ane coe Saag), Ra, QTR, TP 77-12 Wind-Wave Propagation ; Over Flooded, Vegetated Land rh) by Frederick E. Camfield TECHNICAL PAPER NO. 77-120 OCTOBER 1977 eee \ Ea \ a i \ Te oer U.S. ARMY, CORPS OF ENGINEERS COASTAL ENGINEERING GE a RESEARCH CENTER YU 50 Kingman Building a Fort Belvoir, Va. 22060 no. 17-o 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. WVU 0 0301 nnagaa MBL/WH ul A UNCLASSIFIED SECURITY CLASSIFICATION OF THIS PAGE (When Data Entered) READ INSTRUCTIONS REPORT DOCUMENTATION PAGE BEFORE COMPLETING FORM 1. REPORT NUMBER 3. RECIPIENT’S CATALOG NUMBER WM?) 77olZ 4. TITLE (and Subtitle) 5. TYPE OF REPORT & PERIOD COVERED WIND-WAVE PROPAGATION OVER FLOODED, VEGETATED LAND Technical Paper 6. PERFORMING ORG. REPORT NUMBER 8. CONTRACT OR GRANT NUMBER(S) 7. AUTHOR(s) Frederick E. Camfield 10. PROGRAM ELEMENT, PROJECT, TASK 9. PERFORMING ORGANIZATION NAME AND ADDRESS AREA & WORK UNIT NUMBERS Department of the Army Coastal Engineering Research Center (CEREN-CD) Kingman Building, Fort Belvoir, Virginia 22060 F31234 12. REPORT DATE October 1977 13. NUMBER OF PAGES 42 1S. SECURITY CLASS. (of this report) 11. CONTROLLING OFFICE NAME AND ADDRESS Department of the Army Coastal Engineering Research Center Kingman Building, Fort Belvoir, Virginia 22060 14. MONITORING AGENCY NAME & ADDRESS(if different from Controlling Office) UNCLASSIFIED 15a. DECL ASSIFICATION/ DOWNGRADING SCHEDULE Approved for public release, distribution unlimited. 16. DISTRIBUTION STATEMENT (of this Report) 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) Wind waves Shallow water Wave generation ABSTRACT (Continue an reverse sides if necessary 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 friction because of dense vegetation. DD pues 1473 ~—s EDITION OF # NOV 65 1S OBSOLETE UNCLASSIFIED SECURITY CLASSIFICATION OF THIS PAGE (When Data Entered) | yorsaeres ee a i eeeiuimen, |: Ley 7, ey Feit Ribena dee is Hee 3 4 sok tes ere hones Se hy gaa a ey apne yy alates ae REiAy eal st ee ae : ¥ Updos ne me Vy oft ‘ d i | 4 ae fie D fa belnen ne ype Saab fare) Al rhe ipa hick 2 al Gamage, siliassphiN A PREFACE This report describes a method for estimating wind-wave growth and decay over flooded areas where there is a major friction effect 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 Devel- opment Workshop, indicating a need for technical guidelines for predicting wind-wave generation over flooded coastal areas. The work was carried out 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. A condensed des- cription of the method is presented in CERC Coastal Engineering Technical Aid No. 77-6 (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, 70th Congress, approved 31 July 1945, as supplemented by Public Law 172, 88th Congress, approved 7 November 1963. JOHN H. COUSINS Colonel, Corps of Engineers Commander and Director CONTENTS CONVERSION FACTORS, U.S. CUSTOMARY TO METRIC (SI). . SYMBOLS AND DEFINIMIONS 3 3 205 3 ee I IINARRODUCMUONS 6 odo ohe os 6 OMB GNS oto 6.6 5° O10 Il WAVE FOREGASTING CURVES <3. 2 6s 3 ee we #8 Iba CALCULATION TORVADIUSDED SEEM CHI ss) septic) 01 oy) to Notte) meter Determinatvon of Friction) RAGCOM) al eien ce yere tenne 2. Seliectilon of Fetch Segment . - 2 2. 2. 25 © « « 3 IV WAVE Vv WAVE . Adjustment of Fetch Length . GROWTH IN SHALLOW WATER. . 2. © © « © o «© © o @ «@ DECAY IN SHALLOW WATER . . . . -« « «© « «© « «© « @ IS WEN) ROLE Mego Gi GG ioo, 6) 686 0\)6 26 OG 0050) 0 2. Wave Period. VI OV 0" 0 e eo: oe. ne? ce e e ° ° Ce) Oy er en ple) SUMMARY AND CONCLUSIONS . 2... «© « o © © © © © © © © GINS ORE (CIID S Gs id VG 6 6 OS 1695610 GF olo oO o Forecasting Forecasting Forecasting 5 feet). . Forecasting 10 feet) Forecasting 15! feet) <. Forecasting 20 feet) Forecasting 25 feet) . Forecasting 30 feet) . Forecasting 35 feet) . 