AD-A247 562 TECHNICAL REPORT CERC-92-1 IIrIIy CrIpsIII11l11 LABORATORY STUDY OF A DYNAMI f EBERM REVETMENT by Donald L. Ward, John P. Ahrens .Coastal Engineering Research Center _ DEPARTMENT OF THE ARMY Waterways Experiment Station, Corps of Engineers I3909 Halls Ferry Road, Vicksburg, Mississippi 39180-6199 * S ELECTE -------- January 1992 Final Report Approved For Public Release; Distribution Is Unlimited - o2... 06915 Prepared for DEPARTMENT OF THE ARMY US Army Corps of Engineers Washington, DC 20314-1000 Under Work Unit 32432
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AD-A247 562 TECHNICAL REPORT CERC-92-1
IIrIIy CrIpsIII11l11 LABORATORY STUDY OF A DYNAMI
f EBERM REVETMENT
by
Donald L. Ward, John P. Ahrens
.Coastal Engineering Research Center
_ DEPARTMENT OF THE ARMYWaterways Experiment Station, Corps of Engineers
I3909 Halls Ferry Road, Vicksburg, Mississippi 39180-6199
* S ELECTE--------
January 1992Final Report
Approved For Public Release; Distribution Is Unlimited
- o2... 06915
Prepared for DEPARTMENT OF THE ARMYUS Army Corps of EngineersWashington, DC 20314-1000
Under Work Unit 32432
Destroy this report when no longer needed. Do not returnit to the originator.
The findings in this report are not to be construed as an officialDepartment of the Army position unless so designated
by other authorized documents.
The contents of this report are not to be used foradvertising, publication, or promotional purposes.
Citation of trade names does not constitute anofficial endorsement or approval of the use of
such commercial products.
REPORT DOCUMENTATION PAGE FormN oAoIAo
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9. AGENCY USE ONLY (Leave blank) 2. REPORT DATE 3. REPORT TYPE AND DATES COVERED
January 1992 Final report Jul 88 to Jan 914. TITLE AND SUBTITLE S. FUNDING NUMBERS
Laboratory Study of a Dynamic Berm Revetment
6. AUTHOR(S)
Donald L. WardJohn P. Ahrens
7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES) S. PERFORMING ORGANIZATION
USAE Waterways Experiment Station REPORT NUMBER
Coastal Engineering Research Center Technical Report3909 Halls Ferry Road CERC-92-1Vicksburg, MS 39180-6199
9. SPONSORING / MONITORING AGENCY NAME(S) AND ADDRESS(ES) 10. SPONSORING / MONITORINGAGENCY REPORT NUMBER
US Army Corps of EngineersWashington, DC 20314-1000
11. SUPPLEMENTARY NOTES
Available from National Technical Information Service, 5285 Port Royal Road,Springfield, VA 22161
12a. DISTRIBUTION /AVAILABILITY STATEMENT 12b. DISTRIBUTION CODE
Approved for public release; distribution is unlimited
13. ABSTRACT (Maxirnum 200 words)
A series of physical model tests was conducted in a Wave flume to studythe effect of dynamic rubble protection in front of a vertical bulkhead. Thisrubble berm revetment was placed in front of a bulkhead with stone sized suchthat wave action would reshape the berm into an equilibrium profile. Profilesof the berm were taken before and after testing, and berms classified asofsafe," "intermediate," or "failed" depending on protection provided to thebulkhead. Regression analysis was used to determine the quantity of stonenecessary to provide protection for a wide range of storm conditions. Anequation also was developed to quantify the reflected wave energy and thus
determine the energy dissipation by the berm.
14. SUBJECT TERMS 15. NUMBER OF PAGESBerm Revetment 78Bulkhead 16. PRICE CODEDynamic revetment
17. SECURITY CLASSIFICATION 18. SECURITY CLASSIFICATION 19. SECURITY CLASSIFICATION 20. LIMITATION OF ABSTRACTOF REPORT OF THIS PAGE OF ABSTRACT
UNCI.ASS IF I ED I.NCIASS I FIElNSN 75-f0-01 280 5500 Standard Forr- 298 (Rev 2-89)
£ by ANS, S6 i 8
PREFACE
The investigation described in thia report was authorizod as a part of
tho Civil Works Research and Development Program by Iloadquartors, US Army
CorpnJ of Engineers (1IQUSACE), Work was porformed under Work Unft 32432,
"Design of Revetments and Seawalls," at the Coastal Engineering Research
Canter (CERC), US Army Engineer Waterways Exporiment Station (WES), The
IIQUSACE Technical Monitors were Messrs. John 1, Lockhart, Jr,; John G.
1lousloy; James E, Crows; and Robert 11, Campbell, Dr, C. Linwood Vincent wa1h
CERC Program Monitor.
The study was conducted by personnel of CERC under the general direction
of Dr. James R. Houston, Chief, CERC, and Mr. Charles C, Calhoun, Jr.,
Assistant Chief, CERC, Direct supervision was provided by Messrs. C. E,
Chatham, Chief, Wave Dynamics Division (WDD), and D, Donald Davidson, Chief,
Wave Research Branch (WRB), WDD, CERC. This report was prepared by
Messrs. Donald L. Ward, Principal Investigator, WRB, and John P. Ahrens,
Research Oceanographer, WRB. The model was operated by Mr. Willie G. Duboso,
Engineering Technician, WRB. Assistance with data analysis and graphics was
provided by Mr, John M. Ileggins, Computer Technician, :;RB, This report was
typed by Ms. Myra E. Willis, WRB, and edited by Ms. Lee T. Byrne, Information
Technology Laboratory, WES.
COL Larry B. Fulton, EN, was Commander and Director of WES during report
publication, Dr. Robert W. Whalin was Technical Director,
, ao /~a 'z
- AoOUIL10f or
NTIS G1RA&IDTIC TAB 0
Unanlouflc* E33u~ttirLo-o-
Availability Codes
Net
SpeoLi.
P " Av L a d
CONTENTS
Page
PREFACE...................................1
PART I: INTRODUCTION ........................... 3
Background...............................3Problem.................................4Purpose.................................4
PART II: DEFINITION OF TEST PARAMETERS .................. 5
Wave and Spectral Parameters......................5Material Parameters .......................... 7Berm Parameters ............................ 7
PART III: FLUME SETUP, TEST CONDITIONS, AND RESULTS. ............ 9
Flume Setup...............................9Test Conditions............................11Results.................................11
PART IV: DISCUSSION............................16
Critical Mass Analysis........................17Wave Reflection and Energy Dissipation................22
PART V: SUMMARY AND CONCLUSIONS.....................24
REFERENCES................................25
APPENDIX A: TABLE OF PROFILE SOUNDINGS ................... Al
APPENDIX B: INITIAL AND EQUILIBRIUM PROFILES ................ Bl
APPENDIX C: NOTATION. ........................... C
2
LABORATORY STUDY OF A DYNAMIC BZM REVTMI N'TL
PART I: INTRODUCTION
1, Conventional revetments are designed to be statically stable; that
is, no motion of the armor storae is anticipated, Stones in the armor layer
are sized and placed such that their weight and interlocking will preclude
movement during wave attack, In contrast, a dynamic rovetment is designed to
allow wave action to rearrange the stones into an equilibrium profile.
