GOTO 649 LET eRtZz=o LET A9=@ LET R(31=9 LET A8=2 LET R(41=@ GOTO 649 LET A9=0 LET R(3)=2 LET R{[4]=@ LET R(4]=@8 LET R=CRC1 HR(2)+R(3J+Rl41) 7/4 LET H5= (CA5+A6+A8) /3+CAT+A9) /2 PRINT ” PRINT A6 PRINT “™ PRINT PROENT «% PRINT A8 PRINT ™ PRINT AS PRINT ” PRINT AS PRINT PRINT “™ PRINT PRINT “™ PRINT PRINT NEXT K RETURN R=; yk INCIDENT WAVE [Sir REFLECTION COEFFICIENT:” "SRC1) Geta grt Se hlal HEIGHT=",H5," FT." 795 REM SUBPROGRAM TO PRINT OUT DATA USED IN THE CALCULATION 706 REM OF THE MAXIMUM WAVE AMPLITUDES AT EACH STATION. TO 2EOR Ye Loe s 715 PRINT 724 PRINT “ STATION NUMBER:"”,Y 7130 FOR Yi=t5 TO V5+D1 732 DEF FNRCX)= INTC X*15C30+.5)/19400 740 PRINT FNRCG(Y,Y1)), 75@ NEXT YI 760 NEXT Y 773 RETURN 8ae@ STOP 1830 REM SUBPROGRAM TO CALCULATE CONVERSION FACTOR FROM VOLTS 1331 REM TO FEET. START AT STILL WATER AND INCREASE THE DEPTH 1@02 REM OF THE PROBES BY @.05 FEET. 1018 LET Y4=9 1014 LET Y3=6 1216 FOR Nt=!1 TO 18 1018 CALL (1,D,F) 1926 IF D >= Q THEN 1642 1938 GOTO 1018 1946 CALL (1,D,F) 1050 IF D <= Q THEN 19728 1068 GOTO 1046 1473 FOR Ne=1 TO 5@ 1086 CALL (1,D,F) 1098 LET G{NI,N23=D 1108 NEXT N2 1165 LET Y=0 1110 FOR X=15 TO 50 1126 LET Y=Y+G({NI,X) 1136 NEXT X 1148 LET Y3=Y/36+Y3 1145 IF Nl=! 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