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(ydw) M paads pulm (aie when the fetch is sufficiently long. From Figure 1, the relationship between the wave height, H and the water depth, d, is given by sm? 0.75 & sm —~ 9. 146|84 (3) uz U2 Where the bottom-friction factor fr= 0.01, and for a given water depth and windspeed, when the initial wave height, Hj, at the beginning of the fetch is less than H,g,, the wave will continue to grow in height as it travels across the fetch, with the height approaching the maximum sig- nificant wave height, Hg7,- When H; is greater than Hgy, it is as- sumed that the wave height will decay as the wave travels across the fetch, with the wave height again approaching the maximum significant wave height, Hg,,,- Where the bottom-friction factor fr> 0.01, i.e., vegetation in- creases the frictional resistance, and the initial wave height H; < Hg, it is assumed that the wave will grow to a final height less than or equal to H,,, but that the growth will occur at a slower rate than where ff = 0.01 (Fig. 13). Although the higher frictional resistance may limit the wave growth in this case to a height less than H,,,, no data are available on this effect. Therefore, as a conservative estimate, it is assumed that the maximum significant wave height will have the same value, Hgy, as where fp = 0.0L. Where the bottom-friction factor ff> 0.01, and the initial wave height H; > Hs,, it is assumed that the wave will decay to a final height less than or equal to H,,,, but that the decay will occur at a greater rate than where 18 0.01 (Fig. 14). Although the higher fric- tional resistance may cause the wave to decay to a height less than Hgm, no data are presently available. Therefore, the maximum significant wave height is assumed equal to the value of Hg, where fr = 0.01 as before. Where the bottom-friction factor f¢> 0.01, the fetch length is ad- justed to an equivalent fetch, F,, which has a bottom friction fr = 0.01, and which will give the same growth or decay as the actual fetch with the actual bottom friction. The relationship between the adjusted fetch, F,; and the actual fetch distance, Ax, is shown in Figures 13 and 14. After the adjusted fetch, F,, has been determined, Figures 1 to 12 are used to predict wave growth or decay (see Secs. IV and V). 1. Determination of Friction Factor. Limited data are available to define friction factors for water wave motion through dense stands of marsh grass, brush, or trees. Saville (1952) presented marsh correction factors for correcting values of wave setup over flooded marsh areas. However, this was not extended to pro- vide corrections for wave heights. Whitaker, et al. (1975) developed numerical simulations of water level changes which considered the effects 23 Wave Growth, fr = 0.01 as Actual Wave Growth, fr¢> 0.01 ———_—_—_ > X Figure 13. Wave growth. Wave Decay, f¢= 0.01 Actual Wave Decay, f¢>0.0l ———___> X Figure 14. Wave decay. 24 = of vegetation in terms of drag coefficients, but effects on wave heights were not considered. Wayne (1975) investigated the decay of very low amplitude waves, from 0.126 to 0.744 foot high (0.038 to 0.227 meter), traveling short distances over grass. However, he did not establish the dependence of wave period, initial wave height, and water depth, or con- sider the combination of wind stress with bottom friction. To obtain values of bottom-friction factors, estimates of Manning's roughness coefficient, mn, have been made using values shown in Chow (1959) for flow over flood plains. These estimated roughness coefficients are conservative; i.e., they are expected to give predicted