WHOL DOCUMENT COLLECTION (Ad-AGT 828) A Method for Estimating Wind-Wave Growth and Decay in Shallow Water With High Values of Bottom Friction by Frederick E. Camfield COASTAL ENGINEERING TECHNICAL AID NO. 77-6 OCTOBER 1977 U.S. ARMY, CORPS OF ENGINEERS COASTAL ENGINEERING RESEARCH CENTER Kingman Building } Fort Belvoir, Va. 22060 Reprint or republication of any of this material shall give appropriate credit to the U.S. Army Coastal Engineering Research Center. Limited free distribution within the United States of single copies of this publication has been made by this Center. Additional copies are available from: National Technical Information Service ATTN: Operations Division 5285 Port Royal Road Springfield, Virginia 22151 The findings in this report are not to be construed as an official Department of the Army position unless so designated by other authorized documents. AMI WHO] DOCUMENT COLLECTION {I ) ij OL ny o09N30 UNCLASSIFIED SECURITY CLASSIFICATION OF THIS PAGE (When Data Entered) READ INSTRUCTIONS REPORT DOCUMENTATION PAGE BEFORE COMPLETING FORM 1. REPORT NUMBER 2. GOVT ACCESSION NO, 3. RECIPIENT'S CATALOG NUMBER CETA 77-6 4. TITLE (and Subtitle) 5. TYPE OF REPORT & PERIOD COVERED A METHOD FOR ESTIMATING WIND-WAVE GROWTH AND Coastal Engineering DECAY IN SHALLOW WATER WITH HIGH VALUES OF Technical Aid 7. AUTHOR(s) 8. CONTRACT OR GRANT NUMBER(s) Frederick E. Camfield 9. PERFORMING ORGANIZATION NAME AND ADDRESS 10. PROGRAM ELEMENT, PROJECT, TASK AREA & WORK UNIT NUMBERS Department of the Army Coastal Engineering Research Center (CEREN-CD) F31234 Kingman Building, Fort Belvoir, Virginia 22060 11. CONTROLLING OFFICE NAME AND ADDRESS 12. REPORT DATE Department of the Army October 1977 Coastal Engineering Research Center 13. NUMBER OF PAGES os of 27 Kingman Building, Fort Belvoir, Virginia 22060 wa? d 59 14. MONITORING AGENCY NAME & ADDRESS(i/f different from Controlling Office) 15. SECURITY CLASS. (of thie report) UNCLASSIFIED 1Sa. DECLASSIFICATION/ DOWNGRADING SCHEDULE DISTRIBUTION STATEMENT (of this Report) Approved for public release, distribution unlimited. DISTRIBUTION STATEMENT (of the abstract entered in Block 20, if different from Report) . SUPPLEMENTARY NOTES KEY WORDS (Continue on reverse side if necessary and identify by block number) melt A @ROQITY A \ _ Flew o4 Wave generation Wind waves -Y Shallow water ABSTRACT (Continue on reverse side ff neceasary and identify by block number) This report presents an approximate method for estimating wind-wave growth and decay over flooded areas where there is a major effect from bottom fric- tion because of dense vegetation. DD ANectie 1473 ~—s Epi TIon OF 1 NOV 65 1S OBSOLETE UNCLASSIFIED SECURITY CLASSIFICATION OF THIS PAGE (When Data Entered) PREFACE This report describes a method for estimating wind-wave growth and decay over flooded areas where there is a major effect from bottom fric- tion because of dense vegetation. The report was initiated in response to a request from the U.S. Army Engineer Division, Lower Mississippi Valley, New Orleans District, at the Division's 14 September 1976 Research and Development Workshop, indicating a need for technical guide- lines for predicting wind-wave generation over flooded coastal areas. The work was conducted under the coastal construction program of the U.S. Army Coastal Engineering Research Center (CERC). These technical guidelines are an extension of the procedures given in the Shore Protection Manual (SPM) (U.S. Army, Corps of Engineers, Coastal Engineering Research Center, 1975). The design curves in