(ics aoa Rah Esq Ge. Qove
TP 81-1
Estimation of Wave Reflection and Energy Dissipation Coefficients for Beaches, Revetments, and Breakwaters
by William N. Seelig and John P. Ahrens
/ “WHO! | DOCUMENT \. COLLECTION
TECHNICAL PAPER NO. 81-1 FEBRUARY 1981
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ESTIMATION OF WAVE REFLECTION AND ENERGY DISSIPATION COEFFICIENTS FOR BEACHES, REVETMENTS, AND BREAKWATERS
7. AUTHOR(s)
William N. Seelig John P. Ahrens
10. PROGRAM ELEMENT, PROJECT, TASK AREA & WORK UNIT NUMBERS
9. PERFORMING ORGANIZATION NAME AND ADDRESS Department of the Army Coastal Engineering Research Center (CERRE-CS) Kingman Building, Fort Belvoir, Virginia 22060
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- KEY WORDS (Continue on reverse side if necessary and identify by block number)
Beaches Irregular waves Revetments Breakwaters Monochromatic waves Wave reflection Energy dissipation
ABSTRACT (Continue en reverse side if neceseary and identify by block number)
More than 4,000 laboratory measurements of wave reflection from beaches, revetments, and breakwaters are used to develop methods for predicting wave reflection and energy dissipation coefficients. Both monochromatic and irregular wave conditions are considered and the prediction techniques apply to both breaking and nonbreaking wave conditions.
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PREFACE
This report is published to provide coastal engineers empirical formulas for predicting wave reflection coefficients for beaches, revetments, and break- waters. The techniques were developed using laboratory data from a number of sources covering a wide range of conditions for both monochromatic and irregular waves. The work was carried out under the coastal processes program of the U.S. Army Coastal Engineering Research Center (CERC).
This report was prepared by William N. Seelig, Hydraulic Engineer, and John P. Ahrens, Oceanographer, both of the Coastal Processes and Structures Branch, under the general supervision of Dr. K.M. Sorensen. J. McTamany, Coastal Oceanography Branch, provided the nonlinear regression analysis used to determine empirical coefficients developed in this report.
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.
vy :
TED E. BISHOP Colonel, Corps of Engineers Commander and Director
CONTENTS
CONVERSION FACTORS, U.S. CUSTOMARY TO METRIC (SI). ....
SHAMIKONLS) ANID) IDIIFIONIEITIONS 6 9 6 6 0 500060 0 OOo i JOININODUGEUGN o 60 0 6 6 0 0 0 0 oo OOOO OO ILI TIC RVMIUOIRS, INNING 6866 6 6 OOo IEIEIE PCP UGMIINAYNL, IWAGoISGLOUIBSs 6 6 6 66 6 6 0 oO 6 60 6 oo IV PAGTORS) INBEUENGING WAVE REBEECTTONS 25 3 3 3
Vv TYPES OF STRUCTURES AND RANGE OF CONDITIONS TESTED .
VI TECHNIQUES FOR PREDICTING REFLECTION AND ENERGY IDS SIMEON COMMPICIONNINSs° 6 06 6 560100000000 1. Modification of the Wave by the Structure .....
2. Breaking of the Toe or Seaward of the Structure .. . 3, IWmtlmenee oF Guirtaee RowmsanesSo 6665656500060 4. Influence of Multiple Layers of Armor... . 5. Wave Reflection from Sand Beaches ..... . 6. Rubble-Mound Breakwaters. . .......+. +226 -s 7. Spectral Resolution of Wave Reflection. .... . 8. Reflection Coefficient Prediction Equations .... . VIL IDONSDPZILI, IROWIDAES 5 5 o 6 6000000000000 06 VIIL SUMMINRN 5 5,0 0 0 0 0000560000060 56 000 I ICIV ARVO (CIID) Gg 5 6 60 0 0 oO Oo 0 APPENDIX A LABORATORY WAVE REFLECTION DATA. ........ B METHOD OF MEASURING WAVE REFLECTION COEFFICIENTS C NONLINEAR WAVELENGTHS AND WAVE SPEED .. . TABLES Sources of data and range of conditions ...... Correction factor due to multiple layers of armor ........ SUMMA OE SGMAIEICMS seo preclleEAy Kao o 5 0 6 6 bo FIGURES
Wave reflection and transmission from a coastal structure
Wave gage array used to measure wave reflection ...... -»
20
24
- 10
10
ial
12
13
CONTENTS FIGURES-—-Continued
Relation between wave reflection, transmission, and Gligoiinelenom CoOeimeEeleneiles 5 ovo. 6 6 d-00 060 o 16 0 0 6 OOo 4
A comparison of wave reflection coefficients for a 1 on 2.5 slope and various equations to predict reflection coefficients ....
3 25 @ wwMcrICMm OF SeEMCEMEe BIDA 5 565566 56565600005 6 6
Joint effect on one layer of armor and Hy /H, on the reflection cochiiicient reduction=fackors. IG) sce it toee Wemsy Nene vin terete yay tet
Observed versus predicted reflection coefficients for a revetment ENicnoneGtel) Aliela\. Cinvai Mee ACNE TECMES “Go Gn bra woleolG a so l0 6 6 allo
Wave reflection coefficients from laboratory beaches. ......
Predicted rubble-mound breakwater wave reflection and transmission coehiveients.) i si ns ceeece we ee eee ee Oe
Wave reflection coefficient as a function of wave frequency for an irregular wave condition with breaking waves. ........
Wave reflection coefficient as a function of wave frequency for an irregular nonbreaking wave condition. .........+4+e+6-.
Predicted wave reflection coefficients for smooth impermeable SUOMaAS Walleln wo) OweceOpplinye oo 5660006
Wave reflection coefficients for a smooth revetment and revetments Wath wonemandmstwomlayerSmOtrrarmOmStONe) jis icy: 1) isiesll cnt ememnrcnne
Iz
18
19
21
22
D3)
23
26
27
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 2524 millimeters 2-54 centimeters square inches 62452 Square centimeters cubic inches 16.39 cubic centimeters feet 30.48 centimeters 0.3048 meters square feet 0.0929 Square meters cubic feet 0.0283 cubic meters yards 0.9144 meters Square yards 0.836 square meters cubic yards 0.7646 cubic meters miles 1.6093 kilometers square miles 259.0 hectares knots 1.852 kilometers per hour acres 0.4047 hectares foot-pounds 1.3558 newton meters millibars UoOLOY sz 2OTe 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 (angel) 0.01745 radians Fahrenheit degrees 5/9 Celsius degrees or Kelvins
To obtain Celsius (C) temperature readings from Fahrenheit (F) readings, use formula: C = (5/9) (F -32).
To obtain Kelvin (K) readings, use formula: K = (5/9) (F -32) + 273.15.
SYMBOLS AND DEFINITIONS
incident wave amplitude at a spectral line
reflected wave amplitude at a spectral line
real and imaginary spectral coefficients from an FFT analysis representative armor diameter = (w/y) 1/3
water depth at the toe of the structure
acceleration due to gravity
a representative breaking wave height at the toe of the structure incident wave height (use Hg for irregular waves)
deepwater wave height
reflected wave height
significant wave height
transmitted wave height
wave dissipation coefficient
wave reflection coefficient
wave transmission coefficient
wave number = 27/L
wavelength at the toe of the structure
deepwater wavelength from linear theory = gT?/(2n)
offshore slope seaward of the structure
number of layers of armor
wave runup
Reynolds number
wave period (use period of peak energy density for irregular waves) period of peak energy density
weight of armor material
empirical wave reflection parameters specific weight of armor unit material wave gage spacing
average root-mean-square surface water level angle of the seaward structure face kinematic viscosity of water
surf similarity parameter = tan 6/VHj/Lo
ase vata, 4
4 it an ee q a ‘ae )
rennin oA “ae Ae ae we se ox ee ee ae sera
ESTIMATION OF WAVE REFLECTION AND ENERGY DISSIPATION COEFFICIENTS FOR BEACHES, REVETMENTS, AND BREAKWATERS
by Willtam N. Seeltg and John P. Ahrens
I. INTRODUCTION
When a wave encounters a coastal structure or beach, a part of the wave energy is dissipated. The remaining energy is reflected seaward except in the case of a permeable or overtopped structure (Fig. 1), which allows transmission of a part of the energy to the leeward side. Wave reflection may have undesir- able effects because the reflected waves are superimposed on the incident waves to increase the magnitude of water particle velocities and water level fluctu- ations seaward of the structure. These enhanced motions may be a hazard to navigation or may undesirably alter sediment transport patterns. This report presents methods for estimating wave reflection coefficients for beaches, revetments, and breakwaters of waves approaching the structure at a normal angle of incidence (wave crests are parallel to the structure axis).