wave heights somewhat greater than the wave heights which actually occur. The re- duction in wind stress due to vegetation extending above the water sur- face is not considered. The roughness coefficient, n, can be related to the friction factor, fr, for flow over ground by combining the Darcy-Weisbach and Manning equations for energy losses due to friction. However, the Manning equation contains a dimensional coefficient which varies depending on the dimensional units in the equation. Using a foot- pound-second system of units, _ 8.60.5 4 fp- SO, (4) where d is in feet and g is in feet per second squared. (In a meter- kilogram-second system of units, the coefficient 3.60 in equation (4) be- comes 8.0.) The roughness coefficient, n, will vary as a function of depth due to the height of the vegetation in relation to the depth of the water. In addition, the friction factor, fF, has an inverse relationship to the water depth, d, as shown by equation (4). Values of fy, used for various kinds of ground cover are shown as curves A to D in‘ Figure 15. Curve A is for a sandy bottom (Bretschneider, 1952, 1958, 1970), where ff is assumed constant. This curve was used for developing the curves in Figures 1 to 12. Curve B is for coastal areas with thick stands of grass. The grass will create a high resistance at low water levels where the depth of water and the height of the grass are nearly equal, but the resistance will rapidly diminish as the water depth becomes greater than the grass height; i.e., the water particle motion under the wave is in- fluenced less by the vegetation. Curve C is for higher levels of veg- etation, such as brush or low bushy trees which extend above the height of the grass. The friction effects will be higher for curve C than for curve B because the higher vegetation will have more influence on the water particle motion for any given water depth. Curve D is for tall trees which will always be higher than the depth of water. The frictional resistance will be high at all water levels, and somewhat higher at low water levels because of the added resistance from the ground. Curve D is for a relatively close spacing of trees; e.g., a second-growth pine forest which will give the highest frictional resistance. A more scat- tered spacing of trees would give a lower value of ff for a given 25 Friction Factor (f,) 1.40 1.205 1.00 == 0.80 0.60 ——_ === = Figure 15. Bottom friction factors. 26 water depth. Chow (1959) provides values of n for a wider range of conditions. 2. Selection of Fetch Segment. The growth or decay of a wave at any point is dependent on the water depth, the wave height and period at that point, the bottom friction, and the windspeed. For the method used here, the windspeed is assumed to be constant across a fetch. To accurately predict the growth or decay of waves traversing a particular fetch, it is necessary to divide the fetch into segments according to water depth and bottom friction. The bottom friction may vary substantially as a function of water depth for a par- ticular type of vegetation (Fig. 15). Figures 1 to 12 were derived for a constant depth; any variation in depth is assumed to be very gradual, and these figures are applied to an average depth across a fetch segment. Therefore, the total variation in depth across a fetch must be considered. The type of vegetation may also vary, With sections of marsh grass, brush, trees, or shallow lagoons. Wave heights will normally vary across a fetch and, as the decay factors will depend on the wave height, new decay factors must be calculated if the wave height varies excessively. A fetch segment is also considered to be much longer than a single wavelength. Dividing the fetch into