U.S. Army
Coastal Engineering
Research
DATA LIBRARY|
Woods Hole Oceanographic Institution
SHORE PROTECTION |
MANUAL
ENVIRO ITAL RESEARCH & TECHNOLOGY. INC
Volume |
Go.
ex 1
Ngee OF
formato 4e@ Oe BALL
DEPARTMENT OF THE ARMY
CORPS OF ENGINEERS
= N 1975
Reprint or republication of any of this material shall give appropriate
credit to the U.S. Army Coastal Engineering Research Center.
U.S. Army Coastal Engineering Research Center
Kingman Building
Fort Belvoir, Virginia 22060
SONOMA COAST, CALIFORNIA (GOAT ROCK) — 10 December 1958
0 0301 OOb07b9 3
SAMAR REAVER
SHORE PROTECTION
MANUAL
VOLUME L
( Chapters 1 Through 4 )
U.S. ARMY COASTAL ENGINEERING RESEARCH CENTER
1975
Second Edition
For sale by the Superintendent of Documents, U.S. Government Printing Office
Washington, D.C. 20402 - Price $15.05 per 3-part set. (sold in sets only)
Stock Number 008-022-00077-1 Catalog Number D 103.42/6:SH7/V.1-3
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PREFACE
The U.S. Army Coastal Engineering Research Center, (CERC) formerly the Beach Erosion Board, has,
since 1930, conducted studies on shore processes and methods of shore protection. CERC continues an
extensive research and development program to improve both shore protection and offshore engineering
techniques. The scientific and engineering aspects of coastal processes and offshore structures are in the
developmental stage and the requirement for improved techniques for use in design and engineering of
coastal structures is evident. This need was met in 1954, to the extent of available knowledge, by
publication of Technical Report Number 4, “Shore Protection, Planning and Design” (TR 4); revised
editions thereof appeared in 1957, 1961, and 1966.
Significant advances in knowledge and capability have been made since the 1966 revision. This Shore
Protection Manual (SPM) incorporates new material with appropriate information extracted from TR 4,
and expands coverage within the shore protection field. This SPM is a replacement volume covering
guidelines and techniques for functional and structural design for shore protection works. Accordingly,
further editions of TR 4 are not planned.
The Shore Protection Manual is in three volumes. Volume I describes the physical environment in the
coastal zone starting with an introduction of coastal engineering, continuing with discussions of mechanics
of wave motion, wave and water level predicitons, and finally littoral processes.
Volume II translates the interaction of the physical environment and coastal structures into design
parameters for use in the solution of coastal engineering problems. It discusses planning, analysis, structural
features, and structural design as related to physical factors, and shows an example of a coastal engineering
problem which utilizes the technical content of material presented in all three volumes.
Volume III contains four appendixes including a glossary of coastal engineering terms, a list of symbols,
tables and plates, and a subject index.
R. A. Jachowski, Chief, Design Branch, Engineering Development Division, was the project engineer
responsible for preparation and assemblage of the text, under the general supervision of G. M. Watts, Chief,
Engineering Development Division. At the time of approval for publication by the Coastal Engineering
Research Board, LTC Don S. McCoy was Commander and Director and Thorndike Saville, Jr. was Technical
Director. Members of the Coastal Engineering Research Board were: MG John W. Morris (President),
MG Daniel A. Raymond, MG Ernest Graves, Jr., BG George B. Fink, Dean Morrough P. O’Brien, Dr. Arthur
T. Ippen, and Prof. Robert G. Dean. The board members were intimately involved in both the planning and
review of this manual.
Preparation of this manual includes the contribution, review and suggestions of numerous engineers,
scientists, technical and support personnel. Present members of the CERC staff who made significant
technical contributions to this manual are: R. H. Allen, B. R. Bodine, M. T. Czerniak, A. E. DeWall, D. B.
Duane, C. J. Galvin, R. J. Hallermeier, D. L. Harris, R. A. Jachowski, W. R. James, O. M. Madsen, P. C.
Pritchett, A. C. Rayner, R. L. Rector, R. P. Savage, T. Saville, Jr., P. N. Stoa, P. G. Teleki, G. M. Watts, J.
R. Weggel and D. W. Woodard. Technical editor for this manuscript was R. H. Allen. Typing and composing
were done by M. L. Vrooman and C. M. Lowe; and drafting by H. J. Bruder and J. S. Rivas. LCDR K. E.
Fusch was responsible for the completion of the final manuscript.
The manual format and binding were selected to optimize its use by scientists and engineers as a learning
text as well as a field and office engineering reference. Chapters include a bibliography. The binding
facilitates text and chart removal for separate use or rebinding in loose leaf form.
Comments or suggestions on material in this manual are invited.
This report is published under authority of Public Law 166, 79th Congress, approved July 31, 1945, as
supplemented by Public law 172, seth Congress, approved November 7, 1963.
CHAPTER
5
5.1
5.2
3.3
5.4
APPENDIX
DTOD>
TABLE OF CONTENTS
VOLUME II
PLANNING ANALYSIS
GENERAL
SEAWALLS, BULKHEADS, AND REVETMETS
PROTECTIVE BEACHES . ,
SAND DUNES . :
SAND BYPASSING.
GROINS .
JETTIES . 0 .
BREAKWATERS— SHORE- CONNECTED 3
BREAKWATERS—OFFSHORE . .
ENVIRONMENTAL CONSIDERATIONS
REFERENCES AND SELECTED BIBLIOGRAPHY
STRUCTURAL FEATURES
INTRODUCTION
SEAWALLS, BULKHEADS, AND REVETMENTS.
PROTECTIVE BEACHES .
SAND DUNES . .
SAND BYPASSING,
GROINS .
JETTIES . : :
BREAKWATERS— SHORE- CONNECTED c
BREAKWATERS—OFFSHORE
CONSTRUCTION MATERIALS. . . .
MISCELLANEOUS DESIGN PRACTICES . .
REFERENCES AND SELECTED BIBLIOGRAPHY
STRUCTURAL DESIGN—PHYSICAL FACTORS
WAVE CHARACTERISTICS .
WAVE RUNUP, OVERTOPPING, AND 1 TRANSMISSION.
WAVE FORCES.
VELOCITY FORCES-STABILITY 0 OF CHANNEL REVET MENTS.
IMPACT FORCES .
ICE FORCES.
EARTH FORCES . .
REFERENCES AND SELECTED BIBLIOGRAPHY
ENGINEERING ANALYSIS—CASE STUDY
INTRODUCTION
DESIGN PROBLEM CLACULATIONS- ARTIFICIAL OFFSHORE ISLAND .
REFERENCES .
VOLUME III
GLOSSARY OF TERMS
LIST OF SYMBOLS. . .
MISCELLANEOUS TABLES AND PLATES.
SUBJECT INDEX
PAGE
5-1
5-3
5-7
5-21
5-24
5-31
5-46
5-49
5-50
5-57
5-58
6-1
6-1
6-16
6-36
6-54.
6-76
6-84
6-88
6-96
6-96
6-98
6-101
7-1
7-15
7-63
7-203
7-204
7-206
7-208
7-214
8-1
8-2
8-132
A-1
B-1
C-1
D-1
SECTION
Ge!
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CONDMNRWN
TABLE OF CONTENTS
VOLUME I
CHAPTER 1 - INTRODUCTION TO COASTAL ENGINEERING
INTRODUCTION TO THE SHORE PROTECTION MANUAL.
iHE SHORE ZONE) ©: 9. © =. :
NATURAL BEACH PROTECTION
NATURAL PROTECTIVE DUNES... . :
BARRIER BEACHES, LAGOONS AND INLETS.
STORM ATTACK... . :
ORIGIN AND MOVEMENT OF BEACH SANDS :
THE SEA IN MOTION.
TIDES AND WINDS.
WAVES. . .
CURRENTS AND SURGES.
TIDAL CURRENTS .
THE BEHAVIOR OF BEACHES.
BEACH COMPOSITION. é
BEACH CHARACTERISTICS.
BREAKERS .. . “
EFFECTS OF WIND WAVES.
LITTORAL TRANSPORT . Ales -
EFFECT OF INLETS ON BARRIER. BEACHES.
IMPACT OF STORMS .
BEACH STABILITY.
EFFECTS OF MAN ON THE SHORE.
ENCROACHMENT ON THE SEA.
NATURAL PROTECTION... .
SHORE PROTECTION METHODS . J 44 :
BULKHEADS, SEAWALLS AND REVETMENTS .
BREAKWATERS.
GROINS .
CSIPIEINENSSS 9G 5S
BEACH RESTORATION. AND. NOURISHMENT.
CONSERVATION OF SAND .
CHAPTER 2 - MECHANICS OF WAVE MOTION
INTRODUCTION .
WAVE MECHANICS .
GENERAL.
NO dh
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Die
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ne et) a: Se¥ Lote” or fet jar) Ye" tom tee 'e) Koh te
NNN Pd
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252
3255
234
5 S15)
WAVE FUNDAMENTALS AND CLASSIFICATION OF WAVES.
ELEMENTARY PROGRESSIVE WAVE THEORY
(Small-Amplitude Wave Theory). 3
Wave Celerity, Length and Period .
The Sinusoidal Wave Profile.
Some Useful Functions. :
Local Fluid Velocities and Accelerations :
Water Particle Displacements .
Subsurface Pressure.
Velocity of a Wave Group .
Wave Energy and Power. ae
Summary - Linear Wave Theory .
HIGHER ORDER WAVE THEORIES... . °
STOKES' PROGRESSIVE, SECOND-ORDER WAVE THEORY.
Wave Celerity, Penaett and Surface Profile.
Water Particle Velocities and Displacements.
Mass Transport Velocity.
Subsurface Pressure.
Maximum Steepness of Progressive Waves :
Comparison of the First- and Second-Order Theories: .
CNOIDAL WAVES. ....
SOLITARY WAVE THEORY . .
STREAM FUNCTION WAVE THEORY.
WAVE REFRACTION.
INTRODUCTION . . a
GENERAL - REFRACTION BY. BATHYMETRY 6 c :
Procedures in Refraction ae Cons euncetont -
Orthogonal Method. we 5
Procedure when a is Less than 80 Degrees 5
Procedure when a is Greater than 80 Degrees - The
R/J Method . c
Refraction Fan Diagrams. 5
Other Graphical Methods of Refraction Analysis 3
Computer Methods for Refraction Analysis . 5
Interpretation of Results and Diagram Limitations.
Refraction of Ocean Waves.
WAVE DIFFRACTION .
TUNLEROVO WIC MON G 6 6 oc 0
DIFFRACTION CALCULATIONS 5 4
Waves Passing a Single Breakwater.
Waves Passing a Gap of Width Less than Bae
Wavelengths at Normal Incidence. =
Waves Passing a Gap of Width Greater hen Five
Wavelengths at Normal Incidence.
Diffraction at a Gap-Oblique Incidence .
REFRACTION AND DIFFRACTION COMBINED.
vi
SECTION
WNW
NN bY
Picea
DUWAWWN
WDWWQHnn
WNWW
AWWW NWN
WAVE REFLECTION.
GENERAL. . . .
REFLECT ION FROM IMPERMEABLE, VERTICAL WALLS.
(Linear Theory). . . c
REFLECTIONS IN AN ENCLOSED BASIN -
WAVE REFLECTION FROM BEACHES .
BREAKING WAVES .
DEEP WATER...
SHOALING WATER .
REFERENCES AND SELECTED BIBLIOGRAPHY .
CHAPTER 3 - WAVE AND WATER LEVEL PREDICTIONS
INTRODUCTION .
CHARACTERISTICS OF OCEAN WAVES ......
SIGNIFICANT WAVE HEIGHT AND PERIOD .
WAVE HEIGHT VARIABILITY.
ENERGY SPECTRA OF WAVES. .. .
DIRECTIONAL SPECTRA OF WAVES .
WAVE FIELD...
DEVELOPMENT OF A WAVE FIELD. 5 Oa
VERIFICATION OF WAVE HINDCASTING .
DECAY OF A WAVE FIELD.
WIND INFORMATION NEEDED FOR WAVE PREDICTION.
ESTIMATING THE WIND CHARACTERISTICS.
DELINEATING A FETCH. ... :
FORECASTS FOR LAKES, BAYS, AND ESTUARIES .
Wind Data. :
Effective Fetch.
SIMPLIFIED WAVE-PREDICTION MODELS. .. .
SMB METHOD FOR PREDICTING WAVES IN DEEP WATER. Sane
EFFECTS OF MOVING STORMS AND A VARIABLE WIND SPEED
AND DIRECTION.
VERIFICATION OF SIMPLIFIED WAVE " HINDCAST PROCEDURES.
ESTIMATING WAVE DECAY IN DEEP WATER.
WAVE FORECASTING FOR SHALLOW WATER .
FORECASTING CURVES. . .
DECAY IN LAKES, BAYS, AND "ESTUARIES.
HURRICANE WAVES. :
DESCRIPTION OF HURRICANE WAVES .
MODEL WIND AND PRESSURE FIELDS FOR HURRICANES.
PREDICTION TECHNIQUE... . °
vil
SECT ION
3.
3
3
3
3
Se
Sr
3
3
3
3
3
FPAHHPA HH AHH AH
SPAHAA HAHAH FPP HP HAHA HHH H
WATERS LEVE Te HU GUAT EONS iy. uucrr- are iinaienit-)l-0-lnn tet
ASTRONOMICAL TIDES . 56 .
TSUNAMIS . .
LAKE LEVELS.
SEMCHE Semen
WAVE SETUP . . :
STORM SURGE AND. WIND SETUP :
General.
Storms .
Factors of Storm ence Generation.
Initial Water Level.
Storm Surge Prediction .
REFERENCES AND SELECTED BIBLIOGRAPHY .
CHAPTER 4 - LITTORAL PROCESSES
TNHENOIDUIERION Gg Go 6G 6 6 © 5 a o Oo a
DEFINITIONS.
Beach Profile.
Areal View .
ENVIRONMENTAL FACTORS.
Waves.
Currents . é :
Tides and Surges 4
Winds. é aC
Geologic actors A
Other Factors.
CHANGES IN THE LITTORAL | ZONE 3
LITTORAL MATERIALS .
CLASSIFICATION .
Size and Size Pavaneters :
Composition. 5
Other Gharcrerisites:
SAND AND GRAVEL. . .
COHESIVE MATERIALS . .
CONSOLIDATED MATERIAL.
Rock . oe
Beach Rock .
Organic Reefs.
OCCURRENCE OF LITTORAL MATERIALS ON ‘U. S. COASTS.
Atlantic Coast .
Gulf Coast .
Pacific Coast.
Alaska .
Hawaii .
Great Lakes.
SAMPLING LITTORAL MATERIALS.
vill
(leer ans
COMMA MOONINNINAD
AMRWNYONOODH EO
WWWWWWWDNTNNnnunwn
I
Ww
i]
i
aS,
uw
UL ts Th let
DADADWUN RR RRR ee
Pa RHR ERR EEA
SECTION
oS
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ALPHA A HAHAH aH
PPHHAHHA HAHAHA
PHP HHP AHAAAHAAHAAHAHHL
5c!
SPAT)
SIZE ANALYSES.
Sieve Analysis .
Settling Tube.
LITTORAL WAVE CONDITIONS .
EFFECT OF WAVE CONDITIONS ON SEDIMENT "TRANSPORT.
FACTORS DETERMINING LITTORAL WAVE CLIMATE.
Offshore Wave Climate. A
Effect of Bottom ag Bee
Winds and Storms .
INSHORE WAVE CLIMATE . 5 :
Mean Value Data on U.S. Riteoral Wave cimaeee 5
Mean vs. Extreme Conditions.
OFFICE STUDY OF WAVE CLIMATE .
EFFECT OF EXTREME EVENTS .
NEARSHORE CURRENTS .
WAVE- INDUCED WATER MOTION. :
FLUID MOTION IN BREAKING WAVES .
ONSHORE-OFFSHORE CURRENTS.
Onshore-Offshore Exchange.
Diffuse Return Flow.
Rip Currents . -
LONGSHORE CURRENTS .
Velocity and Flow Rate .
Velocity Prediction.
SUMMARY.
LITTORAL TRANSPORT .
INTRODUCTION . -
Importance of Littoral Transport -
Zones of Transport .
Protslesiy. 6
Profile Accuracy. . A
ONSHORE- OFFSHORE TRANSPORT :
Sediment Effects . :
Initiation of Sediment Motion. :
Seaward Limit of Significant Transport :
Beach Erosion and Recovery .
Bar-Berm Prediction
Slope of the Foreshore .
LONGSHORE TRANSPORT RATE .
Definitions and Methods.
Energy Flux Method .
Energy Flux Example (Method 3).
Empirical Prediction of Gross Longshore Transport
Rate (Method 4).
Method 4 Example .
Bal Me ang Bhan eae
UOUPHHPAHHHHWNW
COUMMNUMNWWN OO
SECTION
ROLE OF FOREDUNES IN SHORE PROCESSES .
BACKGROUND . . . Scat ee lot Sor at ee oa
ROLE OF FOREDUNES. ‘
Prevention of Overtopping.
Reservoir of Beach Sand.
Long-Term Effects.
SEDIMENT BUDGET.
DEFINITIONS.
Sediment Budget.
Elements of Sediment Badeee!
Sediment Budget Boundaries .
SOURCE OF LITTORAL MATERIAL.
Rivers . :
Erosion of Shoes Veaae Cliffs .
Transport from Offshore Slope.
Windblown Sediment Sources .
Carbonate Production .
Beach Replenishment. . .
SINKS FOR LITTORAL MATERIALS ;
Inlets and Lagoons .
Overwash . 66
Backshore and Dune etowane ,
Offshore Slopes.
Submarine Canyons.
Deflation.
Carbonate Loss . :
Mining and Dredging. . . 5
CONVECTION OF LITTORAL MATERIALS :
RELATIVE CHANGE IN SEA LEVEL .
SUMMARY OF SEDIMENT BUDGET .
ENGINEERING STUDY OF LITTORAL PROCESSES.
OFFICE STUDY .
Sources of Data
Interpretation of Enpreiane Detain C
FIELD STUDY. -
Wave Data @alieceacin :
Sediment Sampling.
Surveys.
hGAaCerS ena.
SED IMENT TRANSPORT CALCULATIONS.
Longshore Transport Rate .
Onshore-Offshore Motion.
Sediment Budget.
REFERENCES AND SELECTED BIBLIOGRAPHY .
PAGE
4-111
4-111
4-113
4-113
4-115
4-115
4-116
4-116
4-116
4-117
4-119
4-119
4-119
4-121
Aaa
4-123
4-123
4-123
4-124
4-124
4-124
4-124
Ae y
4-127
4-129
4-129
4-129
4-131
4-131
4-131
4-139
4-139
4-139
4-140
4-147
4-147
4-147
4-147
4-150
4-152
4-152
4-153
4-154
4-155
LIST OF FIGURES
Beach Profile - Related Terms. -
Sand Dunes Along the South Shore of Lake Michigan.
Sand Dunes, Honeyman State Park, Oregon.
Barrier Beach Island Developed as Recreation Park - “Jones
Beach State Park, Long Island, New York.
Large Waves Breaking Over a Breakwater 5
Wave Characteristics .
Schematic Diagram of Storm Wave "Attack on bienah Sidi Dane :
Backshore Damage at Sea Isle City, New Jersey.
Weir Jetty at Masonboro Inlet, North Carolina.
Wrightsville Beach, North Carolina, after Completion of
Beach Restoration and Hurricane Protection Project .
Approximate Distribution of Ocean Surface Wave Energy
Illustrating the Classification of Surface Waves by
Wave Band, Primary Disturbing Force and La
Restoring Force.
Definition of Terms - Elementary, ysinusoidal, ‘Progressive
Wave . a -
Local Fluid Velocities and Accelerations : -
Water Particle Displacements from Mean Position for
Shallow-Water and Deepwater Waves.
Formation of Wave Groups by the Addition of Two " Sinusoids
Having Different Periods . - 5
Summary - Linear (Airy) Wave Theory: - “Wave
Characteristics.
Regions of Validity for Various “Wave Thbontds - - -
Comparison of Second-Order Stokes' Profile with Linear
Profile. 5
Cnoidal Wave Surface Profiles as a Function tok k2.
Cnoidal Wave Surface Profiles as a Function of k?.
Relationship Between k*, H/d and Tvg/d
Relationship Between k? and L?H/d?
Relationships Between k? and L?H/d? andi Between Oe -a)/i,
(y; -d)/H)+ 1 and L*H/d? . . ;
Relationship Between Tvg/d yz/4, H/ye and “L2H/a3 -
Relationship Between C/Veyz , H/yz and L7H/d? .
Functions M and N in Sanntery Wave Theory. =
Wave Refraction at West Hampton Beach, Long Island,
New York . 5 ae age
Refraction Template.
Changes in Wave Direction ands Height Due to ‘Refraction on
Slopes with Straight, Parallel slag Contours.
Use of the Refraction Template . e305 ¢
Refraction Diagram Using R/J Method.
Use of Fan-Type Refraction Diagram . J
Refraction silage a Saree Beach with Parallel Bottom
Contours . - Se DO eG Gb Oo Ge
a
Refraction by a Submarine Ridge and Submarine Canyon .
Refraction Along an Irregular Shoreline.
Wave Incident on a Breakwater.
Wave Diffraction at Channel Islands Harbor Breakwater,
California . Bt ONE AG SIRSEs teers
Wave Diffraction Diagram - 15° Wave Angle.
Wave Diffraction Diagram - 30° Wave Angle.
Wave Diffraction Diagram - 45° Wave Angle.
Wave Diffraction Diagram - 60° Wave Angle.
Wave Diffraction Diagram - 75° Wave Angle.
Wave Diffraction Diagram - 90° Wave Angle.
Wave Diffraction Diagram - 105° Wave Angle .
Wave Diffraction Diagram - 120° Wave Angle .
Wave Diffraction Diagram - 135° Wave Angle .
Wave Diffraction Diagram - 150° Wave Angle .
Wave Diffraction Diagram - 165° Wave Angle .
Wave Diffraction Diagram - 180° Wave Angle .
Diffraction for a Single Breakwater Normal Incidence
Schematic Representation of Wave Diffraction Overlay .
Generalized Diffraction Diagram for a Breakwater Gap
Width of Two Wave Lengths (B/L = 2).
Contours of Equal Diffraction Coefficient; Gap Width =
0.5 Wave Length (B/L = 0.5). 3 fad fend 2a
Contours of Equal Diffraction Coefficients Gap Width =
1 Wave Length (B/L = 1).
Contours of
1.41 Wave
Contours of
1.64 Wave
Contours of
1.78 Wave
Contours of
Equal Diffraction Coefficient: Gap Width =
Lengths (B/L = 1.41)
Equal Diffraction Goefficient; Gap Width =
Lengths (B/L = 1.64) .
Equal Diffraction Coefficient: Gap Width =
Lengths (B/L = 1.78)
Equal Diffraction Coefficient; Gap Width =
2 Wave Lengths (B/L = 2)
Contours of
2.50 Wave
Contours of
2.95 Wave
Contours of
3.82 Wave
Contours of
Equal Diffraction Coefficient: Gap Width =
Lenghts (B/L = 2.50) .
Equal Diffraction Goeseicients Gap Width =
Lengths (B/L = 2.95)
Equal Diffraction Coefficient; Gap Width =
Lengths (B/L = 3.82)
Equal Diffraction Coefficient; pi Width =
5 Wave Lengths (B/L = 5)
Diffraction
for a Breakwater Gap oF Width > ier (B/L 5)
Wave Incidence Oblique to Breakwater Gap .
Diffraction for a Breakwater Gap of One Wave Length “Width |
(¢=10) andielS>)).
Diffraction for a Breakwater Gap of One “Wave Length Width
(¢ = 30 and 45°)
Diffraction for a Breakwater Gap of One “Wave Length ‘Width
(¢ = 60 and 75°)
xil
“fe (abr 2) SSL DSL
1 1
WMWANDAAVANAAHAAAD
SFO ANKNAUABRWN
CAE CARCASS CARGAECA
ONAN ARWN HE
Diffraction Diagram for a Gap of Two Wave Lengths and a
45° Approach Compared with that for a Gap Width Y2 Wave
Lengths with a 90° Approach. A
Single Breakwater - Refraction - Diffraction Combined.
Wave Reflection at Hamlin Beach, New York.
Standing Wave (Clapotis) System - Perfect Reflection from
a Vertical Barrier - Linear Theory .
(H,/Lo5) max vs Beach Slope . ‘
X5 VS Beach Slope for Various values Ho/Lo f
Wave of Limiting Steepness in Deep Water .
Breaker Height Index vs Deep Water Wave Steepness.
Dimensionless Depth at Breaking vs Breaker Steepness .
Spilling Breaking Wave .
Plunging Breaking Wave .
Surging Breaking Wave.
Collapsing Breaking Wave .
Sample Wave Records.
Waves in a Coastal Region. : 5
Theoretical and Gbserved Wave- SHetieht Die emanerens -
Theoretical and Observed Wave-Height Distributions .
Theoretical Wave-Height Distributions. ake
Typical Wave Spectra from the Atlantic Coast .
Observed and Hindcasted Significant Wave Heights vs “Time °
Map of North Atlantic Grid Points, Ocean Weather ee
(OWS) Stations and Argus Island. . . aan’
Surface Synoptic Chart for 0030Z, 27 October 1950.
Sample Plotted Report. - SES ohtlee ole the
Geostrophic Wind Scale .
Possible Fetch Limitations .
Relation of Effective Fetch to Width- Length Ratio’ for!
Rectangular Fetches. : :
Computation of Effective Fetch for irrepular Shoreline :
Deepwater Wave Forecasting Curves (for Fetches of 1 to
1000 miles). ... A-ha ae
Deepwater Wave Forecasting Curves. (for Fetches of 100°
to More than 1000 miles) = x
Location of Wave Hindcasting Stations aad Summary of
Synoptic Meterological Observations (SSMO) Areas .
A Comparison of Shipboard Observations and Hindcasts .
Decay Curves ..
Travel Time of Swell Based on tty = = D/Cg.
Forecasting Curves for Shallow-Water Waress Constant
Depth = 5 Feet . :
Forecasting Curves for Shaddiewe Water waves: hconseant
Depth = 10 Feet.
Forecasting Curves for Shallow- Water aves ‘Gonsvant
Depth = 15 Feet.
xii
PAGE
FIGURE
Forecasting Curves for Shallow-Water Waves; Constant
Depth =i20i Feet 2.0 ea he RT O.s ORO LUA .
Forecasting Curves for Shallow-Water Waves; Constant
Deyo WA Wo pig 6 oo OF Oo 6 ee Oo 6 Gg oO oe
Forecasting Curves for Shallow-Water Waves; Constant
Depth =sS0MRCCt sar-ak-em iltme- eit aati cem fale ren tn
Forecasting Curves for Shallow-Water Waves; Constant
IDYeyohe a Ss) IEE 5 505 Solo ol 5 G 85 51g 3 ole
Forecasting Curves for Shallow-Water Waves; Constant
Depth. =! 40.\Feetssrekracse i. caanioetll Pals tc
Forecasting Curves for Shallow-Water Waves; Constant
Depth = 45 Feet. MY eens ee ee ee eee
Forecasting Curves for Shallow-Water Waves; Constant
Depth = 50 Feet. 5 6 6 oo
Typical Hurricane Wave sage 5
Composite Wave Charts. 5
Pressure and Wind Distribution in iMedes Hurricane.
Isolines of Relative ee aan Wave Height for Slow
Moving Hurricane . -
Relationship for Friction: Loss Over a Bottom of
Constant Depth . - -
Typical Tide Curves Along “Atlantic and Gulf Coasts :
Typical Tide Curves Along Pacific Coasts of the United
States . E 5
Sample Tsunami Records from Tide Gages :
Typical Water Level Variations in Lake Erie.
Long-Wave Surface Profiles .
Storm Surge and Observed Tide Chart. oo 4
High Water Mark Chart for Texas, Hurricane Carian
7-12 September 1961.
Notation and Reference Frame .
Storm Surge Chart. 5 :
Schematic of Forces and Responses “for. Bathystrophic
Approximation. 5
Various Setup Components Over. the iconcinentat “Shelf.
Track for Hurricane Camille, August 1969 .....
Foot Surface Isovels Soebee Hurricane nahi A
August 1969.
Seabed Profile Used for’ Beemalesiie ficamiaialos Acirast? 1969 :
Open Coast Surge eee Hurricane Camille,
August 1969. - -
Preliminary Estimate os Peak paeee :
Shoaling Factors on Gulf Coast .
Shoaling Factors on East Coast .
Correction Factor for Storm Motion .
Comparison of Observed and Computed Peak Surges (for 43.
Storms with a Landfall South of New England from
1893 - 1957) ay) Rep ie) CA eae orto
Surge Profile Along Coast. Hurricane Camille,
August 1969.
XIV
FIGURE
3-57
Lake Surface Contours on Lake Okeechobee, Florida
Hurricane, 26-27 poy Ora 1949.
Grid System . C 3
Lake Erie .
Cross-Section beer ane PoPase “Width: - Ties THe.
Mean Bottom Profile of Lake Erie. ‘ :
Wind Speed and Direction for Lake Erie - “Storm, March 1955.
Wind Setup Hydrograph for Buffalo and Toledo - Stem
March 1955. .
Wind Setup Profile for Lake Erie - “Stora, March 1955.
Typical Profile Changes with Time, Nes EnSRp Gon Beach, N.Y..
Three Types of Shoreline. 3 :
Shoreline Erosion near Shipbottom, 'N. an Suns
Shoreline Accretion and Erosion Near Beach Haven, N. im
Stable Shoreline Near Peahala, N.J... . ;
Fluctuations in Location of Mean Sea Level "Shoreline. on
Seven East Coast Beaches.
Grain Size Scales . -
Example Size Distribution :
Sand Size Distribution Along the v7. S. Actant ic Coast.
Mean Monthly Nearshore Wave aeaaiits for Five Coastal
Segments. 4 :
Mean Monthly Nearshore. Wave peered for, Eine Coastal
Segments. ..
Distribution of Significant Wave “Heights “from Coastal
Wave Gages for l-year Records . se
Nearshore Current System Near LaJolla Sea Getareceaanen
Typical Rip Currents, Ludlam Island, N.J.
Distribution of Longshore Current (elecieics:
Measured versus Predicted Longshore Current Speed .
Coasts in the Vicinity of New York Bight. nets
Three Scales of Profiles, Westhampton, Long oersicle 3
Unit Volume Change versus Time Between Surveys for Profiles
on South Shore of Long Island . 5c 7
Maximum Wave Induced Bottom NeORIEY as a Foe cea AG
Relative Depth. . . -
Maximum Bottom Velpcere Sst) Small Amplitude finean -
Initiation of Ripple Motion .
Wave Conditions HEE Sane Maximum Becton ete Ae
0.5 ft/sec. ‘
Nearshore Bachymetry “with Syne Parallel aaa off
Panama City, Florida. :
Nearshore Bathymetry with Shore- Parallel "Contours and
Linear Bars off Manasquan, N.J. oc :
Slow Accretion of Ridge-and-Runnel at Crane Beach, “Mass...
Rapid Accretion of Ridge-and-Runnel - Lake Michigan .
Typical Berm and Bar Profiles from Prototype Size
Laboratory Wave Tank.
XV
fe Rude eee
uuuwnunb fp HSH
WNHrOUWUON AM
Berm - Bar Criterion Based on Dimensionless Fall Time and
Deep Water Steepness. ;
Berm - Bar Criterion Based on Dimensionless Fall “Time oad
Height to Grain Size Ratio. c 4
Fall Velocity of Quartz Spheres in Water asa Funetien “of
Diameter and Temperature. ome ine
Data Trends - Median Grain Size versus teoncavere Sipe.
Data - Median Grain Size versus Foreshore Slope . :
Longshore Component of Wave Energy Flux in Dimensionless
Form as a Function of Breaker Conditions. 5
Longshore Component of Wave Energy Flux as a Finetion of
Deepwater Wave Conditions .
Transport Rate versus Energy Flux Factor for. Field and
Laboratory Conditions . :
Relationship Between Wave Energy ead Longshore Transport.
Longshore Transport Rate as a Function of Breaker Height
and Breaker Angle . ; :
Longshore Transport Rate as a Function “of Deepwater Wave
Height and Deepwater Angle.
Upper Limit on Longshore Transport Rates.
Typical Barrier Island Profile Shape. : 5
Event Frequency per 100 Years the Stated Level is 5 Equalled
or Exceeded on the Open Coast, South Padre Island, Texas.
Basic Example of Sediment Budget. 0 ¢
Erosion Within Littoral Zone a Te Uniform Retreat of an
Iidealaizedserotivle sens.
Sediment Trapped Inside old Drum Inlet, N. C.
Overwash on Portsmouth Island, N.C. . .
Growth of a Spit in to Deep Water, Sandy “Hook, N. oe
Dunes Migrating Inland Near Laguna Point, California.
Materials Budget for Littoral Zone. 4
Summary of Example Problem Conditions and Results :
Variation of y with Distance Along Spit .
Growth of Sandy Hook, N.J., 1835-1932 . c 3
Transport Directions at New Buffalo Harbor Jetty on
Lake Michigan . .
Sand Accumulation at Point ‘Mugu, “California qe
Tombolo and Pocket Beach at Greyhound Rock, California.
A Nodal Zone of Divergence Illustrated by Sand Accumulation
at Groins, South Shore Staten Island, N.Y. ;
South Shore of Long Island, N.Y. Savin van Closed,
Partially Closed and Open Inlets. :
Four Types of Barrier Islands Offset.
Fire Island Inlet, New York - Overlapping Offset.
Old Drum Inlet, North Carolina - Negligible Offset.
xvi
LIST OF TABLES
Shoreline Characteristics .
Distribution of Wave Heights in a Short Train of Waves.
Example Computations of Values of C /C, for Refraction
Analysis. Sete Od
Correction for Sea-Air Temperature.
Wind-Speed Adjustment, Nearshore.
Values of K, or (H/H3). ss 3 :
Computations for Wind Waves Over ithe: Continental ‘Shelf.
Tidal Ranges. : -
Fluctuations in Water Levels - esa lakes: aren
(1860 through 1970)
Short-Period Fluctuations in take Levels) ae Selected
Gage Sites. .
Highest and Lowest Water hovers : é
Systems of Units for Storm Surge Computations -
Manual Surge Computations .
Seasonal Profile Changes on Southern California Beaches .
Density of Littoral Materials .
Minerals Occurring in Beach Sand. - :
Mean Wave Height at Coastal Localities of Conterminous
United States . -
Storm-Induced Beach Changes c
Longshore Transport Rates from U. S. Coasts. :
Longshore Energy Flux Pg, for a Bos Periodic Wave. in
Any Specified Depth . 5 0%
Approximate Formulas for Computing Longshore Energy Pine
Factor, Peg, Entering the Surf Zone . Bends
Assumptions for Pos Formulas in Table 4-8 . AEE
Deepwater Wave Heights, in Percent by Direction, Bee East
Facing Coast of Inland Sea. c 5 -
Computed Longshore Transport for East - Facing ‘Coast “of
Inland Sea. : A
Estimate of Gross Longshore Transport Rate “for Shore. of
Inland Sea.
Classification of Boerne in ane eet rane sadeaeet
Budget. .
Sand Budget of the Littoral Zone .
Xvil
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43 ocapiteaieieab hoe
49ubi a ny)
CHAPTER 1
INTRODUCTION
TO
COASTAL ENGINEERING
HALEIWA BEACH, OAHU, HAWAII — 23 August 1970
CHAPTER 1
INTRODUCTION TO COASTAL ENGINEERING
1.1 INTRODUCTION TO THE SHORE PROTECTION MANUAL
This Shore Protection Manual has been prepared to assemble in a
single three-volume publication coastal-engineering practices for shore
protection. ''Coastal Engineering" is defined as the application of the
physical and engineering sciences to the planning, design, and construc-
tion of works to modify or control the interaction of the air, sea, and
land in the coastal zone for the benefit of man and for the enhancement
of natural shoreline resources. "Shore protection," as used in this
Manual, applies to works designed to stabilize the shores of large bodies
of water where wave action is the principal cause of erosion. Much of
the material is applicable to the protection of navigation channels and
harbors.
The nature and degree of required shore-protection measures vary
widely at different localities. Proper solution of any specific problem
requires systematic and thorough study. The first requisite for such
study is a clear definition of the problem and the objectives sought.
The first factor to be determined is the cause of the problem. Ordinarily
there will be more than one method of obtaining the immediate objective.
Therefore, the long-term effects of each method should be studied. The
immediate and long-term effects of each method should be evaluated not
only within the problem area, but also in adjacent shore areas. All
physical and environmental effects, advantageous and detrimental, should
be considered in comparing annual costs and benefits to determine the
justification of protection methods.
Detailed summaries of applicable methods, techniques, and useful data
pertinent to the solution of shore protection problems have been included
in this Manual.
By replacing Shore Protection, Planning and Destgn with the Shore
Protection Manual, CERC is providing coastal engineers with an improved
tool for solving shore-protection problems. The Manual is designed as
an advanced text, but contains sufficient introductory material to allow
a person with an engineering background to obtain an understanding of
coastal phenomena and to solve related engineering problems.
Chapter 1 presents a basic introduction to the subject. Chapter 2,
"Mechanics of Wave Motion," treats wave theories, wave refraction and
diffraction, wave reflection, and breaking waves. Chapter 3, "Wave and
Water Level. Predictions," discusses wave forecasting, hurricane waves,
storm surge, and water level fluctuations. Chapter 4, "Littoral Proc-
esses,\' treats the characteristics and sources of littoral material
nearshore currents, littoral transport, and sand budget techniques.
Chapter 5, "Planning Analyses," treats the functional planning of shore-
protection measures. Chapter 6, "Structural Features," illustrates the
functional design of various structures. Chapter 7, "Structural
Design--Physical Factors," treats the effects of environmental forces
on the design of protective works. Chapter 8, "Engineering Analysis--
Case Study,"' presents a series of calculations for the preliminary
design of an offshore island facility in the mouth of the Delaware
Bay.
Each chapter contains its own bibliography. This Manual concludes
with four appendixes. Because the meanings of coastal engineering terms
differ from place to place, the reader is urged to use Appendix A,
Glossary of Terms, that defines the terms used in this Manual. Appen-
dix B lists the symbols used. Appendix C is a collection of miscella-
neous tables and plates that supplement the material in the chapters.
Appendix D is the subject index.
1,2 THE SHORE ZONE
Table 1-1 summarizes regional shoreline characteristics. The infor-
mation obtained from the "Report on the National Shoreline Study," by the
Department of the Army, Corps of Engineers (1971), indicates that of the
total 84,240 miles of U.S. shoreline, there are 34,520 miles (41 percent)
of exposed shoreline and 49,720 miles (59 percent) of sheltered shoreline
(i.e., in bays, estuaries and lagoons). About 20,500 miles of the shore-
line (or 24 percent of the total) are eroding. Of the total length of
shoreline, exclusive of Alaska (36,940 miles), about 12,150 miles (33
percent) have beaches; the remaining 24,790 miles have no beach.
1,21 NATURAL BEACH PROTECTION
Where the land meets the ocean at a sandy beach, the shore has
natural defenses against attack by waves, currents and storms. First
of these defenses is the sloping nearshore bottom that causes waves to
break offshore, dissipating their energy over the surf zone. The pro-
cess of breaking often creates an offshore bar in front of the beach
that helps to trip following waves. The broken waves re-form to break
again, and may do this several times before finally rushing up the beach
foreshore. At the top of wave uprush a ridge of sand is formed. Beyond
this ridge, or crest of the berm, lies the flat beach berm that is
reached only by higher storm waves. A beach profile and its related
terminology are shown in Figure 1-1.