II. LITERATURE REVIEW
Previous investigators have experimentally and analytically studied wave energy dissipation and reflection characteristics for a variety of structures. Various prediction techniques have been proposed to estimate reflection coef- ficients for specific types of energy dissipation. Miche (1951) proposed a wave reflection-coefficient prediction technique that is often quoted in lit- erature (e.g., Sec. 2.54 in U.S. Army, Corps of Engineers, Coastal Engineering Research Center, 1977). He assumed that there is some critical deepwater wave steepness below which the reflection coefficient is a constant. For conditions where wave steepness is greater than the critical value, the reflection coef- ficient is proportional to the ratio of the wave steepness to the critical value of wave steepness. Predictions using Miche's approach give the right order of magnitude estimate of the reflection coefficient, but as Ursell, Dean, and Yu (1960) illustrated, predictions may be conservative by a factor of 2.
Moraes (1970) has performed some of the most extensive laboratory tests to date on monochromatic wave reflection from a variety of smooth and rough slopes.
$j ____ Hj Incident Waves
ee Hy Transmitted Waves
Reflected Waves Hr
Kr = Hp / Hj Kt = Ht/Hj 1
Figure 1. Wave reflection and transmission from a coastal structure.
Madsen and White (1976) made a number of additional carefully controlled reflection measurements for smooth and rough steep-sloped structures under nonbreaking wave action. Based on these data, they developed an analytical- empirical model for predicting reflection coefficients for rough slopes with nonbreaking waves.
Battjes (1974) used Moraes' data to develop an equation for predicting reflection coefficients for smooth slopes where the slope induces wave breaking. This technique is conservative for nonbreaking (surging) waves. Ahrens (1980) has made a number of irregular wave reflection coefficient measurements for overtopped and nonovertopped plane smooth slopes.
A number of wave reflection measurements for laboratory breakwaters have been made. Seelig (1980) investigated rubble-mound and caisson breakwaters using monochromatic and irregular waves. Brunn, Gunbak, and Kjelstrup (1979) measured reflection coefficients for rubble-mound breakwaters and proposed an empirical prediction technique. Additional breakwater reflection data are available in Debok and Sollitt (1978) and Sollitt and Cross (1976). Madsen and White (1976) give a procedure for predicting reflection from rubble-mound breakwaters for nonbreaking waves.
Chesnutt and Galvin (1974) and Chesnutt (1978) have made some of the most detailed measurements available of wave reflection from laboratory sand beaches. Little prototype data are available; however, Munk, et al. (1963) and Suhayda (1974) reported reflection measurements for beaches exposed to extremely low steepness swell waves.
IIJ. EXPERIMENTAL TECHNIQUES
The primary emphasis of this report is on the reanalysis of existing data from a number of published sources. However, some additional laboratory data were taken to supplement the sources; these data are reported in Appendix A.
Goda and Suzuki's (1976) method was used to determine wave reflection coef- ficients. This method was selected because with the test setup used it gave consistent results which are as reliable as obtainable with other currently used procedures. Experience with this technique suggests that the error is on the order of 5 percent. A typical wave gage setup is illustrated in Figure 2, and a detailed discussion of the analysis method given in Appendix B. The test procedure uses three gages, located a minimum of 6 meters seaward of a test
Incident Woves Reflected Woves SS <ag——_—_—__—_—_—_———_—_—__ |—_-_—- AL = 125 cm ————
O2:90cm pe a AL = 35cm Wave Gages Tank Bottom
Figure 2. Wave gage array used to measure wave reflection.
10
structure, to collect simultaneous wave records (incident and reflected waves superimposed), each containing 4,096 data points at a sampling interval of one- sixteenth of a second. A fast Fourier transform (FFT) analysis is made of each record, and each gage pair gives an estimate of the reflection coefficient sub- ject to the criteria discussed in Appendix B. The mean of the three estimates is taken as representative at each spectral line, and an energy-weighted aver- age is taken to characterize reflection for the entire spectrum of irregular waves. The significant incident wave height, H,, for irregular waves (Goda and Suzuki, 1976) is defined as
4 Nrms
18 Tl ape 3 (1)
where bane is the average root-mean-square (rms) water surface displacement of the wave records at the three gages, and K;, the reflection coefficient.
Data collection in this study emphasized obtaining additional data on wave reflection on smooth slopes and examining the influence of one or more layers of armor on reducing the reflection coefficient. Monochromatic and irregular waves were tested.
For monochromatic wave conditions (sinusoidal wave generator blade motion), the wave reflection measurement technique was slightly modified. The wave- form for monochromatic waves is described by a Fourier series with the entire waveform moving at the speed of the primary wave (Dr. R. Dean, University of Delaware, personal communication, 1980). This allows the wave energy appearing in harmonics of the primary wave to be considered in determining the reflec- tion coefficient (App. B).
IV. FACTORS INFLUENCING WAVE REFLECTION
The conversion of wave energy concept is useful for defining the interre- lation between the wave reflection, dissipation, and transmission coefficients. Assuming that the water depth remains constant seaward and leeward of the struc- ture the partition of wave energy is given by
l= Ki + K2 + Ke (2)
where Kr is the reflection coefficient, Ka the ratio of wave energy lost through dissipation to the total incident wave energy, and K, a transmission coefficient including transmission through a permeable structure and trans- mission by overtopping for a low-crested structure. In an idealized monochro- matic wave situation where there are no transfers of wave energy to other wave frequencies,
H Kr SS Tal (3) aL and H a (4) Hy
where Hj, H,, and H, are the incident, reflected, and transmitted wave heights, respectively (see Fig. 1).
Rearranging equation (2) gives 2 1 - (K§ + Ke) (5)
which clearly shows that any process that increases the sum K2 + Ke) will cause the reflection coefficient to decrease. Figure 3 illustrates equation (5) and the nonlinear relation of the variables. Note that for a given value of the transmission coefficient the reflection coefficient may be very sensitive to
the amount of energy dissipation. In addition, with no transmission large values of energy dissipation will allow the reflection coefficient to be rela- tively large. For example, with 90-percent energy dissipation and no trans— mission, the reflection coefficient is 0.31 (see Fig. 3).
Pct Wave Energy Dissipated 10 20 40 60 80 90 98 98 OY 99.9
O-Pct Energy issipation
s) D | | |
0 ORO re 04 0.6 O:60:9 0.98 0.99 0.999
2 KG
Figure 3. Relation between wave reflection, transmission, and dissipation coefficients.
[2
V. TYPES OF STRUCTURES AND RANGE OF CONDITIONS TESTED
Table 1 summarizes the sources of wave reflection coefficients for struc-— tures and beaches and the range of conditions tested. Three types of structure are considered: smooth, impermeable slopes with no overtopping; revetments armored with one or more layers of riprap with no overtopping; and rubble-mound breakwaters armored with stone or dolos.
The water depth at the toe of the structure, d,, is taken as a character- istic water depth, g is the acceleration due to gravity, and a representative armor unit diameter, d, is determined from
2-3)" ©
where W is the armor weight, and y the specific weight of the armor mate- rial. A measure of the wave breaker height that could occur at the toe of the structure, Hp, is given by Goda (1975) as
d gb, = O17 Lo{1-0 = exp [-4.712 = (1.0 + 15 n!-233)]h (7) lo)
where Lo is the deepwater wavelength given by linear wave theory, and m the tangent of the slope of the seabed seaward of the structure.
Other variables summarized in Table 1 include dimensionless ratios using Hj, the incident wave height (significant height for irregular waves) at the toe of the structure; T, the wave period (period of peak energy density for irregular waves); and L, the wavelength at the toe of the structure.
Only those tests with fully turbulent hydraulic conditions are considered in order to minimize the influence of viscous effects (Jonsson, 1966). The Reynolds number, Re, proposed by Madsen and White (1976),
R2 20 Re“ T vu tand 8) where R is the wave runup and v the kinematic viscosity of water (about 0.009 square centimeter per second at 20° Celsius), is used to establish which tests are fully turbulent. For smooth slopes only those tests with Re > 3 x 10" are analyzed; for rough slopes only tests with Re > 10* are considered (Jonsson, 1966; Madsen and White, 1976).
VI. TECHNIQUES FOR PREDICTING REFLECTION AND ENERGY DISSIPATION COEFFICIENTS
Section IV showed the strong dependence of the magnitude of the reflection coefficient on the amount of wave energy dissipated (also on the amount of wave energy transmitted in the case of a permeable or overtopped structure). In this section, factors that influence the reflection coefficient are systematically investigated, and empirical prediction formulas are developed. Types of wave energy dissipation considered include losses in energy due to structure-induced wave breaking and wave modification, breaking at the toe of a structure or in the surf zone seaward of the structure, structure surface roughness, and internal flow in permeable sections of a structure.