segments, the segment distance (length in the direction of wave travel) is determined so that, first, Ad < 0.25 dz > (5) where Ad is the change in depth over the seginent distance and d; is the depth at the seaward or beginning edge of the segment; second, Afp< 0.25 fp; (6) where Af is the change in the bottom-friction factor over the seg- ment distance and fr; is the friction factor at the seaward or begin- ning edge of the segment; and third, after the change in wave height across the segment, AH, has been determined, NE < O45 Re (7) where H; is the initial wave height at the seaward edge of the segment. Wave decay is also assumed to be small (less than 0.1 H;) over a single wavelength. The segment distance may require adjustment; i.e., a shorter segment may be necessary if the computed value of AH is unacceptable. The numerical coefficients used on the right-hand side of equations (5), (6), and (7) are arbitrarily chosen. Smaller coefficients would be ex- pected to increase the calculations required for a solution; larger co- efficients would be expected to produce a greater error in the estimated wave height obtained. 27 Because of the storm conditions producing the surge, it is assumed that waves will always exist at the seaward edge of the initial segment (GLo@on ln ee 0), and values of wind velocity, U, greater than, e.g., 50 miles per hour (73 feet or 22.4 meters per second) will be expected. In case where H; = 0, the methods presented here do not apply. Initial wave growth would consist of very low amplitude waves with. very short periods. These initial waves will only be influenced by vegetation if the vegetation extends above the wave surface. That case has not been investigated, and cannot be treated by the method discussed in this report. 3. Adjustment of Fetch Length. The adjusted fetch length of the segment is determined by comparing the decay factor, Kr, for the actual bottom friction of the segment (from Fig. 15) with fie decay factor obtained for the friction factor f¢ = 0.01 which was used for the wave predictions in Figures 1 to 12. Using the method of Bretschneider (1954) (Fig. 16), the decay factor, Ke, is obtained by using the two factors fr H; Ax/d* and 2nd/(gT?), wees Ax is the actual segment distance in feet in the direction of wave motion. The decay factor, Kp, was defined by Bretschneider as tg = eee ee (8) ff He Ax a 167 Ke4 — eS, Bt il d2 3 3 [sin 2a) gr? where Kg is the shoaling coefficient defined as H/HS in Table C-1 of the SPM. By noting that Tae 2 (9) where Lo is the deepwater wavelength, and K, and sinh (21d/L) are functions of d/L, and therefore of 2nd/(gT2), it can be shown that when 2md/(gT*) is held constant that the right-hand side of equa- tion (8) is then a function of fp He Ax/d? only. Bretschneider (1954) used the work of Putnam and Johnson (1949) to define a friction factor for an impermeable sandy bottom as fr = 0.01. The curves in Figures 1 to 12 were developed for that friction factor. The wave decay is proportional to (1 - Ky)- To compare wave decay over a fetch segment with high bottom friction to wave decay over a fetch segment where the bottom friction ff = 0.01 (Hz, d, and U being the same), a proportionality factor, a, is defined by 28 (pG6l ‘sapiauydsjasg 4a4j0) 000'1 008 009005 00» 00¢ 002 00! 