1.22 NATURAL PROTECTIVE DUNES
Winds blowing inland over the foreshore and berm move sand behind the
beach to form dunes. (See Figures 1-2 and 1-3.) Grass, and sometimes
bushes and trees, grow on the dunes, and the dunes become a natural levee
against sea attack. Dumes are the final natural protection line against
wave attack, and are also a reservoir for storage of sand against storm
waves.
Table 1-1. Shoreline Characteristics
es os a =—
Region Total Exposed Non-Eroding i Without
H Beach
Nard Fes) (miles) | (miles) | i i (miles) i i
North
Atlantic
South B 11,020
Atlantic-
Gulf
Lower
Mississippi
Texas Gulf
Great Lakes
California
North
Pacific
From: Saaeem on National Shoreline Study, Department of the Army, Corps of Rnginders, Aust i97i. Engineers, August 1971.
Coosto!l area
Beoch or shore Nearshore zone
(defines ores of neorshore currents)
Bockshore ° ° Inshore or shoreface
Breokers
Beach scarp J
Crest of berm
Bottom
Figure 1-1. Beach Profile - Related Terms
is
Figure 1-3. Sand Dune, Honeyman State Park, Oregon
1-4
1.23 BARRIER BEACHES, LAGOONS AND INLETS
In some areas, an additional natural protection for the mainland is
provided in the form of barrier beaches. (See Figure 1-4.) Nearly all
of the U.S. east coast from Long Island to Mexico is comprised of bar-
rier beaches. These are long narrow islands or spits lying parallel to
the shoreline. Barrier beaches generally enclose shallow lagoons that
separate the mainland from the ocean. During severe storms these barrier
beaches absorb the brunt of the wave attack. When barrier-beach dunes
are breached, the result may be the cutting of an inlet. The inlet per-
mits sand to enter the lagoon and settle to the bottom, removing sand
from the beach.
1.24 STORM ATTACK
During storms, strong winds generate high waves. Storm surge and
waves may raise the water level near the shore. If storm surge does
occur, large waves can then pass over the offshore bar formation without
breaking. If the storm occurs at high tide, storm surge super-elevates
the water, and some waves may break on the beach or even at the base of
the dunes. After a storm or storm season, natural defenses may again be
re-formed by normal wave and wind action.
1.25 ORIGIN AND MOVEMENT OF BEACH SANDS
Most of the sands of the beaches and nearshore slopes are normally
small, resistant rock particles that have traveled many miles from in-
land mountains. When the sand reaches the shore, it is moved alongshore
by waves and littoral currents. This alongshore transport is a constant
process, and great volumes may be transported. In most coastal segments
the direction of movement changes as direction of wave attack changes.
1.3 THE SEA IN MOTION
1.31 TIDES AND WINDS
The motions of the sea originate in the gravitational effects of the
sun, the moon, and earth; and from air movements or winds caused by dif-
ferential heating of the earth.
The moon, and to a lesser extent the sun, creates ocean tides by
gravitational forces. These forces of attraction, and the fact that the
sun, moon, and earth are always in motion with relation to each other,
cause waters of ocean basins to be set in motion. These tidal motions of
water masses are a form of very long period wave motion, resulting in a
rise and fall of the water surface at a point. There are normally two
tides per day, but some localities have only one per day.
1.32 WAVES
The familiar waves of the ocean:are wind waves generated by winds
blowing over water. They may vary in size from ripples on a pond to
5
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large ocean waves as high as 100 feet. (See Figure 1-5.) Wind waves
cause most of the damage to the ocean coasts. Another type of wave,
the tsunami, is created by earthquakes or other tectonic disturbances
on the ocean bottom. Tsunamis have caused spectacular damage at times,
but fortunately, major tsunamis do not occur frequently.
Wind waves are of the type known as oscillatory waves, and are
usually defined by their height, length, and period. (See Figure 1-6.)
Wave height is the vertical distance from the top of the crest to the
bottom of the trough. Wavelength is the horizontal distance between
successive crests. Wave period is the time between successive crests
passing a given point.
As waves propagate in water, only the form and part of the energy
of the waves move forward; the water particles remain.
The height, length, and period of wind waves are determined by the
feteh (the distance the wind blows over the sea in generating the waves),
the wind speed, the length of time the wind blows, and the decay distance
(the distance the wave travels after leaving the generating area). Gener-
ally, the longer the fetch, the stronger the wind; and the longer the
time the wind blows, the larger the waves. The water depth, if shallow
enough, will also affect the size of wave generated. The wind simul-
taneously generates waves of many heights, lengths, and periods as it
blows over the sea.
If winds of a local storm blow toward the shore, the generated
waves will reach the beach in nearly the form in which they are gener-
ated. Under these conditions, the waves are steep; that is, the wave-
length is 10 to 20 times the wave height. Such waves are called seas.
If waves are generated by a distant storm, they may travel through
hundreds or even thousands of miles of calm areas before reaching the
shore. Under these conditions, waves decay - short, steep waves are
eliminated, and only relatively long, low waves reach the shore. Such
waves have lengths from 30 to more than 500 times the wave height, and
are called swell.
1.33 CURRENTS AND SURGES
Currents are created in oceans and adjacent bays and lagoons when
water in one area becomes higher than water in another area. Water in
the higher area flows toward the lower area, creating a current. Some
causes of differences in the elevation of the water surface in the oceans
are tides, wind, waves breaking on a beach, and streams. Changes in water
temperature or salinity cause changes in water density that may also pro-
duce currents.
Wind creates currents because, as it blows over the water surface,
it creates a stress on surface water particles, and starts these parti-
cles moving in the direction in which the wind is blowing. Thus, a sur-
face current is created. When such a current reaches a barrier, such as
coastline, water tends to pile up against the land. In this way, wind
fe”
(Portland Cement Associat son)
Figure 1-5. Large Waves Breaking over a Breakwater
Direction of Wave Travel
L= Wovelength
Wave Crest ~~» ~~ a = Wove Height
Le Crest Length——=—
Region
~~~ Wave Trough
Stillwoter Level
Trough Length—
Region d= Depth
Ocean Bottom :
SRRAMAMAAARAAAAARAARA AAA
Figure 1-6, Wave Characteristics
setup or storm surges are created by strong winds, The height of storm
surge depends on wind velocity and direction, fetch, water depth, and
nearshore slope. In violent storms, storm surge may raise sea level at
the shore as much as 20 feet. In the United States, larger surges occur
on the Gulf coast because of the shallower and broader shelf off that
coast compared to the shelves off the Atlantic and Pacific coasts. Storm
surges may also be increased by a funneling effect in converging estuaries.
When waves approach the beach at an angle, they create a current in
shallow water parallel to the shore, known as the longshore current. This
current, under certain conditions, may turn and run out to sea in what is
known as a rtp current,
1,34 TIDAL CURRENTS
If water level rises and falls at an area, then water must flow into
and out of the area, Significant currents generated by tides occur at in-
lets to lagoons and bays or at entrances to harbors. At such constricted
places, tidal currents generally flow in when the tide is rising (flood
tide) and flow out as the tide falls (ebb tide). Exceptions can occur
at times of high river discharge or strong winds, or when density cur-
rents are an important part of the current system.
In addition to creating currents, tides constantly change the level
at which waves attack the beach.
1.4 THE BEHAVIOR OF BEACHES
1.41 BEACH COMPOSITION
The size and character of sediments on a beach are related to forces
to which the beach is exposed and the type of material available at the
shore. Most beaches are composed of fine or coarse sand and, in some
areas, of small stones called shingle or gravel. This material is sup-
plied to the beach zone by streams, by erosion of the shores caused by
waves and currents and, in some cases, by onshore movement of material
from deeper water. Clay and silt do not usually remain on ocean beaches
because the waves create such turbulence in the water along the shore
that these fine materials are kept in suspension, It is only after mov-
ing away from the beaches into quieter or deeper water that these fine
particles settle out and deposit on the bottom.
1.42 BEACH CHARACTERISTICS
Characteristics of a beach are usually described in terms of average
size of the sand particles that make up the beach, range and distribution
of sizes of those particles, sand composition, elevation and width of berm,
slope or steepness of the foreshore, the existence (or lack) of a bar, and
the general slope of the inshore zone fronting the beach, Generally, the
larger the sand particles the steeper the beach slope. Beaches with
gently sloping foreshores and inshore zones usually have a preponderance
i=<9
of the finer sizes of sand. Daytona Beach, Florida, is a good example
of a gently sloping beach composed of fine sand.
1.43 BREAKERS
As a wave moves toward shore, it reaches a depth of water so shallow
that the wave collapses or breaks. This depth is equal to about 1.3 times
the wave height. Thus a wave 3 feet high will break in a depth of about
4 feet. Breaking can occur in several different ways (plunging, spilling,
surging, or collapsing). Breaking results in a dissipation of the energy
of the wave and is manifested by turbulence in the water. This turbulence
stirs up the bottom materials. For most waves, the water travels forward
after breaking as a foaming, turbulent mass, expending most of its remain-
ing energy in a rush up the beach slope.
1.44 EFFECTS OF WIND WAVES
Wind waves affect beaches in two major ways. Short steep waves,
which usually occur during a storm near the coast, tend to tear the
beach down. (See Figure 1-7.) Long swells, which originate from dis-
tant storms, tend to rebuild the beaches. On most beaches, there is a
constant change caused by the tearing away of the beach by local storms
followed by gradual rebuilding by swells. A series of violent local
storms in a short time can result in severe erosion of the shore if
there is not enough time between storms for swells to rebuild the
beaches. Alternate erosion and accretion of beaches may be seasonal
on some beaches; the winter storms tear the beach away, and the summer
swells rebuild it. Beaches may also follow long-term cyclic patterns.
They may erode for several years, and then accrete for several years.
1.45 LITTORAL TRANSPORT
Littoral transport is defined as the movement of sediments in the
nearshore zone by waves and currents and is divided into two general
classes: transport parallel to the shore (longshore transport) and
transport perpendicular to the shore (onshore-offshore transport). This
transport is distinguished from the material moved, which is called
littoral drtft.
Onshore-offshore transport is determined primarily by wave steepness,
sediment size, and beach slope. In general, high steep waves move material
offshore, and low waves of long period (low steepness waves) move material
onshore. This onshore-offshore process associated with storm waves is
illustrated in Figure 1-7.
Longshore transport results from the stirring up of sediment by the
breaking wave, and the movement of this sediment by the component of the
wave in an alongshore direction, and by the longshore current generated
by the breaking wave. The direction of longshore transport is directly
related to the direction of wave approach, and the angle of the wave to
the shore. Thus, due to the variability of wave approach, longshore
He=AK 9,
Dune Crest
Profile A — Normal wave action
Profile B — Initial attack of
storm waves
Crest
Lowering Profile C — sor wave attac : = ia
noel of foredune ea prt
Recession lo
~ ACCRETION ater
Profile A
Profile D — After storm wave attack
Normal wave action
ACCRETION cad =
bs
Profile A
Figure 1-7, Schematic Diagram of Storm Wave Attack on Beach and Dune
transport direction can vary from season to season, day to day or hour
to hour. These reversals of transport direction are quite common for
most United States shores. Direction may vary at random, but in most
areas the net effect is seasonal.
The rate of longshore transport is dependent on both angle of wave
approach, and wave energy. Thus, high storm waves will generally move
more material per unit time than low waves. However, if low waves
exist for a much longer time than do high waves, the low waves may be
more significant in moving sand than the high waves.
Because reversals in transport direction occur, and because different
types of waves transport material at different rates, two components of
the longshore transport rate become important. The first is the net rate,
the net amount of material passing a particular point in the predominant
direction during an average year. The second component is the gross rate,
the total of all material moving past a given point in a year regardless
of direction. Most shores consistently have a net annual longshore trans-
port in one direction. Determining the direction and average net and
gross annual amount of longshore transport is important in developing
shore protection plans.
In landlocked water of limited extent, such as the Great Lakes, a
longshore transport rate in one direction can normally be expected to be
no more than about 150,000 cubic yards per year. For open ocean coasts,
the net rate of transport may vary from 100,000 to more than 2 million
cubic yards per year. The rate depends on the local shore conditions
and shore alignment as well as the energy and direction of wave action.
1.46 EFFECT OF INLETS ON BARRIER BEACHES
Inlets may have significant effects on adjacent shores by interrupt-
ing the longshore transport and trapping onshore-offshore moving sand.
On ebb current, sand moved to the inlet by waves is carried a short dis-
tance out to sea and deposited on an outer bar. When this bar becomes
large enough, the waves begin to break on it, and sand again begins to
move over the bar back toward the beach. On the flood tide, when water
flows through the inlet into the lagoon, sand in the inlet is carried a
short distance into the lagoon and deposited. This process creates shoals
in the landward end of the inlet known as middleground shoals or tnner
bars. Later, ebb flows may bring some of the material in these shoals
back to the ocean, but some is always lost from the stream of littoral
drift and thus from the downdrift beaches. In this way, tidal inlets
may store sand and reduce the supply of sand to adjacent shorelines.
1.47 IMPACT OF STORMS
Hurricanes or severe storms moving over the ocean near the shore may
greatly change beaches. Strong winds of a storm often create a storm surge.
This surge raises the water level and exposes to wave attack higher parts
of the beach not ordinarily vulnerable to waves. Such storms also generate
teli2
large, steep waves. These waves carry large quantities of sand from the
beach to the nearshore bottom. Land structures, inadequately protected
and located too close to the water, are then subjected to the forces of
waves and may be damaged or destroyed. Low-lying areas next to the ocean,
lagoons, and bays are often flooded by storm surge. Storm surges are
especially damaging if they occur concurrently with astronomical high tide,
Beach berms are built naturally by waves to about the highest eleva-
tion reached by normal storm waves. Berms tend to absorb the wave energy;
however, overtopping permits waves to reach the dunes or bluffs in back of
the beach and damage unprotected upland features.
When storm waves erode the berm and carry the sand offshore, the pro-
tective value of the berm is reduced and large waves can overtop the beach.
The width of the berm at the time of a storm is thus an important factor
in the amount of upland damage a storm can inflict.
Notwithstanding changes in the beach that result from storm-wave
attack, a gently sloping beach of adequate width and height is the most
effective method known for dissipating wave energy.
1.48 BEACH STABILITY
Although a beach may be temporarily eroded by storm waves and later
partly or wholly restored by swells, and erosion and accretion patterns
may occur seasonally, the long-range condition of the beach - whether
eroding, stable or accreting - depends on the rates of supply and loss
of littoral material. The shore accretes or progrades when the rate of
supply exceeds the rate of loss. The shore is considered stable (even
though subject to storm and seasonal changes) when the long-term rates
of supply and loss are equal.
1.5 EFFECTS OF MAN ON THE SHORE
1.51 ENCROACHMENT ON THE SEA
During the early days of the United States, natural beach processes
continued to mold the shore as in ages past. As the country developed,
activity in the shore area was confined principally to harbor areas.
Between harbor areas, development along the shore progressed slowly as
small, isolated, fishing villages. As the national economy grew, im-
provements in transportation brought more people to the beaches. Gradu-
ally, extensive housing, commercial, recreational and resort developments
replaced fishing villages as the predominant coastal manmade features.
Examples of this development are Atlantic City and Miami Beach.
Numerous factors control the growth of development at beach areas,
but undoubtedly the beach environment is the development's basic asset.
The desire of visitors, residents, and industries to find accommodations
as close to the ocean as possible has resulted in man's encroachment on
the sea.
I-13
There are places where the beach has been gradually widened, as well
as narrowed, by natural processes over the years. This is evidenced by
lighthouses and other structures that once stood on the beach, but now
stand hundreds of feet inland.
In their eagerness to be as close as possible to the water, developers
and property owners often forget that land comes and goes, and that land
which nature provides at one time may later be reclaimed by the sea. Yet
once the seaward limit of a development is established, this line must be
held if large investments are to be preserved. This type of encroachment
has resulted in great monetary losses due to storm damage, and in ever-
increasing costs of shore protection.
1.52 NATURAL PROTECTION
While the sloping beach and beach berm are the outer line of defense
to absorb most of the wave energy, dunes are the last zone of defense in
absorbing the energy of storm waves that overtop the berm. Although dunes
erode during severe storms, they are often substantial enough to afford
complete protection to the land behind them. Even when breached by waves
of a severe storm, dunes may gradually rebuild naturally to provide pro-
tection during future storms. Continuing encroachment on the sea with
manmade development has often taken place without proper regard for the
protection provided by dunes. Large dune areas have been leveled to make
way for real estate developments, or have been lowered to permit easy
access to the beach. Where there is inadequate dune or similar protection
against storm waves, the storm waters may wash over low-lying land, mov-
ing or destroying everything in their path, as illustrated by Figure 1-8.
1.53 SHORE PROTECTION METHODS
Where beaches and dunes protect shore developments, additional pro-
tective works may not be required. However, when natural forces do
create erosion, storm waves may overtop the beach and damage backshore
structures. Manmade structures must then provide protection. In gene-
ral, measures designed to stabilize the shore fall into two classes:
structures to prevent waves from reaching erodible material (seawalls,
bulkheads, revetments); and an artificial supply of beach sand to make
up for a deficiency in sand supply through natural processes. Other
manmade structures, such as groins and jetties, are used to retard the
longshore transport of littoral drift. These may be used in conjunction
with seawalls or beachfills or both.
Separate protection for short reaches of eroding shores (as an indi-
vidual lot frontage) within a larger zone of eroding shore, is difficult
and costly. Such protection often fails at its flanks as the adjacent
unprotected shores continue to recede. Partial or inadequate protective
measures may even accelerate erosion of adjacent shores. Coordinated
action under a comprehensive plan that considers erosion processes over
the full length of the regional shore compartment is much more effective
and economical.
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1,54 BULKHEADS, SEAWALLS AND REVETMENTS
Protection on the upper part of the beach which fronts backshore
development is required as a partial substitute for the natural pro-.
tection that is lost when the dunes are destroyed. Shorefront owners
have resorted to shore armoring by wave-resistant walls of various
types. A vertical wall in this location is known as a bulkhead, and
serves as a secondary line of defense in major storms. Bulkheads are
constructed of steel, timber, or concrete piling. For ocean-exposed
locations, bulkheads do not provide a long-term solution, because a
more substantial wall is required as the beach continues to recede and
larger waves reach the structure. Unless combined with other types of
protection, the bulkhead must be enlarged into a massive seawall capable
of withstanding the direct onslaught of the waves. Seawalls may have
vertical, curved or stepped faces. While seawalls may protect the up-
land, they can create a local problem. Downward forces of water created
by waves striking the wall, can rapidly remove sand from in front of the
wall. A stone apron is often necessary to prevent excessive scouring
and undermining.
A revetment armors the slope face of a dune or bluff. It is usually
composed of one or more layers of stone or is of concrete construction.
This sloping protection dissipates wave energy with less damaging effect
on the beach than waves striking vertical walls.
1.55 BREAKWATERS
Beaches and bluffs or dunes can be protected by an offshore break-
water that reduces the wave energy reaching the shore. However, offshore
breakwaters are usually more costly than onshore structures, and are sel-
dom built solely for shore protection. Offshore breakwaters are construc-
ted mainly for navigation purposes. A breakwater protecting a harbor
area provides shelter for boats. Breakwaters have both beneficial and
detrimental effects on the shore. All breakwaters reduce or eliminate
wave action and thus protect the shore immediately behind them. Whether
offshore or shore-connected, the elimination of wave action reduces long-
shore transport, obstructing the movement of sand along the shore and
starving the downdrift beaches.
At a harbor breakwater, the sand stream generally can be restored by
pumping sand through a pipeline from the side where sand accumulates to
the eroded downdrift side. This type of operation has been in use for
many years at Santa Barbara, California.
Even without a shore arm, an offshore breakwater reduces wave action
and creates quiet water between it and the shore. In the absence of wave
action to move sand, it is deposited and builds the shore seaward toward
the breakwater. The buildup serves as a barrier which also blocks the
movement of littoral materials. If the offshore breakwater is placed
immediately updrift from a navigation opening, the structure impounds
sand, prevents it from entering the navigation channel, and affords
1-16
shelter for a floating dredge to pump the impounded material across the
navigation opening back onto the downdrift.beach. This method is used
at Channel Island Harbor near Port Hueneme, California.
1.56 GROINS
The groin is a barrier-type structure that extends from the backshore
into the littoral zone. The basic purposes of a groin are to interrupt
longshore sand movement, to accumulate sand on the shore, or to retard
sand losses. Trapping of sand by a groin is done at the expense of the
adjacent downdrift shore unless the groin or groin system is artificially
filled with sand to its entrapment capacity from other sources. To reduce
the potential for damage to property downdrift of a groin, some limitation
must be imposed on the amount of sand permitted to be impounded on the
updrift side. Since more and more shores are being protected, and less
and less sand is available as natural supply, it is now desirable, and
frequently necessary, to place sand artificially to fill the area between
the groins, thereby ensuring a more or less uninterrupted passage of the
sand to the downdrift shores.
Groins have been constructed in various configurations using timber,
steel, concrete or rock. Groins can be classified as high or low, long
or short, permeable or impermeable, and fixed or adjustable.
A high groin, extending through the breaking zone for ordinary or
moderate storm waves, initially entraps nearly all of the longshore
moving sand within that intercepted area until the areal pattern or sur-
face profile of the accumulated sand mass allows sand to pass around the
seaward end of the groin to downdrift shores. Low groins (top profile no
higher than that of desired beach dimensions) function like high groins,
except that sand also passes over the top of the structure. Permeable
groins permit some of the wave energy and moving sand to pass through the
structure.
1.57 JETTIES
Jetties are generally employed at inlets in connection with naviga-
tion improvements. When sand being transported along the coast by waves
and currents arrives at an inlet, it flows inward on the flood tide to
form an inner bar, and outward on the ebb tide to form an outer bar.
Both formations are harmful to navigation through the inlet, and must be
controlled to maintain an adequate navigation channel. The jetty is sim-
ilar to the groin in that it traps sand moving along the beach. Jetties
are usually constructed of steel, concrete, or rock. The jetty type
depends on foundation conditions, wave climate, and economic considera-
tions. Jetties are much larger than groins, since jetties sometimes ex-
tend from the shoreline seaward to a depth equivalent to the channel
depth desired for navigation purposes. To be efficient in maintaining
the channel, the jetty must be high enough to completely obstruct sand
movement.
Wg
Jetties aid navigation by reducing movement of sand into the channel,
by stabilizing the location of the channel, and by shielding vessels from
waves. Sand is impounded at the updrift jetty, and the supply of sand to
the shore downdrift from the inlet is reduced, thus causing erosion of
that shore. Before the installation of a jetty, nature supplied sand by
transporting it across the inlet intermittently along the outer bar to the
downdrift shore.
To eliminate undesirable downdrift erosion, some projects provide for
dredging the sand impounded by the updrift jetty and pumping it through a
pipeline (bypassing the inlet) to the eroding beach. This provides an
intermittent flow of sand to nourish the downdrift beach, and also prevents
shoaling of the entrance channel.
A more recent development for sand bypassing provides a low section
or weir in the updrift jetty over which sand moves into a sheltered pre-
dredged, deposition basin. By dredging the basin periodically, deposition
in the channel is reduced or eliminated. The dredged material is normally
pumped across the inlet to provide nourishment for the downdrift shore.
A weir jetty of this type at Masonboro Inlet, North Carolina, is shown in
Figure 1-9,
1.58 BEACH RESTORATION AND NOURISHMENT
As previously stated, beaches are very effective in dissipating wave
energy. When maintained to adequate dimensions, they afford protection
for the adjoining backshore. Therefore, a protective beach is classed as
a shore-protection structure. When studying an erosion problem, it is gen-
erally advisable to investigate the feasibility of mechanically or hydrau-
lically placing borrow material on the shore to form and maintain an ade-
quate protective beach. The method of placing beach fill to ensure sand
supply at the required replenishment rate is important. Where stabiliza-
tion of an eroding beach is the problem, suitable beach material may be
stockpiled at the updrift sector of the problem area. The establishment
and periodic replenishment of such a stockpile is termed artificial
beach nourtshment. To restore an eroded beach and stabilize it at the
restored position, fill is placed directly along the eroded sector, and
then the beach is artificially nourished by the stockpiling method.
When conditions are suitable for artificial nourishment, long reaches
of shore may be protected by this method at a relatively low cost per
linear foot of protected shore. An equally important advantage is that
artificial nourishment directly remedies the basic cause of most erosion
problems - a deficiency in natural sand supply - and benefits rather than
damages the adjacent shore. An added consideration is that the widened
beach has value as a recreation feature. A project for beach restoration
with an artificial dune for protection against hurricane wave action, com-
pleted in 1965 at Wrightsville Beach, North Carolina, is shown in Figure
1-10.
1-18
February 1966
Figure 1-9, Weir Jetty at Masonboro Inlet, North Carolina
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Sometimes structures must be provided to protect dunes, to maintain
a specific beach dimension, or to reduce nourishment requirements. In
each case, the cost of such structures must be weighed against the bene-
fits they would provide. Thus, measures to provide and keep a wider pro-
tective and recreational beach for a short section of an eroding shore
would require excessive nourishment without supplemental structures such
as groins to reduce the rate of loss of material from the widened beach.
A long, high terminal groin or jetty is frequently justified at the down-
drift end of a beach restoration project to reduce losses of fill into an
inlet and to stabilize the lip of the inlet.
1.6 CONSERVATION OF SAND
Experience and study have demonstrated that sand from dunes, beaches,
and nearshore areas is the best material available naturally in suitable
form to protect shores. Where sand is available in abundant quantities,
protective measures are greatly simplified and reduced in cost. When
dunes and broad, gently sloping beaches can no longer be provided, it is
necessary to resort to alternative structures, and the recreational attrac-
tion of the seashore is lost or greatly diminished.
Sand is a diminishing natural resource. Sand was once available to
our shores in adequate supply from streams, rivers and glaciers, and by
coastal erosion. Now cultural development in the watershed areas and along
previously eroding shores has progressed to a stage where large areas of
our coast now receive little or no sand through natural geological pro-
cesses. Continued cultural development in both inland and shore areas
tends to further reduce coastal erosion with resulting reduction in sand
supply to the shore. It thus becomes apparent that sand must be conserved.
This does not mean local hoarding of beach sand at the expense of adjoin-
ing areas, but rather the elimination of wasteful practices and the pre-
vention of losses from the shore zone whenever feasible,
Fortunately, nature has provided extensive stores of beach sand in
bays, lagoons, estuaries and offshore areas that can be used as a source
of beach and dune replenishment where the ecological balance will not be
disrupted. Massive dune deposits are also available at some locations,
though these must be used with caution to avoid exposing the area to flood
hazard. These sources are not always located in the proper places for
economic utilization, nor will they last forever. When they are gone, we
must face increasing costs for the preservation of our shores. Offshore
sand deposits will probably become the most important source in the future.
Mechanical bypassing of sand at coastal inlets is one means of conser-
vation that will come into increasing practice. Mining of beach sand for
commercial purposes, formerly a common procedure, is rapidly being reduced
as coastal communities learn the need for regulating this practice. Modern
hopper dredges, used for channel maintenance in coastal inlets, are being
equipped with a pump-out capability so their loads can be discharged near
the shore instead of being dumped at sea. On the California coast, where
large volumes of sand are lost into deep submarine canyons near the shore,
feel
facilities are being considered that will trap the sand before it reaches
the canyon and transport it mechanically to a point where it can resume
normal longshore transport. Dune planting with appropriate grasses and
shrubs reduces landward windborne losses and aids in dune preservation.
Sand conservation is an important factor in the preservation of our
seacoasts, and must be included in long-range planning. Protection of
our seacoasts is not a simple problem; neither is it insurmountable. It
is a task and a responsibility that has increased tremendously in impor-
tance in the past 50 years, and is destined to become a necessity in future
years. While the cost will mount as time passes, it will be possible
through careful planning, adequate management, and sound engineering to
do the job properly and economically.
lee
CHAPTER 2
MECHANICS
OF
WAVE MOTION
LEO CARRILLO STATE BEACH, CALIFORNIA — July 1968
CHAPTER 2
MECHANICS OF WAVE MOTION
2.1 INTRODUCTION
The effects of water waves are of paramount importance in the field
of coastal engineering. Waves are the major factor in determining the
geometry and composition of beaches, and significantly influence planning
and design of harbors, waterways, shore protection measures, coastal
structures, and other coastal works. Surface waves generally derive their
energy from the winds. A significant amount of this wave energy is finally
dissipated in the nearshore region and on the beaches.
Waves provide an important energy source for forming beaches; assort-
ing bottom sediments on the shoreface; transporting bottom materials on~
shore, offshore, and alongshore; and for causing many of the forces to
which coastal structures are subjected. An adequate understanding of the
fundamental physical processes in surface wave generation and propagation
must precede any attempt to understand complex water motion in the near-
shore areas of large bodies of water. Consequently, an understanding of
the mechanics of wave motion is essential in the planning and design of
coastal works.
This chapter presents an introduction to surface wave theories.
Surface and water particle motion, wave energy, and theories used in
describing wave transformation due to interaction with the bottom and
with structures are described. The purpose is to provide an elementary
physical and mathematical understanding of wave motion, and to indicate
limitations of selected theories. A number of wave theories have been
omitted. References are cited to provide information on theories not
discussed and to supplement the theories presented.
The reader is cautioned that man's ability to describe wave phenomena
is limited, especially when the region under consideration is the coastal
zone. Thus, the results obtained from the wave theories presented should
be carefully interpreted for application to actual design of coastal struc-
tures or description of the coastal environment.
2.2 WAVE MECHANICS
2.21 GENERAL
Waves in the ocean often appear as a confused and constantly changing
sea of crests and troughs on the water surface because of the irregularity
of wave shape and the variability in the direction of propagation. This
is particularly true while the waves are under the influence of the wind.
The direction of wave propagation can be assessed as an average of the
directions of individual waves. A description of the sea surface is
difficult because of the interaction between individual waves. Faster
waves overtake and pass through slower ones from various directions.
Waves sometimes reinforce or cancel each other by this interaction, and
often collide with each other and are transformed into turbulence, and
aa
spray. When waves move out of the area where they are directly affected
by the wind, they assume a more ordered state with the appearance of
definite crests and troughs and with a more rhythmic rise and fall. These
waves may travel hundreds or thousands of miles after leaving the area in
which they were generated. Wave energy is dissipated internally within
the fluid by interaction with the air above, by turbulence on breaking,
and at the bottom in shallow depths.
Waves which reach coastal regions expend a large part of their energy
in the nearshore region. As the wave nears the shore, wave energy may be
dissipated as heat through turbulent fluid motion induced by breaking and
through bottom friction and percolation. While the heat is of little
concern to the coastal engineer, breaking is important since it affects
both beaches and manmade shore structures. Thus, shore protection measures
and coastal structure designs are dependent on the ability to predict wave
forms and fluid motion beneath waves, and on the reliability of such
predictions. Prediction methods generally have been based on simple waves
where elementary mathematical functions can be used to describe wave motion.
For some situations, simple mathematical formulas predict wave conditions
well, but for other situations predictions may be unsatisfactory for
engineering applications. Many theoretical concepts have evolved in the
past two centuries for describing complex sea waves; however, complete
agreement between theory and observation is not always found.
In general, actual water-wave phenomena are complex and difficult
to describe mathematically because of nonlinearities, three-dimensional
characteristics and apparent random behavior. However, there are two
classical theories, one developed by Airy (1845) and the other by Stokes
(1880), that describe simple waves. The Airy and Stokes theories gener-
ally predict wave behavior better where water depth relative to wavelength
is not too small. For shallow water, a cnoidal wave theory often provides
an acceptable approximation of simple waves. For very shallow water near
the breaker zone, solitary wave theory satisfactorily predicts certain
features of the wave behavior. These theories will be described according
to their fundamental characteristics together with the mathematical equa-
tions which describe wave behavior. Many other wave theories have been
presented in the literature which, for some specific situations, may pre-
dict wave behavior more satisfactorily than the theories presented here.
These other theories are not included, since it is beyond the scope of
this Manual to cover all theories.
The most elementary wave theory, referred to as small-amplitude or
linear wave theory, was developed by Airy (1845). It is of fundamental
importance since it not only is easy to apply, but is reliable over a
large segment of the whole wave regime. Mathematically, the Airy theory
can be considered a first approximation of a complete theoretical descrip-
tion of wave behavior. A more complete theoretical description of waves
may be obtained as the sum of an infinite number of successive approxima-
tions, where each additional term in the series is a correction to preced-
ing terms. For some situations, waves are better described by these
higher order theories which are usually referred to as finite amplitude
2a
theories. The first finite amplitude theory, known as the trochoidal
theory, was developed by Gerstner (1802). It is so called because the
free surface or wave profile is a trochoid. This theory is mentioned
only because of its classical interest. It is not recommended for appli-
cation, since the water particle motion predicted is not that observed in
nature. The trochoidal theory does, however, predict wave profiles quite
accurately. Stokes (1880) developed a finite-amplitude theory which is
more satisfactory than the trochodial theory. Only the second-order Stokes'
equations will be presented, but the use of higher order approximations is
sometimes justified for the solution of practical problems.
For shallow-water regions, cnoidal wave theory, originally developed
by Korteweg and De Vries (1895), predicts rather well the waveform and
associated motions for some conditions. However, cnoidal wave theory has
received little attention with respect to actual application in the solu-
tion of engineering problems. This may be due to the difficulties in
making computations. Recently, the work involved in using cnoidal wave
theory has been substantially reduced by introduction of graphical and
tabular forms of functions. (Wiegel, 1960), (Masch and Wiegel, 1961.)
Application of the theory is still quite involved. At the limit of cnoidal
wave theory, certain aspects of wave behavior may be described satisfacto-
rily by solitary wave theory. Unlike cnoidal wave theory, the solitary
wave theory is easy to use since it reduces to functions which may be
evaluated without recourse to special tables.
Development of individual wave theories is omitted, and only the
results are presented since the purpose is to present only that infor-
mation which may be useful for the solution of practical engineering
problems. Many publications are available such as Wiegel (1964), Kinsman
(1965), and Ippen (1966a), which cover in detail the development of some of
the theories mentioned above as well as others. The mathematics used here
generally will be restricted to elementary arithmetic and algebraic opera-
tions. Emphasis is placed on selection of an appropriate theory in accord-
ance with its application and limitations.
Numerous example problems are provided to illustrate the theory
involved and to provide some practice in using the appropriate equations
or graphical and tabular functions. Some of the sample computations give
more significant digits than are warranted for practical applications.
For instance, a wave height could be determined to be 10.243 feet for
certain conditions based on purely theoretical considerations. This
accuracy is unwarranted because of the uncertainty in the basic data used
and the assumption that the theory is representative of real waves. A
practical estimate of the wave height given above would be 10 feet. When
calculating real waves, the final answer should be rounded off.
2.22 WAVE FUNDAMENTALS AND CLASSIFICATION OF WAVES
Any adequate physical description of a water wave involves both its
surface form and the fluid motion beneath the wave. A wave which can be
described in simple mathematical terms is called a simple wave. Waves
2-5
which are difficult to describe in form or motion, and which may be com-
prised of several components are termed complex waves. Sinusoidal or
simple harmonic waves are examples of simple waves since their surface
profile can be described by a single sine or cosine function. A wave is
periodic if its motion and surface profile recur in equal intervals of
time. A wave form which moves relative to a fluid is called a progressive
wave; the direction in which it moves is termed the direction of wave
propagation. If a wave form merely moves up and down at a fixed position,
it is called a complete standing wave or a clapotis. A progressive wave
is said to be a wave of permanent form if it is propagated without experi-
encing any changes in free surface configuration.
Water waves are considered oscillatory or nearly oscillatory if the
water particle motion is described by orbits, which are closed or nearly-
closed for each wave period. Linear, or Airy, theory describes pure
oscillatory waves. Most finite amplitude wave theories describe nearly
oscillatory waves since the fluid is moved a small amount in the direction
of wave advance by each successive wave. This motion is termed mass
transport of the waves. When water particles advance with the wave, and
do not return to their original position, the wave is called a wave of
translation. A solitary wave is an example of a wave of translation.
It is important to distinguish between various types of water waves
that may be generated and propagated. One way to classify waves is by
wave period T (the time for a wave to travel a distance of one wave
length), or by the reciprocal of T, the wave frequency f. One illustra-
tion of classification by period or frequency is given by Kinsman (1965)
and shown in Figure 2-1. The figure shows the relative amount of energy
contained in ocean waves having a particular frequency. Of primary concern
are those waves referred to as gravity waves in Figure 2-1, having periods
from 1 to 30 seconds. A narrower range of wave periods, from 5 to 15
seconds, is usually more important in coastal engineering problems. Waves
in this range are referred to as gravity waves since gravity is the
principal restoring force; that is, the force due to gravity attempts to
bring the fluid back to its equilibrium position. Figure 2-1 also shows
that a large amount of the total wave energy is associated with waves
classified as gravity waves; hence gravity waves are extremely important
in dealing with the design of coastal and offshore structures.
Gravity waves can be further separated into two states:
(a) seas, when the waves are under the influence of wind in a
generating area, and
(b) swell, when the waves move out of the generating area and
are no longer subjected to significant wind action.
Seas are usually made up of steeper waves with shorter periods and
lengths, and the surface appears much more confused than for swell. Swell
behaves much like a free wave, i.e., free from the disturbing force that
caused it, while seas consist to some extent of forced waves, 1.e., waves
on which the disturbing force is applied continuously.
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Ocean waves are complex. Many aspects of fluid mechanics necessary
to a complete discussion have only a minor influence on solving most
coastal engineering problems, Thus, a simplified theory which omits most
of the complicating factors is useful. The assumptions made in developing
the simple theory should be understood, because not all of the assumptions
are justified in all problems. When an assumption is not valid in a
particular problem, a more complete theory should be employed.
The most restrictive of common assumptions is that waves are small
perturbations on the surface of a fluid which is otherwise at rest. This
leads to a wave theory which is variously called, small-amplitude theory,
linear theory, or Airy theory. The small-amplitude theory provides in-
sight for all periodic wave behavior and a description of the periodic flow
adequate for most practical problems. This theory is unable to account for
mass transport due to waves (Section 2.253 Mass Transport Velocity), or the
fact that wave crests depart further from the mean water level than do the
troughs. A more general theory, usually called the finite amplitude, or
nonlinear wave theory is required to account for these phenomena as well as
most interactions between waves and other flows. The nonlinear wave theory
also permits a more accurate evaluation of some wave properties than can be
obtained with linear theory.
Several assumptions, commonly made in developing a simple wave theory
are listed below.
(a) The fluid is homogeneous and incompressible; therefore, the
density p is a constant.
(b) Surface tension can be neglected.
(c) Coriolis effect can be neglected.
(d) Pressure at the free surface is uniform and constant.
(e) The fluid is ideal or inviscid (lacks viscosity).
(f) The particular wave being considered does not interact with
any other water motions.
(g) The bed is a horizontal, fixed, impermeable boundary which
implies that the vertical velocity at the bed is zero.
(h) The wave amplitude is small and wave form is invariant in
time and space.
(i) Waves are plane or long crested (two-dimensional).
The first three are acceptable for virtually all coastal engineering
problems. It will be necessary to relax assumptions (d), (e), and (f)
for some specialized problems not considered in this Manual. Relaxing
the three final assumptions is essential in many problems, and is con-
sidered later in this chapter.
2-6
In applying assumption (g) to waves in water of varying depth encoun-
tered when waves approach a beach the local depth is usually used. This
can be rigorously justified, but not without difficulty, for most practical
cases in which the bottom slope is flatter than about 1 on 10. A progres-
sive wave moving into shallow water will change its shape significantly.