13
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14
1. Modification of the Wave by the Structure (Smooth Slopes).
For a structure with a toe water depth-to-wave height ratio greater than five and wave steepness much less than one-seventh, the interaction of the wave and structure will have dominant control on the magnitude of the reflection coefficient. Miche (1951) proposed that the reflection coefficient for this situation is proportional to the ratio of a critical wave steepness to the inci- dent wave steepness. The critical steepness is
(=) -(2)" sin79 (9) ib 9 T T o/erit
where Ho is the deepwater wave height, and © the angle the structure slope makes with the horizontal, in radians. Miche's equation gives conservative results. For example, it overpredicts monochromatic wave reflection from a 1 on 15 slope by a factor of 2 (Ursell, Dean, and Yu, 1960).
Battjes (1974) recommends the equation,
tan@ K, = 0-1 E23 — = mates Le (10) Lo which can be written as 0.1 tan2@ Ky = Ho i (@aIb) its
Battjes (1974) is assuming an equation similar to the formula proposed by Miche (1951) where the critical steepness is
H:
(=) = 0.1 tan?6 (12) o/crit
This criterion gives lower and more realistic values of the reflection coeffi- cient than Miche (1951) and is especially useful for & < 2.3 where breaking is induced by the structure (for plunging breakers). Figure 4 shows the compari- son between the equations of Battjes (1974) and Miche (1951).
The following revised equation,
Kre=)tanh (Os) 62), (13)
is recommended to give a conservative prediction of reflection coefficients. At small values of the surf similarity parameter (& < 2.3),
Oniaae= tanh (Ones) (14)
and equation (13) gives the same results as equation (10). At larger values of the surf similarity parameter, £€, equation (13) asymptotically approaches 1.0 and gives an upper bound closer to the data than equation (10) (see Fig. 4).
15
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077!'H yf) uD) = G v
(O86!) suasyy Aq 0}0q
(,3 10) yuo) = 7»
3
(1S61) 949!W
. =e | 23!0= (p261) salyjog
20
£0
v0
GO!»
90
20
80
60
O'l
16
An improved equation for predicting reflection coefficients with less error in the estimates is
Ky = — = (15)
where a and £8 are empirical coefficients determined from the laboratory data (e.g., Fig. 4). The value of 8 increases as the slope becomes flatter and is larger for irregular waves than for monochromatic waves (Fig. 5). For slopes with cot® < 6, the suggested prediction coefficients are a = 1.0 and
8 = 5.5 with the equation,
MOEN 52 + 8
Ky whichever (16)
or : is smaller
K, = a tanh (0.1 £2)
@ Irregular Waves ( Ahrens, 1980)
O Monochromatic Waves (Ursell, Dean, and Yu, 1960; Moraes, 1970; This Study )
ce)
O f 2 B.S BGR BO sitive 1s 14 16 16 cot 8
Figure 5. 8 as a function of structure slope.
2. Breaking at the Toe or Seaward of the Structure.
If the water depth at the toe of the structure is less than five times the incident wave height or if the wave steepness is large, significant additional wave energy loss may result from wave steepness/water depth-limited breaking. The dimensionless ratio describing this type loss is the ratio of the incident wave height to the maximum possible breaker height, (H;/Hp) » where Hp is given by equation (7). This ratio includes the influence of offshore slope, water depth at the toe of the structure, and wave steepness, and gives a meas-— ure of breaking at the toe. The suggested empirical coefficient to account for this type energy loss in predicting reflection coefficients is
: Hi Lod a = exp {—- 0.5 il (aL)
for use with equation (16), where a is a reflection coefficient reduction factor.
3. Influence of Surface Roughness.
Armor units placed on the surface of a smooth structure will increase the amount of energy loss in a wave encountering the structure, thereby reducing the amount of wave reflection. The suggested prediction equation for a revet-— ment with one layer or armor rock with representative diameter, d, is
los H- a = exp ee coté - 0.5 (=) | (18)
for use with equation (16), where L is the wavelength at the toe of the struc- ture. This equation was developed from the data in Table 1.
Figure 6 illustrates the joint influence of a relative armor roughness parameter, Va/L coté, and a relative breaking height parameter, H;/Hp, on the reflection coefficient reduction factor, a. An examination of equation (18) and Figure 6 indicates that if all other factors remain fixed, the reflection coefficient will decrease as the ratio of the stone size to wavelength, d/L, increases, as the cot® increases (the slope becomes flatter), or as the ratio of the incident wave height to the breaking wave height, (H;/H,), increases. Figure 7 shows a comparison between predicted reflection coefficients using equations (18) and (16) versus observed reflection coefficients for monochro- matic and irregular waves on a 1 on 2.5 slope armored with one layer of stone with d/dg = 0.15. The correlation coefficient is 0.98 for monochromatic waves and 0.94 for irregular waves.
The ratio of armor stone diameter to incident wave height, d/H;, on the average has little influence on the reflection coefficient for one layer of armor, so this parameter is not included in equation (18). Some deviation from equation (18) occurs where stone size is much larger than wave height and resulting predictions are conservative. For example, where the stone size-to- wave height ratio is greater than 2.0, equations (16) and (18) overpredict reflection coefficients by an average of 6 percent.
1.0 —————=) 0.9 0.8
0.7
[o) Ww —_+- ——__ + ——_,___]
0.001 0.01 0.1 1.0
J a/c cot @ Figure 6. Joint effect on one layer of armor and Hy /Hy, on the reflection coefficient reduction factor, a.
18
Kr) predicted
Irregular Waves, Correlation Coefficient = 0.94
0 0.1 0.2 0.3 0.4 0.5 0.6 K,, observed
0,6 0.5
0.4
2 w
predicted
ia)
0.2
0.1 Monochromatic Waves, Correlation Coefficient = 0.98 0 0 0.1 0.2 0.3 0.4 0.5 0.6 Kr, observed Figure 7.
Observed versus predicted reflection coefficients for a revetment armored with one layer of stone.
19
4. Influence of Multiple Layers of Armor.
As the number of layers, n, of armor on a revetment increases, the amount
of wave energy dissipated increases and the reflection coefficient decreases. In addition, as the size of the stone increases relative to the wave height, the roughness becomes more effective and the reflection coefficient decreases.
Table 2 gives selected values of a correction factor, a', where
1.3 Hy: a = a' exp [eedon - o.5( 2) ] (19)
Table 2. Correction factor due to multiple layers of armor.!
d/H, <0.75 0.78 O75 tt 2.0 0.69 22,0 0.49 | Teo) = 2.55 d/l, 2 O15, 0,004 « desu <A 0803):
for multiple layers of armor. These coefficients were obtained by taking the average of the ratios of the measured reflection coefficients for two, three, and four layers of armor to predicted coefficients for a slope with one layer of armor. Only one slope, cot® = 2.5, and stone size-to-water depth ratio, d/d, = 0.15, was tested.
5. Wave Reflection from Sand Beaches.
Chesnutt (1978) has the most extensive data set of wave reflection coeffi- cients from laboratory sand beaches. Unfortunately, there are little prototype data available. Chesnutt and Galvin (1974) and Chesnutt (1978) found that many
factors influence the magnitude of the reflection coefficient. Their data suggest that
2 eo oe aS SS (20) 2 Eo ap B
can be used to estimate reflection coefficients with the beach slope at the stillwater level intercept used to determine &€. Use a = 1.0 for conservative
estimates of K, and a = 0.5 to give predictions of the average reflection coefficient measured throughout a test (Fig. 8).
20
0.6
0.5
0.4
Kr 0.3
0.2
Ronge of Kr
0 OSD 1.0 1.5 2.0 ; 2.5 3.0 3.5
3
Figure 8. Wave reflection coefficients from laboratory beaches (from Chesnutt, 1978).
6. Rubble-Mound Breakwaters.
An upper limit or conservative estimate of Ky for breakwaters armored with rock or dolos may be obtained using
a ge
kK, = — ro g2 +B
s; a = 0.6, 8 = 6.6 (21)
Ninety-five percent of all observed laboratory breakwater wave reflection coefficients fall below this prediction equation for data sets c, d, g, and h outlined in Table l.
More reliable predictions of wave reflection coefficients for rubble-mound breakwaters may be made using the method of Madsen and White (1976) (also see Seelig, 1979). Equations (16) and (18) should be used with the Madsen and White (1976) method to estimate energy dissipation on the seaward face of the breakwater caused by the outer layer of armor units. Figure 9 shows sample laboratory measurements (Sollitt and Cross, 1976) and predicted reflection and transmission coefficients for a rubble-mound breakwater. Observed and predicted reflection coefficients have the best agreement for wave conditions in the tur- bulent zone, but deviate where the Reynolds number becomes less than 10+ due to laboratory scale effects.
2|
Possible Laboratory Scale Effects Turbulent Zone
Measurements Predictions
Hy/L
Figure 9. Predicted rubble-mound breakwater wave reflection and transmission coefficients (laboratory data from Sollitt and Cross, 1976).