08 09 0S Ob *szoqzoey AVIOQ 2P xglHy of 02 0! QT omnsty 09 06 Ob oO¢€ 02 90 SO +0 Wil! £0 1000 100 20°0 soo O10 020 Ovo 090 060 S60 860 660 6660 66660 29 Ch = Se oid) 2. Cae: ae where K 01 is the decay factor defined for f,= 0.01 and K is the decay factor defined for the actual bottom friction with water depth, d, initial wave height, H windspeed, U, and segment length, Ax, re- maining the same. 1? For wave growth, a segment with bottom friction fr > 0.01 would require a longer segment length for waves to grow to a given height than the length that would be required where fr= 0.01. Therefore, the seg- ment with greater bottom friction is equivalent to a shorter segment with fr = 0.01. As an approximation, the shorter adjusted segment length, Fg, is defined as Fo =F On Aex 5 (11) where Ax is the actual segment length (see Fig. 13). For wave decay, a fetch segment with bottom friction fr> 0.01 would require a shorter length for waves to decay to a given height than would be required where fr= 0.01. Therefore, the segment with higher bottom friction is equivalent to a longer segment with fre= 0.01 (see Fig. 14). The longer adjusted segment length, F in this case is defined as a? F, = Gy, Ax 5 (12) where i} ae Ce ads ng ae, Ol AGEN Ga Guo nih IV. WAVE GROWTH IN SHALLOW WATER From Figure 1 or equation (3), for any given water depth, windspeed, and fetch length, a maximum significant wave height, Hgm, which would be generated can be defined. If the initial wave height, H;, at the seaward or beginning edge of the fetch segment is less than Hg;,,, it is assumed that the wave will grow to a higher height as discussed previously. To determine the wave growth, it is necessary to first determine an equivalent fetch length, Fe, for the initial wave. This is obtained directly from Figures 1 to 12 using the given windspeed and water depth. Secondly, the adjusted fetch, Fg, is determined using equations (10) and (11) and Figure 16. The total fetch is then given as Fe. Peer (Beier (14) Re-entering Figures 1 to 12 with the fetch length, F, the windspeed, U, and water depth, d, the final wave height at the end of the fetch segment, He, is determined. This is shown schematically in Figure 17 and in the following example problem. ee kK kK * kK RK K K k * * * * * EXAMPLE PROBLEM 1 * * * * * * * * * * * & GIVEN: A wave passes into shallow water over a flooded coastal area. ihiemwatemidepth), 1d. at) the seaward edge of the areas 25yteet) (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 10,000 feet (3,050 meters). The wave height, Hz, 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); the wave period is 3.2 seconds. The windspeed is 70 miles per hour (102.7 feet or 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; O525 (25) = Sof eu Ndv=9 255-3 LOM Eee t= 025 dae. Since this does not meet the condition of equation (5), 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 KMS 253.5 We 5 fest < 0.25 6p . At the 23-foot depth (from Fig. 15, curve B), fr = 0.080 and at the 18-foot depth (curve B), fF = 0.095 Afr = 0.095 - 0.080 = 0.015 0.25 fry 0.25 (0.080) = 0.020 Afp = 0.25 fey . Equations (5) and (6) 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 segment; i.e., 3| ay Step in Calculations s- Procedure Figure 17. Schematic of wave growth calculation. 32 23 + 18 d = 5 = 20.5 feet . For a uniform type of vegetation, the friction factor will vary as a function of water depth (Fig. 15). 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., _ 0.080 + 0.095 _ fr i ae aie: a 0.088 . For d= 20.5 feet, H; = 3 feet, and U = 70 miles per hour from Figures 1 or 6 Gal G62 3 AD.8 SS eee BB 0.0626 UZ (102.7)2 SH) eSZereeixS —— = ——— = 0.00916 u2 (102.7) 2 and from Figure 1 af wale U 2 Veen GODS Fe = 12.2 7- = 12.2 “37°— = 4,000 feet . For fp = 0.01, Spies’ MBONOA Ixus xm5)0007. a7 ae a d2 20.52 ; ‘ for fp= 0.088,. fe H; Ax fixes OOS exis ic S000) Mt d2 20.52 For the period, t = 3.2 seconds, and d = 20.5 feet, 21d t: 2m (20.5) Boney gts) 32).2 (3.2)2 For 2md/(gT*) = 0.391 (from Fig. 16) SS) iH] ms 2 Kol 0.996 for fe = 0.01 and fe Hy syd OS 557 Kpq = 0.965 for fp = 0.088 and fp H; Ax/d? = 3.14 . From equation (10), _ 2 Sf 0 dhemonsociiy oncom RAs) bite, Sy LS ekerat ih) Le OR OSS i OMOSSI a an: Y Fy = @ Ax = 0.114 (5,000) = 570 feet; and FOS ES) + Rn =) 45000 a o70u— 4) a7 Obpteet yy: 20.5 feet, U = 70 miles per hour, and F = 4,570 feet (from 3.17 feet and T = 3.31 seconds alg iT INSU) Sodio G'S Woodly Giese < Wa50 |slo o This satisfies the requirements of equation (7), 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, so O25 5d =s0). 