Because stonaa are allowed to move, a smaller stone size is us@d for the
dynamic revotment than for the static revetmant, but dynamic rovetmants
require a larger quantity of stone to allow for the reshaping of the revetment
into an equilibrium profile, The dynamic revetmont is effective becaune the
largo mass of stone near the still-water level (SWL) disrupts the wave action
and dissipates wave energy. Although dynamic rovetments require a larger
quantity of stone, these costs may be offset by the typically lower cost of
smaller stone, and, because size i loess critical, a more cogt-tiffective ufo
may be made of quarry output, In addition, smaller stone is le . expensive to
handle, and, since initial placement is not critical, dynamic rovotments moy
be dumped in place rather than the stones being individually placod,
2. The concept of a rubble breakwator having a dynamic response to
wave attack is not now Per Bruun has commented frequently about the high
stability of "5" shape profiles of some vary old breakwaters in Plymouth,
England, and Cherbourg, France (Bruun and Johannesson 1976), and the borm
breakwater concept developed by William Baird (Baird and Hall 1984, Hall
1987) is an adaptation of this "S" profile. The idea of a dynamic rovetment,
however, seems to be of more recent origin, Van Hijum and Pilnrczyk (1982)
and Pilarczyk and den Boer (1903) presunt data and summarize some of tihe
Dutch experience with gravel beaches and cobble-sized revetmtitz, and
research has boon initiated in England on the response of shingle beaches
to wave action (Channoll, Stevenson, and Brown 1985; Powell 1988), Recent
research in The Netherlands and England is motivated by a need for fundamental
understanding of shingle beaches, how they might be nourishod, and if slingle
3
benches could be umad in some situations instead of a traditionnl sutatically
stable riprap revotment,
3. In the United States, Johnson (1987) found that gravel beaches and
dumped rubble are frequently cost-offectiv alternatives to using sand for
beach nourishment and placed mtone for rovetments, respectively, Johrneon's
findings were obtained from extensive experience on Lakes Michigan and
Superior, where fluctuating water levels created enormous problems for con-
ventional shoreline protection, This experience indicated dynamic revetments
were not vulnerable to too scour, overtopping, or flanking, Advantages cited
by Johnson for coarse material on beaches include a long residence time and tn
abiLity to stay in the vicinity of the water line, Other advantango are
similar to those noted by Baird and Hall (1984), i.e., ease of placomont and
lower unit cost.
Problam
4, Comprehensive rosoarch efforts conducted recently in The Netherlands
resulted in detailed and quantitative findings on dynamic stability (van der
Moor 1988), Although the findings were based on extensive laboratory work and
data analysis, the data are of limited applicability in the United States
because van der Moor's tests were conducted in relatively deep water, who~rof
most US Army Corps of Engineers (USACE) probloms involving shoroline erosion
and protection are in shallow water, There is no design guidance on tho use
of dynamic revetments for coastal protection that is applicable to the shallow
waters typically encountered In USACE projects,
5. The purpose of this study was to determine how dumped stone might
protect a vertical bulkhead in shallow water, and particularly to determine a
means of calculating the minimum quantity of stone necessary to provide
a !quate protection (the "critical mass"). However, the information on
reshaping, equilibrium profile, and dynamic stability is of general
applicability and should prove of value to a wide range of coastal projectg,
4
PART II: DEFINITION OF TEST PARAMETERS
6. Inconsistencies among authors in notations, dofinitions of paramno-
torm, and the mothods by which a value for a parametor is obtained groatly
complicate the task of comparing results from different studies, In this
report, notations will follow guidelines publishod by the International
Association for llydvaulic Research in its "List of Sea State I'aramForrs"
(1986) where applicaSle, Additional parameters, definitions, and method uod
to determine the value of certain parameters are given in the following
section,
Wnve and gnegtrnl Pargmetarn
7. Wave heights used in this report are the heights of the zeroth
moment (l(o)* and are obtained as four times the square root of the zeroth
moment of the potential energy spectrun, The h1's of the incident spectra
are separated from the im's of the reflecoted spectra by the method of Coda
and Suzuki (1976), using a threo-gage array, Two arrays are used, one to
measure the H.'s near the wave generator (Array 1) and one near the
structure too (Array 2),
8, Peak period (Tp) is the wave period agsociated with the highest
energy density of the spectrum, This was obtained by dividing the spectrum
into 256 bands and finding tho period causing the highest onergy donoity over
11 adjacent bandwidths,
9, Peak period was used to estimate the desired length of A t st run by
multiplying the dosired number of waves by the peak period,
10. Reflection coofficint is commonly defined as the ratio of
reflected wave height to incident wave height, This is clearly innpproprinto
when incident and reflected wave heights are doecribed by different spectra,
Reflection coefficients were therefore determined by the energy of the
respective spectra, following the method of Coda and Suzuki (1976),
* For convenience, symbols and abbreviations Are listed in the Notation
(Appendix C).
whare Kr i the refloction coufficient and M x and I nro tha enargy of
the rerflected and incidont spetcra, roepectiv ly,
1i, Ieflaeted wave he hit in obtLained ns the product of refloaectill
coefficient and incident wave height,
Hr • Kr Hmo (2)
where II, in reflected wave height,
12, Wave heights and periods are frequently reported in other investi-
gations in terms of significant wave height (11.) and average wave period (T.),
where !11 in the average of the one-third highest waven. Both 1I1 and T y
are included in the data in this report to simplify comparison with oLher in-
veatigationo, Because the measured II, includes both incident and reflected
wave energy, the incident 1l, is estimated from the reflection coefftcient as
1 1(t) (3)
whore r1(j) is the incident significant wave height and Its(L) is the
comtbined incident and reflected significant wave height. I1 (,) was determined
as the average from the three gages in the array, Average wave periods in
this report were determined as
O(4)
where in, and in2 are the zeroth and second moments of the incident
potentiial energy Apectrum, respecCively,
13. The spectral width or peakednes determined from the wave record is
given as Qp , doefinod by Coda (1970) as
(Q2 . 3f] d 5
where f is frequency and S(f) is the wave spectral density function for
the given frequency.