the SPM are limited to waves passing over a sandy bottom. The guidelines presented in this report are discussed at greater length in CERC Technical Paper No. 77-12 (Camfield, 1977). This report was prepared by Dr. Frederick E. Camfield, Hydraulic Engineer, under the general supervision of R.A. Jachowski, Chief, Coastal Design Criteria Branch. Comments on this publication are invited. Approved for publication in accordance with Public Law 166, 79th Congress, approved 31 July 1945, as supplemented by Public Law 172, 88th Congress, approved 7 November 1963. He OHN H. COUSINS Colonel, Corps of Engineers Commander and Director CONTENTS Page CONVERS TON EAGTORS US. CUSTOMARY OPMEMTRUCG (SiS) en jenn nnn SNANUIONAS, ZANID) IDISIEMONIEIEIOINS «od na Gyo 5 60 6 bh ae 0 6 8 Ss Be 6 7 I TN PRODU GI TO Nich sheath sare upregeen att tat) sso te tice ete trea eman etre Ceara 9 II PERCHPADIUSIMEN perv ienemst rete loat atelhtstli et esaaien (sata itsy i mitt eet ee ane () III WAVENGRO Wiis sate ieee aa rem rsnivech oytluteyig Yevineet as ney! aye eam Lau a) es at aN IV WAVIESDIE CAN? (aul rani mirm iets cu iis) (ar sete) meena tan cu om ahah ll tamOman Sei csp cellPe pae.G) V SAMIR IE DESIGN TINO 1G Solyeyeio ve) fo Gl ooo 0 6 @ 6) d lo oe DB FIGURES Forecasting curves for wave height, water depths constant .... 11 Forecasting curves for wave period, water depths constant .... 12 Forecasting curves for shallow-water waves (constant Cols y oye) OWE ISTE =X21 ci) Ae AW AU ATOM BUM AG MS nid Ilene MM AME OMg at ullig ig 46, ILS) Forecasting curves for shallow-water waves (constant de piehi= AOR ES Sit) Wey cum uice mrenumera hotiere a veil Siyatiss nus iP aurea N Peres ana a a a LK Forecasting curves for shallow-water waves (constant depthr= St Peete: Mel a tetas roi Tella sa ter trees emosMnc SHR roIn yetleecy al euu pane) Forecasting curves for shallow-water waves (constant GNC) ON CIOURES 5 PAUL ABELEXERE) | Wee cual CIMA ENN IG thy Toure Al gs lalye 1a" fo, oo! 6! on 64m 0 6 LO Forecasting curves for shallow-water waves (constant : col oes Meet as ye ou XeH edi odinibondicy aieintoomkos Glee lay an TeGO Golo oo le ote wo bd Forecasting curves for shallow-water waves (constant alepin SSO) skeet 5 os O16 b 80 6 6 ao 10 6 00 0 0 0 Oo 0 0 6G Forecasting curves for shallow-water waves (constant rel joel oye neo ine (ne) Cm Ame Seat a Walia lomo Gb Gl udiiasto oa pero io! le Forecasting curves for shallow-water waves (constant depth = 40 feet ioc te oleh arian nel ceuelin US Aide) oe mere ERC nuee ly CULL erro enna) Forecasting curves for shallow-water waves (constant depth 3:45.) eet yy ties ea aay ee ese an Eee eal Pe ct ee oe 12 13 14 15 CONTENTS F IGURES-Continued Forecasting curves for shallow-water waves (constant depth = 50 feet) Bottom-friction factors Decay factors . Schematic of wave decay calculations. ...... Page 22 24 25 Zul CONVERSION FACTORS, U.S. CUSTOMARY TO METRIC (SI) UNITS OF MEASUREMENT U.S. customary units of measurement used in this report can be converted to metric (SI) units as follows: —————— Multiply by To obtain inches 25.4 millimeters 2.54 centimeters square inches 6.452 square centimeters cubic inches 16. 39 cubic centimeters feet 30.48 centimeters 0.3048 meters square feet 0.0929 square meters cubic feet 0.0283 cubic meters yards 0.9144 meters square yards 0. 836 square meters cubic yards 0.7646 cubic meters miles 1.6093 kilometers square miles 259.0 hectares knots IS SOe kilometers per hour acres 0.4047 hectares foot-pounds LESSONS newton. meters millibars ONS gOme kilograms per square centimeter ounces 28.35 grams pounds 453.6 grams 0.4536 kilograms ton, long 1.0160 metric tons ton, short 0.9072 metric tons degrees (angle) 0.1745 radians Fahrenheit degrees 5/9 Celsius degrees or Kelvins! 