7. Spectral Resolution of Wave Reflection.
The significant wave height and period of peak energy density are used to characterize irregular wave conditions in this report. However, a more detailed analysis shows that the reflection coefficient varies as a function of wave frequency for irregular waves. Figure 10 illustrates the decrease in reflection coefficient as a function of wave frequency that is typical of waves breaking on a smooth impermeable 1/2 slope (&€ < 2.3). Nonbreaking waves have a different characteristic shape of the reflection coefficient as a function of wave fre- quency. K,; increases as a function of f for frequencies higher than the frequency of peak energy density (Fig. 11). The shift to high frequencies seems to occur because wave energy is transferred from low to higher frequencies due to nonlinear effects when the waves interact with the structure. Note that this energy shift may produce a range of wave frequencies in which more wave energy is moving away from the structure than is incident to the structure, and the local reflection coefficient may be larger than 1.0 over this range of fre- quencies. Caution should be used when trying to obtain information from the highest frequency part of the spectrum above approximately the 95-percent cumu- lative energy density level because the signal-to-noise ratio is low and the wave speed is poorly known (Mansard and Funke, 1979).
8. Reflection Coefficient Prediction Equations.
Table 3 summarizes the equations recommended for estimating reflection coefficients for slopes, revetments, rubble-mound breakwaters, and beaches.
22
Incident Wove
Spectrem Cot 8 = 2.0
ds/Hs = 4.2
0.8 ds/gTp* = 0.035 Hs /gTp* = 0.0084 = 2.18 0.6 ¢ Kr = 0.45
Reflection Smooth Slope
Coefficient
2 >
S ~
Reflection Coefficient and Dimensionless Spectral Energy Density
0.2 0.4 0.6 0.8 1.0 1.2 1.4 f (hz)
Figure 10. Wave reflection coefficient as a function of wave frequency for an irregular wave condition with breaking waves.
1.2 @
2 $1.0 e 2 @ g 5 oS C) és 0.8 e f=) S$ e@ 0 oO ® 5 a E cot 0 =2.5 ra ds/Hs = 5.9 =i ds/QTp* = 0.0084 Ss 212 ie Sr0N4 Hs/gTp = 0.0014 Oe €=4.3 cs Kis Olay ra
0.2 Smooth Slope
0 0.2 0.4 0.6 0.8 1.0 1.2 f (hz)
Figure 11. Wave reflection coefficient as a function of wave frequency for an irregular nonbreaking wave condition.
23
“Iy pue Jy jo UOFIV[NITBS aTqey{eir e1ow e 30z (6/6T)
83}T99g 10 (9/61) 23FUM pue uaspeEW asp ‘ty JO |3eUTISe aATIBAIaSUOD B BAaATD
SEO
O9IBUFISA BATFIPAIVSUOD IOJ Q’T = 0 9Sp
°SJBUTISS SATIBAIVSUOD B 10F O°T = ,»% 9S)
*T < U IOZ Z PTqe]_ wo1rzy pajeutyjsa ,v
T > W@ 103 O°T = ,0
zeT Tews { (,9T°0) yuer ©
S °3Td 10 12Aa 9+9 q Bt G'S ie G°0 - 9209 a L°Tt- | dxe ,o] -yozya Se = evt\ Fa P| z 9g t) uofjenbe uoz{o};pejag
pt
819 BAyveIq puNow-aTqqny
sayoveg
SJUusMIDADY
ad43q a1njoniqjs5
TH QO = wW pue ¢g < — 103 O'T = 0 ®p (24) uot3zenbe worz Fy 8 juamu0) out Ds
8ut}ItTpead 1037 suotjenbs jo Areuuns
"€ OTqeL
24
VII. EXAMPLE PROBLEMS
The following example problems illustrate the methods of predicting reflec- tion coefficients presented in this report.
kok kk kk Kk KOK OK OK O&K * & EXAMPLE PROBLEM 1% * * * XK KK KKAKKKAKE
GIVEN: A smooth impermeable revetment (nonovertopped) has a toe water depth,
d, = 7.62 meters, a slope cot® = 2.0, and the offshore slope is m= 0.02.
FIND: The wave reflection coefficient and fraction of wave energy dissipated for a wave with H; = 3.05 meters and T = 10 seconds. Illustrate the influ- ence of wave height and period on K, and show the effect of reducing the slope to cot@ = 5.
SOLUTION: From equation (7),
d Hy -8OMl7 be {1.0 - exp E 4.712 Tal opal 1.33)}} Oo
= 0.17 (1.56 x 10? ) {2 - exp Le AID soa fee 2(a plo\(ORO2)) as -32)]\ = 5.85 m
From equation (17)
tané g = eRe = O2_ = 3.58 E 3.05 lly 156 and from equation (15) = .807 (3.58) Ky = a WSUS = 0.56
E2 + B Glss) 2 ses
The energy dissipation coefficient for this example is K4 = 0.69, or 69 per- cent of the incident wave energy is dissipated (from Fig. 3). Other reflection coefficient calculations for 5-, 10-, and 20-second periods for wave heights between 0.3 and 4.4 meters are summarized in Figure 12. Predictions are also shown for a structure with cot6§ = 5. Figure 12 illustrates the influence of wave height, period, and structure slope on Ky
ZS)
dg = 7.6m cot @=2.0 Cor © = 50) —-S—=
O 0.5 1.0 1.5 2.0 Cs) 3.0 S)9) 4.0 4.5
Figure 12. Predicted wave reflection coefficients for smooth impermeable slopes with no overtopping.
% + + a
kk kK kk kk kk kk kk * XEXAMPLE PROBLEM 2% * * * * RK RR RK
GIVEN: The wave conditions in example problem 1.
FIND: The wave reflection coefficients if one layer (n = 1) or two layers (n = 2) of 4,500-kilogram (5 tons) rock at 2,700 kilograms per cubic meter (169 pounds per cubic foot) were added as armor to the revetment with coté@ = 2 Og
SOLUTION: The armor material in this example has Wy) ne 1/3 ¢ -(#) =| oan pian using equation (6). For the case of T = 10 seconds and H = 3.05 meters, equation (18) gives
Hi: hog exp j-.7 2 coté —- 0.5 (5)
1.3 is 1.19 3.05 is exp [a1 ae OW) O55 as) = 0,556
and from equation (16)
fo) if
(°] i]
26
The energy dissipation coefficient from Figure 2 is K4 = 0.86, 86-percent dissi- pation or 17 percent more dissipation than occurred for the smooth slope (see example problem 1). Sample predicted reflection coefficients are given in Figure 13. The preliminary information in Table 2 suggests that further re- duction in the reflection coefficients could be achieved by adding a second layer of armor (n = 2) for wave heights less than 3 meters Giilceeel Sir
1.0 p=TOks cot @=2.0 ree dg = 7.6m 0.8 W = 4,500 kg 0.7 Kr 0.6 0.5 0.4 Os) 0.2 0.1 O OFS 1.0 1.5 20 2.5 3.0 33,5) 4.0 4.5 H; (m)
Figure 13. Wave reflection coefficients for a smooth revetment and revetments with one and two layers of armor stone.
VIII. SUMMARY
Methods for predicting wave reflection and dissipation coefficients for beaches, nonovertopped revetments, and breakwaters are presented. Types of wave energy dissipation considered are wave breaking induced by the structure, wave breaking at the toe of the structure, turbulence produced by wave inter- action with the outer layer of armor, and flow through additional layers of armor. These techniques are combined with the method of Madsen and White (1976) to estimate reflection and transmission coefficients for permeable rubble-mound breakwaters. Factors considered when making reflection coeffi- cient estimates include structure slope, water depth at the toe of the struc- ture, offshore slope, incident wave height and period, the size and number of layers of armor units, and the type of structure. Techniques presented apply to breaking and nonbreaking (surging) waves and can be used for monochromatic and irregular wave conditions.
27
LITERATURE CITED
AHRENS, J.P., Unpublished irregular wave reflection data, U.S. Army, Corps of Engineers, Coastal Engineering Research Center, Fort Belvoir, Va., 1980.
AHRENS, J.P., and SEELIG, W.N., “Wave Runup on a Riprap Protected Dike," Report for the Detroit District, U.S. Army, Corps of Engineers, Coastal Engineering Research Center, Fort Belvoir, Va., unpublished, Apr. 1980.
BATTJES, J.A., "A Computation of Set-Up, Longshore Currents, Run-Up and Over- topping Due to Wind-Generated Waves," Ph.D. Dissertation, Delft University of Technology, The Netherlands, July 1974.
BRUUN, P., GUNBAK, A.R., and KJELSTRUP, S., "Design of Mound Breakwaters," Report No. 6, The University of Trondheim, Division of Port and Ocean Engineering, Trondheim, Norway, Oct. 1979.
CHESNUTT, C.B., "Analysis of Results from 10 Movable-Bed Experiments," Vol. VIII, MR 77-7, Laboratory Effects tn Beach Studtes, U.S. Army, Corps of Engineers, Coastal Engineering Research Center, Fort Belvoir, Va., June 1978.