6
Material Parameters
14. Small sizes of stone, such as those used in the current study, are
frequently measured by sieve analysis. Larger stones are described by their
nominal diameter dn defined as
d. (6)
where W is the weight of the stone and w, is the specific weight of the
material. The nominal diameter of the median stone weight is d.(50)•
Berm Parameters
15. Figure 1 illustrates the major berm parameters prior to a test run.
Berm width WB is the horizontal length of the berm as it was constructed at
the beginning of a test. Berm height hB is the average vertical distance
from the SWL to the horizontal berm surface at the beginning of a test.
16. A typical after-test profile is shown in Figure 2. Berm crest
height h; and berm crest length lc are the vertical and horizontal dis-
tances respectively from the intersection of the SWL and the equilibrium pro-
file to the conspicuous berm crest formed by the wave runup. Erosion depth
he and erosion length 1e are the depth and horizontal distance, respec-
tively, of the revetment toe from the intersection of the SWL and the
equilibrium profile.
17. Revetment Response Category (RRC) is a simple evaluation of the
performance of the revetment during a test where "F" indicates the revetment
failed, "S" indicates the revetment was safe, and "I" indicates an inter-
mediate condition. For these tests, a failure was defined as exposure of the
bulkhead, whereas a safe condition indicated that neither sand nor water
overtopped the bulkhead. These RRC's are described in more detail in
paragraph 30.
7
1 0 0 TE G --
0 80
ELL 70 h
60CSWL.9- 60 ,
so
40 60 80 100 120 140
0 20 40
Horizontal Distance trom Bulkhead (cm)
Figure i. Berm parameters from the pretest profile
100
" concrete slope
post-test profile20 3000 Waves
00
E
0
U- 70 -
o F60 F ----
,- -SWL
50
40 --100 120 14
0 20 40 60 80 140
Horiz~nta( Distance from Bulkhead (cm)
Figure 2. Berm parameters from the posttest profile
8
PART III: FLUME SETUP, TEST CONDITIONS, AND RESULTS
Flume Setup
18. Model tests were conducted in the USACE Coastal Engineering
Research Center's (CERC's) 0.46-m-wide by 0.91-m-high by 45.73-m-long glass-
walled wave flume (Figure 3) using an undistorted Froude scale (Stevens et al.
1942) of 1:16 (model:prototype). Irregular waves representing Joint North Sea
Wave Project (JONSWAP) spectra (Hasselmann et al. 1973) were generated by a
hydraulically actuated piston-type wave maker. Wave data were collected for
each run using two arrays, each consisting of three electronically driven
resistance-type wave gages. Wave signal generation and data acquisition were
controlled using a DEC MicroVAX I computer, and data analysis was performed on
a DEC VAX 750 and 3600.
ARRAY 1 ARRAY 2
WAVE GAGES
P ISTO N TYPEWAVE GENERATOR 300
2 - 1 43 -- -
45 7 . . . . . . .. . .
Note: Distorted scale: lV = 5H
All measurements in meters.
Figure 3. General la}out of wave flume
19. The test sections were placed approximately 35.4 m from the wave
board. Figure 4 shows a typical initial and equilibrium profile for a dynamic
revetment. All initial profiles except for Test 4 had a horizontal berm and a
seaward face on a slope of 1:1 (vertical:horizontal). Test 4 used the
equilibrium profile from Test 3 as a starting profile to determine how
sensitive the equilibrium profile was to initial conditions (see paragraph
25). A bulkhead was simulated in the model using a plywood board to
terminate the rubble on the landward side, located at 0.0 on the horizontal
axis in the profile figures.
20. Profiles shown in the figures are the average of five profile
9
100
Plan 1 Test I
90 LEGEND
0 Concrete slope
" -_ Pre test prore
E 00 .- : Post lest profileC, 3000 Waves
0
n 70
0 "
60 SWL.60
so\ <
0 20 40 60 80 1 V) 120 140
Horizontal Distance from Bulkhead (cm)
Figure 4. Typical pre- and posttest profiles
surveys taken along the length of the test section. Surveys were made by
taking soundings with a rod attached to a 15-mm-diam disc by a ball and socket
connection. Soundings were taken every 3.05 cm along the length of the test
section. Very little across-tank variation in the profile was observed during
these tests.
21. A dense limestone was used for the rubble in this study. Because
of its small size, the stone was graded by sieve size. Table I summarizes the
gradations and specific gravity of the stone used.
Table 1
Characteristics of Stone Used in This Study
Tests TestsCumulative 1-22 23-26Percent Sieve Size Sieve SizePassing mm mm
2 4.8 3.1
15 5.6 4.350 8.1 5.6
85 11.2 7.398 12.7 9.3
Specific gravity: 2.68 2.72
10
Test Conditions
22. Initial test conditions were generated to simulate wave action
similar to that found on Lake Michigan near a site in Chicago at Devon Avenue.
The initial berm width approximated that obtained by substituting these
initial test conditions into the Dutch equations (van der Meer 1988). As
testing progressed, more severe wave conditions were run in the wave tank to
fully test the range of circumstances for which a dynamic revetment would be
suitable. Later tests examined a shorter wave period that may better repre-
sent the conditions for a structure on a body of water smaller than
Lake Michigan.
Results
23. Water depths and wave data collected during the wave flume tests
are given in Table 2, and data for the dynamic revetments are listed in
Table 3. Soundings from the tests are given in Appendix A, and initial and
equilibrium profiles are illustrated in Appendix B.
24. It was found that the initial profile adjusts rapidly to incident
wave conditions. For tests with TP = 2.5 , there was little change in the
profile between 3,000 and 5,000 waves, and for tests where TP = 1.75 , there
was little change between 3,000 and 4,000 waves.
25. The Dutch found the shape of dynamic profiles at equilibrium were
not very sensitive to initial configuration (van der Meer and Pilarczyk 1986).
Verification of this finding was made in a pair of early tests. Figure 5a
shows the initial profiles of Tests 4 and 5, which can be seen to be quite
different. The initial profile for Test 4 is the equilibrium profile at the
completion of Test 3, and the initial profile for Test 5 is the standard
starting profile used in this study with a horizontal berm and a 1:1 seaward
face. Wave conditions for Tests 4 and 5 were nearly identical, and the
equilibrium profiles that were produced were also almost identical, as shown
in Figure 5b. A preliminary conclusion is that the final profile is indepen-
dent of the initial profile as long as the volume of stone remains constant.
This is a very important finding because it would reduce the cost of construc-
tion by allowing rough placement of the stone berm. This conclusion parallels
findings from studies of sand beach profile development. The conservation of
11
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UZ IMM M M - N M w M ~ ~ m m m m m m m m Q
r= 1 . . . . . . . . . . . . . . . . . I . . ..1 3 .