1Po obtain Celsius (Cc) temperature readings from Fahrenheit (F) readings, WSS sommes C= (5/9) Ce 2&2). To obtain Kelvin (K) readings, use formula: K = (5/9) (F -32) + 273.15. SYMBOLS AND DEFINITIONS water depth water depth at seaward or beginning edge of segment fetch length adjusted fetch length for distance across a segment in the direction of wave motion equivalent fetch length for the initial wave at the seaward or beginning edge of the segment bottom-friction factor (Darcy-Weisbach friction factor) bottom-friction factor at seaward or beginning edge of the segment fractional growth factor of equivalent initial wave gravitational acceleration wave height decayed wave height at the end of the fetch equivalent wave height at the end of the fetch wave height at end of fetch wave height at seaward or beginning edge of the segment equivalent initial wave height maximum stable wave height maximum significant wave height which would be generated for a given windspeed and water depth decay factor decay factor when the bottom-friction factor, ff = 0.01 decay factor when the bottom-friction factor, ffs has a value different than 0.01 shoaling coefficient wavelength r Ax SYMBOLS AND DEFINITIONS --Continued fractional reduction of initial wave at the seaward edge of the segment, as compared to the maximum stable wave height wave period windspeed distance in the direction of wave motion factor for reducing fetch length to the adjusted length factor for increasing fetch length to the adjusted length = 1/a incremental change actual distance across a segment in the direction of wave travel A METHOD FOR ESTIMATING WIND-WAVE GROWTH AND DECAY IN SHALLOW WATER WITH HIGH VALUES OF BOTTOM FRICTION by Fredertck E. Camfield I. INTRODUCTION An important factor in the planning and design of works to protect upland property during periods of storm surge involves the prediction of the wave height and period that will prevail at and seaward of the pro- tective works (i.e., levee, dike, seawall, etc.) for the selected design storm. Although improvements are needed, guidelines are available for prediction of the water levels in upland areas that will result from storm surge; however, no guidelines are presently available for computing the wave attenuation for conditions when a storm-generated wave travels a distance across a shallow flooded area where the bottom characteristics in- clude vegetation which causes a moderate to high frictional stress. There- fore, it is necessary to estimate the heights and periods of waves which have traveled across a shallow flooded area. At times the initial heights and periods of the waves may increase; i.e., when the wind stress exceeds the frictional stress of the ground and vegetation underlying the shallow water. The initial wave heights may decay at other times when the fric- tional stress exceeds the wind stress. This report presents a preliminary (approximate) method for estimating the growth or decay of waves traveling through shallow water over areas with a high frictional resistance from vegetation. The method is based on previously developed equations for wave growth over areas with low bottom friction given in the Shore Protection Manual (SPM) (U.S. Army, Corps of Engineers, Coastal Engineering Research Center, 1975)!, and an equation for the decay of gravity waves over areas with a constant