CHESNUTT, C.B., and GALVIN, C.J., "Lab Profile and Reflection Changes for Ho/Lo = 0.02," Proceedings of the 14th Conference on Coastal Engineering, American Society of Civil Engineers, 1974, pp. 958-977(also Reprint 11-74, U.S. Army, Corps of Engineers, Coastal Engineering Research Center, Fort Beiyosiuray | Vales ON Sm OM Omron).
DEAN, R.G., "Evaluation and Development of Water Wave Theories for Engineering Application," SR-1, U.S. Army, Corps of Engineers, Coastal Engineering Research Center, Fort Belvoir, Va., Nov. 1974.
DEBOK, D.H., and SOLLITT, C.K., "A Large Scale Model Study of Placed Stone Breakwaters,'' Oregon State University, Department of Ocean Engineering, Corvallis, Oreg., 1978.
GODA, Y., “Irregular Wave Deformation in the Surf Zone," Coastal Engineering “pe Mejoow, VWOls WS, ISVS, jxpo L3H20.
GODA, Y., and ABE, Y., "Apparent Coefficient of Partial Reflection of Finite Amplitude Waves," Report of the Port and Harbor Research Institute, Japan, Woo 75 WOo 35 Sapte, IGS.
GODA, Y., and SUZUKI,Y., "Estimation of Incident and Reflected Waves in Random Wave Experiments," Proceedings of the 15th Conference on Coastal Engineering, American Society of Civil Engineers, 1976, pp. 828-845.
GUNBAK, A.R., ‘Rubble Mound Breakwater,'' Report No. 1, The University of Trondheim, Division of Port and Ocean Engineering, Trondheim, Norway, 1979.
HYDRAULICS RESEARCH STATION, "High Island Water Scheme - Hong Kong," Report EX 532, Wallingford, Berkshire, England, Oct. 1970.
JONSSON, I.G., "Wave Boundary Layers and Friction Factors," Proceedings of the
10th Conference on Coastal Engineering, American Society of Civil Engineers, 1966, pp. 127-148.
28
MADSEN, O.S., and WHITE, S.M., "Reflection and Transmission Characteristics of Porous Rubble-Mound Breakwaters," MR 76-5, U.S. Army, Corps of Engineers, Coastal Engineering Research Center, Fort Belvoir, Va., Mar. 1976.
MANSARD, E.P.D., and FUNKE, E.R., "The Measurement of Incident and Reflected Spectra Using a Least Squares Method," Proceedings of the 17th Internattonal Conference on Coastal Engineering, American Society of Civil Engineers, 1979.
MICHE, M., "The Reflecting Power of Maritime Works Exposed to Action of the Waves," Annals des Ponts et Chaussees, June 1951 (partial translation in Bulletin No. 2, Vol. 7, U.S. Army, Corps of Engineers, Beach Erosion Board, Washington, D.C., Apr. 1953).
MORAES, C.D., "Experiment of Wave Reflection on Impermeable Shores," Proceedings of the 12th Conference on Coastal Engineering, American Society of Civil Engi- neers, Vol. I, 1970, pp. 509-521.
MUNK, W.H., et al., "Directional Recording of Swell from Distant Storms," Philosophical Transactions of the Royal Soetety of London, Series A, Vol. 225, No. 1062, 1962, pp. 505-584.
SEELIG, W.N., "Estimation of Wave Transmission Coefficient for Permeable Break- waters,'’ CETA 79-6, U.S. Army, Corps of Engineers, Coastal Engineering Research Center, Fort Belvoir, Va., Oct. 1979.
SEELIG, W.N., "Two-Dimensional Tests of Wave Transmission and Reflection Characteristics of Laboratory Breakwaters,'’ TR 80-1, U.S. Army, Corps of Engineers, Coastal Engineering Research Center. Fort Belvoir, Va., June 1980.
SOLLITT, C.K., and CROSS, R.H. III, "Wave Reflection and Transmission at Permeable Breakwaters,'' TP 76-8, U.S. Army, Corps of Engineers, Coastal Engineering Research Center, Fort Belvoir, Va., July 1976.
SUHAYDA, J.N., "Standing Waves on Beaches," Journal of Geophysteal Research, Vol. 79, No. 21, July 1974, pp. 3065-3071.
URSELL, R., DEAN, R.G., and YU, Y.S., "Forced Small-Amplitude Water Waves: A Comparison of Theory and Experiment," Journal of Fluid Mechanics, Vol. 7, Peele Jano OO pp ESS —o2t
U.S. ARMY, CORPS OF ENGINEERS, COASTAI, ENGINEERING RESEARCH CENTER, Shore
Protectton Manual, 3d ed., Vols. I, II, and III, Stock No. 008-022-00113-1, U.S. Government Printing Office, Washington, D.C., 1977, 1,262 pp.
29
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Fein ae ee i
ifedoan? ak, hain
. o9 eon
ety aun e 3 marban veh re — tao cee (g AVOT spi % iy Y ue AIGA,” Dag “TG = ees ue ath a cee ‘ee Sau" tLe AR pork: ‘ cia ee
"dae 4 Vida ae €
hal ae : om say i ; ; og sew na oe BBA comply +, eT HE Y bar neabrigic ons sven. ih) MS. AR ety vg 4 S409. WAKA Bs hs PELOO~ 250-1 O4- Ao mo Se Lt Liga +) iy a pues , if Tapas phic he
ChB) an og ie yi . ah ct eee SAV Sheds ; 4 Aue 4 » 4 j vol .. ’ A i ; A? ’ » #unr f ahd is Oy M Gc ny “ irre ro iy | 1 PORE a Ta ara yer eng LAE Vee iva vee
Pithis' dy bivaie rT . me, a i | 4 rrr TPG ad: wet " } tio) AVALON A. bt A tos : i
APPENDIX A LABORATORY WAVE REFLECTION DATA
This appendix includes tables of wave reflection data (Tables A-1 to A-7) obtained as a part of this study. The following variables are used:
ID - an identification code assigned to each data run
H - the incident wave height (centimeter); the significant wave height for irregular waves
T - the wave period (second), the period of peak energy density for irregular waves
SURF - the surf similarity parameter = tané/VH,/gT?
H/HB - the incident wave height divided by the maximum breaker height expected at the toe of the structure
D/H - water depth divided by incident wave height KR - reflection coefficient
QP -— the spectral peakedness parameter for irregular wave conditions
3|
Table A-l. Wave reflection from a 1/15.0 smooth slope (monochromatic waves).
WAVE REFLECTION FROM A 1/15,0 SLOPE WITH 0 LAYERS OF ARBOUR A STONE DLAMFTER UF 0,00 CM WATER DEPTH 2 20S Ge
ad) HCC) T(SEC) SURF H/HB DY Aa) KR 8006120001 75 2,00 1°93 095 28.7 ,169 8006120002 130 2.v0 1046 008 1605 e080 8006120003 107% 2,00 1c27 ell 124 0082 8006120004 029 2.90 3090 002 7504 948 8006120005 283 20390 eecd 005 260! o2d3 8006120096 101% 2.20 1094 097 1806 617} 8006120007 1.57 2.50 10748 008 1S5e7 eeS 8006120008 1.78 2.20 1056 off 12e1 082 8000120009 1064 2.20 1055S off $109 .079 8006120010 1.45 2e70 1087 009 14e8 238A 8006120011 lel6 2070 2009 007 1865S 526 8006120012 1070 2.70 1073 010 $207 185 8006120013 06u 3.00 3012 ofa S304 2952 8006120014 1005 3.00 2e4u 06 200e5 0405 8006120015 1041 3,00 2010 008A 1565 0 518 8006120016 1017 3.50 2070 007 1865 564 8006120017 1e6u 3,50 2028 010 1301 457
Table A-2. l\ave reflection from a 1/2.5 smooth slope (monochromatic waves).
@AVE REFLECTIUN FROM A 1/ 2,5 SLOPE WITH 9 LAYERS OF ARMUR A STONE OLAMETER UF 0.00 CM WATER DEPTH & 53.0 cH
ID AC(CM) TCSEC) SURF H/HR D/H RR 8005221248 2087 lee> $070 olf 1A eS 2697 8605221258 oe/7 1e25 od] B25 708 o%16 8y0S221359S 126353 1.25 {079 046 Wes ol? 8005221314 12009 1.25 12450 e4S Ye4 oeS7 8005221424 9048 1.90 2044 o3} Se6 4498 8005221533 8.33 1.50 2061 027 604 124A 8005221342 5.9A 1250 307 020 BoF 704 8005221351 $001 1090 4e32 010 1706 0/56 8005221400 1052 1043 7.42 004 3409 6A 80052e1411 1043 1.93 7065 004 3701 9843 8005221428 7229 1,83 3.39 e2e2 703 e326. 8005221437 120605 1293 2057 037 Wee 0212 8005221447 18.26 eo37 2e77 049 209% 044A 8005221457 14655 2037 3210 039 Beeb 6709 8005221507 8078 e357 4200 72d \ 600 0799 8005221522 4ol4 2.37 5.82 eff - 3208 529 8005221832 3037 2,88 7084 209 {5e7 956 8005291435 13.209 2.98 3.98 o3u Ge? 0007 8005291448 19,88 2.58 cae) 051 2e7 0448 8005291459 1019 3.90 46027 oN Bue ofl] 8005291517 Sot} 3.50 80% 010 1309 68 6005291528 oecS 3,50 6099 el6 8.5 2850
8005291549 10061 3.90 5037 ee? S00 0531
32
Table A-3.