100 LEGEND
* ccete slope
90 pre-test p o fle. test 4
80
0
U-70LA
.0 1
C-z- SWL
.9 60
50 -
40 100 120 140~
0 20 40 60 80 101214
Horizontal Distance from Bulkhead (cm)
a. Pretest profiles
100LEGEND
90 L:j Conc'ete slope
go pot-test profie test 4
080
C07
U-i
>
0 604 6 010 2
HorizOntal Distance fromn Bulkhead (cm)
b. Posttest profiles
Figure 5. Pre- and posttest profiles for Tests 4 and 5
14
stone places an important constraint on this generality since for severe wave
conditions attacking a small revetment, a large portion of the stone can be
thrown landward beyond the bulkhead and lost to the system. Loss of stone can
cause a nonreversible deterioration of the revetment and ultimately failure.
26. Other interesting features of both the Dutch and CERC dynamic
profiles are a pronounced beach crest and a very steep subaerial beach face.
During the CERC study, the dynamic profile would typically reach a slope of
about 45 deg seaward of the beach crest. The steepest beach face segment
observed during this study was 52 deg. The angle of repose for sharp-sided
stone or gravel is approximately 45 deg, and this value is assumed to be about
the limiting value for the beach face slope.
15
PART IV: DISCUSSION
27. One development from van der Meer's (1988) research is a method to
categorize "structures" from breakwaters to sand beaches (Table 4). The
method is based on a stability number similar to the one used extensively by
Hudson and Davidson (1975) in their study of breakwater stability. For
irregular waves, the stability number is defined as
N. = or Wr 1 3 H
where w, is the unit weight of water, with w, = 1.000 g/cm 3 for fresh
water and w, = 1.025 g/cm 3 for seawater. When stone sizes are relatively
small, as in this study, dn(50 ) is determined by sieve analysis. Energy-
based wave parameters are used in this study so the zeroth moment wave height
Ho (measured at Array 2) is used in Equation 7 rather than H. For this
study, the range of stability numbers is from 2.7 to 9.2. Since CERC tests
are run in shallow water where H. is typically greater Hmo , the stability
numbers from this study will be somewhat lower than van der Meer's.
Regardless of differences, tests from this study fall into van der Meer's
"berm breakwater and S-shaped profiles" and "dynamically stable rock slopes"
categories (see Table 4 taken from van der Meer and Pilarczyk (1987).
Table 4
Structure Classification Based on van der Meer and Pilarczyk (1987)
Structure Range of Stability Number
Statically stable breakwaters N, = 1-4
Berm breakwater and S-shaped profiles Nr = 3-6
Dynamically stable rock slopes Nr = 6-20
Gravel beaches N, = 15-500
Sand beaches N, > 300
16
28. Johnson (1987) remarks about the impressive aspect of waves build-
ing high, steep beach faces on Lake Michigan, and this feeling was shared by
laboratory observers watching the rapid development of the beach face during
small-scale tests. The reason for these beach features relates to the high
porosity of the rubble, estimated to be about 45 percent. If the waves are
large enough to mobilize the rubble, then runup will carry some of it upslope,
but the return flow will not carry all of it downslope until the profile has
reached equilibrium. Therefore, the subaerial beach face is about equal to
the angle of repose largely because the runup flow drains away so quickly.
The extent of particle mobilization is measured by the stability number N.,
defined by Equation 7. Height of the berm crest is a conspicuous feature that
can be easily identified and accurately measured. The berm crest height is at
the approximate upper limit of wave runup (Powell 1988). Visual observations
of the tests indicate that the "mature" crest is only occasionally overtopped.
Therefore, berm crest height is a good measure of extreme wave runup height.
Critical Mass Analysis
29. To evaluate economic feasibility of a rubble structure, it is
clearly necessary to determine the minimum amount of stone that will provide
the desired protection. This minimum quantity (volume per unit length of
revetment) is referred to in this study as the critical mass.
30. All of the test results were classified into one of three revetment
response categories. When wave conditions were severe in relation to the
quantity of stone in the revetment, wave action eroded the rubble, usually by
carrying it over the bulkhead, until waves impacted directly against the
bulkhead. This category was designated failure, denoted by "F" in Table 3.
When the amount of stone in a revetment was large in relation to wave
conditions, development of the berm crest had enough room so that neither
stone nor water was carried over the bulkhead. This category was designated
safe, denoted by "S" in Table 3. The third category fell between safe and
failure and occurred when the berm crest buildup extended far enough landward
to reach the bulkhead and there was at least some overtopping of the bulkhead
by both water and stone. This category was designated intermediate, denoted
by "I" in Table 3. The three RRC's are illustrated in Figure 6.
17
100
LEGEND
0 Concrete slopePre-test profile
RRC's
.... "S" Safe80
X "r Intermediate0
E v "F" Failed
.0
60 '
5,ILuj
50
40 - - T I
0 20 40 60 80 100 120 140
Horizontal Distance from Bulkhead (cm)
Figure 6. Typical equilibrium profiles illustrating the safe,
intermediate, and failed revetment response categories
31. To calculate critical mass, it is necessary to estimate three
characteristic dimensions of a dynamic revetment, i.e., berm crest height, h c
berm crest length lc , and erosion length 1 . . Regression analysis was
employed to determine the following equations, which define these character-
istic dimensions as functions of local wave steepness Ho/Lp , where LP is
the wavelength determined by linear wave theory for the depth at the toe and
the peak period.
hT I-0.6'45
h = 0.270 * R 2 =0.96 (8)Hmo , R2 =0.9
l- 0.521= .677 R = 0.92 (9)
H L8
18
e= exp 2.24 * (Hm 0.143 R)2 = 0.64 (10)
Equations 8, 9, and 10 are based on analysis of Tests 1 through 22. R2
values give the portion of the variance explained by the regression analysis.
Tests 23, 24, 25, and 26 were conducted with somewhat smaller stone (see
Table 1) and were withheld from analysis. Figures 7, 8, and 9 show observed
data with regression trends for Equations 8, 9, and 10, respectively.
Although the smaller stone was not included in the analysis, it has been in-
cluded in Figures 7, 8, and 9 to illustrate the applicability of the equations
to other stone sizes. Stone sizes are denoted by symbols in these figures,
with "L" indicating the larger stone and "S" indicating the smaller stone.
32. Characteristic dimensions determined by Equations 8, 9, and 10 may
be used to determine a pseudo-cross-sectional area of the mature revetment A,
where
As = (ds + h) * (le+c) ()
This equation is essentially just length times height. Water depth at the toe
of the revetment dr is selected based on design considerations. The total
volume of the revetment per unit length is then determined from A. for the
desired degree of protection. Total design volume is denoted At (cm3/cm) ,
which includes void space. Figure 10 shows the revetment response category
versus the ratio of At to A. * The two solid lines illustrate values of
At/A3 of 0.67 and 0.46, and indicate that if
A t > 0.67 As (12)
the revetment is safe, and if
A t < 0.46 As (13)
the revetment will fail. Values of At/As between 0.46 and 0.67 are in the
intermediate revetment response category. This guidance is based on labora-
tory tests with a range of stability numbers (Equation 7) from 2.7 to 9.2.