water depth and high bottom friction. The method uses existing shallow-water wave forecasting curves by adjusting fetch lengths to account for higher bottom friction. Simplifying assumptions are used. The water depth is assumed to have only gradual variations, and the frictional resistance is treated as bottom friction. The method presented has not been verified in the field and may not be applicable to other problems relating fric- tional resistance to wave development. Only limited data are available at this time on the effects of high values of bottom friction on wind waves. Friction factors are estimated by comparing vegetation to similar conditions in river channels and on flood plains. The effect of the vegetation on wind stress, the possible effects of motion of the vegetation, and the dense vegetation effects near the water surface which will damp out short-period waves much faster than long-period waves are not considered. The results obtained are considered ly.S. ARMY, CORPS OF ENGINEERS, COASTAL ENGINEERING RESEARCH CENTER, Shore Protectton Manual, 2d ed., Vols. I, II, and III, Stock No. 008-022- 00077-1, U.S. Government Printing Office, Washington, D.C., 1975. 9 conservative; i.e., the predicted wave heights are expected to be slightly higher than the wave heights which actually occur. Wave prediction curves for waves passing through shallow water where the bottom friction, fr» = 0.01, are shown in Figures 1 and 2. For any given windspeed, U, and water depth, d, there is a maximum (depth- limited) significant wave height, H,,, which would be generated (long dashline in Fig. 1). Where H;, the initial wave height at the seaward or beginning edge of the fetch, is less than Hg, the wave would increase in height. Where the bottom friction, fr > 0.01, the wave would not become as high as a wave traveling over a bottom where fr = 0.01, with the segment fetch distance, Ax, being the same in both cases. Therefore, an adjusted fetch, Bo < Ax, would be used to describe the wave, using Figures 1 and 2 which were developed for the case of ff = 0.01. Except for specific water depths, Figures 3 to 12 (after SPM) show the same results as Figures ivande2e Where ele eA Bln the wave would decay. As a value of fr > 0.01 would cause a wave to decay a greater amount than if it were traveling over a bottom where fr = 0.01, an adjusted fetch, Fy > Ax, would be used in this case. The details of this method are discussed by Camfield (1977)2. II. FETCH ADJUSTMENT The fetch should initially be divided into segments so that (a) Ad < 0.25 d; (1) where Ad is the change in depth over the distance across the segment in the direction of wave motion, and d; is the depth at the seaward or beginning edge of the segment; (b) Afr < 0.25 ffi (2) where Afr is the change in the bottom-friction factor over the segment distance,-and fr; is the bottom-friction factor at the beginning edge of the segment; and (c) after computation of the wave height at the end of the fetch, AH < (0)e5) Hy (3) where AH is the change in the wave height over the segment distance and H; is the wave height at the beginning edge of the segment. Each segment of the fetch can then be considered separately using the method indicated. 2CAMFIELD, F.E., "Wind-Wave Propagation over Flooded, Vegetated Land," TP 77-12, U.S. Army, Corps of Engineers, Coastal Engineering Research Centex, Komt Belvoane Vian Oct Oia 10 27/P6 jo sanjon 000'8 000'9 000'» 000'¢ 000'0! 000'2 000's *zuezsuod syzdep 103em ‘AYySTOYy 9AeM LOF soAIND ButysedeL104 “T eIn3Ty 000'2 000'! 