25 118) 4 50 teets. Since Ad = 18 - 13 = 5 feet > 0.25 d;, which does not satisfy equation (5), 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. 15) fr = 0.120 fey = 0.095 at the 18-foot depth as previously shown. Afr = 0.120) - 0.095 = 0-025 = (0.25 fey C This satisfies equation (6) and the solution may proceed. The average depth, d = 16.5 feet, and the average friction factor, fp = 0.108. For d = 16.5 feet and H = 3.17 feet (from Fig. 1) F, = 5,400 feet ; 34 for dy—wlosse feet ype sy SdA treet), ito (0108 Ax — 3000 feet , and T = 3.31 seconds (from Fig. 16) and _ 9.294 gt? Kp.o1 = 9-988 for fp = 0.01 and f-H: .x/d* = 0.349 Kpq = 0-88 for fp = 0.108 and fe H; ye 2) SIT a0 Using equation (10), a = 0.1 and Bg eo xe S Osi (S,000) S00, exec F = Fp + Fg = 5,400 + 300 = 5,700 feet . For d = 16.5 feet (from Figs. 1 and 2) iPS SoBt/ 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, fr SO 186 For d = 14 feet and Hj = 3.27 feet (from Fig. 1), Fe = 7,200 feet; ; LOGE spl detects Hai S27 eee, ff = 0.13 (from Fig. 16) Ax 3.41 seconds, and 2md/(gt*) = 0.235 , 2,000 feet, T Kp.o1 = 0.98 for fp = 0.01 and fp Hy Ax/d* = 0.334 Kfq = 0.80 for ff = 0.13 and ff Hi Ax/d* = 4.34. Using equation (10), a = 0.1 and F, = a Ax = 0.1 (2,000) = 200 feet F =F, + Fy = 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) He = 3.34 feet and T = 3.51 seconds . Note.--For a sandy bottom fr = 0.01, the wave would have increased to a height of approximately 4.26 feet, a 42-percent increase from 35 the initial wave height of 3 feet. For the thick stands of tall grass, the predicted increase in wave height is only 11 percent using the approximate method of solution discussed in this report. eK RK KR KK RK KR KK RK RK KK KK KK RK KR RK RK KK KK KK KK KK KK KK KKK V. WAVE DECAY IN SHALLOW WATER Values of wind-generated significant wave heights and wave periods as a function of windspeed and water depth are shown in Figures 1 to 12. 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, the effects of the bottom friction may be assumed to exceed the effects of the wind stress. Therefore, the wave is assumed to decay, lose height, and over a long distance to approach a height equal to the maximum sig- nificant wave height. The higher waves at the beginning of the fetch will actually repre- sent a spectrum of waves. Waves of various periods could break at dif- ferent points approaching the shallow-water fetch area from deep water. As shown in Figure 16, the method used here predicts that the longer period wind waves would decay faster than shorter period waves passing through the shallow water (see Fig. 18). As discussed previously, the effects of vegetation near the surface may have a strong influence on very short-period waves. However, these waves are considered to be less important for design purposes, and this influence is ignored. Assumed Exponential Decay of Significant Wave ae of Short-Period Waves = —, = =, —— Decay of Long-Period Waves Exponential Growth of Significant Wave Fetch Length, X ———=— Figure 18. Exponential wave growth and decay. 