33. Johnson's (1987) criteria for protecting eroding portions of the
Lake Michigan shoreline was 36 metric tons/metre. Assuming a porosity of
35 percent and a unit weight of 2.64 g/cm3 , this guidance is equal to 21 m3/Im.
Figure 11 compares this guidance with the design volumes determined by
19
4.6 . .
4,4 $
4,2,4,0 |
3,8
14 -LL
21.-
2,4 -
2,2 -L L L LL L2,0-
0,014 01016 0,022 0,026 01030 0034 0,038 0,042 0,0406
LOCAL WAVE STEEPNES, Hmo/Lp
Figure 7. Calculated and observed relative berm crest heightsas a function of local wave steepness
7,0
S 6,0 L, 5
6,5 L
' LL
6,0 LL
LL3,5 -L
5 L L.
310 - I I I I I
0,014 0,018 0,022 0,026 0,03 0,034 0,058 0,042 0,04
LOCAL WAVE STEr'NCSS, Hmo/Lp
Figure 8, Calculated and observed relative berm crest lengthsas a function of local wave steepness
20
4.g4.8
4,7
4.8
4,4 L413 L L
S 4,2LL
4,0L
LL
385- IL L LL
313
3,23.1
0,014 01016 0,022 0,026 01030 0,034 01038 0,042 0,045
LOCAL WAVE STEEPNESS, Hmo/Lp
Figure 9. Calculated and observed relative erosion lengthsas a function of local wave steepness
2,4 t 0,7A
212
2.4
At0,,4-AA
112 i's 4 I'1$.Is(Thousands)
As *(do +. ho)'(I.+ 0) (C to o A)
Figure 10, Total area of berm, A2 , versus calculated A. , withwith observed RRC. Solid lines illustrate that for At < 0.46 as
the revetment will fail
21
100 - _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
90
80
N 70
60
C 50
0 40
j 3030
>
20
10
0-
0.2 0.6 1 1.4 1.8 2.2 2.6 3
Depth of Bulkhead Toe, ds )0 Foil/inter + Inter/Safe 0 C. Johnson
Figure 11. Comparison of stone volume calculated in this reportversus guidance in Johnson (1987) for Great Lakes storm with
TP-l0 sec and HL. - 0.6*d.
Equations 8 through 13. Figure 11 assumes design conditions of Tp = 10 sec
and H - 0.6 * d .
Wave Reflection and Energy Dissipation
34. The reflection coefficient is defined as the square root of the
ratio of reflected wave energy to incident wave energy (Goda and Suzuki 1976).
Wave reflection from dynamic revetments appears to be a function of two
variables, wave steepness and relative void size. Reflection coefficients can
be predicted with the following equation:
1.0Kr =1.0+ CO [ 1.0 (14)
where L. is the deepwater wavelength. Dimensionless regression coefficients
are given by,
22
CO - 23.4
CI = 0.312
C2 - -0 00374
Equation 14 explains about 97 percent of the variance in a sample size of 30,
i.e., R2 = 0.97 and N = 30 . Tests in the failure response category were
not included in this analysis since at failure a substantial part of the
reflection is from the vertical bulkhead. Percentage of incident wave energy
dissipated by a dynamic revetment can be estimated by using Equation 14 and
the relation,
%D = (1.0 - K') * 100% (15)
where %D is the percent energy dissipation. Observed data give reflection
coefficients between 0.27 and 0.50, indicating that dynamic revetments dissi-
pate between 75 and 92 percent of the incident wave energy. By dissipating
over three-quarters of the incident wave energy, dynamic revetments make good
wave absorbers.
23
PART V: SUMMARY AND CONCLUSIONS
35. A series of laboratory tests were conducted to investigate the
response of dynamic revetments to shallow-water wave conditions. Most tests
from this study fall into the category "dynamically stable rock slopes" based
on the Dutch classification system (van der Meer and Pilarczyk 1987). For
this study, the ratio of the wave height to stone dimension is in the range of
roughly 5 to 16. Typically, zero-damage on a conventional riprap revetment
occurs when the wave is about two and a half times larger than the stone
dimension.
36. It was found that the equilibrium dynamic revetment profile was not
sensitive to the initial profile. This finding means that construction costs
can be lowered because special care is not required in the placement of the
stone. The berm crest is a conspicuous feature of the profile and provides a
good indication of the extreme wave runup.
37. The concept of a critical mass for a dynamic revetment is
introduced. Critical mass is the quantity of stone required to protect a unit
length of a vertical bulkhead for a given water depth at the toe and given
wave conditions. This quantity is found to increase with increasing water
depth, zeroth moment wave height, and period of peak energy density.
38. The influence of the initial berm width and berm height above the
still-water level were two of the major variables investigated in this study.
It was found that these parameters play a major role in determining how much
stone is required to protect a vertical bulkhead from direct wave attack.
24
REFERENCES
Baird, W. F., and Hall, K. R. 1984. "The Design of Breakwaters Using
Quarried Stones," Proceedings. 19th Coastal Engineering Conference, Houston,
TX.
Brunn, P., an' Johannesson, P. 1976. "Parameters Affecting Stability of
Rubble Mounds,' Journal of the Waterways., Harbors, and Coastal Engineering
Division, American Society of Civil Engineers, Vol 102, No. WW2.
Channell, A. R., Stevenson, T. A., and Brown, R. 1985. "Runup on Shingle
Beaches," Report No. SR 72, Hydraulic Research Wallingford, Wallingford
England.
Coda, Y. 1970. "Numerical Experiments With Wave Statistics," Report of the
Port and Harbor Research Institute, Ministry of Transportation, Japan, Vol 9,
No. 3.
Coda, Y., and Suzuki, Y. 1976. "Estimation of Incident and Reflected Waves
in Random Wave Experiments," Proceedings, 15th Coastal Engineering Conference,
Honolulu, Hawaii, Vol I, pp 828-845.
Hall, K. R. 1987. "Experimental and Historical Verification of the Perfor-
mance of Naturally Armouring Breakwaters," Proceedings Conference on Berm
Breakwaters, American Society of Civil Engineers, Ottawa, Canada.
Hasselmann, K., Barnett, T. P., Bouws, E., Carlso, H., Cartwright, D. C.,
Enke, K., Ewing, J., Gienapp, H., Hasselmann, D. E., Sell, W., and Walden, H.