2n7a6 008 009 002 005 00% oo€ 002 OS Ob o€¢ 02 o16er9 6 » £ 4 | 20 G0 ¥0 €00'0 S000 200°0 0100 S100 0200 0£0°0 0s0°0 0200 0010 os!0 F 002°0 00¢£°0 *quejZsuod syujidep 10,4em ‘Spotted oAemM LOF SaAINO Butysedet0y “7 oAn3ty 2/46 000'8000'9 000'» : 008 009 08 09 000'01 000'2 000'S 000° o00'2 0001 002 00500» oo¢ Te 7 OS Ob Oo€ o16e29 6 3f/P6 40 sonjoA l2 "(9903 ¢ = yidep que ZsUOdD) SeAeM I92eM-MOT[TeYS OF soAINd BurTyseder10y *¢ ern3Ty 0” 49440 ; ; 422 4a 5 JOHOR AUP) 000 001x 000 01x Ne) 000 1x Oolx o16ée@29 G bv CeeSicae Gl OOO 2 O- b CS Gace, GI ONG 29): -G b CeeGiC mete GI MGW Zo. -G b € G2 = : z t= sR ES = i == PRK po ~ LE LAr Le eis i — if ete Ease : Ae) ~ = LL L. a $7>~ orn sy | ~ V3 vy ILI = RESIS Eee Es Wale ST} $ FS WG Sy ] ~ | ¢ SS al i ee TSF SLT ; {ie alps 2S aS Se |i sy _I™ Ft -+ he $4 ~~ j Ea (eg he Pea a i Ys pee tell [eealesTss 2 a an —|— +— |S Se Se Se t| }— — lay vas re T = SAL | ~ = = ~ =—_ OL ts = SESE ; PAL ——— 08 JET “a ae ie 06 ~ | a = S= Ts 00! 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Se T = t T cs Ss B - . = 001 Se ~~ = = ~LF SS 2 = | | Ss | min Sess SS —= SG = — SS S . >> | = “+> = to = ; >= = as 2S > I Sa | [Ss aah 29S G6=1 9asGe=1 99S QO B8=1 430° 21=H 440 O1=H 8 99SOOl=1 °9SO06=1 4JO'DI=H sas Gua] WO'EI=H 4O1l=H 20 T=9.5sec OB 3 7/-3 2:00 wo wo Tv Mm (ydw) m paads pum 2| 1.5 2) 25. 3 X100 (ofter CERC,1975) Fetch (F) feet Forecasting curves for shallow-water waves (constant depth = 45 feet). Figure 11. "(2993 0S = yadep zueIsSUOD) sonem TOJEM-MOTTeYS LOF S9AINI BuTIsed9L0q “ZT eanBTy G261 ‘9439 184j0) 139 24a { Seal 000'001x 000'01 x i(a) uateg olée829 ¢ b fe G2 Gl ONGH OR 2 69 =¢ b Cd Gi Ol6 8 2 9 G b (Cis Sacer, a OG D2 OE | —SOfe= = 3 % = 3 = E = = 5 = 2 = : ae 001 i ~ ~~ T = = ; | ae NG ; | i = : ; - ' | | i r — — Ss het = = H = = S = i = ~ SSS = = = . : Bo = 0s | 29SOOI=1 99SO6=1 sesgcg=1 440 D1=H 440°21=H 9eSGOl=L 99SGCG=1 JJO9I1=H 99SQO 8=1 vasGy=l] 4JOEl=H 440°SI=H 99SOL=L 22 The bottom friction, fr, can be obtained from Figure 13 for a known type of vegetation. Ihe adjusted fetch distance, Fg, fora segment distance, Ax, is then obtained using values of the decay factor, Kr, from Figure 14 (after Bretschneider, 1954)2. An adjust- ment factor a, where H; < H is defined as sm? 1-K a Leesan (4) 1 = Kfq where Kp,gz is the decay factor for a bottom-friction factor, fr = 0.01, and Ke is the decay factor for the actual bottom-friction factor. The adjusted fetch length, Eee is then given as F, = a Ax . (5) An adjustment factor, oa ,, where Hj; > Hem, is defined as 1-K Op = ale (6) LS LSP, Ol and, for a decaying wave, 9 = Ch IE 6 (7) III. WAVE GROWTH For any given water depth, windspeed, and fetch length, a maximum Significant wave height, Hg, which would be generated can be defined from Figure 1. If the initial wave height, H;, at the seaward or beginning edge of the fetch segment is less than Hey, it is assumed that the wave will increase in height. To find the wave growth, first determine an equivalent fetch length, F,, for the initial wave (obtained directly from Fig. 1 using the given windspeed and water depth). Secondly, the adjusted fetch, Fg, is deter- mined using equations (4) and (5) and Figure 14. The total fetch is then given as | Reeeas mc Res ad (8) Re-entering Figures 1 and 2 with the fetch length, F, and the windspeed, U, and water depth, d, the wave height and period at the end of the fetch segment, Hr and T, are determined. SBRETSCHNEIDER, C.L., "Modification of Wave Height Due to Bottom Friction, Percolation, and Refraction," TM-45, U.S. Army, Corps of Engi- neers, Beach Erosion Board, Washington, D.C., Oct. 1954. 