Effects due to viscosity and vertical velocity on a permeable bottom may
be measurable in some situations, but these effects can be neglected in
most engineering problems.
2.23 ELEMENTARY PROGRESSIVE WAVE THEORY (Small-Amplitude Wave Theory)
The most fundamental description of a simple sinusoidal oscillatory
wave is by its length L (the horizontal distance between corresponding
points on two successive waves); height H (the vertical distance to its
crest from the preceding trough); period T (the time for two successive
crests to pass a given point); and depth d (the distance from the bed
to the stillwater level). (See Appendix B for a list of common symbols.)
Figure 2-2 shows a two-dimensional simple progressive wave propagating
in the positive x-direction. The symbols used here are presented in the
figure. The symbol n denotes the displacement of the water surface
relative to the stillwater level (SWL) and is a function of x and time.
At the wave crest, n is equal to the amplitude of the wave a, or one-
half of the wave height.
Small-amplitude wave theory and some finite-amplitude wave theories
can be developed by introduction of a velocity potential 9(x, z, t). Hori-
zontal and vertical components of the water particle velocities are defined
at a point (x, z) in the fluid as u = 0$/3x and w = 3$¢/3z. The velocity
potential, Laplace's equation, and Bernoulli's dynamic equation together
with the appropriate boundary conditions provide the necessary information
needed in deriving the small-amplitude wave formulas. Such a development
has been shown by Lamb (1932), Eagleson and Dean (See Ippen 1966b), and
others.
2.231 Wave Celerity, Length and Period. The speed at which a wave form
propagates is termed the phase velocity or wave celerity, C. Since the
distance traveled by a wave during one wave period is equal to one wave-
length, the wave celerity can be related to the wave period and length by
poe
= (2-1)
An expression relating the wave celerity to the wavelength and water depth
is given by
[gl ral
C= ata -
ae fan = (2-2)
SABM SATSSOIB0Ig ‘[Teptosnuts ‘ArejusUeTA - swxa], JO UOTIIUTJOq ‘7-7 eAN3TY
b x
p-—=2 ‘wo}jog
$249 8ADM JD Z2/H=D= "(9
vase b/LILb/L2 ‘b/LE =} 405
UMOYS SI a|l|JOJd BADM (0 =X) UIHI40 UaAIB 404 *(q
aah 08S le
(are 5 ied
yBnos} aADM 4D 2/H -= D-
)soon-k'(D :31ON p
ybnosy
yi6ua7
uol}DHDdojg 40 uol}D0I1Q “Ea ph chnieo.
From Equation 2-1, it is seen that 2-2 can be written as
T
C= 5° tanh (=) (2-3)
2 L
TT
The values 27/L and 2n/T are called the wave number k and wave angular
frequency w, respectively. From Equations 2-1 and 2-3 an expression
for wavelength as a function of depth and wave period may be obtained.
T?
i Bo tanh i=) (2-4)
2m
Use of Equation 2-4 involves some difficulty since the unknown L, appears
on both sides of the equation. Tabulated values in Appendix C may be used
to simplify the solution of Equation 2-4.
Gravity waves may also be classified by the depth of water in which
they travel. Classification is made according to the magnitude of d/L
and the resulting limiting values taken by the function tanh(2rd/L).
Classifications are:
Classification tanh (27d/L)
Deep Water S12 5 (5
Transitional 1/25 to 1/2 1/4 to7m tanh (2nd/L)
Shallow Water <a /25 < 1/4 = 2nd/L
In deep water, tanh(27md/L) approaches unity and Equations 2-2 and 2-3
reduce to
° =@
o 2n iT (2-5)
and
a
Cc. = 6
aa 2-6
oeitt 9 (2-6)
Although deep water actually occurs at infinite depth, tanh(2md/L),
for most practical purposes, approaches unity at a much smaller d/L. For
a relative depth of 1/2 (that is, when the depth is one-half the wavelength),
tanh(2md/L) = 0.9964.
Thus, when the relative depth d/L, is greater than 1/2, the wave
characteristics are virtually independent of depth. Deepwater conditions
are indicated by the subscript o as in Lo and Cy. The period T,
remains constant and independent of depth for oscillatory waves; hence the
eae)
subscript is omitted. (Ippen, 1966b, pp 21-24.) If units of feet and
seconds are specified, the constant g/2n is equal to 5.12 ft/sec? and
Coe et = 5.12 T (ft/sec) , (2-7)
2m
and
eisai
L, — Re ==) 11% Te (ft) . (2-8)
If Equation 2-7 is used to compute wave celerity when the relative depth
is d/L= 0.25, the resulting error will be about,9 percent. -It is evi-
dent that a relative depth of 0.5 is a satisfactory boundary separating
deepwater waves from waves in water of transitional depth. If a wave is
traveling in transttional depths, Equations 2-2 and 2-3 must be used with-
out simplification. Care should be exercised to use Equations 2-2 and 2-3
when necessary, that is, when the relative depth is between 1/2 and 1/25.
When the relative water depth becomes shallow, i.e., 2md/L < 1/4 or
d/L < 1/25, Equation 2-2 can be simplified to
G= aly (2-9)
This relation, attributed to Lagrange, is of importance when dealing with
long-period waves, often referred to as long waves. Thus, when a wave
travels in shallow water, wave celerity depends only on water depth.
2.232 The Sinusoidal Wave Profile. The equation describing the free
surface as a function of time t, and horizontal distance x, fora
simple sinusoidal wave can be shown to be
21x 2nt H 21x 2nt (2-10
= — — —] = — cos |{— —- —}, =
ii mae a Ge T 2 L T )
where n is the elevation of the water surface relative to stillwater
level, and H/2 is one-half the wave height equal to the wave amplitude
a. This expression represents a periodic, sinusoidal, progressive wave
traveling in the positive x-direction. For a wave moving in the negative
x-direction, one need only replace the minus sign before 2mt/T with a
plus sign. When (2mx/L - 21t/T) equals 0, 1/2, 1, 31/2, the corresponding
values of) mn, ane H/2>)053— H/2,) ands) 0), respectively.
2.233 Some Useful Functions. It can be shown by dividing Equation 2-3 by
Equation 2-6, and by dividing Equation 2-4 by Equation 2-8 that
Ck ay (28 a
Guo sbabroq oul vebelr S274)
20
If both sides of Equation 2-11 are multiplied by d/L, it becomes:
d d 2nd
he Seat fe (2219)
iE, i Cole
The term d/Lo has been tabulated as a function of d/L by Wiegel (1954),
and is presented in Appendix C on Table C-1. Table C-2 includes d/L as
a function of d/ in addition to other useful functions such as 2nd/L
and tanh(27d/L). These functions simplify the solution of wave problems
described by the linear theory.
An example problem illustrating the use of linear wave theory and
the tables in Appendix C follows:
ae, Pin ake, Sia SM, Damme Past, Pe Fae 2 * * * * EXAMPLE PROBLEM RM ORAR LEDGES E GEESE EE MERE
GIVEN: A wave with a period of T = 10 seconds is propagated shoreward
over a uniformly sloping shelf from a depth of d = 600 feet to a depth
of d = 10 feet.
FIND: The wave celerities C and lengths L corresponding to depths of
d = 600 feet and d = 10 feet.
SOLUTION:
Using Equation 2-8,
L, = 5.12T? = 5.12 (10)? = 512 feet.
For d = 600 feet
= 117198
From Table C-1 it is seen that for values of
d Sil)
= :
Oo
dyad
ie
therefore
daa Al
L = L, = 512 feet (deepwater wave, since © > aie
e-i
By Equation 2-1
For d = 10 feet
d 10
== = S= i= 70.0195.
L 512
Entering Table C-1 with d/L, it is found that,
d
— = 0.05692,
L
hence
10 = Mate
L = ROEeos = 176 feet (vansional depth, since Be < a a :) ;
L 176
CSS Si = 17-6 ft/sec:
ih 10
ise et: Sey Ra ee ee ek Re ete ip eee Pe ee der Fen eon ae: Se es ecg ve te a ae to Ue ee
2.234 Local Fluid Velocities and Accelerations. In wave force studies,
it is often desirable to know the local fluid velocities and accelerations
for various values of z and t during the passage of a wave. The
horizontal component u, and the vertical component w, of the local
fluid velocity are given by:
Hat coshlanteta/Ll (2ux _ 2at\ ae
21 cosh (2nd/L) L T
_ Hef sinh[2r(@z+d/L] |. 2x 2at
a 25 cosh (27d/L) oP ( TL T (2-14)
These equations express the local fluid velocity components any distance
(z + d) above the bottom. The velocities are harmonic in both x and t.
For a given value of the phase angle 6 = (2mx/L - 21t/T), the hyperbolic
functions, cosh and sinh, as functions of z result in an approximate
exponential decay of the magnitude of velocity components with increasing
distance below the free surface. The maximum positive horizontal velocity
2-2
occurs when 6 = 0, 21, etc., while the maximum horizontal velocity in the
negative direction occurs when 6 = 7, 37, etc. On the other hand the
maximum positive vertical velocity occurs when 6 = 1/2, 51/2, etc., and
the maximum vertical velocity in the negative direction occurs when
6 = 31/2, 71/2, etc. (See Figure 2-3.)
The local fluid particle accelerations are obtained from Equations
2-13 and 2-14 by differentiating each equation with respect to t. Thus,
ee: gmH_ cosh [2n(z+d)/L] 2mx nt (2=155)
cosh (27d/L) i T
ee gtH_ sinh [2n(z + d)/L] ; 2nx — 2nt (2-16)
z E cosh (27d/L) L ie
Positive and negative values of the horizontal and vertical fluid
accelerations for various values of 6 = 2mx/L - 21t/T are shown in
Figure 2-3.
The following problem will illustrate the computations required to
determine local fluid velocities and accelerations resulting from wave
motions.
ie UE I ee he ae ie es a Ae eg ae EXAMPLE PROBLEM Be ey A OR ae ORS SE sah) Dee eras
GIVEN: A wave with a period of T = 8 seconds, in a water depth of
d = 50 feet, and a height of H = 18 feet.
FIND: The local horizontal and vertical velocities, u and w, and
accelerations a, and a, at a depth d = 15 feet below the SWL
when 6 = 27x/L -27t/T = 1/3 (60 degrees).
SOLUTION: Calculate
Lo=)5.12T?°= 5.128)? -= 328 fer ,
Soe entisns
pay de Gin [eh |}
From Table C-1 in Appendix C for a value of
d
— = 0.1526 ,
L,
d 2nd
— & 0.1854; cosh — = 1.759 ,
IL; Ie
2-13
Le
—-*p!o-=*p
O=M'+=nN
UOI}DHDdO1dg AADM $0 UOl}JIaIIG
St
Ajisaja9
SUOTIEIOTOOOY pUe SOTITOOTSA PINTY [ed90] “E-Z eansTY
o/LE i o/i 0)
Oeto 20 +e-D ORD =27p!+=%p -—=7n!0
—=M‘o-n O=mM!f-=n +=M!oQ=an O=mM b+
= *p
=a
uol}
pi9}a007
LSTELOEY,N
2-14
hence
50
0.1854
L=
= 270 feet .
Evaluation of the constant terms in Equations 2-13 through 2-16 gives
HeT f 1 18220)
2L__ cosh (2nd/L) 2(270) (1.758) :
gtH 1 (W822) G) bs Ri
iL cosu@nd/k). (70) (1-758) “T
Substitution into Equation 2-13 gives
27(50 — 15
u = 4.88 cosh fees [cos 60°] = 4.88 [cosh (0.8145)] (0.500).
From Table C-1 find
2nd
= — =NOL8i4 5...
L
and by interpolation
cosh (0.8145) = 1.3503,
and
sinh (0.8145) = 0.9074.
Therefore
u = 4.88 (1.3503) (0.500) = 3.29 ft/sec ,
w= 4.88 (0.9074) (0.866) = 3.83 ft/sec ,
a, = 3.84 (1.3503) (0.866) 4.49 ft/sec”,
a, = — 3.84 (0.9074) (0.500) = — 1.74 ft/sec”.
Figure 2-3, a sketch of the local fluid motion, indicates that the
fluid under the crest moves in the direction of wave propagation and
returns during passage of the trough. Linear theory does not predict any
mass transport; hence the sketch shows only an oscillatory fluid motion.
Bea mak: aCe I) meee Ir nee (oR eo eee ei ie Se ae ee ee SR ee ee er oe SE OR ee
2.235 Water Particle Displacements. Another important aspect of linear
wave mechanics deals with the displacements of individual water particles
within the wave. Water particles generally move in elliptical paths in
shallow or transitional water and in circular paths in deep water. If the
2515
mean particle position is considered to be at the center of the ellipse or
circle, then vertical particle displacement with respect to the mean
position cannot exceed one-half the wave height. Thus, since the wave
height is assumed to be small, the displacement of any fluid particle from
its mean position is small. Integration of Equations 2-13 and 2-14 gives
the horizontal and vertical particle displacement from the mean position,
respectively. (See Figure 2-4.)
Thus,
HgT? cosh [21(z+d)/L] 2nx 2nt
ae ee i —— os B=i\
: 4nL cosh (21d/L) ial & } ame
ea HeT? sinh [2n(z+d)/L] 2x 2at (2-18)
4nL cosh (21d/L) iG al
The above equations can be simplified by using the relationship
€ 2
ie) pO oe
it L by
Thus,
H cosh [2n(z+d)/L] . 2nx 2nt
2g ee a 2-1
: 2 sinh (27d/L) ( i 7 i Cn”)
haar H sinh [27(z + d)/L] 2mx i ant
~ "95 BS cinhonditoie AA Le ae GE (2720)
Writing Equations 2-19 and 2-20 in the following forms:
dp. hea wn One £ sinh (2nd/L) i
sin —_— -—- ~— = |— ————]|,
1 T a cosh [2n(z+d)/L
F 2nx 2nt G sinh (27d/L) 2
OS a a = ae ea ee =
i Ei 7 a sinh [2n(z+ d)/L]
and adding, gives:
e? °?
al” weno? (2-21)
in which
es H_ cosh [2n(z+d)/L]
~ 2 ~~ sinh (2nd/L) ° G7!)
Bde H sinh [27(z+ d)/L]
High) Veinth! (Qrd/1)2 2° (2723)
2-\6
SOAeM To1eMdaeq pue I9}eM-MOTTeYS LOF uoTAISOg uvaW WOTF SjUoWedeTdsTq eTITIIeq LOM ‘“P-Z 9aANSTY
rd 7
>= p
cm oe QADM JO}DM-|DUOIJISUDI |
| p JO
QADM JajoMdeaq QADM J8}DM-MO}|DYS
Ore ofn
o=™ =
$33 fF See
p-=Z wo}jog p—=Z wojjog
|
a#V
S}IG4O |DOI4VdII| 3
qd=V
|
"
1 |
rv
| |
Hh
le \
S}1G41Q JDINDIID | |
\
|
|
|
|
|
|
|
|
|
‘i
UOI}ISOd
ube
BAN
Equation 2-21 is the equation of an ellipse with a major (horizontal)
semiaxis equal to A, and a minor (vertical) semiaxis equal to B. The
lengths of A and B are measures of the horizontal and vertical dis-
placements of the water particles. Thus, the water particles are predicted
to move in closed orbits by linear wave theory: i.e., each particle returns
to its initial position after each wave cycle. Morison and Crooke (1953),
compared laboratory measurements of particle orbits with wave theory and
found, as had others, that particle orbits were not completely closed. This
difference between linear theory and observations is due to the mass trans-
port phenomenon which is discussed in a subsequent section.
Examination of Equations 2-22 and 2-23 shows that for deepwater
conditions A and B are equal and particle paths are circular. The
equations become
lal ay d 1
oe oe for — > —. (2-24)
2 IL 2
For shallow-water conditions, the equations become
ee H
pee) 2nd d ;
Bre K = (2-25)
nae H z+t+d L 25
2 d
Thus, in deep water, the water particle orbits are circular. The more
shallow the water, the flatter the ellipse. The amplitude of the water
particle displacement decreases exponentially with depth and in deepwater
regions becomes small relative to the wave height at a depth equal to
one-half the wavelength below the free surface, i.e., when z = - Lo/2.
This is illustrated in Figure 2-4. For shallow regions, horizontal
particle displacement near the bottom can be large. In fact, this is
apparent in offshore regions seaward of the breaker zone where wave action
and turbulence lift bottom sediments into suspension.
The vertical displacement of water particles varies from a minimum of
zero at the bottom to a maximum equal to one-half the wave height at the
surface.
Fey RC cere et Rec ae «Rye ey gai oer Reese Ie EXAMPLE PROBLEM RR ee) ey ee Noe ere
PROVE:
74
(a) & = gl tanh (2)
AL, L IL,
tH cosh [2n(z+d)/L] = Le =|
(b) =
a = i) ante)
2-18
SOLUTION:
(a) Equation 2-3,
Equation 2-1,
Therefore, equating 2-1 and 2-3,
L
== gt tanh Fe}
ap 2n 1k,
and multiplying both sides by (27)2/LT
2m)? 2m)?
Ge go SSE am!)
Crore Li 27 1b
Hence,
(b) Equation 2-13 may be written
PAY, cosh (27d/L) a
gTH_ cosh [2n(z+d)/L] a 7
i cos ae T
1 gH cosh [2n(z+d)/L] G ae
ok cos ;
CB2 cosh (27d/L) IB. It
since
ele
be
Since
fy (288
C= = tanh ( a
mH i cosh [2n(z+d)/L] 2nx 2nt
SS eS cos
T tanh(2zd/L) cosh (27d/L)
219
and since,
abe __ sinh (2nd/L)
IL, cosh (27d/L) ’
therefore,
= 7H cosh [27(z+ d)/L] 2nx Bh 2at
"2 oT disinhiad/i) i Tt)"
ier SR Re OS eae? te des ae ie” se = ee des Cake oe ode eae) ode key i ae ae cael ae eee ae) ier ee) eee ena
ene: I Re Ie ae eS ie Ie ie (aes ae nee a: EXAMPLE PROBLEM Caer ee i ia a Me i FOR rs
GIVEN: A wave in a depth of d = 40 feet, height of H = 10 feet, and a
period of T = 10 seconds. The corresponding deepwater wave height is
H, = 10.45 feet.
FIND:
(a) The horizontal and vertical displacement of a water particle from
its mean position when z = 0, and when z = - d.
(b) The maximum water particle displacement at a depth d = 25 feet when
the wave is in infinitely deep water.
(c) For the deepwater conditions of (b) above, show that the particle
displacements are small relative to the wave height when z = - Lo/2.
SOLUTION:
(a) Lo =) 3-12. T* =. 5.12,(10)7e = 512 feet,
ae 0.0781
1s hee a
From Appendix C, Table C-1
When z = O, Equation 2-22 reduces to
H 1
2 tanh (2nd/L) °
2=20
and Equation 2-23 reduces to
H
Bs =.
2
Thus,
10 1
DN ST a 5
2 (0.6430)
H 10
B= => == = 5.0 feet .
2 2
When z = - d,
H 10
A = ——. = —— = 5.96 feet ,
2 sinh (21d/L) 2 (0.8394)
andpyeBE=s Or
(b) With Hp = 10.45 feet, and z = - 25, evaluate the exponent of e
for use in Equation 2-24, noting that L = Lo>
2nz 2n(— 25)
— } = —— = — 0.307 ,
IL, Bil?
thus
ent 0h 0.736 .
Therefore,
H 2712 10.45
A — B = a e h — Pa (0.736) = 3.85 feet .
The maximum displacement or diameter of the orbit circle would
bee2Z(Ses5)h=" 7-7 0sbeet.
ie
= il
(c) eee Se = abi eee.
2
2nz _ 2nC 256) 2 3.142
it Bie capes
Using Table C-4 of Appendix C,
eo 2 = 0043"
H 22 10.45
ge ee naa (0.043) = 0.225 feet .
ol
Thus the maximum displacement of the particle is 0.45 feet which is small
when compared with the deepwater height, H, = 10.45 feet.
KR Re OR ER oe OR eee ie a ee ee i) es eee ee
2.236 Subsurface Pressure. Subsurface pressure under a wave is the
summation of two contributing components, dynamic and static pressures,
and is given by
has cosh [2n(z+d)/L] H By (ue cl a rs
PO TGhE Sree nea eats aE 7 pee st ips & 82276)
where p/ is the total or absolute pressure, p, is the atmospheric
pressure and p = w/g is the mass density of water (for saltwater,
0 = 2.0 lbs sec*/ft*= 2.0 slugs/ft?; for fresh water, p = 1.94 slugs/ft3).
The first term of Equation 2-26 represents a dynamic component due to
acceleration, while the second term is the static component of pressure.
For convenience, the pressure is usually taken as the gage pressure
defined as
cosh [2n(z+d)/L] H 2nx 2nt (2-27)
= = = - 8) SS = |= : -
Ne a Aaa eo ayy sail. a cpm wees
Equation 2-27 can be written as
cosh [2n(z+ d)/L]
= ee : 2-28
Si Pics Cao m Wee aa
since
H a
2 IL, r
The ratio
. cosh [2n(z+d)/L] (2-29)
z cosh (27d/L) i
is termed the pressure response factor. Hence, Equation 2-28 can be
written as
piS vee ke 2): (2-30)
The pressure response factor K, for the pressure at the bottom when
i, ne ky
i
Cn cosh (27d/L) ° (22)
is tabulated as a function of d/L, and d/L in Tables C-1 and C-2 of
Appendix C.
It is often necessary to determine the height of surface waves based
on subsurface measurements of pressure. For this purpose it is convenient
to rewrite Equation 2-30 as,
e pine) (2-32)
pgk,
where z is the depth below the SWL of the pressure gage, and N isa
correction factor equal to unity if the linear theory applies. Several
empirical studies have found N to be a function of period, depth, wave
amplitude and other factors. In general, N decreases with decreasing
period, being greater than 1.0 for long-period waves and less than 1.0 for
short-period waves.
A complete discussion of the interpretation of pressure gage wave
records is beyond the scope of this Manual. For a more detailed discussion
of the variation of N with wave parameters, the reader is referred to
Draper (1957), Grace (1970), and Esteva and Harris (1971).
ie at RAI. ee oe) ae ie! ie) Se EXAMPLE PROBLEM KEK KOR SEO Bee LMT
GIVEN: An average maximum pressure of p = 2590 lbs/ft2 is measured by a
subsurface pressure gage located 2 feet above the bed in water at d = 40
feet. The average frequency f = 0.0666 cycles per second.
FIND: The height of the wave H assuming that linear theory applies and
the average frequency corresponds to the average wave amplitude.
SOLUTION:
1 1
SS Se d
f 2300666) iat
L, = 5.12T? = 5.12(15)? = 1152 feet ,
d 40
eS eS O0847
EY tise
From Table C-1 of Appendix C, entering with d/Lo>
d
= = WOT .
L
hence
40
| ee ee
(0.07712)
(=)
cosh {| — }= 1.1197 .
IL
Therefore, from Equation 2-29
519 feet ,
and
H cosh [27(z+d)/L] __ cosh [2n(— 38+ 40)/519] _ 1,0003
z cosh (2nd/L) 1.1197 Toicy,
= 0.8934 .
= a = H/2 when the pressure is maximum (under the wave crest),
1.0 since linear theory is assumed valid,
H N(p + 1.0 [2590 + (64.4) (— 38
_ (p + pgz) a [ ( )¢ )] = pages:
pgk, (64.4) (0.8934)
Therefore,
lal = YQAW) 3 Bites
Note that the tabulated value of K in Appendix C, Table C-1, could not
be used since the pressure was not measured at the bottom.
Ce we a ee a ee eS a rc Se ee Fe A Oe eM eR TO ee ee eas to ft
2.237 Velocity of a Wave Group. The speed with which a group of waves
or a wave train travels is generally not identical to the speed with which
individual waves within the group travel. The group speed is termed the
group velocity, C,; the individual wave speed is the phase velocity or
wave celerity given by Equations 2-2 or 2-3. For waves propagating in
deep or transitional water with gravity as the primary restoring force,
the group velocity will be less than the phase velocity. (For those waves
propagated primarily under the influence of surface tension, i.e., capil-
lary waves, the group velocity may exceed the velocity of an individual
wave. )
The concept of group velocity can be described by considering the
interaction of two sinusoidal wave trains moving in the same direction
with slightly different wavelengths and periods. The equation of the
water surface is given by:
H Q2nx 27t H Q2nx 2it
OF seein tai ae COS Het Pet eco (2-33)
where n, and nN, are the contributions of each of the two components.
They may be summed since superposition of solutions is permissible when
linear wave theory is used. For simplicity, the heights of both wave
components have been assumed equal. Since the wavelengths of the two
component waves, L, and L,, have been assumed slightly different, for
some values of x at a given time, the two components will be in phase
and the wave height observed will be 2H; for some other values of x,
the two waves will be completely out of phase and the resultant wave
height will be zero. The surface profile made up of the sum of the two
sinusoidal waves is given by Equation 2-33 and is shown in Figure 2-5.
The waves shown on Figure 2-5 appear to be traveling in groups described
by the equation of the envelope curves:
EL 1 == 1b
Nenvelope = + H cos f Gan Xe 7 (=) , (2-34)
1 oz 1 2
It is the speed of these groups, i.e. the velocity of propagation of
the envelope curves, that represents the group velocity. The limiting
speed of the wave groups as they become large, i.e., as the wavelength,
L,;, approaches L, and consequently the wave period T, approaches T,
is the group velocity and can be shown to be equal to:
Cc = - [ Scher = nC, (2235)
1
gy Oi sinh (4nd/L)
where
1 4nd/L
tS pa
2 sinh nl
In deep water, the term (4nd/L)/sinh(41d/L) is approximately zero and,
eb 1
C, Ser eS C, (deep water) , (2-36)
or the group velocity is one-half the phase velocity. In shallow water,
sinh(4nd/L) ~ 41d/L and,
Cc =
L
ae cen C = wed (shallow water) | (2-37)
2-25
envelope
AMMVANNACAITEINNG
/ |
=O:2/=0' 10) 40!) 0:2: 013) 04 O05 1016 "0:7-70.6) (OS ston (2) ies
x oa t peli :
Di ( ake )- 72) (=, (after Kinsman,1965)
Figure 2-5. Formation of Wave Groups by the Addition of Two Sinusoids
Having Different Periods
hence the group and phase velocities are equal. Thus in shallow water,
because wavecelerity is fully determined by the depth, all component waves
in a wave train will travel at the s-me speed precluding the alternate
reinforcing and cancelling of components. In deep and transitional water,
wave celerity depends on the wavelength; hence slightly longer waves travel
slightly faster, and produce the small phase differences resulting in wave
groups. These waves are said to be dispersive or propagating in a
disperstve medtum, i.e. in a medium where their celerity is dependent on
wavelength.
Outside of shallow water, the phase velocity of gravity waves is
greater than the group velocity, and an observer moving along with a group
of waves at the group velocity will see waves that originate at the rear
of the group move forward through the group traveling at the phase velocity,
and disappear at the front of the wave group.
Group velocity is important because it is with this velocity that wave
energy is propagated.
Although mathematically, the group velocity can be shown rigorously
from the interference of two or more waves (Lamb, 1932), the physical
significance is not as obvious as it is in the method based on the con-
sideration of wave energy. Therefore an additional explanation of group
velocity is provided on wave energy and energy transmission.
2.238 Wave Energy and Power. The total energy of a wave system is the
sum of its kinetic energy and its potential energy. The kinetic energy is
that part of the total energy due to water particle velocities associated
with wave motion. Potential energy is that part of the energy resulting
from part of the fluid mass being above the trough - the wave crest.
According to the Airy theory, if the potential energy is determined
relative to mean water level, and all waves are propagated in the same
direction, potential and kinetic energy components are equal, and the
total wave energy in one wavelength per unit crest width is given by
pgH?*L pgH?L pgH?*L
iss chee + = 2-
k P 16 16 Ban aa
Subscripts k and p refer to kinetic and potential energies. Total
average wave energy per unit surface area, termed the spectfic energy or
energy density, is given by
shee TOU Seg?
E> sae. (2-39)
ene
Wave energy flux is the rate at which energy is transmitted in the
direction of wave propagation across a vertical plane perpendicular to the
direction of wave advance and extending down the entire depth. The average
energy flux per unit wave crest width transmitted across a plane perpen-
dicular to wave advance is
DS Ec = EC, . (2-40)
Energy flux P is frequently called wave power and
1 q 4nd/L
2 sinh (47d/L)
If a plane is taken other than perpendicular to the direction of wave
advance, P=EC, sin >, where $ is the angle between the plane across
which the energy is being transmitted and the direction of wave advance.
For deep and shallow water, Equation 2-40 becomes
E,C, (deep water) . (2-41)
P= EC, = EC (shallow water) . (2-42)
An energy balance for a region through which waves are passing will
reveal, that for steady state, the amount of energy entering the region
will equal the amount leaving the region provided no energy is added or
removed from the system. Therefore, when the waves are moving so that
their crests are parallel to the bottom contours,
Sue = EnC
: 1
or, since = 5?
le E 43
7 o@o = Ene (2-43)
When the wave crests are not parallel to the bottom contours, some parts
of the wave will be traveling at different speeds, the wave will be
refracted and Equation 2-43 does not apply. (See Section 2.3. WAVE
REFRACTION. )
The following problem illustrates some basic principles of wave energy
and energy flux.
eee eae) Cy BR i a des oa ee a) ke EXAMPLE PROBLEM RS el eee ere eee Ry ie We! Sie ae
GIVEN: A deepwater oscillatory wave with a wavelength of L, = 512 feet,
a height of H, = 5 feet and a celerity of C, = 51.2 ft/sec, moving
shoreward with its crest parallel to the depth contours. Any effects
due to reflection from the beach are negligible.
FIND:
(a) Derive a relationship between the wave height in any depth of water
and the wave height in deep water, assuming that wave energy per
unit crest width is conserved as a wave moves from deep water into
shoaling water.
(b) Calculate the wave height for the given wave when the depth is 10
feet.
(c) Determine the rate at which energy per unit crest width is trans-
ported toward the shoreline and the total energy per unit width
delivered to the shore in 1 hour by the given waves.
SOLUTION:
(a) Since the wave crests are parallel to the bottom contours, refraction
does not occur, therefore Hp = Ho: (See Section 2.3. WAVE
REFRACTION. )
From Equation 2-43,
1
2 E,C,
The expressions for E, and E are,
mi pgH’
p= PHO !
ie 8
and
= H?
ry ocaeee
8
where H, represents the wave height in deep water if the wave
were not refracted.
Substituting into the above equation gives,
,2
(pee H
= iC 2 = acre A
a 8 8
(b)
(c)
Therefore,
Xo
G
1
n
(ie)
H’, 2
and since from Equations 2-3 and 2-6
(Cc 2nd
— = tanh |—} ,
CG, IL,
and from Equation 2-35 where
il 4nd/L
n = —{1+ ———],
2 sinh (47d/L)
H i a
Hi” | tanh Qnd/L) (4nd/L) he (ne)
sinh (4nd/L)
where Kg or H/HO is termed the shoaling coefficient.
of H/Hj as a function of d/Lo and d/L
in Tables C-1 and C-2 of Appendix C.
Values
have been tabulated
For the given wave, d/Lo = 10/512 = 0.01953. Either from Table
C-1 or from an evaluation of Equation 2-44 above,
H
= 1,233 :
aap
oO
Therefore,
H = 1.233(5) = 6.165 ft.
The rate at which energy is being transported toward shore is the
wave energy flux.
EC, = nEC..
oO
Since it is easier to evaluate the energy flux in deep water,
the left side of the above equation will be used.
BiiPh i ge 1 pg (H’)? 51.2 1 64 (5)?
Pp =) = C= = eb Dae ote ie = = (5) 512.
De ee 2 8 2 8
ce ft-l .
P = 5120 —— per ft. of wave crest ,
sec
= 5120 ;
[= 550 = 9.31 horsepower per ft. of wave crest .
2-30
This represents the expenditure of
ft- }
aoe -36ab - 1855 Oc 10° tele =
sec
of energy each hour on each foot of beach.
oy ak ee ee ee ee ee Ck ee a Cp et ee ee ee Pe Ce ee ee te a Je
The mean rate of energy transmission associated with waves propagating
into an area of calm water provides a better physical description of the
concept of group velocity. An excellent treatment of this subject is given
by Sverdrup and Munk (1947) and is repeated here.
Quoting from the Beach Erosion Board Technical
Report No. 2, (1942): "As the first wave in the group
advances one wave length, its form induces correspond-
ing velocities in the previously undisturbed water and
the kinetic energy corresponding to those velocities
must be drawn from the energy flowing ahead with the
form. If there is equipartition of energy in the wave,
half of the potential energy which advanced with the
wave must be given over to the kinetic form and the
wave loses height. Advancing another wave length
another half of the potential energy is used to supply
kinetic energy to the undisturbed liquid. The process
continues until the first wave is too small to identify.
The second, third, and subsequent waves move into water
already disturbed and the rate at which they lose height
is less than for the first wave. At the rear of the
group, the potential energy might be imagined as moving
ahead, leaving a flat surface and half of the total
energy behind as kinetic energy. But the velocity
pattern is such that flow converges toward one section
thus developing a crest and diverges from another
section forming a trough. Thus the kinetic energy is
converted into potential and a wave develops in the
wear of the group."
This concept can be interpreted in a quantitative
manner, by taking the following example from R. Gatewood
(Gaillard 1904, p. 50). Suppose that in a very long
trough containing water originally at rest, a plunger
at one end is suddenly set into harmonic motion and
starts generating waves by periodically imparting an
energy E/2 to the water. After a time interval of n
periods there are m waves present. Let m be the posi-
tion of a particular wave in this group such that m=1
refers to the wave which has just been generated by
the plunger, m=(n+1)/2 to the center wave, and m=n to
the wave furthest advanced. Let the waves travel with
constant velocity C, and neglect friction.
Poll
After the first complete stroke one wave will be
present and its energy is E/2. One period later this
wave has advanced one wave length but has left one-
half of its energy or E/4 behind. It now occupies a
previously undisturbed area to which it has brought
energy E/4. In the meantime, a second wave has been
generated, occupying the position next to the plunger
where E/4 was left behind by the first wave. The
energy of this second wave equals E/4 + E/2 = 3E/4.
Repeated applications of this reasoning lead to the
results shown in Table 2-1.
The series number 7 gives the total number of
waves present and equals the time in periods since
the first wave entered the area of calm; the wave
number m gives the position of the wave measured from
the plunger and equals the distance from the plunger
expressed in wave lengths. In any series, n, the
deviation of the energy from the value E/2 is
symmetrical about the center wave. Relative to the
center wave all waves nearer the plunger show an
excess of energy and all waves beyond the center wave
show a deficit. For any two waves at equal distances
from the center wave the excess equals the deficiency.
In every series, n, the energy first decreases slowly
with increasing distance from the plunger, but in the
vicinity of the center wave it decreases rapidly.
Thus, there develops an "energy front'' which advances
with the speed of the central part of the wave system,
that is, with half the wave velocity.
According to the last line in Table 2-1 a definite
pattern develops after a few strokes: the wave closest
to the plunger has an energy E(2"%-1)/2" which approaches
the full amount E, the center wave has an energy E/2,
and the wave which has traveled the greatest distance
has very little energy (E/2”).
Table 2-1. Distribution of Wave Heights in a Short Train of Waves
Wave number, m
With a large number of waves (a large n), energy decreases with
increasing n, and the leading wave will eventually lose its identity.
At the group center, energy increases and decreases rapidly - to nearly
maximum and to nearly zero. Consequently, an energy front is located
at the center wave group for deepwater conditions. If waves had been
examined for shallow rather than deep water, the energy front would
have been found at the leading edge of the group. For any depth, the
ratio of group to phase velocity (C,/C) _ generally defines the energy
front. Also, wave energy is transported in the direction of phase propa-
gation, but moves with the group velocity rather than phase velocity.
2.239 Summary - Linear Wave Theory. Equations describing water surface
profile particle velocities, particle accelerations, and particle displace-
ments for linear (Airy) theory are summarized in Figure 2-6.
2.24 HIGHER ORDER WAVE THEORIES
Solution of the hydroynamic equations for gravity-wave phenomena can
be improved. Each extension of the theories usually produces better
agreement between theoretical and observed wave behavior. The extended
theories can explain phenomena such as mass transport that cannot be
explained by linear theory. If amplitude and period are known precisely,
the extended theories can provide more accurate estimates of such derived
quantities as the velocity and pressure fields due to waves than can
linear theory. In shallow water, the maximum wave height is determined
by depth, and can be estimated without wave records.
When concern is primarily with the oscillating character of waves,
estimates of amplitude and period must be determined from empirical data.
In such problems, the uncertainty about the accurate wave height and
period leads to a greater uncertainty about the ultimate answer than does
neglecting the effect of nonlinear processes. Thus it is unlikely that
the extra work involved in using nonlinear theories is justified.
The engineer must define regions where various wave theories are
valid. Since investigators differ on the limiting conditions for the
several theories, some overlap must be permitted in defining the regions.
Le Mehaute (1969) presented Figure 2-7 to illustrate approximate limits
of validity for several wave theories. Theories discussed here are indi-
cated as are Stokes' third- and fourth-order theories. Dean (1973),
after considering three analytic theories, presents a slightly different
analysis. Dean (1973) and Le Mehaute (1969) agree in recommending cnoidal
theory for shallow-water waves of low steepness, and Stokes' higher order
theories for steep waves in deep water, but differ in regions assigned
to Airy theory. Dean indicates that tabulated stream function theory is
most internally consistent over most of the domain considered. For the
limit of low steepness waves in transitional and deep water, the differ-
ence between stream function theory and Airy theory is small. Additional
wave theories not presented in Figure 2-7 may also be useful in studying
2-35
Pp
Y3LVM d335d
SOTYSTIOJOeIeY) oAeEM - ATOOY]L 9AeM (AATY) AZeOUTT - ATeUUNS
(1/pu22) ysoo
26d - T7p4z) uz] uso ub
(1/Ppi2) yuis
[1/(P+2) “2] yuls
(1/pi2) quis
[1/(P+2Z) 22] ysoo
Q soo
Quis
(1/pz2) ysoo 7
Che [77(p+z)u2]yuis Hub
ie (1/ P22 ) ysoo 1
he [1/(P+2)u2]ysoo Hub
(1/pi2 ) ysoo
Che [1/(p+z) 22] yuls
(1/ Pz) yso9
6 $°° Ta7(p+z) wz] uso
it UIs
ae ee b) Yu! +1] 2
T/Pip
YALVM IVNOILISNVYL
P
gus =
gus (+ +1)
Y3aLVM MOTIVHS
Jl
itH
"9-7 oansTy
aINSsaig BdDJANSGNS
[DO1j40AQ (Q
JDjUOZI40H (D
S}UBWAIDIGSIG 3]91jIDg J3}0M
{DI1j4aA (q
]DJUOZIJOH (D
SUOI}DI9{IIIY BjIIp4IDg 19j0M
{DII}1aA (q
[DyUOZIJOH (0
Ky1o0jaA BfOlt4sDg Jd}0M
Ayio0jaq dnoig
yybua7 aAnmM
Ajisaja9 BAOM
Bl1JOld AADM
H1d30 3AILV 134
726 os A)
0.00004 F
0.00003 \—
die idias
Lia 0.040 tite 0.500
ane dee.