1973. "Measurements of Wind-Wave Growth and Swell Decay During the Joint
North Sea Wave Project (JONSWAP)," Deutshes Hydrographisches Institut,
Hamburg, Germany.
Hudson, R. Y., and Davidson, D. D. 1975. "Reliability of Rubble-Mound Break-water Stability Models," Proceedings Symposium on Model Techniques, American
Society of Civil Engineers, San Francisco, CA.
International Association for Hydraulic Research. 1986. "List of Sea State
Parameters," Supplement to Bulletin No. 52.
Johnson, C. N. 1987. "Rubble Beaches Versus Rubble Revetments," Proceedings
Conference on Coastal Sediments' 87, American Society of Civil Engineers, New
Orleans, LA.
Pilarczyk, K. W., and den Boer, K. 1983. "Stability and Profile Development
of Coarse Materials and Their Application in Coastal Engineering," ProceedingsInternational Conference on Coastal and Port Engineering in Developingi
Countries, Colombo, Sri Lanka; also Delft Hydraulics Laboratory Report 293,
1983, Delft, The Netherlands.
Powell, K. A. 1988. "The Dynamic Response of Shingle Beaches to Random
Waves," Proceedings 21st Confurence on Coastal Engineering, Malaga, Spain.
25
Stevens, J. C., Bardsley, C. E., Lane, E. W., and Straub, L. G. 1942."Hydraulic Models," Manuals on Engineering Practice No. 25, American Societyof Civil Engineers, New York.
van der Meer, J. W. 1988. "Rock Slopes and Gravel Beaches Under WaveAttack," Ph.D. Thesis, Dept. of Civil Engineering, Delft Technical University;also Delft Hydraulics Communication No. 396, 1988, Delft, The Netherlands.
van der Meer, J. W., and Pilarczyk, K. W. 1987. "Dynamic Stability of RockSlopes and Gravel Beaches," Proceedings 20th Conference on CoastalEngineering, Taipei, Taiwan, Nov 1986; also Delft Hydraulics CommunicationNo. 379, 1987, Delft, The Netherlands.
van Hijum, E., and Pilarczyk, K. W. 1982. "Gravel Beaches: EquilibriumProfile and Longshore Transport of Coarse Material under Regular and IrregularWave Attack," Delft Hydraulics Laboratory Publication No. 274, Delft, TheNetherlands.
26
APPENDIX A: TABLE OF PROFILE SOUNDINGS
Al
-4 -4t
LI) - 1 ) . . . . . . . . . . . . . .) .) .~ rQ O .- -r-4 a) 44Q (Zj -4MM')L) Dr 0 0L)C400 . r I r-4 0 0 0r- OVI) tCl fJ- ) ll IDLrV
czU 4 C >4 (1 4 4i 0 aca *c)*c4 Q* *c4 ,, ,, a n,-I r,-1 4 ) 0* b q * O ,' DL) I * 1,c4 l*r4 *CYor, .*L) l
o oF cL :: d - -r -r -r -r-r -r lr -rlo1010%010V)U IL)L)L r L)L)L)L)L)-
u-4c'
a) -4 En -'OI Y I 0 , 00 f )0 14- l ,r4L)%0 1M-4- r 0 ,- 4r ,C,- r NC)C
4 -4 Q)(4- (= CO '- C1 1 4C4C4C I qM M 4 n I" WL W n-I- 1- D O'0'r*L) r*
-4 4 Co -a) (n mJ a)Q m) " N br om0E" L)mmC4r- nC)U)Cir a 0C) Y Im n"-C) 0O
> :" (1 3 r 4C(: 4 r 4-4- N 4'D00C 41 0C4C)0r 1 4Mr-L)- 0 q Da r ,z0a) u 10 > > 1 . . . . . . . . . . . . . . .
> (.4Q Ln cl '-. r- 00 '0 I r-oo "IC1 0.r Ck,4 V) 0 . L('4 C'.a\ O"l - '.o. in- 4- CV)(\1 C CJ 4 0"0 z ) Ln ~a) Z 1 U r C 0L)C40\mMko or 1ko 40C)L "L n0 Nm owlTmm\oI
a) ) > > . . I . . . . . . . . ..0
-4()4 )C t 4CJ O 1 110r 71(N10r)- T . L)MCJ- , .'U tc l 1 1 )M r ' r r r
O c 3 - r -r -r -r /r -r l - ID o\o\ . U)L)L)L )u r , ' r nL)- 3 : 1 1 z
0'-4 CO I
Q)4-4 4'
412
(N 00
"~ -I > > . . . .v -- . . . . . . . . . . . . . ... ... 1 ' ' ~ ' 0 ' 0 0 N . . . . . . . . . . .'~ . . . . . '..L4- C 0 C Ln ElJ U)O U-1 ( NJ 0 00 C) -, (N 00 0' 0 0 .0 In N 4 (D, (3N a, 0 -n a4' -- r xN TTh <!0 N( '0 - 00 '0
F- :s .
0. -'0 -) > . . - ... - . .C .- . .- .- r . C. -. -. r. I. .- r. r. r. -. -. -. C. 4 . . .n . . .- . . . . . . . . .4 . . .
C)I :1 IN( i1 l , I N(1r-I t 1 l ,0 1 1L n -11 l -: -X c
C) W) A -4 ( f0 00 W W W rN00'(N ' O 1-0 100'(ND fl0'I -400) In(N-~-- InO C'.( 4-n(In I
000
-l D.00.C0' -10' ,I ' n0'(n o n (NCN(NCJ'0( r n U) o n n .4'.4'.; 4 0CN :I n CN ),CIO I.r. . -)
a- 0 )-.. N 0- -4 CN 1 CJ( (N -CC CN '0( N(N.0. . . 'O0-4.4 CN ic CNNNN44f m D"M o C0- .4-) If, r) In U-
-CCXu 0Q ID1 nI n- 1
) 1 00 -.- 0-- o 0 000(NCl0' 0(.4(00 (00 r- r--N(N(N0 0 0 10 In ''..00n I 0.
In (N C-- 0010 I0 N C 4 I Oc
C)2 I) I))E -D ID 0'0 -'0~~r'' 0 0N 0'- <tr C'0(N 00<40,Iz '0 002C a,,-C) 0'
(N CA C1 ((( N ((((NN4 In () r-.(N( I")N(NC - ,) QD '.'.f' r 00 W W o CY a,:DC CC-C-. r 044)4 .4
(N'0~~~~~~ ~ ~ ~~~~~~~~ -N -N -0. ' ' ' ' -4C -f -C-C -)1414C -CC -CC -- -0 C C 4..