Table A-4.
Wave reflection from a 1/2.5 slope with one layer of armor
(monochromatic waves). WAVE REPLECTIUN FROM A Ly 2,5 SLOPE
LAYERS UF &RMOW
a STONE DIAMETER UF 7.95 CM
WITH 4
WATER DEPTH &
bo) 8001291 513 8001291322 8001291332 8001291 $44 8y0129135) 6001291408 8001291207 8003291218 6001291227 8001291238 8001291248 6001291258 8001291544 8001291552 8001291001 8001291608 60013291459 8001291508 8001291519 8001291535 8001291416 8001291420 8001291435 8001291449
Wave reflection from a 1/2.5 slope with two
BCC!) 4,93 B8el6
11.9%4
14046
13.56
16040 5006 5.68 7.40 Be
1024
11053 eoed 4.07 9963
14,609 4e2eb 6.54
14,607
21047 3.34 6091
12.99
22ee}
$$.0 ¢™ Tcsec) SURF 1.e% 2081 1.29 2019 1.¢9 108} 1.25 1062 129 1°68 1.29 1054 1.5V 3092 1299 3014 1290 e775 1290 eS 1250 2e3S 1299 2023 1-83 bell 1285 Go23 1.243 2095 1.93 2039 20357 573 2e57 4e10 20357 3209 2.357 2056 2,68 7.83 2.48 5047 2.58 32099 2,88 3205
(monochromatic waves).
WAVE REFLECTION FROM A ty 2,5 SLOPE WITH 2 LAYERS OF -ARMUN
A STUNE DIAMETER UF 9,95 CM
WATER DEPTH 2
Id 80021213501 8002121251 8002121243 8002121c3e 80021212095 8002121213 8002121220 8002121228 8002121158 8002121150 8002121143 8002121134 6002121058 8002121106 8002121114 8002121127 8002121051 Buv02e121044 8002120023 8002120014
WCC) eel! 5095
13.023
14944 4650 9.31
12044
1204] 2e4S 5012
10.34
19.84 3.97 722A
13.67
20006 2090 6015
12028
21.78
53.
T(SEC) 1.e5 1.25 1029 10ed 1.90 1.50 1.50 1.90 1205 12093 1,83 1.43 2.57 2.357 2.357 2.57 2.88 2.08 2,88 2,68
0 cM SURF 4eed 2056 1e7e2 1e65 3e42 2e43 2013 2013 5045 4,04 2.84 230 6027 4.39 Jean 2064 Bo44 § 280 Goll 3208
33
H/HR 0/9 | 045 056 052 062 el2 019 0fu 029 eu 037 07 ela 029 044 el2 023 040 058 ok) 018 a) 058
HsRA 008 eee 050 054 016 031 o4{ e044 e007 015 034 047 010 020 037 054 08 el6 032 056
ip Aa 10.7 605 4Ued Bod BoA Seed 1405 903 Pea 5.9 Se2 Uo? 2307 1103 5.5 Jed 1204 b6e4 326 209 1Se7 To7 Yoel 2e4
74 e4ueS 420 3o? $10 Seo 4_3 4e3 210? 1003 Sel 304 1409 703 3.9 ecb 1403 Aeb 4o3 204
AR 025A oN 74 0124 0089 0098 ella 0 348 0298 0208 0240 0216 eei2 0438 0593 0309 0265 0404 0356 0290 0197 0513 0474 0400 e322
layers of armor
«RQ 0194 e146 el26 ella ec3A 0196 olo9 olo7 0278 ecld ol73 ecu 0246 0219 0194 0 5605 0372 0359 0515
Table A-5. Wave reflection from a 1/2.5 slope with three layers of armor (monochromatic waves).
WAVE REFLECTION FROM & Ly 2,5 SLUPE WITH 3 LAVERS OF. ARMOUR A STUNE IT AME TER UF 7.95 CM WATER DEPTH = 53.0 CM
Ip w(CM) T(SEC) SURF 4/HB VAs) KR 6003281253 2075 125 3e76 010 1%ee 23h 8003281501 7007 1ee5 2026 029 609 158 8003281510 15606 1.25 1.61 057 $05 9148 80032R1244 1eo2 1.59 5.A9 NS BPo7 oes 8003281235 304A 1.90 Ue0S ell 1565 .22A 8003281224 5046 105v Bol 018 Go? 0219 8003281218 8.53 1.50 2060 027 6e4 192 8003281205 10028 1.50 20 34- 34 See =o S80 8003781158 12073 1050 2010 42 Yee 0160 8003281135 12098 1250 2.08 043 Gel e158 8003281052 2097 1.83 5.30 009 17.4 el7a 8003261102 0010 1003 3e7y 018 Ao? 182 B00seb1113 «11045 1283 2070 o3u Heb 0155 6003281125 10079 1.03 202d 049 $02 «6 149 8003281039 2.089 2057 7007 008 1Ace% e207 8003281029 5.62 ees? Oe94 ale 9eol oe} 8003281019 11078 2.37 Bed5S 032 ue 0219 8003281009 15.84 2.37 2098 043 Bo4 oe} 8003280922 1265 2045 11020 204 $2ol 293 8003280931 eedu 2,48 9022 006 2107 ee94 8003280940 5246 2.88 6016 014 Ge? 9 540 8003280950 11267 2.88 Bol 0°30 Bod 05309 80032809857 20059 2.68 BolT 053 2e6 66 500 8003281353 60cb 5,90 6099 16 BeS 45} 8v03281 $20 8,86 $.90 5.87 e022 600 9443 8003281343 12033 3220 4.98 oS Hod 9452
Table A-6. Wave reflection from a 1/2.5 slope with four layers of armor (monochromatic waves). WAVE REFLECTION FROM a Jy 2,5 SLOPE wITH 4 LAYERS OF ARMUK
A STONE DIAMETER UF 7,95 CM WATER DEPTH & 23,0 ¢™
Ip M(CM) T(SeC) SURF H/HA D/H Ke 8004011326 2.59 1,29 4094 209 22.2 262 6004011334 7c? 1.25 2032 eo? 703 e168 6004011343 11.288 1.25 1084 045 405 el14 8004011234 ofa 1.50 8673 02 710% 22 8004011225 1050 1.50 bell 095 3502 ef lb 8004011217 Secu 1,50 Velo olf loed e199 B00unL1Ene 7.027 1.50 ec78 eed 763 elo) 8004011127 1.5] $.63 7044 Ca} $501 0180 8004041136 3015 1,93 5.15 0N9 1608 e103 8004011145 6057 1,05 3057 019 Bel 158 8004011156 12018 1.93 2062 036 4o4 0139 8004011116 05U 2037 1209] 002 630.0 304 8c00u011107 2056 2037 740 007 CMe? »eun 8004011058 5.32 2.57 $013 e14 1900 ,eu4 8004011047 11,14 2,57 3,55 030 4.8 ,e62 8004010958 1.040 2,98 12.018 ofu 3800 ecua 8004011007 200? 2098 10601 005 2507 6275 8004011019 4.08 2,08 6065 012 1105 349 8004011034 100610 2098 4053 026 See = 547 8004011258 2073 3.50 10059 097 19064 589 8004011249 6083 3,50 6069 017 708 9446 6004011307 9264 3.50 50648 024 5eS 0459 8004011510 15013 3.50 4083 033 4.0 429
34
Table A-7. Wave reflection from a 1/2.5 slope with one layer of armor
s(Gingegullarwavies) imac 0a Nuits. Geuee OMNe ee
WAVE REFLECTION FROM & ly 2,5 SLOPE wITH 4 LAYERS OF AHMUR A STONE DIA“FTER UF FeQS CH WATER DEPTH & $60.4 C™ Ip wCCM) T(SEL) SURF H/HBR os" AR Qe IXREGULAR wAaVES
8001220925 be76 1.25 2040 o32 Se4 0209 2c 8001220934 Told 1037 2056 032 Gol 147 209 8001220945 7.53 1293 2083 o3t G00 e225 204 80012209558 7.02 279 1079 044 408 «1% oe? 8001221007 6.0} 1248 ecebl 035 Ue 0259 203 8001221017 7.48 teeS 2029 236 Go 0199 ee 8001221028 Teea 1.51 2043 o3u B00 eehl2 $03 8v01221038 80% 1.51 2053 39 Yoel c257 2e7 8001221048 6.57 1.50 2055 037 ee 0737 303 80012211160 10060 2.09 Go13 039 BoS 6597 105 8001221126 11.64 for! 2e2e 0°50 Jol ef€7% 205 8001221158 11076 1,08 2045 049 Bol e511 ec 8001221148 4el? 5.25 7.51 017 Feb 0495 305 6001221158 7.40 3,28 6203 027 4.9 0213 209 8001221220 9,39 1,79 2.92 038A 99. 63707 60 8001221231 10043 4.57 7007 037 305 ee? 103 8001221241 7052 3,28 5.98 027 He8 733 204
BAVE REFLECTION FRIM A YZ 2.5 SLOFE wIlTH 4 LAYERS OF AN*UR A STONE DIa“ETER UF 7,95 CM WATER CEFIR 3 45.0 CC" Ip h(C™) T(SEC) SURE Hs D/H AR ap IRREGULAR wAVES
