Estimation of wave reflection and energy dissipation coefficients for beaches, revetments, and breakwaters

(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

Approved for public release;
distribution unlimited.

U.S. ARMY, CORPS OF ENGINEERS
COASTAL ENGINEERING
RESEARCH CENTER

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SO Fort Belvoir, Va. 22060

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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

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Department of the Army
Coastal Engineering Research Center (CERRE-CS)
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F31538

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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.

FORM
DD . san 73 1473 EDrTion OF t Nov 65 1S OBSOLETE UNCLASSIFIED

SECURITY CLASSIFICATION OF THIS PAGE (Wren Data Entered)

y De

‘ae em np aeahe i fa

ate Wt 4 Ve eter oii
beat ey Le Se, aa

Me

‘er bak r tin Ly th 1 Rm rere e oaew
7 ve y , ’ ’

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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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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adoTsS G°Z UO T & AOJ SJZUSTOTFJ9OO uOTJOeT JOA BAeM Fo uOSsTAedwod Wy ‘yh sANBTy

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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ak, hain

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— tao cee (g AVOT spi % iy Y
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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

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ie yi . ah ct eee
SAV Sheds ; 4
Aue 4 » 4 j
vol
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| 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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-T3uqg TeyseoD *s*n -- aaded Jeoptuyoey) -- *wo /z : *TIT : <«d [or]
*T861 ‘a0TAIeg uoTIeWAOCZUT
TeoTuy.eL TeuOTIeN WoAZ aTqeTyTeae : °en ‘pyTetT;F3utads { aaquay yoieasay
Suyiseutsug TeIseOD *S*n : “eA ‘SATOATAG Y4aoqY -- “*sueryy *q uyor
pue STT99S *N WeETTTHM Aq / szejemyPeIq pue ‘sjuamjzeAeI ‘sayoeaq 105
SJUeTITFJI09 uotTiedtsstp AZisue pue uoTIeTJeI APM JO uUOTIeUTISY
°N WETTTIM ‘3TTeeS

Le9 T-1T8 *°ou darecne €072OL
"I-18 °ou iteded
TeoTuysey *19que) YyoIeessy BuTiaeuT3uq [eqiseo) sg :SeT1ES “TIT
*d uyor ‘suszyy “Il “*eTITL *I «“SeAemM *g *uoT{DeTJAaI aAeM °G = *SqUaW
-12Asy °h *uotjedtsstp AS1suq °*¢ *slazemyeoIg °Z *soyoeag °]
*suoT Tpuoo
SAeM JeTNSe1IT pue ITJeWoLyouow Buyyeerquou pue BuTyeeI1q yIOq 04
Ajdde senbtuyseq uot oTpeid ayy *szejzeMyeeIq pue ‘squewqoneI *sayoeoq
A0J SqSTITFJ909 uoyjedytsstp ASisue pue uoTAoeTJeq sAeM BuTAOTpead i0F
spoyjem doTeaep 07 pasn aieM sjueweinseam A10ReIOGeT QOOSH ueYyI e10K
*saduetezel TeoTydeaZoTTqtq sepntouy
(1-18 °ou £ aaqUaD YoIessoy BuTI9eU
-}8uy Teyseop *s*n -- aaded Yeotuyosy) -- *mo te 8 Age 8 Gl [ol
"1861 ‘@0TAIeg uot IeWIOZUT
TeoTFUYySe] TeUOTIEN Worzy aTqeTyTeae : sea ‘pyetys3utads { aaquag yoireesoy
Suyaseut3uq TeIseop *s*n : “BA ‘aTOATaG 310g -- *suaeruy gd uyor
pue 3TT299g *N WeTTTTM Aq / szejemyeo1q pue ‘squamqenalz “sayoreq 10;
SJUSETOTJJI09 uoyTjedtsstp ASieue pue uoTIIeTJaI AAPM JO UOTIeWTISY
‘N WETTTIM ‘ST TeeS