H 4 .01 ON U- al \N "I _:I 11) 04) "Iw ( '014 C Z N 0
- L " ) IW N' Q) C) (t A0 (N'N'''-'-01 -f0-fC.4.40'.--I C) 'D <C4 CD Of)0' 0'.- 1 I n 7 _0-(
>.H ' -C- C N ( .N .N .N .N .- .- .- .- .- .- .~ .- .N .N .N .N .N .0 ' . . .C-~l4C 4 .4 .4 .4 .4 .4 ..
4-, InIIn n
w A x0( '0( w(N(N( 0 0'n In (3N4wIn 00N 0 '(N0'0'(N-4( N-r,-4C ( n0 '4N 1C-4 0 r 0.4,lE --N00'( (~ 0(00 C I o-. In(NCO0'0 0 C, 0 C 1(( 0 0(N N N .4O00
oo no c nI o . o o IA In In'.0 fl'4 ( O( 'O 0'4.1---O(ON0'NC- -'- (Ns0(4 00 '.4-.
.- 4 .4',' l ~ -4 C4 (N i c--I C' N 4 (N C N C-C . -1 4 CIA4C(C 0 0' (N (N (N (N C0' w ' 0' 0' w ' In 0~ 07 0 0 -4. 4(n (
- W 1) 0 r- , r- ID 1 U nAI
CN ICnI )C)%0 =>C4C)00 , 0 C r)CO-)L)O 4 )" I -40 n C>r-On '. O C4o
4. Ef r-I --- nL 4C 44O 01 Nr4C)0 )0 C )% 410L)M<NC o()- - l 3,0 r
4- Q) ). . . . . . . . . . . . . . . . .
caU -)0 > 4Clt)C): l% 0M0 nI : 0r f e 40MM0 r ot nIIc 40Mr Dr- 4)4 1 O -r -r -r -r -r -r ,%DD oI . oL nL nL nL nL nV nL nL041
-44) . O . - r--r- r, r- r- r-. r- r- r, r- r- r-. r,- r'- \ -\Z \D . r- '.0 L'.) '.0 i Lrt I ~ -f) .. all Lr -r) -.) Ln -.) -U-ll
a4) [- < I J:
2 n () - C -40 Lr)-, Lf) 1o r-400 - )4- 00 D0 NWMO10ML n 4Mr 4r4- U
C) 0, > M. Lnr Or %D . DL % n- 40 010r AM 0C V V N \r-r00 4- ] . . . . . . . . . . . . . . . .
CO U)4 o > - 40 4"C 4C40 1 4C NC40 4ciC4C4- 0L N0 DMC\r- D 0L)L)LL'A()4 ) r -r -r '-,r ,4 lr ,r ,r r ,r ,. o nL r
EI \D4 ONr )M Cr 4C4" M0 \C 0.)r L)t Nr r -L)0 \r 4C4C -4 -4 0-4 ON
0 0[.0
1-4
-4 r-4 -4ij -
en -
-4-L LI.4 0I4O Q ~ ' ' 0
-(1) VC~ '- f C0 \ ( N \ ' 0 f ' fN0-H'
Htz1. - 4MCJMC NL)r I . - - . 0U Ir-Ir r 4 Mr nO
'V n\DC4r Z' 0r C40 -- 4r -0 r 4- - Dr . r
(1) 'VC ) b) . . . . . . . . . . .
I - C)0 F r -0C A *4 . 1- 0r -C r- 0 - L
r-I f) LnC C) 4 4 l 0 %4C 0 NA400rCV 4r- 4 0O)r 10u
.- ~ ~ ( 4~ ..]r- r- fl- r-. r-. r-. r,-- r- r-. w~ w w ~ .w m r- r, ID 10 100 '.D ID0 1' 0 ID ID .0 0 '0 Ir Lr 'n rn Ln 'n f) f
wU (U > > U0 Y )C n T 3 ooC 4O (3% OLn 4 0 Ol W r(4r InI nC Do WrDUN, 4-
[7- <3
C D- 0''0 r-r-oD n IDDID 0 IDr- I r D-QO 4 0 r~0 "D -'C 01W C f- D Ln 17 I 'lr- 0 . -j .l . .
11I 4o >lo oo oo o0 oo oo oc 0o oc oo oo o oo oo o o r l o( l l 4 -a) (U) < --1 r-r -r -r - l ol r n r nL)L)L nL r r nv
0.. H .
a)I
c0 a) 0 o rnm-A) N L)L)% )o o r I D C 1 1 Da nIC r 00 OI : Df04 CL w J 04C Z ) 00 Dk 11C :N t)IIC N0 lr D r IC 4 0o oo oI 1()(u0 > . . . . . . . . . . . . . . . . . . . . . . .
o - -4, Lr nX 1 n< 1-1" 1 1- 11 1m r)r C nC)M M( 1 I 1 1 N N I qC q
. . . . .'4 0 4 . n~~ .40'.. . .. .0' 4 r ~ . . .
n " w- -- -a,4 \D' -In'--D,' I D-C D0 1Q C (114-
C:a01 E . . . . . . . . . .
(v
,u0 jo Nr No Dm mw l nU)4m1) cj (1o
a)41L.m2 o% nL ni U L - nU nL
'0- I- Lj > . . . . . . . . 4C~. '~. . .t~*
0- (U r, -r'r- rl 1 l DI nL I' 1' T<
0 tA "11 .
-. 0 -4 U ' -4 " " C 4 4 ^ r- o - r- w w , (710 C) '0'0lW (A . . . . . . . . . . . . . . . . . .