23 Friction Factor (f,) |.20E |. 00 === 0.80 0.60 =SnE Figure 13. Bottom-friction factors. 24 (pG6l ‘Japiauyrsjasg 49440) 00011008 009005 00 oo¢ 002 "sZojeF ABDAQG ‘pI omn3sty 02 90 SO v0 €0 20 t 1000 100 20°0 so'0 010 020 Ovo 09°0 080 060 S60 860 660 6660 66660 25 IV. WAVE DECAY If the initial significant wave height, at the seaward or beginning edge of a segment of fetch, exceeds the maximum significant wave height for the given water depth of the segment of fetch and the given windspeed, it may be assumed that the effects of the bottom friction will exceed the effects of the wind stress. Therefore, the wave will decay, will lose height, and over a long distance will approach a wave height equal to the maximum significant wave height. The method of determining the decayed wave height is shown in Figure 15. The following steps are used to predict the decay of a wave: (a) Determine the maximum significant wave height that would be generated for a given windspeed and water depth, assuming an unlimited fetch and using Figure 1. (b) Determine the fractional reduction, R;, repre- sented by the initial wave at the seaward edge of the segment of fetch under consideration. This is given by = j8ls ed atu (9) Hm - Hem where H, is the maximum stable wave height given as Hy, = 0o78 as (10) (c) Determine the equivalent initial wave height, Hj., for wave growth by Hye = Ri Hom - (11) (d) Determine the equivalent fetch length, F,, for the wave height, Hze. (e) Determine an adjusted fetch length, Fy, for the segment length, Ax, using equations (6) and (7). (f) Determine the total fetch, F, from equation (8). (g) Determine an equivalent wave height, H,, for the total fetch and the given windspeed and water depth. (h) Calculate the fractional growth by He ste. (12) -Hsm) = Hm-Hi Gi (Hm-Hsm) cy Step in Calculations \—" Procedure Schematic of wave decay calculations. 27 (i) Calculate the decayed wave height at the end of the fetch by ep en @ Se Gla > ea) © (13) As a conservative estimate, it is assumed that the wave period remains constant as the wave decays. V. SAMPLE DESIGN PROBLEMS The following examples demonstrate the use of the techniques discussed in this report in the sclution of design problems. Refer to the SPM (U.S. Army, Corps of Engineers, Coastal Engineering Research Center, 1975)! for other information related to the total design problem (e.g., wave theory, storm surges, wave setup, wave breaking, runup, etc.). xR Kk kK kK kK K Kk * Kk * * EXAMPLE PROBLEM 1 * * * * * * * * & * ® & * * GIVEN: A wave passes into shallow water over a flooded coastal area. The water depth, d;, at the seaward edge of the area is 23 feet (7 meters), and at the landward edge of the area the depth is 13 feet (4 meters). The distance across the area in the direction of wave motion is 19,000 feet (3,050 meters). The wave height, H;, at the seaward edge of the area is limited by large sandbars seaward of the area being considered and is 3 feet (0.91 meter), and the wave period is 3.2 seconds. The windspeed is 70 miles per hour (31.3 meters per second). The flooded area is covered with thick stands of tall grass. FIND: The height and period of the significant wave at the landward edge of the segment. SOLUTION: 0.25 d, O545 (23) 2 So7/5 veer eS Zo 3 ils es tO soot > O25 Glo « Since this does not meet the condition of equation (1), the area should be divided into two fetch segments. Assuming a uniform variation in depth, take the first segment as a distance Ax = 5,000 feet with a depth variation from 23 to 18 feet. Then Ad = 23 - 18 = 5 feet < 0.25 d; .