Lc9 I-18 *ou datgcn: €070L
*T-18 °ou ieded
TeotTuyosy, *19qUa) Yyoieessy BuTiveuTsuq TeqIseo) *s°n :SeTies *TII
ed uyor ‘suoryy “II “TTL °I °seaeM °9 “UuOTIDeTJaI aAeM *G = *SqUem
-JeaARY 4h euoTjzedtsstp ASisuq °¢ *slsjzemyeolg °7 *soyoeeg °T
*suoT3Tpuoo
aAemM AeTNSei1i~ pue ITJeWoOIY.ouoCM BuTYeaiquou pue BuTyeeaq yYyI0q 0F
Ajdde sanbtuyseq uototpeid sayy *sieqjemyeeiq pue ‘squewjeae1 ‘sayoeeq
1OJF SJUSTOTFJI09 uoTIedTsstp AZiseua pue uot IeTJeI |aaem BuTWOTpead ao0z
spoyjom doTaaap oj pesn eiam sjuaweinseaw A10je10GeT CQOOSY UeYI e10W
*saouetleyjel Teotydeiso0TTqtq sepntouy
(1-18 °ou £ iaqUaD YdIeaSey BuTI28eU
-T8uq Teyseog *S*q -- aeded [eotuydel) -- -wo /7 : “TTF : *d [04]
"1861 ‘e0TAIeg uot eWIOZUT
Teotuyoesy TeuoTjeN WolZ sTqeTTeae : *ea ‘pyTetz3utads { aaquayg yoieessy
SutiseuTsuy TeIseoD *S*n : “eA SATOATOGY YA0gY -- “sueayy *q uyor
pue 81728285 °N WertTTIM Aq / sazeqemyeoIq pue ‘squewjeAe1 ‘sayoeaq 105
SqueToTsyye0o uotjedtsstp ASisua pue uoTIIeTJeI APM JO UOTIeUTISY
°N WETTTIM ‘8TTEeS

Le9 I-18 *ou dargsn: €O0¢2OL
*I-1g °ou iteded
Teotuysey °19}Ua) Yyoleassy BuTiseuTsuy Teqseo) “*s°g :SeTies *TII
*d uyor Ssueiryy “Il “eTITL *I “seaemM °9 sUuOTIIAaTFAaI |AeM *G = *SqUeU
-JaARy °h suoTjedtsstp Asisuq °¢ *slaqemyeoig °7 *soyoeeg °{
*suoT }puoo
aAem AeTNZelit pue IT}eEWoOTYOUOW BuTHYeaAquou pue BuTyeaIq YIOG 03
Atdde senbtuyseq uotzItperd eyy “sisqemyeeiq pue ‘squemqeAer ‘sayoraq
IOJ SqUSTITJJaA09 uoTJedtTsstp AZisue pue uoTIIeTJaI aAeM BUTIOTpeiad 105
spoyjzew doTeaep 07 pasn aiam sjuoweinseaw AiOjVeIOGeT QOOSY ueYyI e10W
*saouetejel TeoTYyderZ0TTqQTq SepnTouy
(I-18 °oOu £ 1aqueD YoIeaSey BuTIsZ—eU
-T8ug Teqseon *S*n -- aeded [eozuyosey) -- ‘wo /Z : “TTT : *d [04]
"1861 ‘Sa0FAITeG UOT IeMIOZUT
TeoFuyoIL TeEUCTIEN WOIZ aeTqeTtTeae : “eA ‘SpTeTFsutadg { Aaquag yoIeasoy
SuTisaeuzsuq TeIseOD *S*n : “eA ‘SATOATAGY WAIOY -- *sueTyY ‘*q uUYor
pue 3172995 “*N WeTTTIM Aq / siaqzeMyYeeIq pue ‘squeWjZeAeT ‘sayoeeq 10Z
SqueTOTJJI09 uot edTsstTp ABSisua pue UoOTIIeTJeI BAPM JO UOT IeWTISY
"N WETTTIM ‘3T Tees

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