=z = 0.050 “a = 2.550
f Ti = 0.0155 —1, = 0.0792 ?
gt
Shallow ic ees mmm iL water Deep water
0.04
0.02
0.01
0.008
0.006
0.004
0.002
2 y : Se:
gate: : Hf nn a]
0.0008 Et Jali? GEGRoant seeaseeas fa Fee
f Pott ‘ tt Past Enaaa zi
0.0006 a =
0.0004 z au
= nae
a
0.0002 a S a =n
0.0001 ee
0.00008 F ea : sat
i SREY : ELT
0.00006 j= J it
sie Hild Se esetuvesee . doitiatii BEGiii Er
0.004:0.006 0.0 i i h z 0.3 04
0.0004 0.001 0.002
0.01 0.02 0.04 0.06 0.1 . 0.6 0.8 1.0 4.0 60 8.0100
+ (ft/sec?) (after Le Méhauté, 1969)
Figure 2-7. Regions of Validity for Various Wave Theories
0.02
0.002
40.001
H (ft/sec?)
T?
wave phenomena. For given values of H, d and T, Figure 2-7 may be used
as a guide in selecting an appropriate theory. The magnitude of the
Ursell parameter Up shown in the figure may be used to establish the
boundaries of regions where a particular wave theory should be used.
The Ursell parameter is defined by
(2-45)
For linear theory to predict accurately the wave characteristics, both
wave steepness, H/gT*, and the Ursell parameter must be small as shown in
Figure 2-7.
2.25 STOKES' PROGRESSIVE, SECOND-ORDER WAVE THEORY
Wave formulas presented in the preceding sections on linear wave
theory are based on the assumption that the motions are so small that the
free surface can be described to the first order of approximation by
Equation 2-10:
H 21x 2nt H ‘ :
=i 7COS) a ;
n 5 L T A Coal Gh 2 eas
More specifically, it is assumed that wave amplitude is small, and the
contribution made to the solution by higher order terms is negligible. A
more general expression would be:
n = acos(@) + a’B, (L,d) cos (26)
(2-46)
+ a°B, (L,d) cos (38) +-++a"B (L,d) cos (nd) ,
where a = H/2, for first- and second-orders, but a < H/2 for orders
higher than the second, and B,, Bz etc. are specified functions of the
wavelength L, and depth d.
Linear theory considers only the first term on the right side of
Equation 2-46. To consider additional terms represents a higher order of
approximation of the free surface profile. The order of the approxima-
tion is determined by the highest order term of the series considered.
Thus, the ordinate of the free surface to the third order is defined by
the first three terms in Equation 2-46.
When the use of a higher order theory is warranted, wave tables, such
as those prepared by Skjelbreia (1959), and Skjelbreia and Hendrickson
(1962), should be used to reduce the possibility of numerical errors made
in using the equations. Although Stokes (1880) first developed equations
for finite amplitude waves, the equations presented here are those of
Miche (1944).
2.251 Wave Celerity, Length, and Surface Profile. It can be shown that,
for second-order theories, expressions for wave celerity and wavelength
are identical to those obtained by linear theory. Therefore,
C= et tanh a (2-3)
2n L :
and
2
L= gr" tanh (#). (2-4)
2n lt
The above equations, corrected to the third order, are given by:
2 2
a eu aa 2nd et 7H 5 + 2 cosh ed + 2 cosh? (4nd/L) (2-47)
16, if 8 sinh* (2md/L)
2 2 2
ae le Aes 2nd A 7H 5 + 2 cosh Cae) + 2 cosh? (4nd/L) (2-48)
2n 1b L 8 sinh* (2nd/L)
The equation of the free surface for second-order theory is
2 1y as
(2-49)
mH?\ cosh (21d/L) 4nx as
—j} ——— _ ]2 + h (4nd SS
(2 sinh? (2nd/L) Soe Man elo SORT \ Taioktar
For deep water, (d/L > 1/2) Equation 2-49 becomes,
H H?
ae 2 cos BEEP aE) <b ee 4x _ 4nt\ (2-50)
2 ara 4L, L a
2.252 Water Particle Velocities and Displacements. The periodic x and z
components of the water particle velocities to the second order are given
by
chs HgT cosh [2n(z + d)/L] 2nx be 2nt
OL suacesh(@ndice. foe T
(2-51)
3 (=) cosh [4n(z + d)/L] (= =
+ 7 —————— cos |— — >
"Tk sinh* (27d/L) L T
| TH ¢ sinh (2a + D/L) (ze * 2
L sinh (27d/L) iL, Tt
(2-52)
atl sinh [4n(z + d)/L]_ [4x zy
4 =| sini*@ndjity - (\ EO OR
Second-order equations for water-particle displacements from their
Mean position for a finite amplitude wave are:
g = — Het? cosh [2n(e + A)/L] a id in) att 1
4nL cosh (2nd/L) iG i 8L_ sinh? (27d/L)
(2-53)
1 — 3 ea an (ls EO cD
2 sinh? (2nd/L) L iP Ue }) 2 jsinh? (2nd/L)
and
2 HgT? sinh [2n(z + d)/L] 2nx 2nt
1b, T
7 +4nb cosh (2nd/L) Piel) ter
(2-54)
3 nH? sinh [4n(z + d)/L] ie |
— a co. _ —_— .
ib T
16 L _ sinh*(2nd/L) ca
2.253 Mass Transport Velocity. The last term in Equation 2-53 is of
particular interest; it is not periodic, but is the product of time and
a constant depending on the given wave period and depth. The term predicts
a continuously increasing net particle displacement in the direction of
wave propagation. The distance a particle is displaced during one wave
period when divided by the wave period gives a mean drift velocity, U(z),
called the mass transport velocity. Thus,
(2-55)
ae) nHY Cc cosh [4n(z + d)/L]
EE Ne |) Pah? (Araht)
Equation 2-53 indicates that there is a net transport of fluid by
waves in the direction of wave propagation. If the mass transport, indi-
cated by Equation 2-55 leads to an accumulation of mass in any region, the
free surface must rise, thus generating a pressure gradient. A current,
formed in response to this pressure gradient, will reestablish the distri-
bution of mass. Studies of mass transport, theoretical and experimental,
have been conducted by Longuet-Higgins (1953, 1960), Mitchim (1940), Miche
(1944), Ursell (1953), and Russell and Osorio (1958). Their findings
indicate that the vertical distribution of the mass transport velocity is
modified so that the net transport of water across a vertical plane is
zero.
2.254 Subsurface Pressure. The pressure at any distance below the fluid
surface is given by
H_ cosh [2n(z + d)/L] 2x ant)
Da dseks saceshiadilune ea: fee
A 2) aH? tanh (2md/L) |cosh [4n(z + d)/L] a 1 i
8 ae L sinh? (27d/L) sinh? (27d/L) 3 iL,
1 nH? tanh (27d/L) h 4n(z + d) =
8 L sinh? (21d/L) 0 ;
2.255 Maximum Steepness of Progressive Waves. A progressive gravity wave
is physically limited in height by depth and wavelength. The upper limit,
or breaking wave height in deep water is a function of the wavelength, and
in shallow and transitional water is a function of both depth and wavelength.
Stokes (1880) predicted theoretically that a wave would remain stable
only if the water particle velocity at the crest was less than the wave
celerity or phase velocity. If the wave height were to become so large
that the water particle velocity at the crest exceeded the wave celerity,
the wave would become unstable and break. Stokes found that a wave having
a crest angle less than 120 degrees would break (angle between two lines
tangent to the surface profile at the wave crest). The possibility of
existence of a wave having a crest angle equal to 120 degrees was shown by
Wilton (1914). Michell (1893) found that in deep water the theoretical
limit for wave steepness was
L, max
Havelock (1918) confirmed Michell's findings.
0.142 = =. (2-57)
Miche (1944) gives the limiting steepness for waves traveling in
depths less than Lj /2 without a change in form as
H H 2
(2) = (| tanh Ga = 0.142 tanh (2) (2-58)
L max L, max L L
Laboratory measurements by Danel (1952) indicate that the above
equation is in close agreement with an envelope curve to laboratory
observations. Additional discussion of breaking waves in deep and
shoaling water is presented in Section 2.6, BREAKING WAVES.
2.256 Comparison of the First- and Second-Order Theories. A comparison
of first- and second-order theories is useful to obtain insight about the
2=sa
choice of a theory for a particular problem. It should be kept in mind
that linear (or first-order) theory applies to a wave which is symmetrical
about stillwater level and has water particles that move in closed orbits.
On the other hand, Stokes' second-order theory predicts a wave form that
is unsymmetrical about the stillwater level but still symmetrical about a
vertical line through the crest, and has water particle orbits which are
open.
e0 KO KR PRR © ee OK eK Kk O* O€: EXAMPLE PROBLEM Fee RD UR ee ee EK Oe ee kee
GIVEN: A wave traveling in water depth of d = 20 feet, with a wavelength
of L = 200 feet and a height of H = 4 feet.
FIND:
(a) Compare the wave profiles given by the first- and second-order
theories.
(b) What is the difference between the first- and second-order
horizontal velocities at the surface under both the crest and
trough?
(c) How far in the direction of wave propagation will a water particle
move from its initial position during one wave period when z = 0?
(d) What is pressure at the bottom under the wave crest as predicted
by both the first- and second-order theories?
(e) What is the wave energy per unit width of crest predicted by the
first-order theory?
SOLUTION:
(a) The first-order profile Equation 2-10 is:
where
and the second-order profile Equation 2-49 is:
t H ane aH? cosh (21d/L) ned h 4nd on
MO na BEE Can edn ces RD A ine
for
d 20
SSS.
L 200
and from Table C-2
2
eash (22!) = 1,2040 ,
5
ss
|
ee
Il
=
en
~I
=)
a
1899s,
ie)
ie}
177)
=
—_——
a
| 3
seul,
ll
2
aH? cosh (27d/L) ee 4nd = 0.48
8L sinh? (21d/L) L
Therefore
n = 2 cos6 + 0.48 cos 26,
Ne 2 = 2.48 fo
where Ne. 2 and Nt,g are the values of n at the crest
a)
Givezycos, 6 = 1, cos 20 := 1) and’ trough (7.e¢ cos .@ = = 1,
cos 26 = 1) according to second-order theory.
Figure 2-8 shows the surface profile n as a function of 6. The
second-order profile is more peaked at the crest and flatter at the
trough than the first-order profile. The height of the crest above
SWL is greater than one-half the wave height; consequently the
distance below the SWL of the trough is less than one-half the
height. Moreover, for linear theory, the elevation of the water
surface above the SWL is equal to the elevation below the SWL;
however, for second-order theory there is more height above SWL
than below.
(b) For convenience, let
uy = value of u at crest according to first-order theory,
ie value of u at a trough according to first-order theory,
Ug, 2 = value of u at a crest according to second-order theory,
uz 9 = value of u at a trough according to second-order theory.
3
‘aT TJOIg IVOUTT YIM 9TTFOIg ,SeYOIS IeprQ puoses FO uostaeduo) °g-zZ ean3Ty
(suoipou) g
2/S u2 2/€ a Cie e
$994 002 = 1 ConN~
6 “a A SS
4994 Ob = H yo \
4284 O02 = P 4 ‘
‘ALON ahi x
7 N See
Se BA jaA97] JA}OM IHS *
Vf
* 7 Q{|JOd JapsoO pra
m, 4
N A ~~ A
Soe Q[lJOJd JapsO pug a
-_—$_——— bus +
(44) 4
2-42
According to first-order theory, a crest occurs at z = H/ 2s
cos © = 1 and a trough at z = - H/2, cos 6 = - 1. Equation 2-13
therefore implies
_ Hg u cosh [2n(z + d)/L]
ee 2a cosh (27d/L)
with
H
Oy Sp
2
and,
on HgT cosh [2n(z + d)/L]
Pet PAL cosh (27d/L) :
with
H
Z=-—
2
According to second-order theory, a crest occurs at z = Np =
2748) feet.) Gos 0) = cos) 20 1 and a trough at z = ny 9 = - “1.252
feet, cos ‘6° = - Ig.cos 20 1. Equation 2-51 therefore implies
; _ HgT cosh [2zn(z + d)/L]
c, PAL, cosh (27d/L)
: tal 5 cosh [4n(z + d)/L]
it sinh* (27d/L)
with
2 Ng E248 feet
and,
r kei. HgT cosh [22(z + d)/L]
t,2 PAV cosh (27d/L)
“i 2 7H © ~ cosh [47(z + d)/L]
ay sinh*(2nd/L)
with
ZS, p= Sele feet .
Entering Table C-2 with d/L = 0.10, find tanh (2md/L) = 0.5569.
From Equation 2-3 which is true for both first- and second-order
theories,
= 571 (ft/sec)? ,
Eee a (2! _ (32.2) (200) (0.5569)
on L 2n
or
C = 23.9 ft/sec.
As a consequence,
1
2 = = 0.0418 sec/ft .
1, C
Referring again to Table C-2, it is found that when
osh peta] = cosh [27(0.11)] = 1.249 ,
and when
2n(z + d
osh peta) = cosh [27(0.09)] = 1.164 .
Thus, the value of u at a crest and trough respectively according
to first-order theory is
4 1.249
= — (32.2) (0.0418) —— = 2.8
Ue 4 5 ( )¢ ) aan ft/sec ,
1.164
= 32.2) (0.0418 = — 2. §
Uy 4 = ¢ )¢ = 12040 2.6 ft/sec
Entering Table C-2 again, it is found that when
Ze 2.48 feet ,
a ps + ‘|
E
a a + "|
E
= 2 = — 1.52 feet,
cosh [27(0.1124)]
1.260,
cosh [47(0.1124)] = 2.175 .
When
2 -
cosh pete cosh [27(0.0924)] = 1.174,
GSS)
4
ach aa cosh [47(0.0924)]
Thus, the value of u at a crest and trough respectively according
to second-order theory is
4 1.260 3) 47 \ PAs
Sa aie: (32.2) (0.0418) ct 4 Se (23:9) —— = 13:6:8t/sec ,
1.2040 200 0.202
4 174s alan 1.753
u, > = — — (32.2) (0.0418 sy Se poh) Oe ar ee
62 Be et rauan & 23.) 0502 ailehsees
(c) To find the horizontal distance that a particle moves during one
wave period at z = 0, Equation 2-55 can be written as:
= AX(z) _ 7H Be C cosh [4n(z + d)/L]
L 2 sinh? (27d/L)
where AX(z) is the net horizontal distance traveled by a water
particle, z feet below the surface, during one wave period.
For the example problem, when
z=0,
wy cosh (4nd/L) C
BO) st le sinh?(2nd/L) 2
(7H)* cosh (4nd/L) (74)? (1.899) f
2L_sinh?(2nd/L) 2(200) (0.6705)? __ sas
(d) The first-order approximation for pressure under a wave is:
_ pg cosh [2n(z + d)/L] pe
Peay cosh (27d/L) fa poe
when
6 = 0 (ie. the wave crest), cos? = 1,
and when
2n(z +d
z = — d, cosh para) = Eoen(@) = 10 -
Therefore
= B20 ous (2) (2:2 20) = 107 + 1288 = 1395 lbs/ft?
2 1.204 pba Er ieee
at a depth of 20 feet below the SWL. The second-order terms
according to Equation 2-56 are
Sood e sinh?(2nd/L) 3|~
L sinh? (2rd/L)
00 | UW
mH? tanh (2nd/L) | cosh [4n(z + d)/L] ; Py
s
ae! aH? tanh (27d/L) h 4n(z + d)
2 Geen @aimay |e i, ee
Substituting in the equation:
m(4)? (0.5569) 1 1
(1)
Gy 22) 200 (0.6705)?
(0.6705)? 3
1 n(4)* (0.5569)
— — (2) (32.2) —— ——— ee :
p22) cao oem,
2-46
Thus, second-order theory predicts a pressure,
p = 1395 + 14 = 1409 lbs/ft? .
(e) Using Equation 2-38, the energy in one wavelength per unit width
of crest given by the first-order theory is:
2 -
pgH?L nf (2) (32.2) (4)? (200) = 25,800 ft-lbs
8 8 ft
Evaluation of the hydrostatic pressure component (1288 lbs/ft7)
indicates that Airy theory gives a dynamic component of 107 lbs/ft2
while Stokes theory gives 121 lbs/ft2. Stokes theory shows a
dynamic pressure component about 13 percent greater than Airy
theory.
Ae ciee et te) Sisk ie mela ie ie ae OE de ae de. Te eee a ae ee ie ae isis Ae eee ie ae ie er oie
2.26 CNOIDAL WAVES
Long, finite-amplitude waves of permanent form propagating in shallow
water are frequently best described by cnoidal wave theory. The existence
in shallow water of such long waves of permanent form may have first been
recognized by Boussinesq (1877). However, the theory was originally
developed by Korteweg and DeVries (1895). The term enotdal is used since
the wave profile is given by the Jacobian elliptical cosine function
usually designated cn.
In recent years, cnoidal waves have been studied by many investigators.
Wiegel (1960) summarized much of the existing work on cnoidal waves, and
presented the principal results of Korteweg and DeVries (1895) and Keulegan
and Patterson (1940) in a more usable form. Masch and Wiegel (1961)
presented such wave characteristics as- length, celerity and period in
tabular and graphical form, to facilitate application of cnoidal theory.
The approximate range of validity for the cnoidal wave theory as
determined by Laitone (1963) and others is d/L < 1/8, and the Ursell
parameter, L2H/d3 > 26. (See Figure 2-7.) As wavelength becomes long,
and approaches infinity, cnoidal wave theory reduces to the solitary wave
theory which is described in the next section. Also, as the ratio of wave
height to water depth becomes small (infinitesimal wave height), the wave
profile approaches the sinusoidal profile predicted by the linear theory.
Description of local particle velocities, local particle accelerations,
wave energy, and wave power for cnoidal waves is difficult; hence their
description is not included here, but can be obtained in graphical form
from Wiegel (1960, 1964) and Masch (1964).
Wave characteristics are described in parametric form in terms of the
modulus k of the elliptic integrals. While k itself has no physical
significance, it is used to express the relationships between the various
wave parameters. Tabular presentations of the elliptic integrals and
other important functions can be obtained from the above references. The
ordinate of the water surface y, measured above the bottom is given by
t
y, = ¥, + Her? axe a - ‘).1 (2-59a)
where yz is the distance from the bottom to the wave trough, cn is the
elliptic cosine function, K(k) is the complete elliptic integral of the
first kind, and k is the modulus of the elliptic integrals. The argument
of cn* is frequently denoted simply by (), thus, Equation 2-59a above
can be written as
ye Saye Hen s()) (2-59b)
The elliptic cosine is a periodic function where cn*[2K(k) ((x/L) - (t/T))]
has a maximum amplitude equal to unity. The modulus k is defined over
the range between 0 and 1. When k = 0, the wave profile becomes a
sinusoid as in the linear theory, and when k = 1, the wave profile
becomes that of a solitary wave.
The distance from the bottom to the wave trough, yz, as used in
Equations 2-59a and b, is given by
H 16d? H
png fe = ao he -60
See MOIS BOL = = (2-60)
where y, is the distance from the bottom to the crest and E(k) is the
complete elliptic integral of the second kind. Wavelength is given by
[16d?
L= aa kK(k) , (2-61)
and wave period by
l6y, d kK(k)
(2-62)
Bie eter me a
y kW) RG)
Cnoidal waves are periodic and of permanent form thus L = CT.
Pressure under a cnoidal wave at any elevation y, above the bottom
depends on the local fluid velocity, and is therefore complex. However,
it may be approximated in a hydrostatic form as
p = pgly,—y), (2-63)
2-48
that is, the pressure distribution may be assumed to vary linearly from
PgY, at the bed to zero at the surface.
Figures 2-9 and 2-10 show the dimensionless cnoidal wave surface
profiles for various values of the square of the modulus of the elliptic
integrals k*, while Figures 2-11 through 2-15 present dimensionless
plots of the parameters which characterize cnoidal waves. The ordinates
of Figures 2-11 and 2-12 should be read with care, since values of k2
are extremely close to 1.0 (k? = I-10r = 1-051. ="0-99)e. It. isthe
exponent a of k* = 1-10°% that varies along the vertical axis of
Figures 2-11 and 2-12.
Ideally, shoaling computations might best be performed using cnoidal
wave theory since this theory best describes wave motion in relatively
shallow (or shoaling) water. Simple, completely satisfactory procedures
for applying cnoidal wave theory are not available. Although linear wave
theory is often used, cnoidal theory may be applied by using figures such
as 2-9 through 2-15.
The following problem will illustrate the use of these figures.
Gen ee) ede ase adel de. ree eon er se tae de: toe EXAMPLE PROBLEM * * * * * * * * * * * * * *
GIVEN: A wave traveling in water depth of d = 10 feet, with a period of
T = 15 seconds, and a height of H = 2.5 feet.
FIND:
(a) Using cnoidal wave theory, find the wavelength L and compare this
length with the length determined using Airy theory.
(b) Determine the celerity C. Compare this celerity with the celerity
determined using Airy theory.
(c) Determine the distance above the bottom of the wave crest (Ye)
and wave trough (y;).
(d) Determine the wave profile.
SOLUTION:
(a) Calculate
and
24 JO UOIJOUN D SD Sa]1JO1d B9DJINS AADM |OPIOUD “6-2 ainbi4
(0961 ‘186a1M 104)0) aya
os'0
Seen, See aE
Hilti
SSS (AAC ERE
EEN
gone
hat
XEN]
Tet HH =
saavageeguesees 1
fo)
szuze yauangeaua [agen ssece
ro) ro)
Zs
24 40 UOIJDUNY D.SD Sa]1JO1q B9DJ4NS ADM |OPIoUD “Q| -zZ a4nbi4
(096) ‘'|8601Mm 19430) X
wo
SSS
fo eee tt Hee
Sa
a: cea
abe
oor ao < Hs
He sania ne ioatill \
oo eS
ae c LE
FLU \ |
YH
Sere ae
ee Ge:
1,000
400 600
200
100
40 60
10
6s
4
(after Wiegel, 1960)
g/d
Between k2, H/d and T
ionship
Relat
igure 2-1.
F
2-52
100,000
(after Wiegel, 1960)
ae)
i Fal GE (10 i) RSS SST AIST ITE ERG) BREE ae aE BESESSSEa! Ll Mn
His SSiai Bai
ae Seas ae
(SESS SEG RSET (OTF GEER FER BH
1-10~!0
2-53
Figure 2-12. Relationships Between k* and L2 H/d°
(after Wiege!, 1960)
100
a
Figure 2-13. Relationships Between k* and L?H/d® and Between (y,-d)/H, (y;-d)/H +1 and L?H/d°
ae
2-54
1
N\\\ss
\\
ts
MOB RAY PEAT POE LOO UL
aif |
7 1
—— rt
9 GL
(e}
o
N
[O47 (1) 9-271](Q4 KH) +! Ye fey p
eS a aaa)
(4) 44 ‘Ko)
10,000
(after Wiegel, 1960)
10
Jq/d yy/d, H/yy and L?H/d9
Figure 2-14. Relationship Between T
2-55
¢P/H 21 puo 17H * $46 A/ usemjag diysuoljpjay “Gi-2 asnbi4
(0961 ‘19601m 10430) Hz]
SE
56
From Figure 2-11, entering with H/d and Tvg/d, determine the square
of the modulus of the complete elliptical integrals, k2,
ie On
Entering Figure 2-12 with the value of k2 gives
L?H
aaah = 190
or
> oe [190(10)°
At 2.5
L = 27507 tt:
From Airy theory,
2
iL, = er tanh (22) = 266.6 ft.
To check whether the wave conditions are in the range for which
cnoidal wave theory is valid, calculate d/L and the Ursell
parameter, L2H/d?.
Al 10 1
ao ee x0 363- <= OK,
L 275.7 8
L?H 1 H
pol ae ee = 100 S260.
d3 (d/L)? (7)
Therefore, cnoidal theory is applicable.
(b) Wave celerity is given by
=] = —s = 18:38 ft/sec,
while the Airy theory predicts
266.6
C= E = — = 17.77 ft/sec.
ai 15
Thus if it is assumed that the wave period is the same for Cnoidal
and Airy theories then
Conoidal a Lcnoidal al
C, iry lL, iry
2-or
(c)
(d)
The percentage of the wave height above the SWL may be determined
from Figure 2-13. Entering the figure with L2H/d? = 190, the
value of (y, - d)/H is found to be 0.833 or 83.3 percent.
Therefore,
y, = 0.833H +d.
0.833 (2.5) + 10 = 2.083 + 10 = 12.083 feet .
Ye
Also from Figure 2-13,
H
thus
yy = (0.833 — 1) (2.5) + 10 = 9.583 ft.
The dimensionless wave profile is given on Figure 2-9 and is
approximately the one drawn for k* = 1 - 10-*. The results
obtained in (c) above can also be checked by using Figure 2-9.
For the wave profile obtained with k* = 1 - 1074, it is seen that
the SWL is approximately 0.17 H above the wave trough or 0.83 H
below the wave crest.
The results for the wave celerity determined under (b) above
can now be checked with the aid of Figure 2-15. Calculate
ely B28 ee
Yy 9.583 P A
Entering Figure 2-15 with
EAE
zB = C0) 5
and
= = oil -
Yt
it is found that
C
= = 1.083 :
SY¢
Therefore
C = 1.083 (32.2) (9.583) = 19 ft/sec .
The difference between this number and the 18.38 ft/sec calculated
under (b) above is the result of small errors in reading the
curves.
So Sep ce ein ED RL, eS A ie ee SOR Rep ke, KK RL ties YK) eae Ke eK Kw. Ie
2.27 SOLITARY WAVE THEORY
Waves considered in the previous sections were oscillatory or nearly
oscillatory waves. The water particles move backward and forward with the
passage of each wave, and a distinct wave crest and wave trough are evident.
A solitary wave is neither oscillatory nor does it exhibit a trough. In
the pure sense, the solitary wave form lies entirely above the stillwater
level. The solitary wave is a wave of translation relative to the water
mass.
Russell (1838, 1845) first recognized the existence of a solitary
wave. The original theoretical developments were made by Boussinesq
(1872), Lord Rayleigh (1876), and McCowan (1891), and more recently by
Keulegan and Patterson (1940), Keulegan (1948), and Iwasa (1955).
In nature it is difficult to form a truly solitary wave, because at
the trailing edge of the wave there are usually small dispersive waves.
However, long waves such as tsunamis and waves resulting from large dis-
placements of water caused by such phenomena as landslides, and earthquakes
sometimes behave approximately like solitary waves. When an oscillatory
wave moves into shallow water, it may often be approximated by a solitary
wave, (Munk, 1949). As an oscillatory wave moves into shoaling water, the
wave amplitude becomes progressively higher; the crests become shorter and
more pointed, and the trough becomes longer and flatter.
The solitary wave is a limiting case of the cnoidal wave. When k? =
1, K(k) = K(1) =~, and the elliptic cosine reduces to the hyperbolic
secant function, y; = d, and Equation 2-59 reduces to
3) lal
y, = d + H sech? [Ze @- cH},
ens
4 @
or
n = H sech? G— Ce). (2-64)
where the origin of x is at the wave crest. The volume of water within
the wave above the still water level per unit crest width is
16 %
yr ea | (2-65)
An equal amount of water per unit crest length is transported
forward past a vertical plane that is perpendicular to the direction of
wave advance. Several relations have been presented to determine the
celerity of a solitary wave; these equations differ depending on the
degree of approximation. Laboratory measurements by Daily and Stephan
(1953) indicate that the simple expression
C= yere , (2-66)
gives a reasonably accurate approximation to the celerity.
The water particle velocities for a solitary wave, as found by
McCowan (1891) and given by Munk (1949), are
1 + cos (My/d) cosh (Mx/d)
2-67
[cos (My/d) + cosh (Mx/d)]? ’ C )
sin (My/d) sinh (Mx/d)
= ps Eee dee See 2-68
a cM [cos (My/d) + cosh (Mx/d)}* ° ( J
where M and N are the functions of H/d_ shown on Figure 2-16, and y
is measured from the bottom. The expression for horizontal velocity u,
is often used to predict wave forces on marine structures sited in shallow
water. The maximum velocity u,9,, occurs when x and t are both equal
to zero; hence,
CN
via > cas (My/d) © —
Total energy in a solitary wave is about evenly divided between
kinetic and potential energy. Total wave energy per unit crest width is,
8
E= pg H2/2 43/2 (2-70)
3V3°
and the pressure beneath a solitary wave depends upon the local fluid
velocity as does the pressure under a cnoidal wave; however, it may be
approximated by
PrS spe Gn yr (227i)
Equation 2-71 is identical to that used to approximate the pressure
beneath a cnoidal wave,
As a solitary wave moves into shoaling water it eventually becomes
unstable and breaks, McCowan (1891) assumed that a solitary wave breaks
aa
Bee seeunseeneseuat
seustatetuesetae
Ee
HH
d
Relative wave height
1949)
(after Munk
Functions M and N in Solitary Wave Theory
-16.
Figure 2
6
when the water particle velocity at the wave crest becomes equal to the
wave celerity. This occurs when
(3)
“ =HOen CLD
d max
Laboratory investigations have shown that the value of (H/d),,.,, = 0.78
agrees better with observations for oscillatory waves than for solitary
waves. Ippen and Kulin (1954) and Galvin (1969) have shown that the near-
shore slope has a substantial effect on this ratio. Other factors such as
bottom roughness may also be involved. For slopes of 0.0, 0.05, 0.10, and
0.20, Galvin found that H/d ratios were approximately equal to 0.83,
1.05, 1.19, and 1.32, respectively. Thus, it must be concluded that for
some conditions, Equation 2-72 is unsatisfactory for predicting breaking
depth. Further discussion of the breaking of waves with experimental
results is in Section 2.6 - BREAKING WAVES.
2.28 STREAM FUNCTION WAVE THEORY
In recent years, numerical approximations to solutions of hydrodynamic
equations describing wave motion have been proposed and developed by Dean
(1965a, 1965b, 1967) and Monkmeyer (1970). The approach by Dean, termed
a symmetric, stream function theory, is a nonlinear wave theory which is
Similar to higher order Stokes' theories. Both are constructed of sums of
sine or cosine functions that satisfy the original differential equation
(Laplace equation). The theory, however, determines the coefficient of
each higher order term so that a best fit, in the least-squares sense, is
obtained to the theoretically posed, dynamic, free-surface boundary con-
dition. Assumptions made in the theory are identical to those made in the
development of the higher-order Stokes' solutions. Consequently, some of
the same limitations are inherent in the stream function theory; however,
it represents a better solution to the equations used to approximate the
wave phenomena. More important is that the stream function representation
appears to better predict some of the wave phenomena observed in laboratory
wave studies (Dean and LéMehauté, 1970), and may possibly describe naturally
occurring wave phenomena better than other theories.
The long tedious computations involved in evaluating the terms of the
series expansions that make up the higher-order stream function solutions,
make it desirable to use tabular or graphical presentations of the
solutions. These tables, their use and range of validity have been
developed by Dean (1973).
2.3 WAVE REFRACTION
2.31 INTRODUCTION
Equation 2-2 shows that wave celerity depends on the depth of water
in which the wave propagates. If the wave celerity decreases with depth,
wavelength must decrease proportionally. Variation in wave velocity occurs
2-62
along the crest of a wave moving at an angle to underwater contours because
that part of the wave in deeper water is moving faster than the part in
shallower water. This variation causes the wave crest to bend toward align-
ment with the contours. (See Figure 2-17.) This bending effect, called
refraction, depends on the relation of water depth to wavelength. It is
analogous to refraction for other types of waves such as, light and sound.
In practice, refraction is important for several reasons such as:
(1) Refraction, coupled with shoaling, determines the wave height
in any particular water depth for a given set of incident deepwater wave
conditions, that is wave height, period, and direction of propagation in
deep water. Refraction therefore has significant influence on the wave
height and distribution of wave energy along a coast.
(2) The change of wave direction of different parts of the wave
results in convergence or divergence of wave energy, and materially affects
the forces exerted by waves on structures.
(3) Refraction contributes to the alteration of bottom topography
by its effects on the erosion and deposition of beach sediments. Munk and
Traylor (1947) confirmed earlier work by many indicating the possible inter-
relationships between refraction, wave energy distribution along a shore,
and the erosion and deposition of beach materials.
(4) A general description of the nearshore bathymetry of an area
can sometimes be obtained by analyzing aerial photography of wave refraction
patterns. While the techniques for performing such analyses are not well
developed, an experienced observer can obtain a general picture of simple
bottom topography.
In addition to refraction caused by variations in bathymetry, waves
may be refracted by currents, or any other phenomenon which causes one part
of a wave to travel slower or faster than another part. At a coastal inlet,
refraction may be caused by a gradient in the current. Refraction by a
current occurs when waves intersect the current at an angle. The extent
to which the current will refract incident waves depends on the initial
angle between the wave crests and the direction of current flow, the
characteristics of the incident waves, and the strength of the current.
In at least two situations, wave refraction by currents may be of practical
importance. At tidal entrances, ebb currents run counter to incident waves
and consequently increase wave height and steepness. Also, major ocean
currents such as the Gulf Stream may have some effect on the height, length
and direction of approach of waves reaching the coasts. Quantitative
evaluation of the effects of refraction by currents is difficult. Addi-
tional research is needed in this area. No detailed discussion of this
problem will be presented here, but an introduction is presented by
Johnson (1947).
The decrease in wave celerity with decreasing water depth can be con-
sidered an analog to the decrease in the speed of light with an increase in
2-63
Figure 2-17. Wave Refraction at Westhampton Beach, Long Island, New York
2-64
the refractive index of the transmitting medium. Using this analogy,
O'Brien (1942) suggested the use of Snell's law of geometrical optics for
solving the problem of water-wave refraction by changes in depth. The
validity of this approach has been verified experimentally by Chien (1954),
Ralls (1956), and Wiegel and Arnold (1957). Chao (1970) showed analyti-
cally that Fermat's principle and hence Snell's law followed from the
governing hydrodynamic equations, and was a valid approximation when applied
to the refraction problem. Generally, two basic techniques of refraction
analysis are available - graphical and numerical. Several graphical pro-
cedures are available, but fundamentally all methods of refraction analyses
are based on Snell's law.
The assumptions usually made are:
(1) Wave energy between wave rays or orthogonals remains constant.
(Orthogonals are lines drawn perpendicular to the wave crests, and extend
in the direction of wave advance.) (See Figure 2-17.)
(2) Direction of wave advance is perpendicular to the wave crest,
that is, in the direction of the orthogonals.
(3) Speed of a wave of given period at a particular location
depends only on the depth at that location.
(4) Changes in bottom topography are gradual.
(5) Waves are long-crested, constant-period, small-amplitude, and
monochromatic.
(6) Effects of currents, winds, and reflections from beaches, and
underwater topographic variations, are considered negligible.
2.32 GENERAL - REFRACTION BY BATHYMETRY
In water deeper than one-half the wavelength, the hyperbolic tangent
function in the formula
IL, 2nd
c? 2 tanh (=) (2-2)
2n IL:
is nearly equal to unity, and Equation 2-2 reduces to
In this equation, the velocity C,, does not depend on depth; therefore
in those regions deeper than one-half the wavelength (deep water), refrac-
tion by bathymetry will not be significant. Where the water depth is
between 1/2 and 1/25 the wavelength (transitional water), and in the region
where the water depth is less than 1/25 the wavelength (shallow water),
refraction effects may be significant. In transitional water, wave velocity
2-65
must be computed from Equation 2-2; in shallow water, tanh(2nd/L) becomes
nearly equal to 2nd/L and Equation 2-2 reduces to Equation 2-9.
C? = gd or C= (gd) F (2-9)
Both Equations 2-2 and 2-9 show the dependence of wave velocity on depth.
To a first approximation, the total energy in a wave per unit crest width
may be written as
pgH*L
iach
E = (2-38)
It has been noted that not all of the wave energy E is transmitted
forward with the wave; only one-half is transmitted forward in deep water.
The amount of energy transmitted forward for a given wave remains nearly
constant as the wave moves from deep water to the breaker line if energy
dissipation due to bottom friction (Kp = 1.0), percolation and reflected
wave energy is negligible.
In refraction analyses, it is assumed that for a wave advancing toward
shore, no energy flows laterally along a wave crest; that is the transmitted
energy remains constant between orthogonals. In deep water the wave energy
transmitted forward across a plane between two adjacent orthogonals (the
average energy flux) is
bEC (2373)
where by, is the distance between the selected orthogonals in deep water.
The subscript o always refers to deepwater conditions. This power may
be equated to the energy transmitted forward between the same two orthogonals
in shallow water
Pian bE C. (2-74)
where b is the spacing between the orthogonals in the shallower water.
Therefore, (1/2) by E,Cy = nb) EG; or
E i. ffl b, C,
ela Wb et ao
uae z 2-76
H = E . ( tT )
From Equation 2-39,
and combining Equations 2-75 and 2-76,
n> MelGNEN $8
The term Y(1/2) (1/n) (C,/C) is known as the shoaling coeffictent K,
or H/HZ. This shoaling coefficient is a function of wavelength and water
depth. Kg and various other functions of d/L, such as 2md/L, 4nd/L,
tanh(2nd/L), and sinh(4nd/L) are tabulated in Appendix C, (Table C-1 for
even increments of d/L,, and Table C-2 for even increments of d/L).
Equation 2-77 enables determination of wave heights in transitional
or shallow water, knowing the deepwater wave height when the relative
spacing between orthogonals can be determined. The square root of this
relative spacing, vb,/b, is the refraction coefficient Kp.
Various methods may be used for constructing refraction diagrams.
The earliest approaches required the drawing of successive wave crests.
Later approaches permitted the immediate construction of orthogonals,
and also permitted moving from the shore to deep water (Johnson, O'Brien
and Isaacs, 1948), (Arthur, et al., 1952), (Kaplan, 1952) and (Saville
and Kaplan, 1952).
The change of direction of an orthogonal as it passes over relatively
simple hydrography may be approximated by
C
sina, = (=) sin a, (Snell’s law) (2-78)
1
where:
®, is the angle a wave crest (the perpendicular to an orthogonal)
makes with the bottom contour over which the wave is passing,
a, is a similar angle measured as the wave crest (or orthogonal)
passes over the next bottom contour,
Cy is the wave velocity (Equation 2-2) at the depth of the first
contour, and
C, is the wave velocity at the depth of the second contour.
From this equation, a template may be constructed which will show the
angular change in a that occurs as an orthogonal passes over a
particular contour interval, and construct changed-direction orthogonal.
Such a template is shown in Figure 2-18. In application to wave refrac-
tion problems, it is simplest to construct this template on a transparent
material.
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Refraction may be treated analytically at a straight shoreline with
parallel offshore contours, by using Snell's law directly:
C
sin a te) sin a, (2-78a)
where a is the angle between the wave crest and the shoreline, and a,
is the angle between the deepwater wave crest and the shoreline.
For example, if og = 30° and the period and depth of the wave
are such that C/C 5 = 0.5, then
a = sin? [0.5 (0.5)] = 14.5 degrees
cosa = 0.968
and
cosa, = 0.866
b. \* cos a = 0.866 v7)
“= 2) 2) 7 a = 0.945
b cos & 0.968
Figure 2-19 shows the relationships between a, a,, period, depth, and
Kp in graphical form.
2.321 Procedures in Refraction Diagram Construction - Orthogonal Method.
Charts showing the bottom topography of the study area are obtained. Two
or more charts of differing scale may be required, but the procedures are
identical for charts of any scale. Underwater contours are drawn on the
chart, or on a tracing paper overlay, for various depth intervals. The
depth intervals chosen depend on the degree of accuracy desired. If
overlays are used, the shoreline should be traced for reference. In
tracing contours, small irregularities must be smoothed out, since bottom
features that are comparatively small in respect to the wavelength do not
affect the wave appreciably.
The range of wave periods and wave directions to be investigated is
determined by a hindcasting study of historical weather charts or from
other historical records relating to wave period and direction. For each
wave period and direction selected, a separate diagram must be prepared.