800123095A 7,69 1,e5 eeoee 033 So? 9192 2.5 80012310908 728% 1040 2064 029 So? 0195 205 8001231018 Bo.16 1.15 ee) 036 S05 o172 360 8001231024 655 1.97 2072 030 504 2215 Pod 6001231038 8.08 1.10 1094 039 Gel 142 2eb 8001231048 9.55 1,48 2e42 035 oH chee 23 80012351057 9,49 1,48 2e41 035 4,8 eceu 203 8001231108 6,08 1.25 2ele 036 Sea eloo 48 8001231117 6.53 1.25 2014 035 SoS o161 Gee 60043231129 8.50 1.51 2025 03d §$.3 06199 3d 6001251139 9.94 1099 2092 036 Bed cf6M 3oel 8001231150 9.99 1.45 2029 0 3A 45 0237 307 8001231200 9.47 1.48 2035 037 Heb 0235 300 8001231210 10209 1.59 2050 036 Ber e2€SA Be 8001231221 8.75 ero)! 2022 035 Sel 0193 306 8001231236 5.00 1.31 2024 o34 Se2 01% 300 8001231248 8,77 1.41 2.34 o3u Sel e107 207 6001231308 13,95 1,08 2.2% 049 Boe 6397 2,9 8001231310 120628 2.00 2098 044 So7F 05868 408 8001231330 13218 1.56 2015S o4A Bed e2oe Sel 8003231344 5200S 3.01 6033 017 B.0 o49R See 8601231352 6.33 3,10 S047 025 Se4 06919 208 8001231403 11033 1.02 2040 240 Ge0 26H Zot 80012314146 12.8A 3,94 5.48 037 Bod 0954 108 8001231437 13.47 1.08 2026 048 yo2 0306 205 8001250924 11092 12591 2018 044 308 0252 303 8003250935 12056 1.07 2e36 e4u yoo 0292 2e6 6001250955 4.85 3.01 6043 oid 905 0493 Soe 8001251005 Webb 3,01 6082 0IS Mod 0491 500 8001251015 12006 1.02 202A 04S 3e6 e2bA 2co 8001251020 7653 Be.20 583 022 be! e4Th 2e8 8001251056 9o71 1,04 2095 033 4.6 e€6N eel 8001251047 11019 4,41 6059 e32 Ue 06901 106 8001251057 146007 4.20 S060 040 So2 0902 Coe 8001251108 12.07 1.51 2017 04S 3o7% 02°49 363 6001251201 9.75 1.82 2090 033 eo o28N 2el 8001251211 12e7a 1007 2034 045 Bod 0290 205
35
Table A-7. Wave reflection from a 1/2.5 slope with one layer of armor
(irregular waves) .--Continued @ave WEFLECTION FROM & 17 2,5 SLUPE elf 4 Layees C6 admon & STUNG UlartrTER ub 7,99 CH maTeR vtPT™ w 43,0 Cc» 30 e(CR) Tcsetc) Sumer 4 7mA on au ue JYREGULOR waves
Goorz71e89 8.05 1033 2000 02h oo! c22% 200 Booy27t09e 8.92 toS2 = 2021 ot2 $o% of04 Sol 8o01271703 Robb 1.55 2052 oS GeO 0232 Pod Boor271710 «©%7A toa! oe? o33 SoS o1%m 209 Bo0271720 10.58 1.93 2056 o3a Sol ecf32 ou Bo0;27173a %eo7A YoeS 2090 o3? SoS o177 408 Goo1271745 9.00 1.50 Polo 03S SoS of14 Son Bo001271755 11.02 1.38 2032 o3S7 Bob e24G Hod O00;27 P4507 10095 1.4) 2018 o3S7 Beo% ef4M Gov Bo0r271817 10092 104! eel ot? Go®% efShm Ged B00;27182A 11.01 1.99 aocS2 ot? Go® ef46 309 Boor2aiioe 13,02 1.91 2,090 ACL} Bo of55 346 BOO 261117 14012 1.95 20%0 Ory BoA oedA 20m 000;261128 5057 3.01 6050 ela @.9 4nd Sed 8y01281138 8.57 2.79% Go75 ofa bod 857 $05 8001241149 10.52 1.98 done 030 $e0 ccoh ee B00;2A1200 12053 4.00 5.09 o3t GeS 0955 10% BoorAr2iy 15673 4.08 Sole 039 Bod o141 209% Gooy2ey22u 13076 1.5! 2004 245 $o% oF 33 508 O00, 2412$80e 1405) 1.07 2021 e4a vo? of08 2.8 O00126124R Lu052 1.94 3eon2 oS0 Sot of0R eee 8001281508 i5030 Tet 1209S 239 Bod ofS? Geol 2001281320 13.09 2005 2077 039 $o% 0535 109 6907241533 15066 1.08 2otk 24e $eS eofot eee @001261 537 0259 3.01 5095 ole Bo3S o470 Go? Oooy2ay3s7 8990 42 20% 4o?o e2a Go? 0908 208 8091281608 ieeeu 20c5 =e 90 03S Bo$ o20G 2e5 G001281419 14045 3.94 519 ete Bo? 2982 20% 60014290963 12.09 1091 Zell 032 Gee cf?) Jeo 6001290955 15021 1.02 2025 032 @.0 9754 2e0 8001291008 7.57 2.75 $007 019 Feed 0452 50h @00:291020 4.67 2.09 010 o13 1909 o44d $08 001291080 %e2n tet Bo1d 027 $08 o20A 2ot O004291002 ttelt 027 = mo 38 027 oS 941 108 G001291052 12.890 @.0e 5oeS o32 Bol e22M 108 0003291105 12.79 to! old ot? @ol 0221 3e0 Boor2%111% 13020 1,02 2022 242 G@oG of3$ oo @001291128 926 1.77 20% 026 So? of0? 2ei
Mave MEFLECT{uUN FRaN™ A jy 2,5 SLOPE elm 4 LOvewS OF ahhUe o STUKE OLast TER Wb 7,95 CH earen VERT a 57.8 C*
to mCC™) T(Sstc) SURE CYA, V2) an ae P92ISII1e 8.49 Lodi 2005 o33 eS ol?! 20? 7912151129 Fein 1,91 2051 028 oo o174 Be? THV2IS1I40 = 90% 1.25 = 10 98 o3o SoA oiSa Sol 7912131149 %o8n 1.20 92001 035 $o% 152 49 7912131200 Fol2 1027? ott 032 eh 156 Sol 7912191209 %12 1057 Poa? 030 eS o15S Sel MM 2I1S1219 P07 1038 2022 o32 Ho 29% 205 79121312729 %86 1023 2008 030 $o% e200 oF 79121351259 Pean 1.50 2010 033 e0 o0f549 506 7912131297 « %.07 4259 2009 033 600 clom 30% 912131297 8.84 1.41 2025 o32 $o% 158 3ou FIL2L31500 8 9094 1.20 10% o37 $8 150 400 FL2IZISi0e 10.87 1.93 2032 233 So3¥ 0205 205 7T9V2151520 10075 1.9! a3} ott Go4 2200 Coo THL2ISTS3A 11078 1o4! 20%0 °SA 40% of%h eS VOL2ISII4T Ll0eH 1o4l 2007 o3A Go 0194 dee 7912151357 12030 Leo! 2015 038 Go? o211 ed P212131605 12032 1051 2016 o38 Go? o21i 3o8 7912171051 See? 10S! =. 2007 004 Ged 01%] 308 9; 2171141 Te7e 2091 = =Se2e 019 Fed 0450 eS 7912171150 Toba 2.01 $220 219 Pot o4ST bee 791217125) 5.99 2ole 003 ol2 $208 0420 4eS 7932371303 Solu 2.76 6013 032 fre2 e423 Uo? POL2171310 90 9S 1075 = 2eoA2 028 GA eu? eel MO21713a1 90% 1077 2079 028 SoA oe€43 2ol VOL2I7ISSL «dele 3092 ©5057 027 Go% 312 30? POL217L341 «LL6)d¥ 3.02 5055 °27 oF 6351S 108 PI2L719S1 «Godt @.00 Se2e 033 cO0 0933 108 7912171059 12247 1.51 2019 240 @ceS ef2G GeO PIL2I71190 «12093 LeSt 2009 040 GeS e233 ou TOL217L4O§ 15037 @,74 0.593 030 @.4 928 1.7 001091029 So7S 2.98 020 ela = 1Mel 045 Uo2 Boor0%1042 11033 fol? 202 o32 Sel cf ea 0001091055 11058 1o7? 2062 oS2 Gol 257 Pee 9003091109 13039 Lo77? 2od4 o32 Sol cf37 ee Gooyo%120 1402A @,00 Se2e 033 @oQ 0933 203 @00;991132 14.52 4.00 $228 o33 G00 0939 2e3 80010911493 19005 Led! 1098 045 $o% 01469 So? 0003091137 14048 1050 2008 o4a @ce0 0108 300 Boe11000%1 %.02 2.78 022 O04 44a Hed Goo1i00%07 8.94 2.78 022 2044S 309 Bo0,;190918 19.140 1.08 04a e223 300 000110093: 14074 1.08 043 et@2S 209% Goo1100942 15018 1,08 2010 24a e220 301 00011009Se® 10.42 @o3S 5e0® 037 oS$e@ aol 8061101009 10.8) a,00 5oOt o37 0901 209 Ged, to1022 52081 5.01 028 ola 045A Sel G@oorioro5s 5.86 3.0! @o22 ofa 095A Sel GOOLLoLoS@ 16054 leSt «1098 245 0208 500
Table A-/7.