36 As a first approximation, the wave decay may be assumed to occur exponentially at a rate similar to the rate of wave growth; i.e., the wave height will decay from a maximum stable wave height given by the shallow-water condition Hes) 0078 d (15) to a height equal to the maximum significant height in a distance approx- imately equal to the shallow-water fetch length required for a wave to grow from a zero height to the maximum generated significant wave height (the fetch length defined by the long dashlines in Fig. 1). 1. Wave Height. The means of determining the decayed wave height is shown schematic- ally in Figure 19. Steps to predict the decay of a wave are: Hm Ri (Hm-Hsm) = Hm-Hi Hj H Hp mt Hem (a) ) Step in Calculations \—" Procedure O Figure 19. Schematic of wave decay calculations. (a) Determine the maximum significant wave height that would be generated for the given windspeed and water depth, assuming an unlimited fetch, and using Figures 1 to 12 or equation (3). (b) Determine the fractional reduction, Ry represented by the initial wave at the seaward edge of the segment of fetch under consideration given by SY. (16) where H, is the maximum stable wave height given by equation (15), H; the incident wave height at the seaward edge of the fetch segment, and Ho, the maximum significant wave height. (c) Determine the equivalent initial wave height, H;,, for wave growth by Hye = Ry Hem . (17) (d) Determine the equivalent fetch length, Fo; for the wave height, Hye. (e) Determine an adjusted fetch length, F,, for the segment length, Ax, using equations (12) and (13). (£) Determine the total fetch, F, from equation (14). (g) Determine an equivalent wave height, H,, for the total fetch and the given windspeed and water depth. (h) Calculate the fractional growth by Ge Sa (18) (1) Calculate the decayed wave height at the end of the fetch by thy 2 Jal Ge (H, - it) : (19) 2. Wave Period. As waves decay over the fetch segment, the significant wave period also changes. Very long waves decay rapidly; shorter waves may decay very little (see Figs. 16 and 18). This means that the significant wave period may be reduced. As a conservative estimate, it will be assumed that the wave period remains constant. This is a conservative estimate since longer period waves would produce higher runup on a structure, all other variables being the same. x kK kK KK kK kK kK kK kK RK * & * * EXAMPLE 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. 38 The windspeed is 90 miles per hour (132 feet or 40.2 meters per second) and the initial wave height at the seaward edge of the area of 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 a BBB) 3 OD EPs aaa 0.0185 giving (at the intersection of the above line with the long dashline) 87 = 0.0075 U so that the maximum significant wave height 0.0075 WE _ O00 75 (lEea)= = 4.1 feet . g 32.2 Hsm = From equation (15), Hei Om7Sd)=0ei73 (10) =m7n8) cect and from equation (16), the fractional reduction is From equation (17), the equivalent initial wave height Ep I I Vale Si Wodielo ss dha il =) IGS) saceie F from Figure 1, for GE B22 (98) = 2606 Moore 000868 U2 (132) 2 and gd _ 9.0185 U2 the fetch is given by Bo oe =e 39 F = 760 feet for the 90-mile per hour windspeed, so that the equiva- lent fetch is ey Hell) SESE 6 The vegetation does not match any of the curves in Figure 15, 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, fe = 0.20, and from curve C, where d = 10 feet, ff = 0.485. The bottom friction is then taken, in this case, as the average of the two values Ay oo O02) : O48 5 9. ane sone a S Oils fe Hy; 4x 9.01 x 6 x 3,000 ge Me SINE Big ene eo, ee for ff = 0.343, fe Hi Ax 0.343 x 6 x 3,000 E Ge = 667 3 10 feet, Fh °o 8 | iT] KR (oa wn co) Q (e) =) Qa n @ Dp Q Q iT} 21d = 27 (10) = 0.096 eT? g (4.5)2 From Figure 16, Kp.o1 = 0-80 for fp = 0.01 and fp H; Ax/d? = 1.8 = = 2a Kg = 0.105 for fp = 0.343 and fp H; Ax/d* = 61.7 From Equation (13), 1 - K - 0. 