- At the 23-foot depth (from Fig. 13, curve B), fr = 0.080 1y.S. ARMY, CORPS OF ENGINEERS, COASTAL ENGINEERING RESEARCH CENTER, @5 Cateo5 Wo De 28 and at the 18-foot depth (curve B), fre= 0.095 Afr = 0.095 - 0.080 = 0.015 0.25 Fp; = 0.25 (0.080) = 0.020 Afr < 0.25 fey , Equations (4) and (5) are satisfied, so the fetch segment chosen is used. For a uniformly varying depth, the average depth can be taken as the average of the depths at the beginning and the end of the seg- ment; i.e., d = a = 20.5 feet . For a uniform type of vegetation, the friction factor will vary as a function of water depth (see Fig. 13). As an approximation, the average friction factor can be taken as the average of the friction factors at the beginning and the end of the segment; i.e., fp = O80. 2 W025. pas - 2 3 feet, and U = 70 miles per hour (102.7 feet u For d = 20.5 feet, H per second), ody pay 2e2ext2 OSes == = 0.0626 U2 (102.7)? ae S202 8 2H aeong UF (102'47):2 and from Figure 1 ee =e u2 ‘ Ue (OZR) awe Fe = 12.2 F SN NASZ TD Un 4,000 feet . For fr = 0.01, E falleEY ByO Ozu S: 05,0005 gy yer * 5) a2 20.52 for fp = 0.088, 29 ap ST WO es 5000 d* 20.54 é For the period, T = 3.2 seconds, and d = 20.5 feet, 2nd _ 2m (20.5) > = SO ; Ae | DD (Gi ED)Z ve For 2nd/(gT*) = 0.391 (from Fig. 14) Kp.g1 = 0.996 for fp = 0.01 and fe H; Ax/d? = 0.357 Kyq = 0.965 for fp = 0.088 and fp H; Ax/d* = 3.14. From equation (4), a1 KRi07 2 ew0N99G SOR004! O° T=Kee | NSROLIGs OROsSea yews from equation (5), F, = a Ax = 0.114 (5,000) = 570 feet ; from equation (8), Se eS OD as SIO Asi skeehe For d = 20.5 feet, U = 70 miles per hour, and F = 4,570 feet (from Faigsy, 12/3 01% 6) Hr = 3.17 feet and T = 3.31 seconds IN| ES Soll > SS Wott so@Qe < OoS0 lilo « This satisfies the requirements of equation (3), and the solution may proceed to the next segment which is the remaining 5,000 feet of the area, with the water depth varying from 18 to 13 feet. 0.25 d; = 0.25 (18) = 4.5 feet . Since Ad = 18 - 13 = 5 feet > 0.25 d;, which does not satisfy equation (1), a shorter segment is required. For a 3,000-foot segment, assuming a uniform depth variation, the depth will vary from 18 to 15 feet. For the 15-foot depth (using curve B in Fig. 13) = 0.120 fr 0 pi = 0.095 at the 18-foot depth as previously shown. Af p = 0.120 - 9.095 = 0.025 = 0.25 fe; . 30 This satisfies equation (2) and the solution may proceed. The average depth, d = 16.5 feet, and the average friction factor, ff = 0.108. For d = 16.5 feet and H; = 3.17 feet (from Fig. 1). f, = 5,400 feet ; for d = 16.5 feet, H; = 3.17 feet, ieye = 0.108, Ax = 3,000 feet, and T = 3.31 seconds {from Fig. 14), eu) 0,294 Bits Kp,91 = 0-988 for fr = 0.01 and fp H; bx/d* = 0.349 = = ° 2 eS ° ° Keg -= 0.88 for fr 0.108 and fe H; Ax/d Bo V7 Using equation (4), a = 0.1 and Fou] owAxs= "0 1) (5,000) 5=s 300 tect Sh a eS S540 > WO SS, 7/00 2tsce « For d = 16.5 feet (from Figs. 1 and 2), Hr = 3.27-feet and T = 3.41 seconds . The remaining 2,000 feet of the fetch can then be treated as a third segment. The average depth, d = 14 feet, and the average friction factor is ff = 0.13. Fored = 14 feetvandyh, =| 3.27 feet, (rom Fig. 1) a Fo = 7,200 feet . e for d = 14 feet, H; = 3.27 feet, fp => Wedls) (Gerxony Wales 1) Ax = 2,000 feet, T = 3.41 seconds, and 2nd/(gT?) = 0.235 = = Di Kr. 01 = 0.98 for fr = 0.01 and fp H; Ax/d“ = 0.334 Krq = 0.80 for fp = 0.13 and fp Hy Ax/d* = 4.34 . Using equation (4), a = 0.1 and F, = a Ax = 0.1 (2,000) = 100 feet F =F, + F. = 7,200 + 200 = 7,400 feet For d = 14 feet, U = 70 miles per hour, and F = 7,400 feet (from Figs. 1 and 2) 3| Hp = 3.34 feet and T = 351 seconds . NOTE.