WAVE REFLECTION FROM A 1/7 2,5 SLOPE wITH 7 LAYERS OF akrur
Ip
8001101116 8001101127 8001101158 8001101150 8001101205 8001101215 8001101228 8001101242 8001101256 8001101512 8001101328 8001101344 8001101358 8001101413 8001110834 8001110845 8001110858 8001110910 8001110920 68001110930 8001110944 8001130952 Boorisii002 6001111013 8001111020 8001111037 8001111049
& STONE OJAFETER UF 7,95 CM wATER DEPTH = 57.8 CM HCC) T(StC) SURE IRREGULAR wAvES 6075 $,01 5.79 6075 3,01 5.79 10.08 2.98 0.55 10.07 2,48 4.53 15051 12.98 2070 15.20 1.98 2073 160.04 3,94 be} 15.57 3,94 4099 15e3n 2.03 20eh0 15232 2.03 2059 15633 2203 2059 16023 12>! 1088 1ool3 1.5! 1048 160079 1068 2095 8.58 15! 2054 8.95 1091 2053 9016 feed 2005 9090 1053 = 2043 10.20 1,20 1088 10693 1.53 2e32 106031 1.c6 1098 9.68 1.e4 1099 12654 1.51 2015 11004 1.41 2ce07 11006 1041 2el2 Leol3 1.91 2el7 9655 1.e0 1094 9.74 1.30 2008
Booi1lii106
H/HB
e016 016 024 oP 036 035 037 36 04] 044 044 050 030 049 027 038 033 30 03a 0 33 0 36 035 038 038 036 037 036 034
WAVE REFLECTION FROM A ty 2,5 SLOPE
WITH 4
10
Bo01111156 8001111205 8001111216 8001111220 8001111238 8001111251 Boorti1308 8001111310 8001111329 Boo1111342 8001111355 8001111408 8001111419 8001141005 Bv01141019 8001141030 8001141042 6003141054 8001141105 Boo141115
LAYERS OF ARMOUR
& STOKE OJ AMETER UF 7.95 CM wATER DEPTH = o3.C ¢” wC(CM) T(SEC) SURF IRREGULAR wAVES 9.44 1.50 Pell 9058 1.e7 2006 10053 1.38 2015 10023 1.29 10A8 11057 1650 2022 10.30 1ee5 1094 100612 1.350 2004 12.89 1091 eell 1eel4 1.290 2e15 12007 1290 2015 12087 105) 9 ol I 10204 1.24 1095 10012 1.025 1096 13-33 1.91 206 14.54 1.96 2005 15.22 1,08 2e16 8.54 Pw Ae) Ue? 5.03 ove 5.80 10006 1.7? 2078 12266 3.82 5.37 15026 3,96 4085
8001141126
S) I
R/HH
031 032 032 036 033 035 033 038 0 36 035 0 3A 035 035 039 042 04} 019 013 oe? 027 033
N74
Aeb Bee Se? Se? Ue3 Ue4 306 307 38 308 3.8 326 $e6 304 605 605 603 5.4 507 503 5? 609 Ue? §.9 S02 408 601 5.9
dss
607 604 bel 602 505 bel 602 49 Seo2 5.2 4.9 603 bee Ue? 403 u, 704 1102 603 5.0 Yel
KR
e457 0455 e484 0484 ee42 0247 0225 0230 0325 0319 0521 0212 0209 0244 0142 oi44 0135 0185 0155 0190 0115 oi42 0206 0185 0182 0206 0145 0140
QP
Wave reflection from a 1/2.5 slope with one layer of armor
(irregular waves) .--Continued
APPENDIX B METHOD OF MEASURING WAVE REFLECTION COEFFICIENTS
The method of Goda and Suzuki (1976) is used to determine laboratory reflec- tion coefficients for monochromatic and irregular conditions. Also used is the energy balance approach for both types of waves, so that wave energy transfer between frequencies and variable amounts of reflection over a range of fre- quencies can be considered. This approach gives a reflection coefficient that is formally defined as the square root of the ratio of the reflected wave energy to incident wave energy. For an idealized case where no energy transfers occur, the reflection coefficient is the ratio of reflected and incident wave heights. Reflection coefficients are determined by placing two or more gages several wavelengths seaward of the structure. Each pair of gages then gives an esti- mate of reflection coefficients.
In these experiments wave records were sampled simultaneously at three wave gages (Fig. 2) at a rate of 16 times a second to obtain 4,096 data points for each run. An FFT was then performed cn each wave gage record to determine real and imaginary spectral coefficients, A and B, at each spectral line j. Let the subscripts } and » indicate the landward and seaward gages in a pair. The re- flected and incident wave amplitudes for each gage pair for each spectral line are then given by
Eroes 1 Ka> - Ay cos kA2 - B, sin kA2)? + (By + Ay sin kAL - By cos kAL)? (B-1)
1 2lsin kA
Natkelss 1 KA> - A, cos kA& + By sin kAg)* + (Bp - Aj sin kAg - By cos kAL) 2 (B-2)
r 2lsin kA
A,B = spectral coefficients k = wave number = =" @=3)) AX = gage spacing Only gage pairs with 0.05 < = < 0.45 (B-4)
are used in the analysis, and wavelength, L, is determined from linear theory for irregular waves,
2 L= RE cearaln (24), (B-5) 27 L
and may be found using Dean's (1974) stream-function theory for steep monochro- matic waves (see App. C).
All estimates of reflection coefficients found using the above procedure are averaged at each spectral line to give an incident wave amplitude and re- flection coefficient for line j:
38
(az)? = average incident wave amplitude at line j
i j P P ar (k,)2 average reflection coefficient at line j = (2) iL
The reflection coefficient is then determined by taking
(B-6)
Irregular wave information is displayed in the form of band spectra, using 11 lines per band and using a variation of equation (B-6) to determine the reflec- tion coefficient for each band.
In the case of monochromatic waves, a nonlinear waveform is described by a Fourier series with each component moving at the speed of the primary wave, and equation (B-6) is used to determine the reflection coefficient.
5)8)
APPENDIX C NONLINEAR WAVELENGHTS AND WAVE SPEED
In the real-time analysis of wave reflection it is necessary to know the wavelength or wave speed. Linear theory gives excellent predicitons for low steepness waves, but tends to underestimate both length and speed for large waves.
Dean (1974) gives tabular values of wave speed and wavelength for finite height waves that can be approximated by the empirical relation,
ee (C-1)
where L and C are wave speed and wavelength, L, and C, are deepwater wave speed and wavelength determined from linear theory where
eT?
aa (C-2)
Lo = La is the local length determined from linear or Airy theory and a and b are empirical coefficients. Airy wave theory predictions and values of a and b are plotted as a function of d,/Ly in Figure C-l1, where d is the water depth.
Ss
2.0 40 lg 36 16 32 4 LIge halle @ a (RVLe))? oe 2 24 °o 4 “= 1.0 20 a S 08 16 06 2 0.4 8 0.2 be O (0) 0.001 0.0! 0.1 1.0 10 d/Lo
Figure C-1. Coefficients for approximating nonlinear wave speed and wavelength determined from stream-function theory.
40
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