895 ee ity 4 De Wont PuMaGES A Ay ic Kf. 01 1 - 0.80 0.20 from equation (12), Fy = Gy, Ax = 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). Jo) et JRL oe EF = 760 + 13,440 = 14,200 feet 40 For a windspeed of 90 miles per hour and a fetch of 14,200 feet (from Fig. 1) a = 0.0185 (as previously determined) Fars 2h el 20,0) on Ges MaAIGIS2) 2 One 26.24 giving Bulg 0.0071 UZ From which the equivalent wave height, OOO WE WsOOmi Glee @.= g 32.2 = 3.84 feet From equation (18), the fractional growth is The decayed wave height is then given by equation (19) as Sip = Bi, = Gp Gy oki) = eG = 0.987 (58 = Gail) = 4398 Bese | 6 At the end of the fetch segment, the wave height and period are approx- imated by Hp 4.33 feet T = 4.5 seconds kK kK kK RK RK KK KK KK KK KK RK RK KK RK KK RK K RK KK KK KK KK KK KK VI. SUMMARY AND CONCLUSIONS The method presented in this report gives a first approximation for estimating wave heights at the end of a fetch with a high value of bottom friction (e.g., a flooded area with dense stands of grass or brush). Only limited data are available for wave height growth or reduction for waves passing over areas with dense bottom vegetation. The method has not been verified. A substantial amount of data is needed for waves passing over areas of flooded vegetation. The method should be verified and modified as required. The shoaling coefficients (Fig. 16) and the friction factors (Fig. 15) should also be compared with values from actual measurements. 4 LITERATURE CITED BRETSCHNEIDER, C.L., ''Revised Wave Forecasting Relationships ,"" Proceedings of the Second Conference on Coastal Engineering, Council on Wave Research, 1952, pp. 1-5. BRETSCHNEIDER, C.L., ''Modification of Wave Height Due to Bottom Friction, Percolation, and Refraction,'' TM-45, U.S. Army, Corps of Engineers, Beach Erosion Board, Washington, D.C., Oct. 1954. BRETSCHNEIDER, C.L., "Revisions in Wave Forecasting; Deep and Shallow Water," Proceedings of the Sixth Conference on Coastal Engineering, Council on Wave Research, 1958, pp. 30-67. BRETSCHNEIDER, C.L., Department of Ocean Engineering, University of Hawaii, Honolulu, Hawaii, unpublished correspondence, 1970-71. CHOW, V. T., Open-Channel Hydraultes, McGraw-Hill, New York, 1959. IJIMA, T., and TANG, F.L.W., "Numerical Calculation of Wind Waves in Shallow Water," Proceedings of the 10th Conference on Coastal Engt- neering, 1966, pp. 38-45. PUTNAM, J.A., and JOHNSON, J.W., ''The Dissipation of Wave Energy of Bottom Friction," Transactions of the American Geophystecal Unton, Vol. 30, No. 1, 1949, pp. 67-74. SAVILLE, T., Jr., "Wind Setup and Waves in Shallow Water," TM-27, WoSo Army, Corps of Engineers, Beach Erosion Board, Washington, D.C., June 1952. U.S. ARMY, CORPS OF ENGINEERS, COASTAL ENGINEERING RESEARCH CENTER, Shore Protection Manual, 2d ed., Vols. I, II, and III, Stock No. 008- 022-00077-i, U.S. Government Printing Office, Washington, D.C., 1975, 1,160 pp. WAYNE, C.J., "Sea and Marsh-Grasses: Their Effect on Wave Energy and Near-Shore Sand Transport,'' M.S. Thesis, Florida State University, Tallahassee, Fla., Sept. 1975. WHITAKER, R.E., REID, R.O., and VASTANO, A.C., '"'An Analysis of Drag Co- efficient at Hurricane Windspeeds from a Numerical Simulation of Dy- namical Water Level Changes in Lake Okeechobee, Florida," TM-56, U.S. Army, Corps of Engineers, Coastal Engineering Research Center, Fort Belvoir, Va., Oct. 1975. 42 Lz9 ZleLL °ou daigcn° €07OL *ZL-ZL °ou aeded Teo eFuyoe, *1ejUueQ yor1eessy BuTAseuT3uq Teqyseop *S*n :SeFIeS “II “eTIFL *I °SOABM 19ReM MOTTBYS “€ “SeAeEM PpUTM *Z “UOTIeLTEUes eAeM °] *m0330q Apues e ivAo Sutssed SOABEM OF SSAIND uUstsep ay. 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