--For a sandy bottom, f+ = 0.01, the wave would have increased to a height of approximately 4.26 feet, a 42-percent increase from the initial wave height of 3 feet. For thick stands of tall grass, the predicted increase in wave height is only 11 percent using the approx- imate method of solution discussed in this report. x oe ek eK Kk KX kK F * * * * * RYAMPLE PROBLEM 2 * * * * * * * * *¥ ® *¥ * & GIVEN: A coastal area is flooded by a storm surge so that the water depth over the area is 10 feet (3.05 meters). The actual fetch across the area, in the direction of wave travel, is 3,000 feet (914 meters). The area is covered with thick stands of tall grass and a small to moderate amount of brush or low, bushy trees in an even distribution. The wind- speed is 90 miles per hour (132 feet per second or 40.2 meters per second) and the initial wave height at the seaward edge of the area is 6 feet (1.83 meters); the wave period is 4.5 seconds. FIND: The decayed wave height at the end of the fetch. SOLUTION: From the long dashline in Figure 1, for the windspeed of 90 miles per hour and the water depth of 10 feet, gel Anas TO U2 (132)? 0.0185 giving (at the intersection of the above line with the long dashline) gH _ #5 = 0.0075 so that the maximum significant wave height H/OROOMSY USN MOMOOMSMGS2) iene ile, Fi = 322 = 4.1 feet . From equation (10), Hy = 0.78d = 0.78 (10) = 7.8 feet and from equation (9), the fractional reduction is Ba ee 6 ee = 8) 0.486 Hn - Hem 7.8 - 4.1 From equation (11), the equivalent initial wave height Ho = Ry Hom = 0.486 x 4.1 = 1.99 feet ; from Figure 1, for 32 Gil 8252 (G99) = 0.00368 U2 (132)2 and a 25 000s, UZ the fetch is given by F = 760 feet for the 90-mile per hour windspeed, so that the equivalent fetch EP = 760 feet . The vegetation does not match any of the curves in Figure 13, but falls between curves B and C. Assuming that a moderate amount of brush will give a friction effect about halfway between the two curves, from curve B, where d = 10 feet, fr = 0.20, and from curve C, where d = 10 feet, fe = 0.485. The bottom friction is then taken, in this case, as the average of the two values go 2 SUE ere For fF = 0.01, Ep Peek Onn xeoucd 5 0000. iL o-8 d2 102 for fp = 0.343, ff ee Ax _ 0.343 x 6 x 3,000 _ AL d 102 ff? for T = 4.5 seconds and d = 10 feet, 2ndd Li2zn d@Ojm yee. aie” (aes)? From Figure 14, Kp.o1 = 0-80 for fp = 0.01 and fp H; Ax/d? = 1.8 Kyq = 0.105 for fp = 0.343 and fp H; Ax/d* = 61.7 | 33 From equation (6), a = Kfa i 1 - 02105 0-/895 2A ne Seaton Tiara To 0.80 0.20 f.01 . from equation (7), yp 4X = 4.48 (3,000) = 13,440 feet (i.e., the wave decay over 3,000 feet of tall grass with some brush is equal to the wave decay over 13,440 feet of a sand bottom for this water depth and windspeed). The total fetch from equation (8) is F = Fo + Fg = 760 + 13,440 = 14,200 feet. For a windspeed of 90 miles per hour and a fetch of 14,200 feet (from Fig. 1) as 0.0185 (as previously determined) ple yO 2e lA) 200 ar u2 (132) 2 ih giving H == 0.007 . U From which the equivalent wave height, 00071 U2) 000ml 10rs2)14 Oe g 5 So = 3.84 feet . From equation (12), the fractional growth is Gyo We EE eR H Abe al The decayed wave height is then given by equation (13) as ose = Cp Geo ii) 750 = OI (ne = ao) Ss A883 moe . At the end of the fetch segment, the wave height and period are approximated by 4.33 feet Hp T 4.5 seconds . 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