C,;/C2 values for each contour interval may then be marked between contours.
The method of computing C,/C2 is illustrated by Table 2-2; a tabulation
of C,/C, for various contour intervals and wave periods is given in
Table C-4 of Appendix C.
To construct orthogonals from deep to shallow water, the deepwater
direction of wave approach is first determined. A deepwater wave front
(crest) is drawn as a straight line perpendicular to this wave direction,
2-69
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2-70
and suitably spaced orthogonals are drawn perpendicular to this wave front
and parallel to the chosen direction of wave approach. Closely spaced
orthogonals give more detailed results than widely spaced orthogonals.
These lines are extended to the first depth contour shallower than L,/2
where L, (in feet) = 5.12 T2.
TABLE 2-2 EXAMPLE COMPUTATIONS OF VALUES OF
C,/C2 FOR REFRACTION ANALYSIS
T = 10 seconds
Column 1 gives depths corresponding to chart contours. These
would extend from 6 feet to a depth equal to L,/2.
Column 2 is column 1 divided by L, corresponding to the given
period.
Column 3 is the value of tanh 21nd/L found in Table C-1 of
Appendix C, corresponding to the value of d/L, in
column 2. This term is also C/C,-
Column 4 is the quotient of successive terms in column 3.
Column 5 is the reciprocal of column 4.
2.322 Procedure when a is Less than 80 Degrees. Recall that a is the
angle a wave crest makes with the bottom contour. Starting with any one
orthogonal and using the refraction template in Figure 2-18, the following
steps are performed in extending the orthogonal to shore:
(a) Sketch a contour midway between the first two contours to be
crossed, extend the orthogonal to the midcontour, and construct a tangent
to the midcontour at this point.
(b) Lay the line on the template labelled orthogonal along the in-
coming orthogonal with the point marked 1.0 at the intersection of the
orthogonal and midcontour (Figure 2-20 top);
emt
Template Orthogonal’ Line
Contour
Tangent to Mid- (C
Contour
Incoming Orthogonal
Turning Point
Template "Orthogonal" Line
eds Roe Turned Orthogonal
~
‘ Mid- Contour
Tangent to Mid - ; Fs ‘i
Distance ~~ !
C
Contour St
Incoming Orthogonal
Incoming Orthogonal
Turning Point
b
The template has been turned about R until the value ee 1.045
intersects the tangent to the mid-contour. The template “orthogonal” line
lies in the direction of the turned orthogonal. This direction is to be laid
off at some point 'B' on the incoming orthogonal which is equidistant
from the two contours along the incoming and outgoing orthogonals.
Figure 2-20. Use of the Refraction Template
2=+2
(c) Rotate the template about the turning point until the C,/C, value
corresponding to the contour interval being crossed intersects the tangent
to the midcontour. The orthogonal line on the chart now lies in the direc-
tion of the turned orthogonal on the template (Figure 2-20 bottom) ;
(d) Place a triangle along the base of the template and construct
a perpendicular to it so that the intersection of the perpendicular with
the incoming orthogonal is midway between the two contours when the dis-
tances are measured along the incoming orthogonal and the perpendicular
(See Point B in Figure 2-20 bottom). Note that this point is not neces-
sarily on the midcontour line. This line represents the turned orthogonal;
(e) Repeat the above steps for successive contour intervals.
If the orthogonal is being constructed from shallow to deep water, the
same procedure may be used, except that C,/C, values are used instead of
Ge) Ce.
12
A template suitable for attachment to a drafting machine can be made,
Palmer (1957), and may make the procedure simpler if many diagrams are to
be used.
2.323 Procedure when a is Greater than 80 Degrees - The R/J Method. In
any depth, when a becomes greater than 80 degrees, the above procedure
cannot be used. The orthogonal no longer appears to cross the contours,
but tends to run almost parallel to them. In this case, the contour
interval must be crossed in a series of steps. The entire interval is
divided into a series of smaller intervals. At the midpoint of the indi-
vidual subintervals, orthogonal-angle turnings are made.
Referring to Figure 2-21, the interval to be crossed is divided into
segments or boxes by transverse lines. The spacing R, of the transverse
lines is arbitrarily set as a ratio of the distance J, between the con-
tours. For the complete interval to be crossed, C2/C, is computed or
found from Table C-4 of Appendix C. (C,/C,, not C,/C,.)
On the template (Figure 2-18), a graph showing orthogonal angle
turnings Aa, is plotted as a function of the C2/C, value for various
values of the ratio R/J. The Aa value is the angle turned by the in-
coming orthogonal in the center of the subinterval.
The orthogonal is extended to the middle of the box, Aa is read
from the graph, and the orthogonal turned by that angle. The procedure
is repeated for each box in sequence, until a at a plotted or interpo-
lated contour becomes smaller than 80 degrees. At this point, this method
of orthogonal construction must be stopped, and the preceding technique
for a smaller than 80 degrees used, otherwise errors will result.
2.324 Refraction Fan Diagrams. It is often convenient, especially where
sheltering land forms shield a stretch of shore from waves approaching in
certain directions, to construct refraction diagrams from shallow water
2-13
= Distance between contours at turning points, @
= Distance along orthogonal
12 seconds
=o tutte
J
R
"| Fat
L
ie]
Figure 2-21. Refraction Diagram Using R/J Method
toward deep water. In such cases, a sheaf or fan of orthogonals may be
projected seaward in directions some 5 or 10 degrees apart. See Figure
2-22a. With the deepwater directions thus determined by the individual
orthogonals, companion orthogonals may be projected shoreward on either
side of the seaward projected ones to determine the refraction coefficient
for the various directions of wave approach. (See Figure 2-22b.)
2.325 Other Graphical Methods of Refraction Analysis. Another graphical
method for the construction of refraction diagrams is the wave-front
method (Johnson, et al., 1948). This method is particularly applicable
to very long waves where the crest alignment is also desired. The method
is not presented here, where many diagrams are required, because, where
many diagrams are required, it is overbalanced by the advantages of the
orthogonal method. The orthogonal method permits the direct construction
of orthogonals and determination of the refraction coefficient without the
intermediate step of first constructing successive wave crests. Thus,
when the wave crests are not required, significant time is saved by using
the orthogonal method.
2.326 Computer Methods for Refraction Analysis. Harrison and Wilson (1964)
developed a method for the numerical calculation of wave refraction by use
of an electronic computer. Wilson (1966) extended the method so that, in
addition to the numerical calculation, the actual plotting of refraction
diagrams is accomplished automatically by use of a computer. Numerical
methods are a practical means of developing wave refraction diagrams when
an extensive refraction study of an area is required, and when they can
be relied upon to give accurate results. However, the interpretation of
computer output requires care, and the limitations of the particular scheme
used should be considered in the evaluation of the results. For a dis-
cussion of some of these limitations, see Coudert and Raichlen (1970).
For additional references, the reader is referred to the works of Keller
(1958), Mehr (1962), Griswold (1963), Wilson (1966), Lewis, et al., (1967),
Dobson (1967), Hardy (1968), Chao (1970), and Keulegan and Harrison (1970),
in which a number of available computer programs for calculation of refrac-
tion diagrams are presented. Most of these programs are based on an algo-
rithm derived by Munk and Arthur (1951) and, as such, are fundamentally
based on the geometrical optics approximation. (Fermat's Principle.)
2.327 Interpretation of Results and Diagram Limitations. Some general
observations of refraction phenomena are illustrated in Figures 2-23, 24,
and 25. These figures show the effects of several common bottom features
on passing waves. Figure 2-23 shows the effect of a straight beach with
parallel evenly spaced bottom contours on waves approaching from an angle.
Wave crests turn toward alignment with the bottom contours as the waves
approach shore. The refraction effects on waves normally incident on a
beach fronted by a submarine ridge or submarine depression are illustrated
in Figure 2-24a and 2-24b. The ridge tends to focus wave action toward
the section of beach where the ridge line meets the shoreline. The ortho-
gonals in this region are more closely spaced; hencey/b,/b is greater
than 1.0 and the waves are higher than they would be if no refraction
occurred. Conversely, a submarine depression will cause orthogonals to
2-75
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Figure 2-23. Refraction Along a Straight Beach with Parallel
Bottom Contours
cs Pas
ihe AE GPS
7
Orthogonals ontours Orthogonals
(a) (b)
Figure 2-24. Refraction by a Submarine Ridge (a) and Submarine
Canyon (b)
Contours
Contours Orthogonals
Figure 2-25. Refraction Along an Irregular Shoreline
diverge, resulting in low heights at the shore. (b)/b less than 1.0.)
Similarly, heights will be greater at a headland than in a bay. Since
the wave energy contained between two orthogonals is constant, a larger
part of the total energy expended at the shore is focused on projections
from the shoreline; consequently, refraction straightens an irregular
coast. Bottom topography can be inferred from refraction patterns on
aerial photography. The pattern in Figure 2-17 indicates the presence
of a submarine ridge.
Refraction diagrams can provide a measure of changes in waves
approaching a shore. However, the accuracy of refraction diagrams is
limited by the validity of the theory of construction and the accuracy
of depth data. The orthogonal direction change (Equation 2-78) is
derived for straight parallel contours. It is difficult to carry an
orthogonal accurately into shore over complex bottom features (Munk and
Arthur, 1951). Moreover, the equation is derived for small waves moving
over mild slopes.
Dean (1973) considers the combined effects of refraction and shoal-
ing including nonlinearities applied to a slope with depth contours
parallel to the beach but not necessarily of constant slope. He finds
that non-linear effects can significantly increase (in comparison with
linear theory) both amplification and angular turning of waves of low
steepness in deep water.
Strict accuracy for height changes cannot be expected for slopes
steeper than 1:10, although model tests have shown that direction
changes nearly as predicted even over a vertical discontinuity (Wiegel
and Arnold, 1957). Accuracy where orthogonals bend sharply or exhibit
extreme divergence or convergence is questionable because of energy
transfer along the crest. The phenomenon has been studied by Beitinjani
and Brater (1965), Battjes (1968) and Whalin (1971). Where two ortho-
gonals meet, a caustic ‘develops. A caustic is an envelope of ortho-
gonal crossings, caused by convergence of wave energy at the caustic
point. An analysis of wave behavior near a caustic is not available;
however, qualitative analytical results show that wave amplitude decays
exponentially away from a caustic in the shadow zone, and there is a
phase shift of 1/2 across the caustic (Whalin 1971). Wave behavior
near a caustic has also been studied by Pierson (1950), Chao (1970) and
others. Little quantitative information is available for the area
beyond a caustic.
2.328 Refraction of Ocean Waves. Unlike Monochromatic waves, actual
ocean waves are more complicated. Their crest lengths are short; their
form does not remain permanent; and their speed, period, and direction
of propagation vary from wave to wave.
2-76
Pierson (1951), Longuet-Higgins (1957), and Kinsman (1965), have sug-
gested a solution to the ocean-wave refraction problem. The sea surface
waves in deep water become a number of component monochromatic waves, each
with a distinct frequency and direction of propagation. The energy spec-
trum for each component may then be found and the conventional refraction
analysis techniques applied. Near the shore, the wave energy propagated
in a particular direction is approximated as the linear sum of the spectra
of wave components of all frequencies refracted in the given direction from
all of the deepwater directional components.
The work required for this analysis, even for a small number of indi-
vidual components, is laborious and time consuming. More recent research
by Borgman (1969) and Fan and Borgman (1970), has used the idea of direc-
tional spectra which may provide a technique for solving complex refraction
problems more rapidly.
2.4 WAVE DIFFRACTION
2.41 INTRODUCTION
Diffraction of water waves is a phenomenon in which energy is trans-
ferred laterally along a wave crest. It is most noticeable where an other-
wise regular train of waves is interrupted by a barrier such as a breakwater
or an islet. If the lateral transfer of wave energy along a wave crest and
across orthogonals did not occur, straight, long-crested waves passing the
tip of a structure would leave a region of perfect calm in the lee of the
barrier, while beyond the edge of the structure the waves would pass un-
changed in form and height. The line separating two regions would be a
discontinuity. A portion of the area in front of the barrier would, how-
ever, be disturbed by both the incident waves and by those waves reflected
by the barrier. The three regions are shown in Figure 2-26a for the hypo-
thetical case if diffraction did not occur, and in Figure 2-26b for the
actual phenomenon as observed. The direction of the lateral energy trans-
fer is also shown in Figure 2-26a. Energy flow across the discontinuity
is from Region II into Region I. In Region III, the superposition of
incident and reflected waves results in the appearance: of short-crested
waves if the incident waves approach the breakwater obliquely. A partial
standing wave will occur in Region III if the waves approach perpendicular
to the breakwater.
This process is also similar to that for other types of waves, such
as light or sound waves.
Calculation of diffraction effects is important for several reasons.
Wave height distribution in a harbor or sheltered bay is determined to
some degree by the diffraction characteristics of both the natural and
manmade structures affording protection from incident waves. Therefore,
a knowledge of the diffraction process is essential in planning such
facilities. Proper design and location of harbor entrances to reduce
such problems as silting and harbor resonance also require a knowledge
of the effects of wave diffraction. The prediction of wave heights near
2-79
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the shore is affected by diffraction caused by naturally occurring changes
in hydrography. An aerial photograph illustrating the diffraction of
waves by a breakwater is shown in Figure 2-27.
Putnam and Arthur (1948) presented experimental data verifying a
method of solution proposed by Penny and Price (1944) for wave behavior
after passing a single breakwater. Wiegel (1962) used a theoretical
approach to study wave diffraction around a single breakwater. Blue and
Johnson (1949) dealt with the problem of the behavior of waves after
passing through a gap, as between two breakwater arms.
The assumptions usually made in the development of diffraction
theories are:
(1) Water is an ideal fluid, i.e., inviscid and incompressible.
(2) Waves are of small-amplitude and can be described by linear
wave theory.
(3) Flow is irrotational and conforms to a potential function which
satisfies the Laplace equation.
(4) Depth shoreward of the breakwater is constant.
2.42 DIFFRACTION CALCULATIONS
2.421 Waves Passing a Single Breakwater. From a presentation by Wiegel
(1962), diffraction diagrams have been prepared which, for a uniform depth
adjacent to an impervious structure, show lines of equal wave height re-
duction. These diagrams are shown in Figures 2-28 through 2-39; the graph
coordinates are in units of wavelength. Wave height reduction is given in
terms of a diffraction coefficient kK’ which is defined as the ratio of a
wave height H, in the area affected by diffraction to the incident wave
height H;, in the area unaffected by diffraction. Thus, H and H; are
determined by H = K’H;.
The diffraction diagrams shown in Figures 2-28 through 2-39 are con-
structed in polar coordinate form with arcs and rays centered at the struc-
tuge's tip. The arcs are spaced one radius-wavelength unit apart and rays
15 apart . In application, a given diagram must be scaled up or down so
that the particular wavelength corresponds to the scale of the hydrographic
chart being used. Rays and arcs on the refraction diagrams provide a
coordinate system that makes it relatively easy to transfer lines of
constant Kk’ on the scaled diagrams.
When applying the diffraction diagrams to actual problems, the wave-
length must first be determined based on the water depth at the tip of the
structure, The wavelength L, in water depth d,, may be found by com-
puting d,/Ly = de / 5121 and using Appendix C, Table C-1 to find the
corresponding value of d,/L. Dividing d, by d,/L will give the shallow
water wave length L. It is then useful to construct a scaled diffraction
2-8!
Figure 2-27. Wave Diffraction at Channel Islands Harbor Breakwater,
California
2-82
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diagram overlay template to correspond to the hydrographic chart being
used. In constructing this overlay, first determine how long each of its
radius-wavelength units must be. As noted previously, one radius-wavelength
unit on the overlay must be identical to one wavelength on the hydrographic
chart. The next step is to construct and sketch all overlay rays and arcs
on clear plastic or translucent paper. This allows penciling in of the
scaled lines of equal K for each angle of wave approach that may be
considered pertinent to the problem. Thus, after studying the wave field
for one angle of wave approach, K lines may be erased for a subsequent
analysis of a different angle of wave approach.
The diffraction diagrams in Figures 2-28 through 2-39 show the break-
water extending to the right as seen looking toward the area of wave dif-
fraction; however, for some problems the structure may extend to the left.
All diffraction diagrams presented may be reversed by simply turning the
transparency over to the opposite side.
Figure 2-40 illustrates the use of a template overlay. Also indicated
is the angle of wave approach which is measured counterclockwise from the
breakwater. This angle would be measured clockwise from the breakwater if
the diagram were turned over. Figure 2-40 also shows a rectangular coordi-
nate system with distance expressed in units of wavelength. Positive
x direction is measured from the structure's tip along the breakwater and
positive y direction is measured into the diffracted area.
y
Template Overlay
| THSAEE TT N Breakwate
Se ae a FT TT YT ET ET TT A
—~~—— Wave Crests
Figure 2-40. Diffraction for a Single Breakwater Normal Incidence
The following problem illustrates determination of a single wave
height in the diffraction area.
eRe eee eRe) ee Ae ee coe) ae EXAMPLE PROBLEM eo ee ee ee RY Pte) Se) ee re eae
GIVEN: Waves with a period of T = 8 seconds and height of H = 10 feet
impinge upon a breakwater at an angle of 135 degrees. The water depth
at the tip of the breakwater toe is dg = 15 feet. Assume that one inch
on the hydrographic chart being used is equivalent to 133 feet.
2-05
FIND: The wave height at a point P having coordinates in units of wave-
length of x = 3 and y = 4. (Polar coordinates of x and y are r=5 at 53°.)
SOLUTION:
Since dg = 15 feet, T = 8 seconds,
d d 15
SO ee = 004580
Uy bln (5.12) (64)
Using Table C-1 with
d d.
— = — = 0.0458
L, ce)
the corresponding value of
d d.
— or — = 0.0899,
1, L
therefore,
d 1
ee 2 = 167 feet .
( /. 0.0899
Because 1 inch represents 133 feet on the hydrographic chart and
L = 167 feet, the wave-length is 1.26 inches on the chart.
This provides the necessary information for scaling Figure 2-36 to the
hydrographic chart being used. Thus 1.26 inches represents a radius/
wavelength unit.
For this example, point P and those lines of equal K’ situated
nearest P are shown on a schematic overlay, Figure 2-41. This over-
lay is based on Figure 2-36 since the angle of wave approach is 1354
It should be noted that Figure 2-41, being a schematic rather than a
true representation of the overlay, is not drawn to the hydrographic
chart scale calculated in the problem. From Figure 2-41 it is seen
that K’ at point P is approximately 0.086. Thus the diffracted
wave height at this point is
H = K’H; = (0.086) (10) = 0.86 foot say 0.9 foot .
The above calculation indicates that a wave undergoes a substantial
height reduction in the area considered.
Ch Meets tee Cpe oe a te gee FU, 2 CJ, Tet SC, hic, Yume ne tae mm ame tie ec ey ee CY Koo ue a BCP OG? or eo CR CO
OVERLAY
(Figure 2-36)
X30
x and y are measured in units
of wavelength.
(These units vary depending
on the wavelength and the
chart scale.)
180°
Breakwater
Wave Crests
Direction of Wave Approach
Figure 2-41. Schematic Representation of Wave Diffraction Overlay
2.422 Waves Passing a Gap of Width Less than Five Wavelengths at Normal
Incidence. The solution of this problem is more complex than that for a
single breakwater, and it is not possible to construct a single diagram
for all conditions. A separate diagram must be drawn for each ratio of
gap width to wavelength B/L. The diagram for a B/L-ratio of 2 is shown
in Figure 2-42 which also illustrates its use. Figures 2-43 through 2-52
(Johnson, 1953) show lines of equal diffraction coefficient for B/L-ratios
o£ (0-507 1-00), 1.41, 4.64, 2-73, 200,892.50, 2.95), seseeand) 5. 00cm as
sufficient number of diagrams have been included to represent most gap
widths encountered in practice. In all but Figure 2-48 (B/L = 2.00), the
wave crest lines have been omitted. Wave crest lines are usually of use
only for illustrative purposes. They are, however, required for an
accurate estimate of the combined effects of refraction and diffraction.
In such cases, wave crests may be approximated with sufficient accuracy
by circular arcs. For a single breakwater, the arcs will be centered on
the breakwater tip. That part of the wave crest extending into unprotected
water beyond the K’ = 0.5 line may be approximated by a straight line.
For a breakwater gap, crests that are more than eight wavelengths behind
the breakwater may be approximated by an arc centered at the middle of
the gap; crests to about six wavelengths may be approximated by two arcs,
centered on the two ends of the breakwater and may be connected by a
smooth curve (approximated by a circular arc centered at the middle of
the gap). Only one-half of the diffraction diagram is presented on the
figures since the diagrams are symmetrical about the line x/L = 0.
2.423 Waves Passing a Gap of Width Greater Than Five Wavelengths at
Normal Incidence. Where the breakwater gap width is greater than five
wavelengths, the diffraction effects of each wing are nearly independent,
and the diagram (Figure 2-33) for a single breakwater with a 90° wave
approach angle may be used to define the diffraction characteristic in
the lee of both wings (See Figure 2-53.)
2.424 Diffraction at a Gap-Oblique Incidence. When waves approach at an
angle to the axis of a breakwater, the diffracted wave characteristics
differ from those resulting when waves approach normal to the axis. An
approximate determination of diffracted wave characteristics may be
obtained by considering the gap to be as wide as its projection in the
direction of incident wave travel as shown in Figure 2-54. Calculated
diffraction diagrams for wave approach angles of 0°, 15°, 30°, 45°, 60°
and 75° are shown in Figures 2-55, 56 and 57. Use of these diagrams will
give more accurate results than the approximate method. A comparison of
a 45° incident wave using the approximate method and the more exact diagram
method is shown in Figure 2-58.
2.43 REFRACTION AND DIFFRACTION COMBINED
Usually the bottom seaward and shoreward of a breakwater is not
flat; therefore, refraction occurs in addition to diffraction. Although
a general unified theory of the two has not yet been developed, some in-
Sight into the problem is presented by Battjes (1968). An approximate
picture of wave changes may be obtained by: (a) constructing a refraction
2-98
(Z = 1/a) syasueToAemM omy FO yIptm deg
IoVemMyeorIg e OF werserq UOTOeIFFIG Ppozt[ersuey *7p-zZ oAnN3TYy
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8 Z2227777ZZZ/ LZ
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Figure 2-43.
Figure 2-44.
(Johnson, 1952)
Contours of Equal Diffraction Coefficient
Gap Width = 0.5 Wave Length (B/L = 0.5)
._ Diffracted Wove Height
Incident Wave Height
( Johnson,1952)
Contours of Equal Diffraction Coefficient
Gap Width = 1 Wave Length (B/L = 1)
2-100
Bzl.4iL K'=1145 K=10 K'0.8 K'=0.6
f DIRECTION OF
( Johnson, 1952)
INCIDENT WAVE
Figure 2-45. Contours of Equal Diffraction Coefficient
Gap Width = 1.41 Wave Lengths (B/L = 1.41)
K's Diffracted Wave Height
| SEU ~ Incident Wave Height
INCIDENT WAVE
( Johnson, 1952)
Figure 2-46. Contours of Equal Diffraction Coefficient
Gap Width = 1.64 Wave Lengths (B/L = 1.64)
2-01
B=1.78L K=LI95 KEL, =0.8 K=0.6
a DIRECTION OF ( Johnson, 1952)
INCIDENT WAVE
Figure 2-47. Contours of Equal Diffraction Coefficient
Gap Width = 1.78 Wave Lengths (B/L = 1.78)
a
e
:
a .. _Diffracted Wave Height
x = : >
B Taecrion» GE Incident Wave Height
f INCIDENT WAVE
K=L203 K=1.0 K=0.8 K=0.6 WAVE. CRESTS
B=2L 2 a= = 45
Sh eerie sSESREEES
ges StH iat
fo) 8 y Wh 12 a
(Johnson, 1952)
Figure 2-48. Contours of Equal Diffraction Coefficient
Gap Width = 2 Wave Lengths (B/L = 2)
202
DIRECTION OF
—1— pi WAVE (Johnson, 1952)
Figure 2-49. Contours of Equal Diffraction Coefficient
Gap Width = 2.50 Wave Lengths (B/L = 2.50)
\ DIRECTION OF 1_ Diffracted Wove Height
7 INCIDENT WAVE Incident Wave Height
oe K'=1.247 K'=1.2 K'= 1.0 K':0.8
(Johnson, 1952)
Figure 2-50. Contours of Equal Diffraction Coefficient
Gap Width = 2.95 Wave Lengths (B/L = 2.95)
rdeal OX)
1 K=1.0 K’=1.268 E F
| DIRECTION OF
INCIDENT WAVE (Johnson, 1952)
} Figure 2-51. Contours of Equal Diffraction Coefficient
Gap Width = 3.82 Wave Lengths (B/L = 3.82)
BREAKWATER
‘ ._ _Diffracted Wave Height
K = ~Tneident Wave Height
DIRECTION OF
INCIDENT WAVE
J
GAP 0 K'=1L.0 K'z1.0 K'sL.282 K'e1.2
( Johnson, 1952)
Figure 2-52. Contours of Equal Diffraction Coefficient
Gap Width = 5 Wave Lengths (B/L = 5)
2-104
Template Overlays
@CAAAAAAAAAAANAAANANANAN
Breakwater
DWIZZZLLLLLLLLLLLLLLLLL LL 2
Breakwater
Wave Crests
Figure 2-53. Diffraction for a Breakwater Gap of Width > 5L (B/L > 5)
/~e
VILLLLLLLLLLLLD PZLLLLLLLLLLLL,
Breakwater Breakwater
Imaginary Equivalent Gap
Wave Crests
( Johnson, 1952)
Figure 2-54. Wave Incidence Oblique to Breakwater Gap
2-105
et
(Johnson , 1952)
Diffraction for a Breakwater Gap of One Wave
Length Width (¢ = 0 and 15°)
Figure 2-55.
2-106
TN amr
EEE)
COO odes a
EAB:
gssttzeers
ae ee }
20
(Johnson ,1952)
YA
Diffraction for a Breakwater Gap of One
Figure 2-56.
30 and 45°)
Wave Length Width ($9
2107
20
dieataetbee!
Diffraction ne a Breakwater Gap of One
be lee EE ebiere eet
Vee SS oe ented
He ee eee
bal SEN a
Ey te
Pjf ob
Ted A TTT
(Ua a a Sh A/V
ree PID) 67, a NS ages
60 and 75°)
Wavelength Width (9
Figure 2-57.
2-108
Gap Width of 2L,and * Bove : j 2 Breakwater
@=45° (solution of . He Aes
Carr & Stelzriede)
Solid lines
0 2
Figure 2-58.
uek iS
of 1.41 L and g=90°
(solution of Blue & Johnson)
Dotted lines
4
Scale of %/L and y/L y | Imaginary gap with a width
( Johnson , 1952)
Diffraction Diagram for a Gap of Two Wave
Lengths and a 45° Approach Compared with
That for a Gap Width /2 Wavelengths with
a 90° Approach
2-109
diagram shoreward to the breakwater; (b) at this point, constructing a
diffraction diagram carrying successive crests three or four wavelengths
shoreward, if possible; and (c) with the wave crest and wave direction
indicated by the last shoreward wave crest determined from the diffraction
diagram, constructing a new refraction diagram to the breaker line. The
work of Mobarek (1962) on the effect of bottom slope on wave diffraction
indicates that the method presented here is suitable for medium-period
waves. For long-period waves the effect of shoaling (Section 2.32) should
be considered. For the condition when the bottom contours are parallel to
the wave crests, the sloping bottom probably has little effect on diffrac-
tion. A typical refraction-diffraction diagram and the method for deter-
mining combined refraction-diffraction coefficients are shown in Figure
2-59. When a wave crest is not of uniform height, as when a wave is under-
going refraction, a lateral flow of energy - wave diffraction - will occur
along the wave crest. Therefore diffraction can occur without the wave
moving past a structure although the diffraction effects are visually more
dramatic at the structure.
2.5 WAVE REFLECTION
2.51 GENERAL
Water waves may be either partially or totally reflected from both
natural and manmade barriers. (See Figure 2-60.) Wave reflection may
often be as important a consideration as refraction and diffraction in the
design of coastal structures, particularly for structures associated with
development of harbors. Reflection of waves implies a reflection of wave
energy as opposed to energy dissipation. Consequently, multiple reflections
and absence of sufficient energy dissipation within a harbor complex can
result in a buildup of energy which appears as wave agitation and surging
in the harbor. These surface fluctuations may cause excessive motion of
moored ships and other floating facilities, and result in the development
of great strains on mooring lines. Therefore seawalls, bulkheads and
revetments inside of harbors should dissipate rather than reflect incident
wave energy whenever possible. Natural beaches in a harbor are excellent
wave energy dissipaters and proposed harbor modifications which would
decrease beach areas should be carefully evaluated prior to construction.
Hydraulic model studies are often necessary to evaluate such proposed
changes. The importance of wave reflection and its effect on harbor
development are discussed by Bretschneider (1966), Lee (1964), and
LeMehaute (1965); harbor resonance is discussed by Raichlen (1965).
A measure of how much a barrier reflects waves is given by the ratio
of the reflected wave height H,, to the incident wave height H; which
is termed the reflection coefficient yx; hence yx = Hyp/Hj. The magnitude
of x varies from 1.0 for total reflection to 0 for no reflection; how-
ever, a small value of y does not necessarily imply that wave energy is
dissipated by a structure since energy may be transmitted through such
structures as permeable, rubble-mound breakwaters. A transmission co-
efficient may be defined as the ratio of transmitted wave height Hz, to
incident wave height H,;. In general, both the reflection coefficient
73K)
Wave Wr | b, Lines of Equal Diffraction Coefficient (K')
Orthogonals
| a ey |
OR NO ea ya
clea fo ee
at a
ca eee eee
Wave Crests
Over-all refraction- diffraction coefficient is given
by (Ka) (K!) /B, 7,
Where Keg=Refraction coefficient to breakwater.
K! =Diffraction coefficient ot point on
diffracted wave crest from which
orthogonalis drawn.
b,= Orthogonal spacing at diffracted wave
crest.
b,= Orthogonal spacing nearer shore.
Figure 2-59. Single Breakwater - Refraction - Diffraction Combined
enh
ONTARIO.
, f
December 1952
Wave Reflection at Hamlin Beach, New York
and the transmission coefficient will depend upon the geometry and compo-
sition of a structure and the incident wave characteristics such as wave
steepness and relative depth d/L, at the structure site.
2.52 REFLECTION FROM IMPERMEABLE, VERTICAL WALLS (LINEAR THEORY)
Impermeable vertical walls will reflect almost all incident wave
energy unless they are fronted by rubble toe protection or are extremely
rough. The reflection coefficient x is therefore equal to approximately
1.0 and the height of a reflected wave will be equal to the height of the
incident wave. Although some experiments with smooth, vertical, impermeable
walls appear to show a significant decrease of yx with increasing wave
steepness, Domzig (1955), Goda and Abe (1968) have shown that this paradox
probably results from the experimental technique, based on linear wave
theory, used to determine yx. The use of a higher order theory to describe
the water motion in front of the wall gives a reflection coefficient of
1.0 and satisfies the conservation of energy principle.
Wave motion in front of a perfectly reflecting vertical wall subjected
to monochromatic waves moving in a direction perpendicular to the barrier
can be determined by superposing two waves with identical wave numbers,
periods and amplitudes but traveling in opposite directions. The water
surface of the incident wave is given to a first order (linear) approxi-
mation by Equation 2-10,
Hi; 2nx 2nt (2-10)
. = — cos|— — — =
i as ii T
and the reflected wave by,
Consequently, the water surface is given by the sum of nz and ny, OF.
SincessH eH,
which reduces to
2
n = H, cos a Cosy = . (2-79)
Equation 2-79 represents the water. surface for a standing wave or clapotts
which is periodic in time and in x having a maximum height of 2H; when
both cos(27x/L) and cos(2mt/T) equal 1. The water surface profile as a
function of 27x/L for several values of 2mt/T are shown in Figure 2-61.
There are some points (nodes) on the profile where the water surface
2s
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2-114
remains at the SWL for all values of t and other points (antinodes) where
the water particle excursion at the surface, is 2H; or twice the incident
wave height. The equations describing the water particle motion show that
the velocity is always horizontal under the nodes and always vertical under
the antinodes. At intermediate points, the water particles move along
diagonal lines as shown in Figure 2-61. Since water motion at the anti-
nodes is purely vertical, the presence of a vertical wall at any antinode
will not change the flow pattern described since there is no flow across
the vertical barrier and equivalently, there is no flow across a vertical
line passing through an antinode. (For the linear theory discussion here,
the water contained between any two antinodes will remain between those
two antinodes.) Consequently, the flow described here is valid for a
barrier at 27x/L = 0 (x = 0) since there is an antinode at that location.
2.53 REFLECTIONS IN AN ENCLOSED BASIN
Some insight can be obtained about the phenomenon of the resonant
behavior of harbors and other enclosed bodies of water by examining the
standing wave system previously described. The possible resonant oscillat-
tions between two vertical walls can be described by locating the two
barriers so that they are both at antinodes; for example, barriers at
x = 0 and m or x = O and 27, etc. represent possible modes of oscillation.
If the barriers are taken at x = 0 and xX = 7, there is one-half of a wave
in the basin or, if %g is the basin length, 2g = L/2. Since the wave-
length is given by Equation 2-4
T? 2nd
a Seals 2),
1,
the period of this fundamental mode of oscillation is,
Ant, *
g tanh (rd/f,) ao
The next possible resonant mode occurs when there is one complete wave in
the basin (barriers at x = 0 and x = 27) and the next mode when there are
3/2 waves in the basin (barriers at x = 0 and x = 3n/2, etc. In general,
£3 = jL/2, where j = 1, 2, .... . In reality, the length of a natural or
manmade basin 2%, is fixed and the wavelength of the resonant wave con-
tained in the basin will be the variable; hence,
ba fH 420004 (2-81)
may be thought of as defining the wavelengths capable of causing resonance
in a basin of length %g. The general form of Equation 2-80 is found by
2-115
substituting Equation 2-81 into the expression for the wavelength; there-
fore,
Anl, %s
T. = | ——————_ agi pitse Us Pas ONG & 2-82
j jg tanh (m jd/ep) J t )
For an enclosed harbor, of approximately rectangular planform with length,
2B, waves entering through a breakwater gap having a predominant period
close to one of those given by Equation 2-82 for small values of j, may
cause significant agitation unless some effective energy dissipation
mechanism is present. The addition of energy to the basin at the resonant
(or excitation) frequency (f£; = 1/T;) is said to excite the basin.
Equation 2-82 was developed by assuming the end boundaries to be
vertical; however, it is still approximately valid so long as the end
boundaries remain highly reflective to wave motion. Sloping boundaries,
such as beaches, while usually effective energy dissipaters, may be signifi-
cantly reflective if the incident waves are extremely long. The effect of
Sloping boundaries and their reflectivity to waves of differing character-
istics is given in Section 2.54, Wave Reflection from Beaches.
Long-period resonant oscillations in large lakes and other large
enclosed bodies of water are termed setches. The periods of seiches may
range from a few minutes up to several hours, depending upon the geometry
of the particular basin. In general, these basins are shallow with respect
to their length; hence, tanh (mjd/%g) in Equation 2-82 becomes approximately
equal to mjd/%g and
Ts See j = 1) 2)": "(small values)” : (2-83)
Equation 2-83 is termed Merian's equation. In natural basins, complex
geometry and variable depth will make the direct application of Equation
2-83 difficult; however, it may serve as a useful first approximation for
enclosed basins. For basins open at one end, different modes of oscilla-
tion exist since resonance will occur when a node is at the open end of
the basin and the fundamental oscillation occurs when there is one quarter
of a wave in the basin; hence, 2p/ = L/4 for the fundamental mode and
T = 42>/V/gd. In general Lp! = (2; - 1)L/4, and
a es (2-84)
Ta —, j= 1,2,°°:: (small values) . =
i =) Gur Spee
Note that higher modes occur when there are 3, 5, ...., 25 = In GCs 5
quarters of a wave within the basin.
2-16
Fe eit: GR ee ER LK ek Ee EXAMPLE GPROBEEMS 2 tie at Fe Ok ee eee
GIVEN: Lake Erie has a mean depth of d = 61 feet and its length %g is
220 miles or 116,160 feet.
FIND: The fundamental period of oscillation Ty, ae aha aie
SOLUTION: From Equation 2-83 for an enclosed basin,
T =—-—,
2 eed:
ta 26116,160) 1
iS 1 [32.2(61)]”
T. = 52,420 sec. or 14.56 hrs.
1
Considering the variability of the actual lake cross-section, this
result is surprisingly close to the actual observed period of 14.38
hours (Platzman and Rao, 1963). Such close agreement may not always
result.
UR Cae. RG) a) oes Lateran Wo Se er ey Se Se ae ae) re Re ee ee) Oe) ee
Note: Additional discussion of seiching is presented in Section 3.84.
2.54 WAVE REFLECTION FROM BEACHES
The amount of wave energy reflected from a beach depends upon the
roughness, permeability and slope of the beach in addition to the steep-
ness and angle of approach of incident waves. Miche (1951) assumed that
the reflection coefficient for a beach yx, could be described as the
product of two factors by the expression,
», de), Soe, a (2-85)
where X) depends on the roughness and permeability of the beach and is
independent of the slope, while x, depends on the beach slope and the
wave steepness. .
Based on measurements made by Schoemaker and Thijsse (1949), Miche
found that X, = 0.8 for smooth impervious beaches. A value of oa 0.3
Jake
to 0.6 has been recommended for rough slopes and step-faced structures.
The second factor Xo is given by
H H
Baie (*.) > le (2-86a)
Ho/L, L, L, max
1 ze < te 2
ea Sie (2-86b)
(2) o/max
where (Ho/La)max is constant for a particular beach slope and is given by
(=) Pe =| sin?B (2-87)
L, max Us uF
in which 8g is the angle the beach makes with the horizontal (tan B =
beach slope), and H,/L, is the incident wave steepness in deep water.
(Ho/Lo) may can be considered a cut-off steepness; waves steeper than
(Ho/Lo) max Will be only partially reflected while waves with a steepness
less than (Ho/Lo)mgy Will be almost totally reflected. Equation 2-87 is
given in graphical form in Figure 2-62, and Equation 2-86 is shown in
Figure 2-63. These equations and figures are valid for impervious beaches
with waves approaching at a right angle to the beach.
ORE ey es ae sel oe? Re es ie es i, EXAMPLE PROBLEM KK eR OR Ok ke RR Ke Ke ee
GIVEN: A wave with a period of T = 10 second, and a deepwater height of
Ho = 5 feet impinges on an impermeable revetment with a slope of tan 8
= 0.20.
FIND:
(a) Determine the reflection coefficient x.
(b) What will be the steepest incident wave almost totally reflected
from the given revetment?
SOLUTION. Calculate:
tan B = 0.20;
eT? 32.2(100)
boo 5 tert.
& 2m 2m
H,
— = —— = 0.00975 = 0.01
IL Sil
2-118
| (after Miche, 1951)
cotangent B = Tonyenre!
Figure 2-62. (Ho/Lo)max Versus Beach Slope
| (after Miche,1951)
tangent B
cotangent B =
Figure 2-63. X2 Versus Beach Slope for Various Values of Ho/Lo
2-119
Entering Figure 2-63 with cot 8 = 5, and using the curve for Ho/lg =
0.01, a value of x, = 0.41 is found. Assuming that since the beach
is impermeable, Xe 0.8 and
KS X7 xe) — 0:8(0-41)7— 10°35,
The steepest incident wave which will be nearly perfectly reflected from
the given revetment is, from Figure 2-62,
H,
(=) = 0.005.