0 0-4 'O0JC4 14I O0' In 4n ZT )- Ln'0 r-u-o'- ID
0 0.bOEA5
0-4 Cfl a% IN LnWr- W %DW -04-4 ON:tM t n ,O-4
\O bO I00- )a 7 -r-0 ~ -L Dr na n . - nk n-: nk 4L
di)4-)
r. -4 < NC4C 1 4C4C4C4CJC 4C4- 4- - 4- 4 4 -4 0 z00 V)i alL)L)L)L)L l r )Lr nL r r r r n)LrC lL C n r r
C-,
00
-4
-4 -4- --1 r 1- 4- 4- 4- 1- 4- -
0-o
0'-4
co,
-4
rI '4 %DDM100MC -4- -- C40 O nCr10' 4 0 Mc Lnr-4>) 0) 0 2 Q) -o
< tv I-
0Z ) 0)o
0 4bCrC 1 Lf cr- r r0 r- r r Lr1 cr crD 1 i 10 r1 D r1 10 r1 c1. r
r-I - -m .J V l 0)U 00 00 30 00 00 0000 00 00 00 00 00 00-4Q O () r -r r ,r -r -r - -r -r Ir l "r lr -rr
CL F-4Ak
-4
m L 4I - 4L1C4r wA %,D Qa' O~N.O%0 C rwmi-D 4 -4 in'G)OGWrJ N NL) i- 100 N . M- )00O)I j- 4I oo
cd n CJO >Ju C'J , .' *in - M*av, *4*-:* i M M 0
-1W(- )r 0i IU)U)Lf ni nL)i)L)L)U)i
c)
aNC4 Ci o )cAi)- , tL)rL)c Ic -L)o nL)L r -or - tIL-
A4O E-4 < 114LC LC
0- A f
014 E-4 Cn 3
0 Cd VA 4J r-- > 04 C4 C1 4C 4M% C q 000 - % .o 4Car ' M(N- CDM001-
LL. 14 IL3
0 C4 --4 M) .i
C: 00
w I -lCJ M- -4 -4 -4 C14 ,I rl 0 -4~ W~ IN ~'0 r-4 r - r- W
4-)0 J ) 4 r . . . . . . . . . . . . . .7
4 ) (4-4 .-4 Cd -1 -. 0IDIDL)L)LnL nL nL )L nL
04 -4C14 bO3:
r-4 CVJ) bD 1-4 C4 0 00e0r I 00D-40.I -4 O )L 0rL) f )C)O 0-4L)C)L)C 0L
td U) 44. V) > 04C 40 NC I 1 qCqC qC q"C 40 4 C D Dr--:T.-4 00 l
OL E-4 < I E-
0.) cj4)W -- 0r 10 V %C 0MO - tCtr 2 7%e -CJf-ILr D\D0 qc
24 0,b =C4- 4C 00 U . r f qC>O 0r ' nC l )00 0r
0 n04MICN 0C 0M0 NNr 4r C . 1% -1 \D4~.~u 0C'O n00 4'C14 r=J-2
~0
0-4
1-4 ).M MMJ D 0Me M) 10-4 0 N0 f0- -000 :D\DC C D10 -ID0 C Mr
L13 UZ 4-) r Q) \, (D bj .
c.4N 0rv. r- 0 C) o J(c, n %D0 \ Ln -1 r- NC 4 q % 0C4Lnooro ~- o - , r- .o m tr) r- a\
<C~~ -4C-Cq M~ :I4 b02 00C404\DC40 -00 MLr D NUlCN. ) -4 ,014M() - 0(1r
4a) 0) Cl.) '-4 E
C4--4 C a ) .
4Z C DIrC> ,r ML)L) 1.O -0000 00C:r M00 , nCN \V)tLn(1 -0ral0 U E C - nMf , IMMI0)L DI nL)L L)1- - 0r-L)a 0 ZU)-IMC440(1rCO U 4 0 ) .. ..... .....j
r- )0 U < NC 4C4"C qCIC I I 4C 4C 4C 4C -- IC -41 0 .* '\, ID .'%,U) n
a4 IO r-0 r, r- r -r r-r -r r r . - ,rf- - , D'o r f n-t .I , :
N ) -.-1 4~
(v (D r-z 0L)O n0 NC,- ""a f -C --- tU) . 0C4r -0 nd)a "V C4cI40 OE C44- )O 00 010L ' IMC "0 -'0L)MMC )00 0r , n CJ 1-4 -4
u > r . . . . . . . . . . . . .r . . . . . . . ..8
-4 .o
r4CaU4 J > 0 C- % r nr4 r4-t0 nWC4C = . L)%DDD DC)4Y -)L)C 00 r
$4W2- C)4- I n1'0O - = - 0C 4o4L 0-4Z -0, 7 - - - 0V NC0 ) 4)> E . . . . . . . . . . . . . . . .
-1 rA"r , - , jc4c4c4c4m4L - 4a,'D4r - oo or o r nC Y -I0c 4 ) r.> 0 (44L)C -r -r -r ,r -r -r -r . I DI . DL)L)L)L " r r n nU1a
cw F4< ) t
U) -4 V)
0 (-4 Q ca. 44 ' C r- r- r- r- r--r- 00 00 o , rr-r- ID . .o% ol 1 Lr Ln I r U Lr V Lr( Lr'Ir( Lr) Lfl) Lr V)~ -.Ul - .
-4~ bfl
04
00>
-4-
4-19
APPENDIX B: INITIAL AND EQUILIBRIUM PROFILES
o '6 w
0 -W0 (fl
'U 0 00o.~ I
+ 0
0
0
E0
0M
0 0
0
C~
0 0 0 000) c 1- 0 U"
(wo)wouo own~l AoqVUOI~A90
B02
0
0 0Z; - 0 (13
C)zZ 3:-CW c)
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APPENDIX C: NOTATION
A. Pseudo-cross-sectional area of revetment
A t Volume of revetment per unit length
CO Regression coefficient
Cl Regression coefficient
C2 Regression coefficient
dn Nominal stone diameter
dn(50) Nominal stone diameter of median stone size
ds. Depth at revetment toe
E, Energy of incident wave spectrum
ER Energy of reflected wave spectrum
f Wave frequency
hB Berm height
h c Berm crest height
he Berm erosion height
H.. Zeroth-moment wave height
Hr Reflected wave height
H. Significant wave height (average height of highest one-thirdwaves)
Hsl) Incident reflected wave height
Hs(t) Total significant wave height (combined incident and reflected)
Kr Reflection coefficient
ic Berm crest length
1. Berm erosion length
L. Linear wave theory deepwater wavelength
IV Linear wave theory wavelength for d. and Tp
m 0 Zeroth moment of incident potential energy spectrum
M2 Second moment of incident potential energy spectrum
N Sample size
Ns HS or W,1/3H,
d,,50 -! ~ 1) w1/3(c
Cl
Qp Spectral width parameter
R 2 Portion of variance explained by regression analysis
S(f) Wave spectral density function
Tp Peak wave period
T. Average wave period
W Weight of an individual stone
WB Berm width
wr Unit weight of rock
WW Unit weight of water
%D Percent energy dissipation
C2
SUPPLEMENTARY
INFORMATION
DEPARTMENT OF THE ARMYWATERWAYS EXPERIMENT STATION, CORPS OF ENGINEERS
3909 HALLS FERRY ROADVICKSBURG. MISSISSIPPI 39180-6199
REPLY TOATTENTION OF rf A 7/
WESCW-R 10 June 1992
Errata Sheet..409 a $q%5r&No. 1
L4BOR-ATORY STUDY OF A DYNAMIC
BERM REVETMENT
Technical Report CERC-92-1
January 1992
Page 19, line 1, Equation 10: Change superscript -0.143 to+0.143. The equation should read as follows:
l exp 2.24 * LP 0113} , R2 = 0.64 (10)
HYDRAULICS GEOTECHNICAL STRUCTURES ENVIRONMENTAL COASTAL ENGINEERING INFORMATION
LABORATORY LABORATORY LABORATORY LABORATORY RESEARCH CENTER TECHNOLOGY LABORATORY
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