Lo}max
It is interesting to note the effectiveness of flat beaches in dissipat-
ing wave energy by considering the above wave on a beach having a slope
of 0.02 (1:50). From Equation 2-87 (noting that 8 ~ sin 8 * tan B = 0.02),
H,
— = 0.000014 .
L max
X = xX, X, = 0.8(0.0014) = 0.0011,
Hence
or the height of the reflected wave is about 0.1 percent of the incident
wave height.
As indicated by the dependence of reflection coefficient on incident
wave steepness, a beach will selectively dissipate wave energy, dissipat-
ing the energy of relatively short steep waves and reflecting the energy
of the longer, flatter waves.
Ce, Oe, Se RC te Ce Ce, SU ES AE a kM, EM the a Me ie Ue OR a Re a a Ne et OLS
2.6 BREAKING WAVES
2.61 DEEP WATER
The maximum height of a wave travelling in deep water is limited by
a maximum wave steepness for which the wave form can remain stable. Waves
reaching the limiting steepness will begin to break and in so doing, will
dissipate a part of their energy. Based on theoretical considerations,
Michell (1893) found the limiting steepness to be given by,
(2-88)
which occurs when the crest angle as shown in Figure 2-64 is 120°.) this
limiting steepness occurs when the water particle velocity at the wave
crest just equals the wave celerity; a further increase in steepness would
2-120
result in particle velocities at the wave crest greater than the wave
celerity and, consequently, instability.
L
. eer Sree ee ee ee =e
& La ig
Crest angle
Limiting steepness # = 0.142
Figure 2-64. Wave of Limiting Steepness in Deep Water
2.62 © SHOALING WATER
When a wave moves into shoaling water, the limiting steepness which
it can attain decreases, being a function of both the relative depth d/L,
and the beach slope m, perpendicular to the direction of wave advance.
A wave of given deepwater characteristics will move toward a shore until
the water becomes shallow enough to initiate breaking, this depth is
usually denoted as dz, and termed the breaking depth. Munk (1949) derived
several relationships from a modified solitary wave theory relating the
breaker height Hp, the breaking depth d,, the unrefracted deepwater
wave height Hj‘, and the deepwater wavelength L,. His expressions are
given by
a = aaa e ” (2-89)
He) 33 (Es)
and
d,
— = 1,28. (2-90)
Hy
The ratio Hp/HG is frequently termed the breaker height index. Subse-
quent observations and investigations by Iversen (1952, 1953), Galvin
(1969), and Goda (1970) among others, have established that Hp/H, and
dp/Hp, depend on beach slope and on incident wave steepness. Figure 2-65
shows Goda's empirically derived relationships between Hp/H, and Hey Ge
for several beach slopes. Curves shown on the figure are fitted to widely
scattered data; however they illustrate a dependence of H D/HZ on the
beach slope. Empirical relationships between dp/Hp and Pup et? for
various beach slopes are presented in Figure 2-66. It is recommended
that Figures 2-65 and 2-66 be used, rather than Equations 2-89 and 2-90,
for making estimates of the depth at breaking or the maximum breaker
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alee I CI Co Co (EEE o Eee ele PH
HEH aa Se ect BeBe ne ae be le a
++ nae fal gies ia ba is a a
s'0 b'0 £0 20 0
or
2-123
height in a given depth since the figures take into consideration the
observed dependence of dz/Hp and Hp/HZ on beach slope. The curves
in Figure 2-66 are given by
d
Big ie Sate (2-91)
H, b—(aHy/eT*)
where a and b are functions of the beach slope m, and may be approxi-
mated by
a = 1.36g(1 —e-19™) (2-92)
rab a ie Sy 2 (2-93)
(+ e7 19.5m)
Breaking waves have been classified as spilling, plunging or surging
depending on the way in which they break (Patrick and Wiegel, 1955), and
(Wiegel, 1964). Spilling breakers break gradually and are characterized
by whtte water at the crest. (See Figure 2-67.) Plunging breakers curl
over at the crest with a plunging forward of the mass of water at the
crest. (See Figure 2-68.) Surging breakers build up as if to form a
plunging breaker but the base of the wave surges up the beach before the
crest can plunge forward. (See Figure 2-69.) Further subdivision of
breaker types has also been proposed. The term collapsing breaker is
sometimes used (Galvin, 1968) to describe breakers in the transition from
plunging to surging. (See Figure 2-70.) In actuality, the transition
from one breaker type to another is gradual without distinct dividing
lines; however, Patrick and Wiegel (1955) presented ranges of H5/Lo for
several beach slopes for which each type of breaker can be expected to
occur. This information is also presented in Figure 2-65 in the form of
three regions on the Hy/Hg vs. HD/Lo plane. An example illustrating the
estimation of breaker parameters follows.
Hk Re RK ee OR eo Ue, EXAMPLE PROBLEM Se ie te ee Cee eee ie) de ek Bie eee
GIVEN: A beach having a 1:20 slope; a wave with deepwater height of
H, = 5 feet and a period of T = 10 seconds. Assume that a refraction
analysis gives a refraction coefficient, Kp = (bo/b) 1/2 = 1.05 at the
point where breaking is expected to occur.
FIND: The breaker height Hp and the depth dp at which breaking occurs.
SOLUTION: The unrefracted deepwater height Ho can be found from
Ee bs \2
= aka = (**) (See Section 2.32),
oO
2-124
Figure 2-67.
Spilling Breaking
ea " =
: ae
og =
wy: SR cet. - x
a. ee
7 Pi OP ee nme OTe ae
2 ee: ad
Me oe i. .
Figure 2-68. Plunging
2N eo
Breaking Wave
Fn
Rpkle -o, >
os Sd yor yee
he
— Xs
- Men n
Figure 2-69. Surging Breaking Wave
(Ze eS)
hence,
Hi = 1.05(5) = 5.25 feet ,
and since, Lo = 5.12T* (linear wave theory),
Hy 5.25
ee eee OOO
L Bel 21110)?
re)
From Figure 2-65, entering with Hj/Lo = 0.010 and intersecting the curve
for a slope of 1:20 (m = 0.05) results in Hp/H5 lose lbereLore
Hy
= aed /
Hy ae H’ H,
[e)
H, = 1.65(5.25) = 8.66 feet .
To determine the depth at breaking calculate:
Hi, 8.66
St aa aa Ue zeoN
eT 32.2 (10)?
and enter Figure 2-66 for m = 0.050.
d,
== = O40),
Hy,
Thus dp = 0.90 (8.66) = 7.80 feet, and therefore the wave will break
when it is approximately 7.80/(0.05) = 156 feet from the shoreline,
assuming a constant nearshore slope. The initial value selected for
the refraction coefficient should now be checked to determine if it is
correct for the actual breaker location as found in the. solution. If
necessary, a corrected value for the refraction coefficient should be
used and the breaker height recomputed. The example wave will result
in a plunging breaker. (See Figure 2-65.)
C3 CN ea ee ee ee ee ee, a a Se SM A, Re ee ee Re Te ee de ee cS
ale
f
itl Ls) Sig Srestag:
a)
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WORTHINGTON, H.W. and HERBICH, J.B., "A Computer Program to Estimate the
Combined Effects of Refraction and Diffraction of Water Waves,"
Sea Grant Publication, No. 219, Texas A&M University, Aug. 1970.
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CHAPTER 3
WAVE AND
WATER LEVEL
PREDICTIONS
OCEAN CITY, NEW JERSEY — 9 March 1962
CHAPTER 3
WAVE AND WATER-LEVEL PREDICTIONS
3.1 INTRODUCTION
Chapter 2, treated phenomena associated with surface waves as though
each phenomenon could be considered separately without regard to other
phenomena. Surface waves were discussed from the standpoint of motions
and transformations without regard to wave generation. Furthermore, the
water level, stillwater level (SWL), on which the waves propagated was
assumed known.
In this chapter, wave observations are presented to show characteris-
tics of surface waves in nature. The characteristics of real waves are
much less regular than those implied by theory. Also presented are pro-
cedures for representing the complexity of the real sea by a small number
of parameters. Deviations between the actual waves and the parameter
values are discussed.
Theory for wave generation is reviewed to show progress in explaining
and predicting the actual complexity of the sea. Wave prediction is
called hindeastitng when based on past meteorological conditions, and fore-
casting when based on predicted conditions. The same procedures are used
for hindcasting and forecasting; the only difference is the source of
meteorological data. The most advanced prediction techniques currently
available can be used only in a few laboratories, because of the need for
electronic computers, the sophistication of the models, and the need for
correct weather data. However, simplified wave prediction techniques,
suitable for use by field offices or a design group are presented.
While simplified prediction systems will not solve all problems, they
can be used to indicate probable wave conditions for most design studies.
Simplified wave prediction can also be used to obtain statistical wave
data over several years.
Review of prediction theories is presented to give the reader more
perspective for the simplified prediction methods that are presently
available. This will justify confidence in some applications of the
simplified procedures, will aid in recognizing unexpected difficulties
when they occur, and will indicate some conditions in which they are not
adequate.
The graphs in Sections 3.5, Simplified Wave Prediction Models, and
3.6, Wave Forecasting for Shallow Water Areas, may be read with the pre-
cision justified by the underlying theory. The equations were derived
originally from graphs, and do not provide any physical understanding.
Calculations with the graphs should be carried out to tenths or hundredths
where this is feasible, and then rounded off in the final result.
Predictions are compared with available observations wherever possible
to indicate their accuracy. Calibration of techniques applied to a spe-
cific geographic area by comparison with available observations is always
desirable.
The problem of obtaining wind information for wave hindcasting is
discussed, and specific instructions for estimating wind parameters are
given.
Water levels continuously change. Changes due to astronomical tides
are predictable, and are well documented for many areas. Fluctuations due
to meteorological conditions are not as predictable, and are less well
documented.
Many factors govern water levels at a shore during a storm. The
principle factor is the effect of wind blowing over water. Some of the
largest increases in water level are due to severe storms, such as hurri-
canes, which can cause storm surges higher than 25 feet at some locations
on the open coast and even higher water levels in bays and estuaries.
Estimating water levels caused by meteorological conditions is complex,
even for the simplest cases; and unfortunately, the best approaches avail-
able for predicting these water levels are elaborate computational tech-
niques which require the use of large digital computers.
3.2 CHARACTERISTICS OF OCEAN WAVES
The earlier discussion of waves was concerned with idealized, mono-
chromatic waves. The pattern of waves on any body of water exposed to
atmospheric winds generally contains waves of many periods. Typical
records from a recording gage during periods of steep waves (Fig. 3-1)
indicate that heights and periods of real waves are not as constant as is
assumed in theory. Wavelengths and directions of propagation are also
variable. (See Figure 3-2.) The prototype is so complex that some
idealization is required.
3.21 SIGNIFICANT WAVE HEIGHT AND PERIOD
An early idealized description of ocean waves postulated a stgniftcant
hetght and stgnitftcant period, that would represent the characteristics of
the real sea in the form of monochromatic waves.
The representation of a wave field by significant height and period
has the advantage of retaining much of the insight gained from the theo-
retical studies. Its value has been demonstrated in the solution of
engineering problems. For some problems this representation appears
adequate; for others it is useful, but not entirely satisfactory.
To apply the significant wave concept it is necessary to define the
height and period parameters from wave observations. Munk (1944) defined
stgntfieant wave height, as the average hetght of the one-third highest
waves, and stated that it was about equal to the average height of the
waves as estimated by an experienced observer. This definition, while
useful, has some drawbacks in wave-record analysis. It is not always
clear which irregularities in the wave record should be counted to deter-
mine the total number of waves on which to compute the average height of
the one-third highest. The significant wave height is written as H1/3 or
simply Hg.
*spuodses g potazoed
*yooF G°S WYSTOY JUROTFIUBIS ‘eTUTOFITeD ‘Yyoeeg uojuTIUNY (q) spuodses yp potaszed ‘jzaaz S's
Y4STOY PUBOTFIUSTIS ‘pue[s] [eItog ‘TouuN], osptag Avg oyeedesoy) (e) ‘spzoday anem otdueg ‘T[-¢ eins ty
ror a PEEPS
Aa Pea eee er ea ee See
Te dt PN ATT ay VTL TNT DA A ae
ATIC PEN EV TW US VY TY PY
aa ee Ne a eee ee | 2
AEF AS SeG aR SRR eS ae eeeee be cose
anes aa PRS) eee
RR TAN CAT aac
AH MSA EMI BI I i kM
FENCES NPE RE VINE ee PY Lt
BE e allele a sale (ilepel sib lea| delislebaletel |) | els. | Talal
| SERB ES See SPREE RAS Resa a eR
A ESSER ee ERS eee
qutod 3utyeorq sy} [TUN jsOUTe SeABM PUTM [BIOT hq
peainosqo [Toms BuoT smoys oJoyd WYySTI oY] “SaABM pozSeLI-I1OYS FO
uzojjed Ie[nZer1iT ue BuTw1OF ATSNoouez[NWTS sLoYys 9Yyy Sutyoevordde
SUT@I} OAEM OM SMOYS OJOYd 4FOT SUL
“uOTROY [Te ISeOD) e UT SoAeM
*7-¢ oinsTy
The significant wave period obtained by visual observations of waves
is likely to be the average period of 10 to 15 successive prominent waves.
When determined from gage records, the significant period is apt to be the
average period of the subjectively estimated most prominent waves, or the
average period of all waves whose troughs are below and whose crests are
above the mean water level, (zero up crossing method).
3.22 WAVE HEIGHT VARIABILITY
When the heights of individual waves on a wave record are ranked from
the highest to lowest, the frequency of occurrence of waves above any given
value is given to a close approximation by the cumulative form of the
Rayleigh distribution. This fact can be used to estimate the average
height of the one-third highest waves from measurements of a few of the
highest waves, or to estimate the height of a wave of any arbitrary
frequency from a knowledge of the significant wave height. According to
the Rayleigh distribution function, the probability that the _Wave height
H is more than some arbitrary value of H referred to as H is given
by
{
P(H > A) =e Wms (3-1)
where Hymg is a parameter of the distribution, and P(H > fH) is the number
n of waves larger than H divided by the total number N_ of waves in
the record. Thus P has the form n/N. The value Hymg is called the
root-mean-square height and is defined by
1
N 5
H a = » H. . 3-2
foe hime yPartt (3-2)
It was shown in Section 2.238, Wave Energy and Power, that the total energy
per unit surface area is given by
oa si? (3-3)
where H; is the height of successive individual waves, and (E) 4 is the
average energy per unit surface area of all waves considered. Thus Hymes
is a measure of average wave energy. Calculation of Hymg by Equation 3-2
is somewhat less subjective than direct evaluation of the Hg because more
emphasis is placed on the larger, better defined waves. The calculation
can be made more objective by substitution of n/N for P(H>4#H) in
Equation 3-1 and taking natural logarithms of both sides to obtain
2
Ln(n) = Ln(N) — (Ons . (3-4)
3-5
By making the substitutions
y(n) = Ln(n), a = Ln(N), b = — H?., x(n) = Hn) .
TMs?’
Equation 3-4 may be written as
y(n) = a + bx(n). (3-5)
The constants a and b can be found graphically or by fitting a least-
square regression line to the observations. The parameters N and Hyms
may be computed from a and b. The value of N found in this way, is
the value that provides the best fit between the observed distribution of
identified waves and the Rayleigh distribution function. It is generally
a little larger than the number of waves actually identified in the record.
This seems reasonable because some very small waves are generally neglected
in interpreting the record. When the observed wave heights are scaled by
Hyms, that is, made dimensionless by dividing each observed height by
Hymg» then data from all observations may be combined into a single plot.
Points from scaled 15-minute samples are superimposed on Figure 3-3 to
show the scatter to be expected from analyzing individual observations in
this manner.
Data from 72 scaled 15-minute samples representing 11,678 observed
waves have been combined in this manner to produce Figure 3-4. The theo-
retical height appears to be about 5 percent greater than the observed
height for a probability of 0.01 and 15 percent at a probability of 0.0001.
It is possible that the difference between the actual and theoretical
heights of highest waves is due to breaking of the very highest waves
before they reach the coastal wave gages.
Equation 3-1 can be established rigorously for restrictive conditions,
and empirically for a much wider range of conditions. If Equation 3-1 is
accepted as an exact law, the probability density function can be obtained
in the form
2
Sat 2 \Wkpons
f[(H — AH) < H < (A + AE) = (z,,| He Fee . (3-6)
The height of the wave with any given probability n/N of being exceeded
may be determined approximately from curve a _ in Figure 3-5 or from the
equation,
ay %
ee = - Ln (5) (3-7)
Ss
Cumulative Probability (P =n/N)
0.0001
0.0005
0.001
0.05 From fifteen 15-minute records
containing a total of 2,342 waves
0.1 (3,007 waves calculated )
0.5
|.OF
(See discussion below Equation 3-8)
0 ROBEZ4eGs Sie 2.0 2.2 2.4 2.6 2.8 3.0 Ore
1.42
Scaled Height (H/ Heel
Figure 3-3. Theoretical and Observed Wave-Height Distributions.
Observed distributions for 15 individual 15-minute
observations from several Atlantic coast wave gages
are superimposed on the Rayleigh distribution curve
0.0001
0.0005
Cumulative Probability (P=n/N)
0.00!
0.005
0.01
0)
From seventy-two 15-minute samples
containing a total of 11,678 waves
(15,364 calculated waves )
[Ose (4 alse TBS 20562208 24), wie 2.8 3.0 3.2
1.42 Scaled Height (H/Hrms)
Figure 3-4. Theoretical and Observed Wave-Height Distributions.
Observed waves from 72 individual 15-minute
observations from several Atlantic coast wave gages
are superimposed on the Rayleigh distribution curve
SUOTINGTIISTG JYSTOH-SACM TBOTJOLZOOY]L *S-¢ OANSTY
(§"7H/y ) 14618H pajD9s
Vv
9°2 v2 2'2 02 8" 9'|
Q9lv'| = Asoayt SH
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ayy Og ‘uooD4y ysaybiy
ay} JO BHDJaAD |OdIJaOaYY
Oy} S! SY7H/1) payooipul ay
“@AIND aU Aq
PaJDIIPU! JOU} UDY) 4a}0046 sI
SMAH /1y yOu} Ajiiqoqosd ayy sig (0)
v
G0 0
m4 0'|
SO
10°0
$00°0
100°0
(N/U=d) Ajlligqoqosg anlyojnuing
The average height of all waves with heights greater than H (H) can be
obtained from the equation.
2) = == (3-8)
or from curve b in Figure 3-5. By setting fl = 0, all waves are con-
sidered, and it is found that the average wave height is given by
H = 0.886 H (3-9)
rms?
and the significant wave height is given by
H. =/LAlen fw VOR: (3-10)
In the analysis system used by CERC from 1960 to 1970,: and whenever
digital recordings cannot be used, the average period of a few of the best
formed waves is selected as the significant wave period. An estimated
number of equivalent waves in the record is obtained by dividing the
duration of the record by this significant period. The highest waves
are then ranked in order with the highest wave ranked 1. The height of
the wave ranked nearest 0.135 times the total number of waves is taken as
the significant wave height. The derivation of this technique is based on
the assumption that the Rayleigh distribution law is exact. Harris (1970)
showed that this procedure agrees closely with values obtained by more
rigorous procedures which require the use of a computer. These procedures
are described in Section 3.23, Energy Spectra of Waves.
The following problem illustrates the use of the theoretical wave
height distribution curves given in Figure 3-5.
Ee ET OSE Ce Ce ICT OC FC EXAMPLE PROBLEM Me Re eR ORR eS ee ee
GIVEN: Based on an analysis of wave records at a coastal location, the
significant wave height H, was estimated to be 10 feet.
FIND:
(a) Hg (Average of the highest 10 percent of all waves)
(b) H, (Average of the highest 1 percent of all waves)
o=10
SOLUTION: H = H, = 10 feet
Using Equation 3-10
or
H = = — = 7.06ft .
(a) From Figure 3-5, curve b, it is seen that for P = 0.1 (10 percent)
Ai
~ 1.80; Hj = 1.80H,,,, = 1-80(7.06) = 12.7 feet, say 13 feet.
rms
(b) esamilarily for P= 0F0le (ll pexcent)
H
1
w~ 2.36; Bn 2236 ag = 25617206) —" 16e7.1t. say 17 ft.
rms
Note that:
H
125
Et = ser) Hig = 1.27 H, '
H, 10
and
H
1 16.7
=i SOF Ey = 1.67 H. :
H 10
Te ee Se REE TRS RR RR eee CK IE Pee ee ee Oe ROK E ee ee eee
Goodknight and Russell (1963) analyzed wave-gage observations taken
on an oil platform in the Gulf of Mexico during four hurricanes. They
found agreement adequate for engineering application between such important
parameters as Hg, Hi9, Hmax, Hyms, and H, although they did not find
consistently good agreement between measured wave-height distributions
and the entire Rayleigh distribution. Borgman (1972) substantiates this
conclusion from wave observations from other hurricanes. These findings
are consistent with Figures 3-3, and 3-4, based on wave records recently
obtained by CERC from shore-based gages. The CERC data include waves from
both extratropical storms and hurricanes.
3.23 ENERGY SPECTRA OF WAVES
The significant wave analysis, although simple in concept, is diffi-
cult to do objectively, and does not provide all of the information needed
in engineering design.
Soil
It appears from Figure 3-1 that the wave field might be better de-
scribed by a sum of sinusoidal terms. That is, the curves in Figure 3-1
might be better represented by expressions of the type
1) = oe a; cos(w;— $5) , (3-11)
where n(t) is the departure of the water surface from its average posi-
tion as a function of time, aj is the amplitude, wz is the frequency,
and $j is the phase of the jth wave at the time t = 0. The values
of w are arbitrary, and w may be assigned any value within suitable
limits. In analyzing records, however, it is convenient to set
we = 27j/D, where j is an integer and D is the duration of the obser-
vation. The as will be large only for those w+ that are prominent in
the record. When analyzed in this manner, the significant period may be
defined as D/j, where j is the value of j corresponding to the
largest az
It was shown by Kinsman (1965), that the average energy of the wave
train is proportional to the average value of [n(t)]*. This is identical
to o% where o is the standard deviation of the wave record. It can
also be shown that
N
eae (3-12)
Experimental results and calculations based on the Rayleigh distri-
bution function show that the significant wave height is approximately
equal to 4c. Thus, recalling that
H, & v2 ree z
and
H, 4g,
then
or 0.251/2 Ho, (3-13)
or
3 ra Dy NID. Whe (3-14)
The as may be regarded as approximations to the energy spectrum
function E(w) where
E(w) Aw = (3-15)
|e,
3-2
Thus
oT | E(w) dw . (3-16)
The spectrum E(w) permits one to assign specific portions of the
total wave energy to specific frequency intervals, to recognize that two
or more periods may be important in describing the wave field, and to give
an indication of their relative importance. This also permits a first
approximation to the calculation of velocities and accelerations from a
record of the wave height in a complex wave field. Several energy spectra,
computed from coastal zone wave records obtained by CERC, are shown in
Figure 3-6.
The international standard unit for frequency measure is the hertz,
defined as one cycle per second. The units cycles per second and radians
per second are also widely used. One hertz = 2n radians per second.
3.24 DIRECTIONAL SPECTRA OF WAVES
A more complete description of the wave field is required to recognize
that not all waves are traveling in the same direction. This may be
written as,
n(x,y,t) = Z aj cos [wt — 9, —k& cos 4; + y sin 6;)| ; (3-17)
where k = 2n/L, and 05 is the angle between the x axis and the
direction of wave propagation, and 7 is the phase of the jth wave
at t= 0. The energy density E(@,w) represents the concentration of
energy at a particular wave direction 6 , and frequency w , therefore
the total energy is obtained by integrating E(6,w) over all directions
and frequencies, thus
2m co
= i | E(6, w)dwdé . (3-18)
Ono
The concept of directional wave spectra is essential for advanced
wave-prediction models, but technology has not yet reached the point where
directional spectra can be routinely recorded or used in engineering design
studies. Therefore, directional wave spectra are not discussed extensively
here.
3-13
Total Wave Energy
Total Wave Energy
+04
+02
Period (Seconds)
25 10 5 25
0.4 0.0 0.2 0.4
Frequency (Hertz)
Savannah Coast Guard Light Tower
Period (Seconds)
10 5 25
Frequency (Hertz)
Nags Head, N.C.
Figure 3-6.
Typical Wave Spectra from the Atlantic Coast.
The ordinate scale is the fraction of total
wave energy in each frequency band of 0.011
Hertz, (one Hertz is one cycle/second). A
linear frequency scale is shown at the bottom
of each graph and a nonlinear period scale
at the top of each graph.
3-14
3.3 WAVE FIELD
3.31 DEVELOPMENT OF A WAVE FIELD
Various descriptions of the mechanism of wave generation by wind have
been given, and significant progress in explaining the mechanism was re-
ported by Miles (1957) and Phillips (1957). Integraged discussions of the
results of many of the more prominent descriptions of wave generation by
wind are given by Kinsman (1965), Phillips (1966), and Ewing (1971).
Laboratory studies, (Hidy and Plate, 1966) and (Shemdin and Hsu,
1966), carefully designed to match the assumptions made by Miles and by
Phillips show reasonably good agreement with the theoretical predictions.
Summaries of various filed studies, (Inoue, 1966, 1967) demonstrate that
theory provides a reasonable framework for the analysis of observations.
The Miles-Phillips theory as extended and corrected by experimental
data permits the formation of a differential equation governing the
growth of wave energy. This equation can be written in 2 wariety of ways-
(Inoue, 1966, 1967) and Barnett, 1968). This approach will not be discussed
in detail because it requires a large capacity computer and more meteoro—
logical data than is likely to be found except in 2 major forecast center.
A brief discussion of the physical concepts employed in the computer
wave forecast, however, is presented to show the shortcomings and merits
of simpler procedures that can be used in wave forecasting.
Growth and dissipation of wave energy are very sensitive to wave
frequency and wave direction relative to the wind direction. Thus it is
desirable to consider each narrow band of directions and frequencies
separately. A change in wave energy depends on the advection of energy
into and out of a region; transformation of the wind's kinetic energy
into the energy of water waves; dissipation of wave energy into turbulence
and by friction, viscosity and breaking; and transformation of wave energy
at one frequency into wave energy at other frequencies.
Wave energy is discussed in Section 2.258, Wave Energy and Power.
Although it is known that energy transfers from one band of wave frequen-
cies to another do take place, this process is secondary to the transfer
of energy from the atmosphere to the sea, and is not yet well enough under-
stood to justify its consideration in a2 practical wave prediction scheme.
Phillips (1957) showed that the turbulence associated with the flow
of wind near the water would create traveling pressure pulses. These
pulses generate waves traveling at a speed appropriate to the dimensions
of the pressure pulse. Wave growth by this process is most rapid when
the waves are short and when their speed is identical with the component
of the wind velocity in the direction of wave travel. The empirical data
analyzed by Inoue (1966, 1967) indicates that the effect of turbulent
pressure pulses is real, but is only about one-twentieth as large as the
original theory indicated.
3-15
Miles (1957) showed that the waves on the sea surface must be
matched by waves on the bottom surface of the atmosphere. The speed
of air and water must be equal at the water surface. Under most meteoro-
logical conditions, the air speed increases from near 0 to 60 - 90 percent
of the free air value within 66 feet (20 meters) of the water surface.
Within a shear zone of this type, energy is extracted from the mean flow
of the wind and transferred to the waves. The magnitude of this transfer
at any frequency is proportional to the wave energy already present at
that frequency. Growth is normally most rapid at high frequencies. The
energy transfer is also a complex function of the wind profile, the
turbulence of the air stream, and the vector difference between wind and
wave velocities.
The theories of Miles and Phillips predict that waves grow most
rapidly when the component of the wind speed in the direction of wave
propagation is equal to the speed of wave propagation.
The wave generation process discussed by Phillips is very sensitive
to the structure of the turbulence. This is affected significantly by
any existing waves, and the temperature gradient in the air near the
water surface. The turbulence structure in an offshore wind is also
affected by land surface roughness near the shore.
The wave generation process discussed by Miles is very sensitive to
the vertical profile of the wind. This is determined largely by turbulence
in the wind stream, the temperature profile in the air, and by the rough-
ness of the sea surface.
Shorter waves grow most rapidly. Those waves which propagate obliquely
to the wind are favored, for they are better matched to the component of
the wind velocity in the direction of wave propagation than those moving
parallel to the wind. Thus, the first wave pattern to appear for short
fetches and durations consists of two wave trains forming a rhombic
pattern with one diagonal along the direction of the mean wind.
There is a limit to the steepness to which a wave can grow without
breaking. Shorter waves reach their limiting growth rather quickly;
longer waves, which grow more slowly but can obtain greater heights,
then become more prominent. Thus, the apparent direction of propagation
of the two wave trains tends to coalesce with increasing fetch and duration.
The length of the region in which a rhombic pattern is apparent may extend
from a few meters to a few kilometers depending on the width of the basin,
the wind speed, and previously existing waves.
Wave growth is significantly affected by any preexisting waves. The
empirical data analyzed by Inoue (1966, 1967) indicated that the magnitude
of the effect of seas already present is about eight times the value given
in the original Miles (1957) theory. Neglecting this effect in early wave
prediction theories has led to large errors in computing the duration
required for a fully arisen sea. There are many situations in which the
largest waves and the waves growing most rapidly are not being propagated
in the wind direction.
3-16
3.32 VERIFICATION OF WAVE HINDCASTING
Inoue (1967) prepared hindcasts for Weather Station J (located near
53°N, 18°W), for the period 15-28 December, 1959, using a differential
equation embodying the Miles-Phillips theory to predict wave growth. A
comparison of significant wave heights from shipboard observations and by
hindcasting at two separate locations near the weather ships is shown in
Figure 3-7. The location of Ocean Weather Ship J, the mesh points used in
the numerical calculations, and four other locations discussed below are
are shown in Figure 3-8. The calculations required meteorological data
from 519 grid points over the Atlantic Ocean as shown in Figure 3-8. The
agreement between observed and computed values seems to justify a high
level of confidence in the basic prediction model. Observed meteorological
data were interpolated in time and space to provide the required data,
thus these predictions were hindcasts.
Bunting and Moskowitz (1970) and Bunting (1970) have compared fore-
cast wave heights with observations, using the same model with comparable
results.
By 1970, it was generally believed that the major remaining difficulty
in wind wave prediction was the determination of the surface wind field over
the ocean. (Pore and Richardson, 1967), and (Bunting, 1970). It is partly
because of the difficulty in obtaining a satisfactory specification of the
wind field over the sea that simpler wave prediction systems are still
being used operationally. (Pore and Richardson, 1969), (Shields and Burdwell,
1970), and (Francis, 1971.)
3.33 DECAY OF A WAVE FIELD
Wind-energy can be transferred directly to the waves only when the
component of the surface wind in the direction of wave travel exceeds the
speed of wave propagation. Winds may decrease in intensity, pass over
land, or change in direction to such an extent that wave generation ceases,
or the waves may propagate out of the generation area. When any of these
events occurs, the wave field begins to decay. Wave energy travels at a
speed which increases with the wave period. Thus the energy packet leaving
the generating area spreads out over a larger area with increasing time.
The apparent period at the energy front increases and the wave height
decreases. If the winds subside before the sea is fully arisen, the
longer waves may begin to decay while the shorter waves are still growing.
This possibility is recognized in advanced wave prediction techniques.
The hindcast spectra, computed by the Inoue (1967) model and published by
Guthrie (1971) show many examples of this for low swell, as do the aerial
photographs and spectra given by Harris (1971). (See Figures 3-2 and 3-6.)
This swell is frequently overlooked in visual observations and even in the
subjective analysis of pen and ink records from coastal wave gages.
Most coastal areas of the United States are so situated that most of
the waves reaching them are generated in water so deep that depth has no
effect on wave generation. In many of these areas, wave characteristics
3-17
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may be determined by first analyzing meteorological data to find deep-
water conditions. Then, by analyzing refraction (Section 2.32, General -
Refraction by Bathymetry.), the changes in wave characteristics as the
wave moves through shallow water to the shore may be found. In other
areas, in particular along the North Atlantic coast, where the bathymetry
is complex, refraction procedure results are frequently difficult to inter-
pret, and the conversion of deepwater wave data to shallow-water and near-
shore data becomes laborious and sometimes inaccurate.
Along the Gulf coast and in many inland lakes, generation of waves by
wind is appreciably affected by water depth. In addition, the nature and
extent of transitional and shallow-water regions complicate ordinary re-
fraction analysis by introducing a bottom-friction factor and associated
wave energy dissipation.
3.4 WIND INFORMATION NEEDED FOR WAVE PREDICTION
Wave prediction from first principles, as described above, requires
very detailed specification of the wind field near the water surface. This
is generally developed in two steps: (1) Estimation of the mean free air
wind speed and direction, (This step may be omitted for reservoirs and small
lakes if surface wind observations are available.), and (2) Estimation of
the mean surface wind speed and direction.
When the full wave generation process is considered, a large capacity
computer must be used for the calculations, and fairly complex procedures
may be used for determining the wind field. Engineers who require wave
hindcasts for only a few locations, and perhaps for only a few dates must
employ simpler techniques. A brief discussion of the processes involved
in determining the surface wind and techniques suitable for use in deter-
mining the characteristics of the wind field needed for the simplified wave
prediction model described in Section 3.5, Simplified Wave Prediction Models,
are given in this section. These procedures will be accurate (within 20
percent) about two-thirds of the time. The following discussion provides
guidance for recognizing cases in which the simplified procedures are not
appropriate. Errors resulting from disregarding the exceptional situations
tend to be random. Thus climatological summaries, based on hindcast data,
may be much more accurate than the individual values that go into them.
Wind reports from ships at sea are generally estimates based on the
appearance of the waves, the drifting of smoke, or the flapping of flags-
although some are anemometer measurements. Actually, even if all ships
were equipped with several aneometers, the wind field over the sea would
still not be known in sufficient detail or precision to permit full
exploitation of modern theories for wave generation.
Fortunately, estimates of the surface wind field that are usefully
accurate most of the time can be based on the isobaric pattern of synoptic
weather charts.
Horizontal pressure gradients arise in the atmosphere primarily
because of density differences, which in turn are generated primarily by
temperature differences. Wind results from nature's efforts to eliminate
the pressure gradients, but is modified by many other factors.
The pressure gradient is nearly always in approximate equilibrium with
the acceleration produced by the rotation of the earth. The geostrophic
wind is defined by assuming that exact equilibrium exists, and is given by
Ue (3-19)
where U, is the wind speed, pg the density of the air, f the coriolis
parameter, f = 2w sind, where w = 7.292 x 10° radians/second and 9
is the latitude, and dp/dn is the horizontal gradient of atmospheric
pressure. A graphic solution of this equation is given in Figure 3-11,
Section 3.41, Estimating the Wind Characteristics. The geostrophic wind
blows parallel to the isobars with low pressure to the left, when looking
in the direction toward which the wind is blowing, in the Northern
Hemisphere, and low pressure to the right in the Southern Hemisphere.
Geostrophic wind is usually the best simple estimate of the true wind in
the free atmosphere.
When the trajectories of air particles are curved, equilibrium wind
speed is called gradient wind. Gradient wind is stronger than geostrophic
wind for flow around a high pressure area,and weaker than geostrophic wind
for flow around low pressure. The magnitude of the difference between
geostrophic and gradient winds is determined by the curvature of the tra-
jectories. If the pressure pattern does not change with time and friction
is neglected, trajectories are parallel with the isobars. The isobar curva-
ture can be measured from a single weather map, but at least two maps must
be used to estimate trajectory curvature. There is a tendency by some
analysts to equate the isobars and trajectories at all times, and to
compute the gradient wind correction from the isobar curvature. When the
curvature is small, but the pressure is changing, this tendency may lead
to incorrect adjustments. Corrections to the geostrophic wind that can-
not be determined from a single weather map are usually neglected, even
though they may be more important than the isobaric curvature effect.
The equilibrium state is further disturbed near the surface of the
earth by friction. Friction causes the wind to cross the isobars toward
low pressure at a speed lower than the wind speed in the free air. Over
water, the average surface wind speed is generally about 60 to 75 percent
of the free air value, and wind crosses the isobars at an angle of 10 to
20 degrees. In individual situations, the magnitude of the ratio between
the surface wind speed and the computed free air speed may vary from 20 to
more than 100 percent, and the crossing angle may vary from 0° to more than
90°. The magnitude of these changes is determined by the vertical tempera-
ture profile and the turbulent viscosity in the atmosphere.
o-2il
3.41 ESTIMATING THE WIND CHARACTERISTICS
To predict wave properties from meteorological data by any of the
simplified techniques, it is necessary to:
(a) Estimate the mean surface wind speed and direction, as dis-
cussed in Section 3.4, Wind Information Needed for Wave Prediction;
(b) delineate a fetch over which the wind is reasonably constant
in speed and direction, and measure the fetch length, and
(c) estimate wind duration over the fetch.
These determinations may be made in many ways depending on the loca-
tion and the type of meterological data available. For restricted bodies
of water, such as lakes, the fetch length is often the distance from the
forecasting point to the opposite shore measured along the wind direction.
There is no decay distance, and it is often possible to use observational
data to determine wind speeds and durations.
When forecasting for oceans or other large bodies of water, the most
common form of meteorological data used is the synoptic surface weather
chart. (Synoptte means that the charts are drawn by analysis of many
individual items of meteorological data obtained simultaneously over a
wide area.) These charts depict lines of equal atmospheric pressure,
called isobars. Wind estimates at sea, based on an analysis of the sea-
level atmoshperic pressure are generally more reliable than wind observa-
tions because pressure, unlike wind, can be measured accurately on a
moving ship. Pressures are recorded in millibars, 1,000 dynes per square
centimeter. One thousand millibars (a bar) equals 29.53 inches of mercury
and is 98.7 percent of normal atmoshperic pressure.
A simplified surface chart for the Pacific Ocean is shown in Figure
3-9, which is drawn for 27 October 1950 at 0030Z (0030 Greenwich mean
time). Note the area labelled L in the right center of the chart, and
the area labelled H in the lower left corner of the chart. These are
low- and high-pressure areas; the pressures increase moving out from L
(isobars 972, 975, etc.) and decrease moving out from H (isobars 1026,
IOZS, Cees )c
Scattered about the chart are small arrow shafts with a varying num-
ber of feathers or barbs. The direction of a shaft shows the direction
of the wind; each one-half feather represents a unit of 5 knots (2.5
meters/second) in wind speed. Thus, in Figure 3-9 near the point 35°N.
latitude, 135°W. longitude, there are three such arrows, two with 3
feathers which indicate a wind force of 31 to 35 knots (15 to 17.5 meters/
second), and one with 3 feathers indicating a force of 26 to 30 knots
(13 to 15 meters/second).
On an actual chart, much more meteorological data than wind speed and
direction are shown for each station. This is accomplished by the use of
3-22
coded symbols, letters, and numbers placed at definite points in relation
to the station dot. A sample model report, showing the amount of informa-
tion possible to report on a chart, is shown in Figure 3-10. Not all of
the data shown on this plot are included in each report, and not all of
the data in the report are plotted on each map.
Figure 3-11 may be used to facilitate computation of the geostrophic
wind speed. A measure of the average pressure gradient over the area is
required. Most synoptic charts are drawn with either a 3- or 4-millibar
spacing. Sometimes when isobars are crowded, intermediate isobars are
omitted. Either of these standard spacings is adequate as a measure of
the geographical distance between isobars. Using Figure 3-11, the distance
between isobars on a chart is measured in degrees of latitude (an average
spacing over a fetch is ordinarily used), and the latitude position of the
fetch is determined. Using the spacing as ordinate and location as abscissa,
the plotted or interpolated slant line at the intersection of these two
values gives the geostrophic wind speed. For example, in Figure 3-9, a
chart with 3-millibar isobar spacing, the average isobar spacing (measured
normal to the isobars) over F,, located at 37°N. latitude, is 0.70° of
latitude. Using the scales on the bottom and right side of Figure 3-11,
a geostrophic wind of 67 knots is found.
Geostrophic wind speeds are generally higher than surface wind speeds.
The following instructions, U.S. Fleet Weather Facility Manual (1966), are
recommended for obtaining estimates of the surface wind speeds over the
open sea from the geostrophic wind speeds:
(a) For moderately curved to straight isobars - no correction is
applied.
(b) For great anticyclonic (clockwise movement about a high pressure
center in Northern Hemisphere and counter-clockwise in Southern Hemisphere)
curvature - add 10 percent to the geostrophic wind speed.
(c) For great cyclonic (counter-clockwise movement about a low
pressure center in Northern Hemisphere and clockwise in Southern Hemisphere)
curvature - subtract 10 percent from the geostrophic wind speed.
Frequently the curvature correction can be neglected since isobars
over a fetch are often relatively straight. The gradient wind can always
be computed if more refined computations are desired.
To correct for air mass stability, the sea-air temperature difference
must be computed. This can be done from ship reports in or near the fetch
area, aided by climatic charts of average monthly sea surface temperatures
when data are too scarce. The correction to be applied is given in Table
3-1. (U.S. Fleet Weather Facility Manual, 1966.)
Over oceans, the surface winds generally cross the isobars toward low
pressure at an angle of 10° to 20°.
3-24
Wind speed (23 to 27 knots )___ ¢¢ Cloud type (Altostratus)
Cloud type (Dense cirrus BP
in patches) C 7
Cu Borometric pressure in
True direction from which tenths of millibors reduced
wind is blowing —=——— =——=— dd to sea level. Initial 9 or 10
and the decimal point are
Current air temperature Ppp ——-— omitted (247= 1024.7 mb)
(CEA) eee es ee eee Tal
Pressure change in 3 hours
Total amount of cloud proceeding observation
(Completely covered )_______ N —vD pp ——— (28 = 2.8mb)
Visibility in miles and | — 247 Characteristic of barograph
fraction (3/4 mile)___-_»_»__ VV_~3/4 if trace (Falling or steady,
28 ~ then rising, or rising, then
K¥ +
Present weather. BS
ww
q——-—Fising more quickly)
Continuous light snow ____ _ 30 _---6 *4 Plus or minus sign showing
whether pressure is higher
Temperature of dew 2 45 ——-—or lower then 3 hours ago
Pomtn SOc Fee reed 2 Ta Td
Time precipitation began
Nh Ry —— or ended (4=3 to 4 hrs. ago)
Cloud type ( Fractocumulus)___Cy
Fraction of sky covered by W ——— Post weather ( Rain)
Height at base of cloud low or middle cloud
(2=300 to 599 feet)__.___ _ h (6=7 or 8 tenths) Amount of precipitation
RR ——— (45: 0.45 inches)
NOTE: The letter symbols for each weather
element are shown above.
Courtesy United States Weather Bureau
abridged from W.M.O. Code
Figure 3-10. Sample Plotted Report
where
= (O&C
Ap =
An
Sone
0.2625 radians/ hour
W = angular velocity of earth,
= latitude in degrees
3mb and 4mb
= jsobar spacing measured in
g
degrees latitude
= 1013.3 mb
gm/cm?
-3
= 1,247X10
= Coriolis parameter = 2Wsin@
SINNER \ ANS a =
SAMAR TA NCA
il : AN ,
f SOA
EAA
mo fe hos OS © oon rondo
ro) Or 16) Oc — —-— —- NN ANNAM
eet ea saaibag - Bune: Joqos| qu¢
pause Latitude
(after Bretschneider ,1952a)
Geostrophic Wind Scale
Figure 3-11.
3-26
Table 3-1. Correction for Sea-Air Temperature*
Sea Temperature minus
Air Temperature
0 or negative
0 to 10
10 to 20
20 or above
Ratio of Surface Wind Speed to
Geostrophic Wind Speed, U/Ug
*Pore and Richardson (1969) and Hasse and Wagner (1971)
report recent studies designed to refine the above table.
Neither were able to find enough high quality observa-
tions of large differences between air and sea tempera-
tures to rigorously establish any effect of the sea-air
temperature difference on the ratio of the surface wind
speed to the gradient wind speed over the ocean. Both
recommended the use of a constant value near 0.6 for
most routine work.
If there are several observed wind reports within the fetch region,
and these consistently deviate in the same manner from wind speeds arrived
at by the instructions given above, an average between the reported values
and those computed by the above instructions will usually be the best
estimate.
Over the Great Lakes and some coastal regions, large temperature
inversions (temperature increasing with elevation) may be observed.
Bellaire (1965) reports air temperatures more than 15°C (27°F) greater
than the water temperature in May 1964. When air temperature is much
greater than that of the sea, all turbulent motion in the lower atmosphere
is suppressed, and the wind near the surface has little relation to the
wind estimate determined from a synoptic weather chart. Near mountainous
coasts and particularly in fjords, the wind near the sea is often channeled
to flow parallel to the mountains. The local temperature contrast between
snow-covered mountains and relatively warm open water may have more control
over the wind near the water than the isobaric pressure pattern from
weather maps. In these cases, the wind determined from the pressure
analysis on a weather map has little if any value for wave prediction.
For these exceptional cases, there is no valid substitute for wind
observations.
3.42 DELINEATING A FETCH
The fetch has been defined subjectively as a region in which the wind
speed and direction are reasonably constant. Confidence in the computed
results begins to deteriorate slightly when wind direction variations
exceed 15°, and deteriorates significantly when direction deviations
exceeding 45° are accepted in the fetch area. The computed results are
sensitive to changes in wind speed as small as 1 knot (0.5 meter/second),
but it is not possible to estimate the wind speed over any sizable region
aed
with this precision. For practical wave predictions it is usually satis-
factory to regard the wind speed as reasonably constant if variations do
not exceed 5 knots (2.5 meters/second) from the mean. A coastline upwind
from the point of interest always limits a fetch. An upwind limit to the
fetch may also be provided by curvature or spreading of the isobars as
indicated in Figure 3-12 (Shields and Burdwell, 1970), or by a definite
shift in wind direction. Frequently the discontinuity at a weather front
will limit a fetch, although this is not always so,
1012 1016
1020
1020
Figure 3-12. Possible Fetch Limitations
Estimates of the duration of the wind are also needed for wave predic-
tion. Computed results, especially for short durations and high wind speeds
may be sensitive to differences of only a few minutes in the duration. Com-
plete synoptic weather charts are prepared only at 6-hour intervals. Thus
interpolation between charts to determine the duration may be necessary.
Linear interpolation is adequate for most uses, and, when not obviously
incorrect, is usually the best procedure.
3-28
3.43 FORECASTS FOR LAKES, BAYS, AND ESTUARIES
3.431 Wind Data. The techniques referred to for determination of wind
speeds and directions from isobaric patterns apply generally to ocean
areas. The friction that causes winds to spiral when crossing isobars
and to have a velocity lower than geostrophic or gradient winds is more
variable over land areas. When a fetch is close to land, this variability
will alter anticipated wind directions and velocities. In enclosed or
semienclosed bodies of water, such as lakes and bays, wind speeds and
directions should be taken from actual weather station reports whenever
possible.
In enclosed bodies of water, or in other areas where the wind blows
off the land, differing frictional effects of land and water should be
considered, and indicated wind speeds should be adjusted for these effects.
Studies by Myers (1954) and Graham and Nunn (1959) indicate recommended
adjustments in wind speeds. (See Table 3-2.) The adjustment factor may
vary considerably depending on the shoreline frictional characteristics.
This adjustment is used only for short fetches such as those in reservoirs
and small lakes.
Often, over small or well-defined fetch areas, it is not convenient
or even possible to utilize surface charts to determine wind characteris-
tics. Where wind records exist for locations in or near a fetch area,
these may be utilized. The accuracy of the forecast will depend on the
completeness of the records, the extent of fetch, and the wave prediction
technique employed. Where wind duration records are not available, local
wind speed reports may still be utilized to forecast waves assuming un-
limited durations, that is, wave growth is limited by the available fetch.
Wave characteristics deduced in this way are only qualitative.
Table 3-2. Wind-Speed Adjustment, Nearshore
Wind Direction Location of Wind Station Ratio*
Onshore 2 to 3 miles offshore
Onshore At coast
Onshore 5S to 10 miles inland
Offshore At coast
Offshore 10 miles offshore
(Graham and Nunn, 1959)
*Ratio of wind speed at location to overwater wind speed
(both at 30-ft. level).
3.432 Effective Fetch. The effect of fetch width or limiting ocean wave
growth in a generating area may usually be neglected since nearly all ocean
fetches have widths about as large as their lengths. In inland waters
(bays, rivers, lakes, and reservoirs), fetches are limited by land forms
surrounding the body of water. Fetches that are long in comparison to
width are frequently found, and the fetch width may become quite important,
resulting in wave generation significantly lower than that expected from
the same generating conditions over more open waters.
S29
Saville (1954) proposed a method to determine the effect of fetch
width on wave generation. Figure 3-13, based on this method, indicates
the effective fetch for a relatively uniform fetch width. The following
problem demonstrates the use of Figure 3-13.
RR OR OR Ree ee. EXAMPLE PROBLEM * * * * * * * * * * * * * *
GIVEN: Consider a channel with a fetch length F = 20 miles, a width
W = 5 miles, an average depth d = 35 feet, and a windspeed U = 50 mph
along the long axis.
FIND: Estimate the significant wave height H,, and the significant
wave period T,.
SOLUTION: Compute W/F = 5/20 = 0.25
From Figure 3-13 for W/F = 0.25, Fr/F = 0.45
Compute Fr = 0.45 x 20 = 9 miles or 47,500 feet.
Using the forecasting relations given in Section 3.6, Wave Forecasting
for Shallow Water, for a fetch of 47,500 feet and a wind speed of
50 mph and an average uniform depth of 35 feet, the significant wave
height may be determined from Figure 3-27 to be H, = 5.2 feet, say
5 feet and the significant wave period will be Dewir ae seconds, say
5 seconds.
Mel OKs eK oe OK ie ete Fe ee a de ee ie Oe. ey eS dee ie: ee ae) oe a eee es oe eee
The preceding example presents a simplified method of determining the
effective fetch. Shorelines are usually irregular, and the uniform-width
method indicated in Figure 3-13 is not applicable. A more general method
must be applied. This method is based on the concept that the width of a
fetch in reservoirs normally places a very definite restriction on the
length of the effective fetch; the less the width-length ratio, the shorter
the effective fetch. A procedure for determining the effective fetch
distance is illustrated in Figure 3-14. It consists of constructing 15
radials from the wave station at intervals of 6° (limited by an angle of
45° on either side of the wind direction) and extending these radials
until they first intersect the shoreline. The component of length of
each radial in a direction parallel to the wind direction is measured and
multiplied by the cosine of the angle between the radial and the wind
direction. The resulting values for each radial are summed and divided
by the sum of the cosines of all the individual angles. This method is
based on the following assumptions:
(a) Wind moving over a water surface transfers energy to the
water surface in the direction of the wind and in all directions within
45° on either side of the wind direction.
(b) The wind transfers a unit amount of energy to the water along
the central radial in the direction of the wind and along any other radial
an amount modified by the cosine of the angle between the radial and the
wind direction.
3-30
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bo a _ 155.46 _
] Be : fe. ae 13.512 =I1.51 Units
“gf Where based on map scale
| One Unit = 1714 feet
/ i 1714 _
Fate =/1.51 X 5280-2 Miles
44 Denison Dam
Ae
ey, Ny TY
6000 12000
Scale in feet
(U.S. Army, B.E.B. Tech. Memo No. 132,1962)
Figure 3-14. Computation of Effective Fetch for Irregular Shoreline
(c) Waves are completely absorbed at shorelines.
Fetch distances determined in this manner usually are less than those
based on maximum straight-line distances over open water. This is true
because the width of the fetch places restrictions on the total amount of
energy transferred from wind to water until the fetch width exceeds twice
the fetch length.
While 6° spacing of the radials is used in this example, any other
angular spacing could be used in the same procedure.
3.5 SIMPLIFIED WAVE-PREDICTION MODELS
Use of the wave-prediction models discussed in Section 3.3, Wave
Field, requires an enormous computational effort and more meteorological
data than one is likely to find outside of a major forecasting center.
The Fleet Numerical Weather Center, Monterey, California began using this
model on an experimental basis for a small part of the globe early in 1972.
Expansion to larger regions is planned. Wave prediction begins with a com-
putation of the existing wave field (often called a zero-time prediction),
and continues with a calculation of the effects of predicted winds on the
waves. A few years after this system is operational, it should be possible
to supply the needs for wave-hindcast statistics by compilations of zero-
time predictions. In the meantime, engineers who require wave statistics
derived by hindcasting techniques for design consideration must accept
simpler techniques.
Computational effort required for the model discussed in Section 3.31,
Development of a Wave Field, can be greatly reduced by the use of simpli-
fied assumptions with only a slight loss in accuracy for wave height cal-
culations, but sometimes with significant loss of detail on the distribu-
tion of wave energy with frequency. One commonly used approach is to
assume that both duration and fetch are large enough to permit an equi-
librium state between the mean wind, turbulence, and waves. If this
condition exists, all other variables are determined by the wind speed.
Pierson and Moskowitz (1964) consider three analytic expressions which
satisfy all of the theoretical constraints for an equilibrium spectrum.
Empirical data, described by Moskowitz (1964) were used to show that the
most satisfactory of these is
-B (w3/w*)
E(w) dw = (ag?/w5) e BK, dw , (3-20)
where a and £g are dimensionless constants, a = 8.1 x 10,°, 8 = 0.74
and wo = g/U, where g is the acceleration of gravity and U is the
wind speed reported by weather ships, and w is the wave frequency
considered.
Equation 3-20 may be expressed in many other forms. Bretschneider
(1959, 1963) gave an equivalent form, but with different values for a
and §. A similar expression was also given by Roll and Fischer (1956).
The condition in which waves are in equilibrium with the wind is called a
3-339
fully arisen sea. The assumption of a universal form for the fully arisen
sea permits the computation of other wave characteristics such as total
wave energy, Significant wave height, and period of maximum energy. The
equilibrium state between wind and waves rarely occurs in the ocean, and
may never occur for higher wind speeds.
A more general model may be constructed by assuming that the sea is
calm when the wind begins to blow. Integration of the equations governing
wave growth then permits the consideration of changes in the shape of the
spectrum with increasing fetch and duration. If enough wave and wind
records are available, empirical data may be analyzed to provide similar
information. Pierson, Neumann, and James (1955) introduced this type of
wave prediction scheme based almost entirely on empirical data. Inoue
(1966, 1967) repeated this exercise in a manner more consistent with the
Miles-Phillips theory using a differential equation for wave growth. Inoue
was a member of Pierson's group when this work was carried out, and his
prediction scheme may be regarded as a replacement for the earlier
Pierson-Neuman-James (PNJ) wave prediction model. The topic has been
extended by Silvester and Vongvisessomjai (1971) and others.
These simplified wave prediction schemes are based on the implicit
assumption that the waves being considered are due entirely to a wind
blowing at constant speed and direction for an overwater distance called
the fetch and for a time period called the duration.
f In principle it would be possible to consider some effects of variable
wind velocity by tracing each wave train. Once waves leave a generating
area and become swell, the wave energy is then propagated according to the
group velocity. The total energy at a point, and the square of the signif-
icant wave height could be obtained by adding contributions from individual
wave trains. Without a computer, this procedure is too laborious, and
theoretically inaccurate.
A more practical procedure is to relax the restrictions implied by
derivation of these schemes. Thus wind direction may be considered con-
stant if it varies from the mean by less than some finite value, say 30°.
Wind speed may be considered constant if it varies from the mean by less
than + 5 knots (2.5 meters/second) or ¥% barb on the weather map. This
assumption is not much greater than the uncertainty inherent in wind
reports from ships. In this procedure, average values are used and are
assumed constant over the fetch area, and for a particular duration.
The theoretical spectra for the partially arisen sea can be used
to develop formulas for such wave parameters as total energy, significant
wave height and period of maximum energy density.
Similar formulas can also be developed empirically from wind and wave
observations. A quasi-empirical - quasi-theoretical procedure was used by
Sverdrup and Munk (1947) to construct the first widely used wave predic-
tion system. The Sverdrup-Munk prediction curves were revised by
Bretschneider (1952a, 1958) with additional empirical data. Thus, this
prediction system is often called the Sverdrup-Munk-Bretschneider (SMB)
method. It is the most convenient wave prediction system to use when a
limited amount of data and time are available.
3-34
3.51 SMB METHOD FOR PREDICTING WAVES IN DEEP WATER
Revisions of earlier SMB forecasting curves are seen in Figures 3-15
and 3-16. The curves represent dimensional plots from the empirical
equations,
F 0.42
gf _ 0.283 tanh [pox (F] | (3-21)
U U
F 0.25
ae 1.20 tanh on (®) | (3-22)
2nU U
and,
F\\’ F 2
gt = Kexp [a(n (i) —B ln (i)* c + Din (i) We23)
U
where
exp {x} e{*}
In log,
K 6.5882
A = 0.0161
B = 0.3692
C = 2.2024
and
D> 058798).
With these relationships, the significant wave height Hp, and significant
wave period Ty, at the end of a fetch may be estimated when wind speed,
fetch length, and duration of wind over the fetch are given. Estimation
of wind parameters is discussed in Section 3.4, Wind Information Needed
for Wave Prediction, following the evaluation of simplified prediction
techniques. In using Figures 3-15 or 3-16 the estimated wind velocity U,
the fetch length F, and the estimated wind duration t, when a fetch
first appears on a weather chart, are tabulated. Figure 3-15 or 3-16 is
then entered with the value of U, using the scale on the left if U is
in knots, or the scale on the right if U is in statute miles per hour.
This U line is then followed from the left side of the graph across to
its intersection with the fetch length or F line, or the duration t
line, whichever comes first.
( Statute Miles)
Fetch Length
20 25 30
150 200 250 300 400 500 600 700 8009001,000
ST D
AOR SESS
Sweex iw
40 50 60 70 80 90100
=
eS
TN NTN
ad
«0
+
Sr
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We Vale
A AA Palate in ld
10 400 500 600 700 8009001,000
COS EDA SS:
i are a We
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y qi
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SEN
REINS
SENG
SCs)
iS
ESO NEES CRS GENS SSN
NSS SSI NGS
AK INTETIN
SSNS, |
ERS Ene
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100 STS LISTENER
{|
Nw,
IN
WIRE EL
Esaun a eave
it Aer |
ue ae PL
3
Significant Ht. ( ft.)
ENED
HAS
Ni
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SR EXNESANE TORN ANS
esearch Center from
neering Ri
equations developed by Chorles L. Bretschneider, ( Bretschneider,
Compiled by U.S. Army, Coastal Engi
ation )
MMUNIC:
1970-71, private co
Significant Period (sec.)
Deepwater Wave Forecasting Curves as a Function of Wind Speed, Fetch
Length, and Wind Duration (for Fetches 1 to 1,000 miles)
Figure 3-15.
Fetch Length (Statute Miles)
! 200) 2 000 2,500 3,000
4,000 5,000 6,000 8,000 10,000
[7
200 250 300 400 500 600 700800 1,000
Significant Period ( sec.)
Significant Ht. ( ft.)
———-—- Min. Duration (hrs.)
= Constant
Sees (2 pC
RT ars ca ae
eH OF
ot
S__o
<
£ Tm es
Wet te
= ; ia a Se a nfesssberboe
‘S. = ae SSS
Of Sip (amr eels
SE Pr iG
tte ee man ae al ai
ere a rt -! 4 4
at Eth is ae
ime a a es
po ;
t
79 sec
Y |
ay) FE
Wh iil
iy
ering Research Center from
quations developed by Charles L. Bretschneider, (Bretschneider,
4,000 5,0006,000 8,000 10,000 Compiled by U.S. Army, Coastal Engin
1,500 2,000 2,500 3,000 | ,
150 200 250 300 400 500 600 700800 1,000
nication
equ
1970-71, private commu
Fetch Length (Nautical Miles)
(for Fetches of 100 to more than 1,000 miles)
Deepwater Wave Forecasting Curves as a Function of Wind Speed, Fetch
Length, and Wind Duration
Figure 3-16.
Hel Fe Fe RS ate (eb HS OK OR RR ak: EXAMPLE PROBLEM ee RR we FR OR & ee
GIVEN: A wind speed U = 35 knots (40 mph) and a duration t = 10 hours.
FIND: The significant wave height Hp, and the significant period Ty,
at the fetch front for:
(a) A fetch length, F = 200 nautical miles and
(b) a fetch length, F = 80 nautical miles.
SOLUTION:
(a) Enter Figure 3-15 from the left side at U = 35 knots and move
horizontally across the figure from the left toward the right, until
intersecting the dashed line for a duration of 10 hours that comes
before the line indicating a fetch length of 200 nautical miles. At
the 10-hour duration line F = 92 nautical miles; this is the minimum
fetch seh, tox ithiis ;case:
With U = 35 knots, t = 10 hours, and F = 200 nautical miles then
Hy = 13.1 feet, Ty = 8.0 seconds, t,, equals 10 hours, and Fy = 92
nautical miles.
(b) Entering Figure 3-15 as above, when F = 80 nautical miles
and t = 10 hours, then the heights, periods, minimum duration and
fetch would be Hp = 12.6 feet, Tp = 7.8 seconds, and t,, = 9.0 hours.
The minimum duration t,, is 9 hours, corresponding to the miles
which limit generation, and comes before a duration of 10 hours.
In this example problem, the wave pattern in (a) is limited by the
duration; the wave pattern in (b) is limited by the fetch.
KURT OK Ke te ke ey eS TS ide ee ee ERT RS KE ade aes Ue ae, “EE Se. ee Cae de) ie Sie Toe ae ee
When a series of surface synoptic weather charts (Fig. 3-9) are used
to determine wave patterns, the values of U, F and t_ can be tabulated
for the first chart. For the same fetch on a later chart drawn for a time
Z,_ after the-first chart, U>) Fy and | t)| arevagain: tabulateds: Using
the subscript 2 to refer to those of the second chart and subscript 1
to refer to those of the first chart, if U, = U,, the above procedures
should be followed using either to = t, + 2 or Fo. If, however, U, # U);
certain additional assumptions must be made before using the forecasting
curves.
A change in wind speed from U, to Uj in a time Z between charts may
be assumed to take place instantaneously at a time Z/2. Waves due to Uj}
may then be calculated by assuming that the first chart minimum duration
time has been lengthened by an amount Z/2 or that its minimum fetch has
been changed by AF/2, where AF represents the change in fetch length
between weather charts. Since at the assumed abrupt change in wind speed,
3-38
the energy imparted to the waves by U,, with a minimum duration t,, + 2/2
for a minimum fetch F, + F/2, does not change, Up, will be assumed to
impart energy to waves which already contain energy due to Uj.
Plotted on Figures 3-15 and 3-16 are dotted lines of constant H2T2 which
are considered lines of constant wave energy. To a first approximation,
deepwater wave energy is given by
2
psH lL, 5.12 pg (HT)
EO = —— = —=— :: 3-24
e 8 8 ia
If energy had been imparted to the waves by U, acting alone, these waves
would be of length and height given in Figures 3-15 or 3-16 by the inter-
section of the U, ordinate with the constant energy line (plotted or
interpolated) corresponding to energy imparted by Uj, with a minimum
duration of t,, + Z/2 or a minimum fetch F,, + F/2. By increasing
the minimum duration at this point by an amount Z/2 or by changing the
minimum fetch by an amount AF/2, wave conditions under U, at the
time of the second chart may be approximated.
For example, if the wind speed increases so that U2 = 40 knots, and
with U, = 35 knots, t,, = 10 hours, F,,, = 92 nautical miles, t,) + Z/2 =
13 hours; an interpolated (by eye) dotted line of constant H2T2 would be
followed up to the U2 = 40-knot line where the duration = 6.5 hours. To
this value Z/2 or 3 hours is added and then moving horizontally along
the line U, = 40 knots to t = 6.5 + 3.0 = 9.5 hours, it is found that
Hpo = 15.6 feet, Tp, = 8.7 seconds, ty». = 9.5 hours, and F,5 = 95 nautical
miles. If the measured fetch F 2 had been less than 95 nautical miles,
this length of fetch would limit the growth of waves. Although the pre-
ceding discussion would indicate that AF should be calculated, in practice
this need not be done; the results obtained through calculation of AF
would be found by reading off wave heights at the intersection of Uz and
Fo if Fo is limiting. Therefore, if F2 had equalled 85 nautical miles,
in this case less than 95 miles and therefore limiting, at the intersection
of U, = 40 knots, then Hp, = 15.0 feet, Tyo = 8.5 seconds, Ty» = 8.8 hours,
and F,,. = 85 nautical miles. Note this important distinction: t,, F, and F,
are calculated by use of Figure 3-15. Some of the measured and calculated
values will be the same, but not all of them.
If the wind velocity Uz is less than Uj), the procedures followed
are nearly the same. From the intersection of U, and t,, + ZL 2re
constant energy line is followed to its intersection, if there is one,
3-39
with either U, or F, whichever comes first from the left side of the
figure. If U comes first, Z/2 is added to the duration at this point,
and the U2 ordinate is followed to either this new duration or to the
F, whichever is first from the left side of the chart. (Compare with the
preceding paragraph.) At this point, Hpo, Tro, tyo, and F,,. inclusive,
are read off. If the constant energy line had intersected F, before U,,
it is only necessary to drop down along the F, abscissa to its intersec-
tion vith U,, and at this point read Hpo, Tyo, ty, and F,5. (This pro-
cedure could be used for many cases in which U, is greater than Uj.)
The major differences in technique are used when Up is less than Uj
and the H2T2 = constant line from the intersection of U, and tm) + Z/2
does not intersect either U, or F,. Forecasting theory used here pre-
dicts that waves due to a constant wind blowing over an unlimited fetch
for an unlimited duration will eventually reach limiting height and period
distributions beyond which growth will not continue. In Figure 3-16 the
limit of this state is delineated by the line labelled maximum condition.
To the right of this line, it is assumed that any energy transport to the
waves by the wind is compensated by wave breaking, hence no wave growth
occurs.
3.52 EFFECTS OF MOVING STORMS AND A VARIABLE WIND SPEED AND DIRECTION
In principle, it should be possible to extend the Inoue differential
equation for wave growth to highly irregular conditions, but no experi-
mental verification of this concept has been published. Kaplan (1953)
and Wilson (1955) have proposed techniques for applying the simplified
prediction techniques to variablewind fields and changing fetches. The
procedures appear reasonable and these techniques are used, although no
Statistics are available for verification.
3.53 VERIFICATION OF SIMPLIFIED WAVE HINDCAST PROCEDURES
Comparisons of hindcast wave heights and observed wave heights,
similar to Figure 3-7, have been given by Jacobs (1965) for the PNJ wave-
prediction system, by Bates (1949) and Isaacs and Saville (1949) for the
early Sverdrup-Munk method, by Kaplan and Saville (1954), for the early
SMB method, and by Bretschneider (1965) for a later revision. The basic
data from which the prediction curves were derived, summarized by
Bretschneider (1951), also indicate the range of variation that may be
expected.
It is generally believed that much of the discrepancy between observed
and predicted waves is random, and that statistical summaries of observa-
tions and predicted values will agree much better than the individual
observations. Saville (1954) and Pierson, Neumann and James (1955) give
summaries of results from a systematic program for deepwater hindcasting
waves at the four locations shown in Figure 3-17. The U.S. Naval Weather
3-40
Figure 3-17. Location of Wave Hindcasting Stations and Summary of
Synoptic Meterological Observations (SSMO) Areas
a> 4]
Service Command (1970) provides summaries of shipboard wave observations,
Summary of Synoptic Meteorological Observations (SSMO) for the hatched
areas indicated in Figure 3-17. Cumulative distribution functions for
wave heights as determined by both hindcasting techniques and the ship-
board observations are given in Figure 3-18. The average of the two
forecasting methods agrees reasonably well with the shipboard observations.
3.54 ESTIMATING WAVE DECAY IN DEEP WATER
Figures 3-19 and 3-20 are used to estimate wave characteristics after
the waves have left the fetch area but are still travelling in deep water.
With Figure 3-19, and given Hp, Ty, F, and D (the decay distance), it is
possible to compute the ratios
decayed wave height _ Hp decayed wave period _ Tp
22228 Fr eo
fetch wave height Hp on” fetch wave period Tr
With Figure 3-20, it is possible to compute wave travel time between
a fetch and a coast, knowing the decayed wave period Ty and the decay
distance D.
This travel time tp is determined by dividing the decay distance
by the deepwater group velocity for waves having a period equal to the
decayed period Tp. These values enable the estimation of arrival times
for waves at the end of the decay distance.
Waves, after leaving a generating area, will generally follow a great-
circle path toward a coast. However, sufficient accuracy is usually
obtained by assuming wave travel in a straight line on the synoptic chart.
Decay distance is found by measuring the straight line distance between
the front of a fetch and the point for which the forecast is being made.
If a forecast is being made for a coastal area, the effects of shoaling,
refraction, bottom friction and percolation will have to be considered in
translating the deepwater forecast to the shore.
3.6 WAVE FORECASTING FOR SHALLOW WATER
3.61 FORECASTING CURVES
Water depth affects wave generation. For a given set of wind and
fetch conditions, wave heights will be smaller and wave periods shorter if
generation takes place in transitional or shallow water rather than in deep
water. Several forecasting approaches have been made; the method given by
Bretschneider as modified using the results of Ijima and Tang (1966) is
presented here. Bretschneider and Reid (1953) consider bottom friction
and percolation in the permeable sea bottom.
There is no single theoretical development for determining the actual
growth of waves generated by winds blowing over relatively shallow water.
The numerical method presented here is based on successive approximations
3-42
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Decay distance D, (Nautical miles)
8 10 4 14 16 18
Wave period T,, at end of decay (seconds)
Figure 3-20. Travel Time of Swell Based on ty = D/Cg
3-45
20
22
4600
4400
4200
4000
3800
3600
3400
3200
3000
2800
2600
2400
2200
2000
1800
1600
1400
1200
1000
800
600
400
200
Decay distance D, (Statute miles)
in which wave energy is added due to wind stress and subtracted due to
bottom friction and percolation. This method uses deepwater forecasting
relationships originally developed by Sverdrup and Munk (1947) and revised
by Bretschneider (1951) to determine the energy added due to wind stress.
Wave energy lost due to bottom friction and percolation is determined from
the relationships developed by Bretschneider and Reid (1953). Resultant
wave heights and periods are obtained by combining the above relationships
by numerical methods. The basic assumptions applicable to development of
deepwater wave-generation relationships (Bretschneider, 1952b) as well as
development of relationships for bottom friction loss (Putnam and Johnson,
1949) and percolation loss (Putnam, 1949) apply.
The choice of an appropriate bottom friction factor fr for use in
the forecasting technique is a matter of judgement; a value of fr = 0.01
has been used for the preparation of Figures 3-21 through 3-30 which are
forecasting curves for shallow-water areas of constant depth. These curves,
which may be used like Figures 3-15 and 3-16, are given by the equations:
F 0.42
0.0125 (5)
U
eH gd 0.75
aa = OMe tanh | 0.530 i tanh ¢ ——~ _“___ (3-25)
U U gd \ 975 ‘
tanh |0.530 ()
and
(' =
OO
T d \0.375 2
=e = 1.20 tanh oss (=) | tanh si ae EE ,(3-26)
0.375
tanh [oss a |
which in deep water reduce to Equations 3-21 and 3-22 respectively.
CO eet a eT ek Ce er AP Ta me Ke ko ede) Ok, EXAMPLE PROBLEM * * * * * * * * *¥ *¥ * kk &
GIVEN: Fetch, F = 80,000 feet, wind speed, U = 50 mph., water depth,
d = 35 feet (average constant depth), bottom friction factor fp = 0.01
(assumed).
FIND: Wave height H and wave period T.
Wind Speed (U) mph
I te aa 4 5 67 8910 (Smee oS as 4aSe67) e¥SN10
X 1,000 Fetch (F) feet X 10,000
Figure 3-21. Forecasting Curves for Shallow-Water Waves
Constant Depth = 5 feet.
Wind Speed (U) mph
| 1.5 ZagcoNeS: Ae Oni fan Onl. 1.5 252505 4= =5) 6eere 6091110
X 1,000 Fetch (F) feet X 10,000
Figure 3-22. Forecasting Curves for Shallow-Water Waves
Constant Depth = 10 feet.
Wind Speed (U) mph
i=
! 15 (419) (aus) AS) 4 3) (657 78) 9n10 ES fe) (Ke)
x
3
X 1,000 Fetch (F) feet 10,000
Figure 3-23. Forecasting Curves for Shallow-Water Waves
Constant Depth = 15 feet.
Wind Speed (U) mph
| 15 (a Paks) BS Ce ey GY ze fehl) te) ete
X
Figure 3-24. Forecasting Curves for Shallow-Water Waves
Constant Depth = 29) feet.
3-48
6.5 sec.
ST &) io)
2 25 3
aS 7 GE) Srl
4
5
alia:
a
}
w)
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“4
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own
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Wee p
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OD
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w Mm
ydw (7) peeds pu!
udw (nm) peeds pum
4
X 10,000
30 feet.
3-49
Fetch (F) feet
Forecasting Curves for Shallow-Water Waves
Constant Depth
X 1,000
-26.
Figure 3
Wind Speed (U) mph
Wind Speed (U) mph
(ley) CNS VTS Io) ite) (Ay 7258)
x
Figure 3-27. Forecasting Curves for Shallow-Water Waves
Froure: 3
Constant Depth = 35 feet.
PAIS ahaa bmelsy onal Sweet ous). KO [Sores eal estes
X 1,000 Fetch (F) feet X 10,000
8.
bo
Forecasting Curves for Shallow-Water Waves
Constant Pepth = 40 feet.
3-50
Gy 7 Se SiO
T=7.0 sec
T=6.0 sec. H=/1.0 ft.
Wind Speed (U) mph
—- ~
I 15 2225S 4 5 6 #7 6)910 15 2 25.3 4 516 7 18) 910
X 1,000 Fetch (F) feet xX 10,000
Figure 3-29. Forecasting Curves for Shallow-Water Waves
Constant Depth = 45 feet.
Wind Speed (U) mph
I 1.5 a Gas! 4) 5) NG) 7 Se9ho 15 2 eto nS 45265 Sno lO
X 1,000 Fetch (F) feet X 10,000
Figure 3-30. Forecasting Curves for Shallow-Water Waves
Constant Depth = 50 feet.
3-5|
SOLUTION: From Figure 3-27 for constant depth, d = 35 feet,
for
F = 80,000 feet,
and
U = 50 mph.
Then
H = 5.9 feet, say 6 feet,
and
T = S-l)seconds, say 5 seconds.
»
eK Fe Kee: ie ke Ke a Kel KD ie eee: ie see Fee) “Fe OR See, OR) ark aa oe) Ce
3.62 DECAY IN LAKES, BAYS, AND ESTUARIES
Section 3.33. Decay of A Wave Field applies to water areas contiguous
with land as well as those in the open ocean. Most fetches in inland
waters will be limited at the front and at the rear by a land mass and
decay distances will usually be relatively small or nonexistent.
3.7 HURRICANE WAVES
When predicting wave generation by hurricanes, the determirution of
fetch and duration from a wind field is more difficult than for more normal
weather conditions discussed earlier. The large changes in wind speed and
direction with both location and time cause the difficulty. Estimation of
the free air wind field must be approached through mathematical] models,
because of the scarcity of observations in severe storms. However, the
vertical temperature profile and atmospheric turbulence characteristics
associated with hurricanes differ less from cne storm to another than for
other types of storms. Thus the relation between the free air winds and
the surface winds is less variable for hurricanes than for other storms.
3.71 DESCRIPTION OF HURRICANE WAVES
In hurricanes, fetch areas in which wind speed and direction remain
reasonably constant are always small; a fully arisen sea state never
develops. In the high-wind zones of a storm, however, long-period waves
which can outrun the storm may be developed within fetches of 10 to 20
miles and over durations of 1 to 2 hours. The wave field in front, or to
either side, of the storm center will consist of a locally generated sea,
and a swell from other regions of the storm. Samples of wave spectra,
obtained during hurricane Agnes, 1972, are shown in Figure 3-31. Most
of the spectra display evidence of two or three distinct wave trains; thus,
the physical implications of a stgnificant wave period is not clear.
Other hurricane wave spectra computed with an analog spectrum analyzer
from wave records obtained during Hurricane Donna, 1959, have been published
by Bretschneider (1963). Most of these spectra also contained two distinct
peaks.
Total Wave Energy
Total Wave Energy
Period (Seconds)
10 5 25
+ 0.4
+0.2
+0.0
0.0 0.2 0.4 0.0 0.2 0.4
Frequency (Hertz)
Period (Seconds)
10 5 220
0.4
0.2
0.0
0.0 0.2 0.4 0.0 0.2 0.4
Frequency (Hertz)
Figure 3-31. Typical Hurricane Wave Spectra. Typical Hurricane
Wave Spectra from the Atlantic Coast of the United
States. The ordinate scale is the fraction of
total wave energy in each frequency band of 0.0011
Hertz (one Hertz is one cycle/second). A linear
frequency scale is shown at bottom of each graph
and a non-linear period scale at top of each graph.
An indication of the distribution of waves throughout a hurricane can
be obtained by plotting composite charts of shipboard wave observations.
The position of a report is determined by its distance from the storm
center and its direction from the storm track. Changes in storm intensity
and shape are often small enough to permit all observations obtained during
a period of 24 to 36 hours to be plotted on a single chart. Several plots
of this type from Pore (1957) are given in Figure 3-32. Additional data
of the same type have been presented by Arakawa and Suda (1953), Pore (1957)
and Harris (1962).
Goodknight and Russell (1963) give a tabulation of the significant
height and period for waves recorded on an 0il drilling platform in approxi-
mately 33 feet of water, 1.5 miles from shore near Burrwood, Louisiana
during hurricanes Audrey, 1957, and Ella, 1950, and tropical storms Bertha,
1957, and Esther, 1957. These wave records were used to evaluate the
applicability of the Rayleigh distribution function (Section 3.22. Wave
Height Variability) to hurricane statistics for wave heights and periods.
They concluded that the Rayleigh distribution function is adequate for
deriving the ratios between H,, Hi9, H, etc., with sufficient accuracy
for engineering design, but that its acceptance as a basic law for wave
height distributions is questionable.
3.72 MODEL WIND AND PRESSURE FIELDS FOR HURRICANES
Many mathematical models have been proposed. for use in studying
hurricanes. Each is designed to simulate some aspect of the storm as
accurately as possible without making excessively large errors in describ-
ing other aspects of the storm. Each model leads to a slightly different
specification of the surface wind field. Available wind data are suffi-
cient to show that some models duplicate certain aspects of the wind field
better than certain other models; but there are not enough data for a
determination of a best model for all purposes.
One of the simplest and earliest models for the hurricane wind field
is the Rankin vortex. For this model, it is assumed that
UH Ke for 2 = R
KR? (3-27)
CSS ere Se
r
where K is a constant, R is the radial distance from the storm center
to the region of maximum wind speed, and r is the radial distance from
the storm center to any specified point in the storm system.
This model can be improved by adding a translational component to
account for storm movement and a term producing cross-isobar flow toward
the storm center.
Extensions of this model are still being used in some engineering
studies (Collins and Viehman, 1971). This model gives an artificial
discontinuity in the horizontal gradient of the wind speed at the radius
of maximum winds, and does not reproduce the well-known area of calm
winds near the storm center.
3-54
STORM DIR STORM DIR \
é
he
a
" ue af Yo We
is "
TZ DS connie 1955 aw a Na y,
“2 EZ mat 2S o UG 11-0730E THRU 2 Tad A
wy jut" C CONNIE 1955 7 Aue i A Sy ae
fA “ aval AUG 12-0730E a ‘
2 “Gly AUG. 10-0130E THRU 45 Lag Ee Nee is ee
ye SS) on AUG. 11-0130E Fis,
s :
PERIOD SCALE IN SEC
© 10 20
3 \' PERIOD SCALE IN SEC
y, ss 0 10 20
= Xu /
ah Bey
: ‘36H Ses, 106 at 4 Tao Ret -|
Ry, . \ a
1
y= \ ‘ } 3259 ai ic
s = bs ain’ XS sy ne? = Ny i
wy P 20 ; ee ee emis a
2. i, x i 360 of eco", I ie Netiay UNG -
ie u Withee De
si!" " 3 ey * a mh. ame, A 15 —5
ves u I ai \p ‘ 2 fo Pay
eae a Oi aE a aN yi qy Se me
= 7 z 14 it po
ab aiak Vig hoe 72 © Sel Te Reve
eae fe fs pa \o ee a2 p10 y nwa Se f=”
We 4 =v J w :
“fy SS SS
y : ‘ :
a S
y STORM DIR STORM DIR
Ee lao
Oe PY rr f£ DIANE 1955
‘ A
*s 10 \ ‘ AUG 16. 1330 THRU
SeaPe aaa re Lh AUG. 17-0730E
BON x NIN, ay a
Ci, | een es wri PERIOD SCALE IN SEC | :
sho AUG. 15-1330 THRU w (A © 10 20 = a ep 2
4 me AUG 16-0730E af Mi, Neer 3 i ¥
i)
P A PERIOD SCALE IN SEC. *¢~ of I x y I 0 .
z 0 10 20 ho? Als Me dad \,
“aN a a fs bi 5 7 AS sais 57 a vv
se os ea a or bo roll Es al rte
pe 10 ® uf 1. {e ~ Wy we \o 374
180 13y a 4 7 yen Wy ee wl “I ant js + 4
240 " 3 a
a y 4 2 13yie a
30 ie _ an ‘of AN \s uf - " S o¥ - Ke ts) iss F |
420 Lae yl 5 xu ze e fs 5.3 4; mw 4 toy
‘ 4s ~ ha Wien 7
480 i a ae fio xo i
See aN a SS fs + a} 7
we Ao , af fo 10 = “a
« hs - 19 ®
# \
2, ~ id s ai ii yf
= Lo ts ry] 5
s . .
vt : 0] ¥ = y a
a a fio 7X aft 1of
y “(a
(after Pore, 1957)
The wave height in feet is plotted
beside the arrow indicating direction from which the waves
came. The length of the arrow is proportional to the wave
period. Dashed arrow indicates unknown period. Distances
are marked along the radii at intervals of 60 nautical miles.
Figure 3-32. Compositive Wave Charts.
A more widely used model was given by Myers (1954). A concise mathe-
matical description of this model is given by Harris (1958) as follows:
Piaf ==
(e)
———=e , (3-28)
P B
z R
gr 1 Ros
= as (B, See ae Be (3-29)
where p is the pressure at a point located at a distance r from the
storm center, p, is the central pressure, oe is the pressure at the
outskirts of the storm, pg is the density of air, and Ugr is the
gradient wind speed. Agreement between this model and the characteristics
of a well-observed hurricane is shown in Figure 3-33. The insert map gives
the storm track; dots indicate the observed pressure at several stations in
the vicinity of Lake Okeechobee, Florida; the solid line (Fig: 3-33a) gives
the theoretical pressure profile fitted to three points within 50 miles of
the storm center. The corresponding theoretical wind profile is given by
the upper curve of Figure 3-33b. Observed winds at one station are indi-
cated by dots below this curve. A solid line has been drawn through these
dots by eye to obtain a smooth profile. The observed wind speed varies in
a systematic way from about 65 percent of the computed wind speed at the
outer edge to almost 90 percent of the predicted value near the zone of
maximum wind speed. Reasonably good agreement between the theoretical and
observed wind speeds has been obtained in only a few storms. This lack of
agreement between the theoretical and observed winds is due in part to the
elementary nature of the model, but perhaps equally to the lack of accurate
wind records near the center of hurricanes.
Parameters obtained from fitting this model to a large number of storms
were given by Myers (1954). Parameters for these other storms (and for
additional storms) are given by Harris (1958). Equation 3-29 will require
some form of correction for a moving storm.
This model is purely empirical, but it has been used extensively and
it provides reasonable agreement with observations in many storms. Other
equally valid models could be derived; however, alternative models should
not be adopted without extensive testing.
In the northern hemisphere, wind speeds to the right of the storm
track are always higher than those on the left, and a correction is needed
when any stationary storm model is being used for a moving storm. The
effect of storm motion on the wind field decreases with distance from the
zone of highest wind speeds. Thus the vectorial addition of storm motion
to the wind field of the stationary storm is not satisfactory. Jelesnianski
(1966) suggests the following simple form for this correction,
= —— Vz. (3-30)
g
a
°
Wind Speed (MP.H)
Curve drawn th rough «
data
Upper curve is given by: =o
Ue R
= + fUg.= > (Fr Fo) & e* 20
a
0) 10 20 30 40 50 60 700 10 20 30 40 50 60
Distance From Pressure Center (Statute Miles) Distance From Wind Center (Statute Miles)
a ’ b
Hurricane August 26-27, 1949 (from Harris, 1958)
Figure 3-33. Pressure and Wind Distribution in Model Hurricane. Plotted
dots represents observations
where Vp is the velocity of the storm center, and U yt) is the
convective term which is to be added vectorially to the wind velocity at
each value of r. Wilson (1955, 1961) and Bretschneider (1959, 1972) have
suggested other correction terms.
3.73 PREDICTION TECHNIQUE
For a slowly moving hurricane, the following formulas can be used to
obtain an estimate of the deepwater significant wave height and period at
the point of maximum wind:
RAp
— 0.208 a Vp
H. = 16.5¢100 |1+——__] , (3-31)
VUR
and
RA
x00 |, , 11044 YE
TS 8.6 e 1+ SSS 3-32
Ss VUp > ( )
where
H, = deepwater significant wave height in feet
T, = the corresponding significant wave period in seconds
R = radius of maximum wind in nautical miles
Ap = P, - Po» Where p, is the normal pressure of 29.92 inches
of mercury, and Po is the central pressure of the hurricane
V, = The forward speed of the hurricane in knots
Up = The maximum sustained wind speed in knots, calculated for
30 feet above the mean sea surface at radius R where
Up = 0.865 U a (For stationary hurricane) (3-33)
Up = | 0).865 Uae + 0.5 Vp (For moving hurricane) (3-34)
Unax = Maximum gradient wind speed in knots 30 feet above the
water surface
Ut) =)5.02868 [7380p -)pie/2 = RGWO.5750)i] (3-35)
f = Coriolis parameter = 2w sing, where w = angular velocity of
earth = 21/24 radians per hour
Latitude (6) 25° 30° 35° 40°
f (rad/hr) OF 2215 SORnZ625) OR SO0Ne MOK So
a = a coefficient depending on the forward speed of the
hurricane and the increase in effective fetch length,
because the hurricane is moving. It is suggested that
for slowly moving hurricane a = 1.0.
Once Ho is determined for the point of maximum wind from Equation
3-31 it is possible to obtain the approximate deepwater significant wave
height H, for other areas of the hurricane by use of Figure 3-34.
The corresponding approximate wave period may be obtained from
Py Se Boils vi, (in seconds) , (3-36)
where H, is in feet (derived from empirical data showing that the wave
steepness H/T? will be about 0.22).
3-58
Figure 3-34.
Isolines of Relative Significant Wave
Height for Slow Moving Hurricane
GIVEN:
FIND;
FOR) Ke RG ok) Cteaeke> ee eS ee
Mp) = AV OA = Poll =
26 knots.
SOLUTION:
Using Equation 3-35
q
U
max
Using Equation 3-34
UR
TR
Using Equation 3-31
H
oO
where the exponent
RAp _
100
and
0-832 =
then
H, =
aL =
Assume for simplicity that a
EXAMPLE PROBLEM
BR IE ee Re ie eee ey eer ced eae)
Consider a hurricane at latitude 35°N. with R = 36 nautical miles,
2.31 inches of mercury, and Vp, forward speed, =
Or
The deepwater significant wave height and period.
0.868 [73(P, ~ 8, ie — R (0.5756)
0.868 [73 (2.31) — 36 (0.575 X 0.300) |
0.868 (110.95 — 6.23)
= 0.865 Uae + 0.5 Vp
0.865 (90.9) + 0.5 (26)
Ree 0.208 aV;,
16.5 a 1 + ———
VUpR
BE ayes
100
2.30
16.5 eos ; :
Il
Il
90.9 knots.
91.6 knots.
|
0.208 X 1X 26
V 91.6
16.5 (2.30) (1.564) = 59.4 feet.
3-60
Using Equation 3-32
229 0.104 a Vp
T = 8.6 e2 1 > ——— | 5
% VUp
where the exponent
RAP 36 (2.31) _
ee ee 10-416
200 200
.104X 1X 26
ei aigecuaiet aye Wak Se
‘ V91.6
T. = 8.6 (1.52) (1.282) = 16.8 seconds.
Alternately, by Equation 3-36, it is seen that
T, = 2.13 V59.4 = 16.4 seconds.
It should be noted that computing the values of wave height and period
to three significant figures does not imply the degree of accuracy of the
method; it is done to reduce the computational error.
Referring to Figure 3-34, H, = 59.4 feet corresponds to the relative
significant wave height of 1.0 at r/R = 1.0, the point of maximum winds
located, for this example, 36 nautical miles to the right of the hurricane
center. At that point the wave height is about 60 feet, and the wave
period T is about 16 seconds. At r/R = 1.0 to the left of the hurricane
center, from Figure 3-34 the ratio of relative significant height is about
0.62, whence H, = 0.62 (59.4) = 36.8 feet. This wave is moving in a direc-
tion opposite to that of the 59.4-foot wave. The significant wave period
for the 36.8-foot wave is: T, = 2.13 736.8 = 12.9 seconds, say 13 seconds.
The most probable maximum wave is assumed to depend on the number of
waves considered applicable to the significant wave, Hp = 59.4 feet. This
number N depends on the length of the section of the hurricane for which
near steady state exists and the forward speed of the hurricane. It has
been found that maximum wave conditions occur over a distance equal to the
radius of maximum wind. The time it takes the radius of maximum wind to
pass a particular point is
t 28 1.38 h 4,970 d
= —S = — = 5 ours = : seconds; -
the number of waves will be
Yo? gem Ge
The most probable maximum waves can be obtained by using
N
H, = 0.707 H, [ og, = (3-39)
The most probable maximum wave is obtained by setting n = 1, and
using Equation 3-39
303
H, = 0.707 (59.4) a{log, a = 100.4 feet, say 100 feet.
Assuming that the 100-foot wave occurred, then the most probable
second highest wave is obtained by setting n = 2, the third from n = 3, etc.
303
1 0.707 (59.4) a|log, ks 94.1 feet, say 94 feet;
303
4 0.707 (59.4) log, Fes 90.2 feet, say 90 feet.
The problem now is to determine the changes in the deepwater waves as
they cross the Continental Shelf, taking into account the combined effects
of bottom friction, refraction, the continued action of the wind, and the
forward speed of the hurricane. This requires numerical integration, using
Table 3-3, Figure 3-35, and refraction diagrams. It is also necessary to
obtain an effective fetch length, by use of
H
2
gob | hae (3-40
SiR POlOS55 a) & Sy
where
F, is the effective fetch in nautical miles,
H, is the deepwater significant wave height in feet,
and
Up is the maximum sustained wind speed in knots.
For this example, using Equation 3-40
59.4
F. = | —————_| = (11.69)° = 137 nautical miles.
e 0.0555 (91.6)
Table 3-3. Values of K, or (H/H‘)
*Units of sec?/ft.
i
i 5
°
x
fo)
°
0
3
4o
SE
S
fo)
re)
fo)
9°
©
ro)
°
=
fo)
°
oo
fo}
0.95 0.90
0.995 0.99 0.98
0.999
0.9999
( Bretschneider, 1957)
ion Loss Over a Bottom of Constant Depth
Relationship for Frict
3-35
igure
F
3-64
For the remaining part of this problem, either the value of F,,;,, equal
to 220 nautical miles as determined from Figure 3-15 for Up = 91.6,
H, = 59.4 can be used with the deepwater forecasting curves, or else
Equation 3-40 can be used, as modified,
/
H, = 0.0555 Up Fl + AF
along with Equation 3-36
T= 9 de Al.
F, is defined below.
The latter being a numerical method is easier to use and more accurate
than the graphical method of using forecasting curves.
The procedure for computing wind waves over the Continental Shelf
will be illustrated by using the bottom profile off the mouth of the
Chesapeake Bay and the standard project hurricane developed for the
Norfolk area. The storm surge computed for the standard project hurricane
and 2.5 feet of astronomical tide are added to the mean low water depths
to obtain the total water depth for wave generation. Refraction is
neglected in this example, i.e., Kp = 1.0. The results of these computa-
tions are given in Table 3-4 followed by examples and explanations.
Column 1 of Table 3-4 is the distance in nautical miles measured
seaward of the entrance to Chesapeake Bay, using increments of 5 nautical
miles for each section.
Column 2, d,, is the depth in feet referred to mean low water at the
shoreward end of each section, denoted by X of Column 1.
Column 3 is the depth d, at the beginning of each section.
Column 4 is the depth d, at the shoreward end of each section.
These depths are the water depths below MLW plus the 2.5-foot astro-
nomical tide plus the hurricane surge and are then rounded off to the
nearest foot.
Column 5, Gen is the average of Columns 3 and 4 to the nearest foot.
Column 6 is the effective fetch Fe (nautical miles), and is
obtained for the first step directly from Equation 3-40. For successive
steps, Fe = FE + AF < 137 n.mi. where Fé is given in Column 14 one line
above in each case (e.g., line X = 40, F, = 80.6 + 5.0 = 85.6) and AF is
5 n.mi. FZ is defined for Column 14.
ra fart a
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3-66
Column 7 is the deepwater significant wave height H, and is
obtained from Equation 3-40;
H, = 0.0555 Up VF, = 5.08 VF, , where Up = 91.6 knots
and Reis obtained from Column 6.
Column 8 is the deepwater significant wave period T, and is
obtained from Equation 3-36:
I = Wal) VH, where H, is obtained from Column 7.
oO
Column 9 is TZ /dp, or Column 8 squared over Column 5.
Column 10 is the shoaling coefficient (H/H5) or K, corresponding
to the value of T3/dr, Column 9, and is obtained from Table 3-3 K,
versus T2/d.
Column 11 is the friction loss parameter and is equal to
4 fH, K,AX be 0.01 H, K, (5) (6,080) 4. 304 H, K, aaa
(d;)? (a7)? (dy ,
where f, is assumed as 0.01, AX = 5(6,080) = 30,400 ft., dp is the
average water depth of the increment AX.
Column 12 is the friction factor Ky, and is obtained from Figure
3-35 where Kr is a function of T?/dp (Column 9) and
fH, K, AX
3 = A (Column 11).
(47)
Column 13 is the equivalent deepwater wave height H, and is
obtained from Ho = HK (the product of Columns 7 and 12).
Column 14 is the equivalent effective fetch length for Hj, and is
obtained from Equation 3-40
H’ 2 H’ 2
/ = |— ° = p50%! h U = 91.6 i ,
" (0.0555 Up ) EWG |e ie a nots (moving hurricane).
Column 15 is obtained by using Equation 3-36 Tf = 2.13 VHD.
is the water depth at shoreward
= Pp \22
Column 16 is (TT) /d,, where d,
end of section AX.
Column 17 is the shoaling coefficient K,, related to the values of
(T5)7/d, (Column 16).
Column 18 is H =H, x Kg (product of Columns 13 and 17).
Column 19 is obtained by using Equation 3-38
4,970 36
N= eu ed a where + = R/Vp = ae = 1.38 hours or 4,970 seconds.
oH ae
Column 20 is oe = 0.707 H vlog .N, H is from Column 18.
After one line of computations across is completed, the next line is
begun, using Ey = Ey > (NE <e IS57/ (pineeee Ee is from Column 14 of the
preceding completed line. For example, consider the line corresponding to
X = 40 being completed. Then the computation for the next line X = 35
is as follows:
je!
e
66.4 from line X = 40, Column 14.
Column 6, F 66.4 + 5 = 71.4 nautical miles for line K = 35.
Compute
Colum 7, H, = 5.08 V71-4 = 42.9 feet.
Column 8, = 2.13 V42.9 = 14.0 seconds.
Column 9, = =) 57 3h.
di 113
2
Column 10, K, = 0.924 (Table 3-3) for values of —2 = 1.73.
d_-
304H, K
Column 11, Ava pe = 2 Ss (304 (42.9) (0.924) = 0.943.
(d.)? G13)?
2
Column 12, Kr = 0.89 (from Figure 3-35) for values of = = 1.73 and
if
A = 0.943.
Column 13, H’ H, Ky = 42.9 (0.89) = 38.2 feet.
~
ela 38.2 |?
Column 14, Fi = {|—2} = |=! = 56.6 nautical miles.
€ 5.08 5.08
3-68
Column 15, 1 2.13 38.2 = 13.2 seconds.
Column 16, — 158)
0.919.
Column 7p
0.919 (38.2) = 35.1 feet, which is the shallow-water height
Column 18, H
for depth d, = 110 feet, corresponding to MLW of 104 feet.
4,970 é
Column 19, N = iy = 377 or the total number of waves applicable to ,
(e] ‘
steady-state significant wave of H = 35.1 feet, say 35 feet .
Column 20, H 35.1 (0.707) Vlog, 377 = 60.4 feet, say 60 feet .
max
Se eK Re RR I Oa a OR RO KO AR ae ee oa Ke ee
The moving fetch model of Wilson (1955) has been adapted for computer
usage by Wilson (1961). The basic equations were modified by Wilson (1966).
The Bretschneider (1959) model for hurricane wave prediction was modified
by Bretschneider (1972). Borgman (1972) used the results of Wilson (1957)
to develop an approach for estimating the maximum wave in a storm which
may be considered as an alternate to that presented here.
3.8 WATER LEVEL FLUCTUATIONS
The focus now changes from wave prediction to water level fluctuations
in oceans and other bodies of water which have periods substantially longer
than those associated with surface waves. Several known physical processes
combine to cause these longer-term variations of the water level.
The expression water level is used to indicate the mean elevation of
the water when averaged over a period of time long enough (about 1 minute)
to eliminate high frequency oscillations caused by surface gravity waves.
In the discussion of gravity waves the water level was also referred to as
the sttllwater level (SWL) to indicate the elevation of the water if all
gravity waves were at rest. In the field, water levels are determined by
measuring water surface elevations in a stilling well. Inflow and outflow
of the well is restricted so that the rapid responses produced by gravity
waves are filtered out, thus reflecting only the mean water elevation.
Water level fluctuations -- classified by the characteristics and
type of motion which take place -- may be identified as:
(a) astronomical tides
tb tsunamis
c) seiches
(d) wave setup
(e) storm surges
(f) climatological variations
(g) secular variations
The first five have periods that range from a few minutes to a few
days; the last two have periods that range from semi-annually to many
years. Although important in long-term changes in water elevations,
climatological and secular variations are not discussed here.
Forces caused by the graviational attraction between the moon, the
sun, and the rotating earth result in periodic level changes in large
bodies of water. The vertical rise and fall resulting from these forces
is called the ttde or astronomical ttde; the horizontal movements of
water are called tidal currents. The responses of water level changes to
the tidal forces are modified in coastal regions because of variations in
depths and lateral boundaries; tides vary substantially from place to
place. Astronomical tide generating forces are well understood, and can
be predicted many years in advance. The response to these forces can be
determined from an analysis of tide gage records. Tide predictions are
routinely made for many locations for which analyzed tide observations
are available. In the United States, tide predictions are made by the
National Ocean Survey, National Oceanographic and Atmospheric Administration
Tsunamis are generated by several mechanisms: submarine earthquakes,
submarine landslides, and underwater volcanos. These waves may travel
distances of more than 5,000 miles across an ocean with speeds at times
exceeding 500 miles per hour. In open oceans, the heights of these waves
are generally unknown but small; heights in coastal regions have been
greater than 100 feet.
Setches are long-period standing waves that continue after the
forces that start them have ceased to act. They occur commonly in
enclosed or partially enclosed basins.
Wave setup is defined as the superelevation of the water surface due
to the onshore mass transport of the water by wave action alone. Isolated
observations have shown that wave setup does occur in the surf zone.
Surges are caused by moving atmospheric pressure jumps and by the
wind stress accompanying moving storm systems. Storm systems are signifi-
cant because of their frequency and potential for causing abnormal water
levels at coastlines. In many coastal regions, maximum storm surges are
produced by severe tropical cyclones called hurricanes.
Prediction of water level changes is complex because many types of
water level fluctuations can occur simultaneously. It is not unusual
for surface wave setup, high astronomical tides, and storm surge to occur
coincidently at the shore on the open coast. It is difficult to determine
how much rise can be attributed to each of these causes. Although astro-
nomical tides can be predicted rather well where levels have been recorded
for a year or more, there are many locations where this information is not
available. Furthermore, the interaction between tides and storm surge in
shallow water is not well defined.
3-70
3.81 ASTRONOMICAL TIDES
Tide is a periodic rising and falling of sea level caused by the
gravitational attraction of the moon, sun, and other astronomical bodies
acting on the rotating earth. Tides follow the moon more closely than they
do the sun. There are usually two high and and two low waters in a tidal
or lunar day. As the lunar day is about 50 minutes longer than the solar
day, tides occur about 50 minutes later each day. Typical tide curves for
various locations along the Atlantic, Gulf, and Pacific coasts of the
United States are shown in Figures 3-36 and 3-37. Along the Atlantic
coast, the two tides each day are of nearly the same height. On the Gulf
coast, the tides are low but in some instances have a pronounced diurnal
inequality. Pacific coast tides compare in height with those on the
Atlantic coast but in most cases have a decided diurnal inequality.
(See Appendix A, Figure A-10.)
The dynamic theory of tides was formulated by Laplace (1775) and
special solutions have been obtained by Doodson and Warburg (1941) among
others. The use of simplified theories for the analysis and prediction
of tides has been described by Schureman (1941), Defant (1961) and Ippen
(1966). The computer program for tide prediction, currently being used
for official tide prediction in the United States is described by Pore
and Cummings (1967).
Data concerning tidal ranges along the seacoasts of the United States
are given to the nearest foot in Table 3-5. Spring ranges are shown for
areas having approximately equal daily tides; diurnal ranges are shown
for areas having either a diurnal tide or a pronounced diurnal inequality.
Detailed data concerning tidal ranges are published annually in Tide Tables,
U.S. Department of Commerce, National Ocean Survey.
3.82 TSUNAMIS
Long period gravity waves generated by such disturbances as earth-
quakes, landslides, volcano eruptions and explosions near the sea surface
are called tsunamis. The Japanese word tsunamt has been adopted to replace
the expression tidal wave to avoid confusion with the astronomical tides.
Most tsunamis are caused by earthquakes that extend at least partly
under the sea, although not all submarine earthquakes produce tsunamis.
Severe tsunamis are rare events.
Tsunamis may be compared to the wave generated by dropping a rock in
a pond. Waves (ripples) move outward from the source region in every
direction. In general, the tsunami wave amplitudes decrease but the
number of individual waves increases with distance from the source region.
Tsunami waves may be reflected, refracted, or diffracted by islands, sea-
mounts, submarine ridges or shores. The longest waves travel across the
deepest part of the sea as shallow-water waves, and may obtain speeds of
several hundred knots. The travel time required for the first tsunami
disturbance to arrive at any location can be determined within a few
percent of the actual travel time by the use of suitable tsunami travel-
time charts.
ANCHORAGE
=a)
——
———
=a ao
ae-
==a808
= OFNON NODWOTNON DWoTNOWDWUANON NODOTNODOTFTNOWHOTNON ZTNON
= z ' -=— ee f) MOMONNNNNBX eX KK = ' '
Curves Along Pacific Coasts of the
(2)
(from National Ocean Survey, NOAA, Tide Tables )
Typical Tide
United States
Lunar data: max. S. declination, 9th; apogee, 10th; last quarter, 13th; on equator, 16th, new moon, 20th; perigee,
22d; max. N. declination, 23d.
Figure 3-37.
Table 3-5. Tidal Ranges
Station Approximate Ranges (feet)
Mean | Diurnal | Spring
Atlantic Coast
Calais, Maine
W. Quoddy Head, Maine
Englishman Bay, Maine
Belfast, Maine
Provincetown, Mass.
Chatham, Mass.
Cuttyhunk, Mass.
Saybrook, Conn.
Montauk Point, N.Y.
Sandy Hook, N.J.
Cape May, N.J.
Cape Henry, Va.
Charleston, S.C.
Savannah, Ga.
Mayport, Fla.
Gulf Coast
Key West, Fla
Apalachicola, Fla.
Atchafalaya Bay, La.
Port Isabel, Tex.
MoOawWUNNN fF HOO
Pacific Coast
Point Loma, Calif.
Cape Mendocino, Calif.
Siuslaw River, Ore.
Columbia River, Wash.
Port Townsend, Wash.
Puget Sound, Wash.
Tsunamis cross the sea as very long waves of low amplitude. A wave-
length of 100 miles and an amplitude of 2 feet is not unreasonable. The
wave may be greatly amplified by shoaling, diffraction, convergence, and
resonance when it reaches land. Sea water has been carried higher than
35 feet above sea level in Hilo, Hawaii by tsunamis. Tide gage records
of the tsunami of 23-26 May 1960 at these locations are shown in Figure
3-38. The tsunami appears as a quasi-periodic oscillation, superimposed
on the normal tide. The characteristic period of the disturbance, as well
as the amplitude, is different at each of the three locations. It is
generally assumed that the recorded disturbance results from forced oscil-
lations of hydraulic basin systems, and that the periods of greatest re-
sponse are determined by basin geometry.
3-74
Tide Gage Record Showing Tsunami
HONOLULU, HAWAIL
May 23-24, 1960
Approx. Hours G.M.T.
LOLS 2) eels) 477 15,7 16. 217) 18 19420421922) 123
Tide Gage Record Showing Tsunami
MOKUOLOE ISLAND, HAWAII
May 23-24, 1960
Approx. Hours G.M.T.
10 11 12 #13 14
Tide Gage Record Showing Tsunami
JOHNSTON ISLAND, HAWAII
May 23-24, 1960
Approx. Hours G.M.T.
74° - 515° 16°17" 18) 19° °20 —21 22:23: 0
(from Symons and Zelter, 1960)
Figure 3-38. Sample Tsunami Records from Tide Gages
SST)
Theoretical and applied research dealing with tsunami problems has
been greatly intensified since 1960. Preisendorfer (1971) lists more than
60 significant theoretical papers published since 1960. The list does not
include observational papers concerned with the warning system.
3.83 LAKE LEVELS
Lakes have insignificant tidal variations, but are subject to sea-
sonal and annual hydrologic changes in water level and to water level
changes caused by wind setup, barometric pressure variations, and seiches.
Additionally some lakes are subject to occasional water level changes by
regulatory control works.
Water surface elevations of the Great Lakes vary irregularly from
year to year. During each year, the water surfaces consistently fall to
their lowest stages during the winter and rise to their highest stages
during the summer. Nearly all precipitation in the watershed areas during
the winter is snow or rainfall transformed to ice. When the temperature
begins to rise there is substantial runoff - thus the higher stages in the
summer. Typical seasonal and yearly changes in water levels for Lake Erie
are shown in Figure 3-39. The maximum and minimum monthly mean stages for
the lakes are summarized in Table 3-6.
Table 3-6. Fluctuations in Water Levels — Great Lakes System (1860 through 1973).
Alltime Monthly Means
Datum* Surface Difference
Factor Elevation
Superior 8/1876 4/1926
Michigan-Huron 6/1886 3/1964
St. Clair£ 6/1973 1/1936
Erie 6/1973 2/1936
Ontario 6/1952 11/1934
Elevations are in feet above mean water level at Father Point, Quebec.
International Great Lakes Datum (IGLD) (1955).
* To convert to U.S. Lake Survey 1935 Datum, add datum factor to IGLD (USLS 1935 = IGLD + datum factor),
t Low water datum is the zero plane on Lake Survey Charts to which charts are referred. Thus the zero (low water)
datum on a USLS Lake Superior chart is 600 feet above mean waterlevel at Father Point, Quebec.
# Lake St. Clair elevations are available only for 1898 to date.
In addition to seasonal and annual fluctuations, the Great Lakes are
subject to occasional seiches of irregular amount and duration. These
sometimes result from a resonant coupling which arises when the propaga-
tion speed of an atmospheric disturbance is nearly equal to the speed of
free waves on a lake. (Ewing, Press and Donn, 1954), (Harris, 1957),
Platzman, 1958, 1965.) The lakes, also and sometimes simultaneously, are
affected by wind stresses which raise the water level at one end and lower
it at the other. These mechanisms may produce changes in water elevation
ranging from a few inches to more than 6 feet. Lake Erie, shallowest of
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the Great Lakes, is subject to greater wind-induced surface fluctuations,
that is, wind setup, than any other Lake. Wind setup is discussed in
Section 3.86, Storm Surge and Wind Setup.
In general, the maximum amount of these irregular changes in lake
level must be determined for each location under consideration. Table 3-7
shows short-period observed maximum and minimum water level elevations at
selected gage sites. More detailed data on seasonal lake levels and wind
setup may be obtained for specific locations from the Lake Survey Center,
National Oceanic and Atmospheric Administration, U.S. Department of
Commerce.
Table 3-7. Short-Period Fluctuations in Lake Levels at Selected Gage Sites
Lake and Gage Location Period of Maximum Recorded
Gage Record Rise in Feet | Fall in Feet
(MSE) * (MSE) *
SUPERIOR at Marquette TIOS—1S 7a
MICHIGAN at Calument Harbor 1903-1971
(Chicago)
HURON at Harbor Beach 1902-1971
ERIE at Buffalo 1900-1971
ERIE at Toledo 1940-1971
ONTARIO at Oswego 1935-19771
*The mean surface elevation (MSE) refers to the water level that represents
the average water elevation for the period of record. It corresponds to
the mean surface elevation given in Table 3-6.
3.84 SEICHES
Seiches are standing waves (Fig. 3-40) of relatively long periods
that occur in lakes, canals, bays and along open seacoasts. Lake seiches
are usually the result of a sudden change, or a series of intermittent-
periodic changes, in atmospheric pressure or wind velocity. Standing
waves in canals can be initiated by suddenly adding or subtracting large
quantities of water. Seiches in bays can be generated by local changes
in atmospheric pressure and wind and by oscillations transmitted through
the mouth of the bay from the open sea. Open-sea seiches can be caused
by changes in atmospheric pressure and wind, or tsunamis. Standing waves
of large amplitude are likely to be generated if the causative force which
sets the water basin in motion is periodic in character, especially if the
period of this force is the same as, or is in resonance with, the natural
or free oscillating period of the basin. (See Section 2.5, Wave Reflection.)
3-78
Free oscillations have periods that are dependent upon the horizontal
and vertical dimensions of the basin, the number of nodes of the standing
wave, that is, lines where deviation cf the free surface from its undis-
turbed value is zero, and friction. ‘ihe period of a true forced-wave
oscillation is the same as the period of the causative force. Forced
oscillations, however, are usually generated by intermittent external
forces, and the period of the oscillation is determined partly by the
period of the external force and partly by the dimensions of the water
basin and the mode of oscillation. Oscillations of this type have been
called forced seiches (Chrystal, 1905) to distinguish them from free
seiches in which the oscillations are free.
(B) CLOSED BASIN (C) OPEN-ENDED BASIN
(1). Fundamental Mode (|). Fundamental Mode
(First Harmonic) (First Harmonic)
Lo
r 2
SS
———
(2). Second Mode (2).Second Mode
(Second Harmonic) (Third Harmonic)
Bi
\
Nu
(a) Profile ES
tetProtfrettiodd (3).Third Mode (3). Third Mode
(b) Water Motion F (Third Harmonic) (Fifth Harmonic)
(A) STANDING WAVES
Surface profiles for oscillating waves
(after Carr ,1953)
Figure 3-40. Long-Wave Surface Profiles
For the simplest form of a standing one-dimensional wave in a closed
rectangular basin with vertical sides and uniform depth (Fig. 3-40(B)),
wave antinodes, that is, lines where deviation of the free surface from
its undisturbed value is a relative maxima or minima, are situated at the
ends (longitudinal seiche) or sides (transverse seiche). The number of
nodes and antinodes in a basin depends on which mode or modes of oscilla-
tion are present. If n = number of nodes along a given basin axis, d =
basin depth, and 2g = basin length along that axis, then Ty, the natural
free oscillating period is given by
i . (3-42)
The fundamental and maximum period (T, for n = 1) becomes
fa ; (3-43)
Equation 3-43 is called Merian's formula. (Sverdrup, Johnson and Fleming,
1942.)
In an open rectangular basin of length %g/ and constant depth d,
the simplest form of a one-dimensional, non-resonant, standing longitudi-
nal wave is one with a node at the opening, antinode at the opposite end,
and n’ nodes in between. (See Figure 3-40(C).) The free oscillation
period Tj, in this case is
Alay
1 i a ell 3-44
"(1+ 2n') Jed’ (3249)
For the fundamental mode (n’ = 0), Tj, becomes
Ved
The basin's total length is occupied by one-fourth of a wave length.
This simplified theory must be modified for most actual basins,
because of the variation in width and depth along the basin axes.
Defant (1961) outlines a method to determine the possible periods for
one-dimensional free oscillations in long narrow lakes of variable width
and depth. Defant's method is useful in engineering work because it permits
computation of periods of oscillation, relative magnitudes of the vertical
displacements along chosen axes, and the positions of nodal and antinodal
lines. This method, applicable only to free oscillations, can be used to
determine the nodes of oscillation of multinodal and uninodal seiches.
The theory for a particular forced oscillation was also derived by Defant
and is discussed by Sverdrup et al (1942). Hunt (1959) discusses some
complexities involved in the hydraulic problems of Lake Erie, and offers
an interim solution to the problem of vertical displacement of water at
the eastern end of the lake. More recently, work has been done by Simpson
and Anderson (1964), Platzman and Rao (1963), and Mortimer (1965).
Rockwell (1966) computed the first five modes of oscillation for each of
the Great Lakes by a procedure based on the work of Platzman and Rao (1965).
Platzman (1972) has developed a method for evaluating natural periods for
basins of general two-dimensional configuration.
3.85 WAVE SETUP
Field observations indicate that part of the variation in mean water
level near shore is a function of the incoming wave field. However these
3-80
observations are insufficient to provide quantitative trends. (Savage,
1957; Fairchild, 1958; Dorrestein, 1962; Galvin and Eagleson, 1965.) A
laboratory study by Saville (1961) indicated that for waves breaking on
a slope there would be a decrease in the mean water level relative to the
stillwater level just prior to breaking, with a maximum depression or set-
down at about the breaking point. This study also indicated that from the
breaking point the mean water surface slopes upward to the point of inter-
section with the shore and has been termed wave setup. Wave setup ts
defined as that super-elevation of the mean water level caused by wave
action alone. This phenomenon is related to a conversion of kinetic
energy of wave motion to a quasi-steady potential energy.
Theoretical studies of wave setup have been made by Dorrestein (1962),
Fortak (1962), Longuet-Higgins and Stewart (1960, 1962, 1963, 1964), Bowen,
Inman, and Simmons (1968), and Hwang and Divoky (1970). Theoretical devel-
opments can account for many of the principal processes, but contain
factors that are often difficult to specify in practical problems.
R.O. Reid (personal communication) has suggested the following approach
for estimating the wave setup at shore, using Longuet-Higgins and Stewart
(1963) theory for the setdown at the breaking zone and solitary wave theory.
The theory for setdown at the breaking zone indicates that
gi? HT
~ 640d, 3? ses
Sp
in which Sp is the setdown at the breaking zone, T is the wave period,
H, is the deepwater significant wave height, dz, is the depth of water
at the breaker point and g is gravity. The laboratory data of Saville
(1961) gives somewhat larger values than those obtained by use of Equation
3-46.
By using relations derived from solitary wave theory relating dp
to the breaker height of the significant wave, Hp, and dp/Hp to Ho)/lo,
the above relation can be converted to
0.536 H, 7?
—————— (3-47)
b gll2 T
Longuet-Higgins and Stewart (1963) show from an analysis of Saville's
data that the wave setup AS between the breaker zone and shore is given
approximately by AS = 0.15 dp. Assuming that dp = 1.28 Hp, this becomes
AS = 0.19 Hp.
The net wave setup at the shore is
Sw = 4S +S, (3-48)
H, %
See Odo 1-282() H, . (3-49)
or
Equation 3-49 provides a conservative estimate for wave setup at the
shore. The difference between laboratory data and theory, however, is not
likely to exceed the uncertainties of field data.
Ce ee ey oe ee Ry EXAMPLE PROBLEM HaK OK OK Ke Se ie ey ae ee
GIVEN: Hp = 20 feet, T = 12 seconds.
FIND: Wave setup, Syp-
SOLUTION: Using Equation 3-49,
H, 2
OND i999 4 H,
gr ;
2
Sy = 0.19 |1— 2.82 AUNT 20
i! ; ee N32 (1 D2 ,
Sw = 3.1 feet. Say 3 feet .
Equation 3-49 is only applicable to normal beach slopes.
WR RK CR dee ie de eo OK KR eR eo OOK KR KR OK a Ok ee eS ee ee
3.86 STORM SURGE AND WIND SETUP
3.861 General. Reliable estimates of water-level changes under storm
conditions are essential for the planning and design of coastal engineer-
ing works. Determination of design water elevations during storms is a
complex problem involving interaction between wind and water, differences
in atmospheric pressure, and effects caused by other mechanisms unrelated
to the storm. Winds are responsible for the largest changes in water
level when considering only the storm-surge generating processes. A wind
blowing over a body of water exerts a horizontal force on the water surface
and induces a surface current in the general direction of the wind. The
force of wind on the water is partly due to inequalities of air pressures
on the windward side of gravity waves, and partly due to shearing stresses
at the water surface. Horizontal currents induced by the wind are impeded
in shallow water areas, thus causing the water level to rise downwind
while at the windward side the water level falls. The term storm surge is
used to indicate departure from normal water level due to the action of
storms. The term wind setup is often used to indicate rises in lakes,
reservoirs and smaller bodies of water. A fall of water level below the
normal level at the upwind side of a basin is generally referred to as
setdown.
Severe storms may produce surges in excess of 25 feet on the open
coast and even higher in bays and estuaries. Generally, setups in lakes
and reservoirs are less, and setdown in these enclosed basins is about
equivalent to the setup. Setdown in open oceans is insignificant because
the volume of water required to produce the setup along the shallow regions
of the coast is small compared to the volume of water in the ocean. How-
ever, setdown may be appreciable when a storm traverses a relatively narrow
3-82
landmass such as southern Florida and moves offshore. High offshore winds
in this case can cause the water level to drop several feet.
Setdown in semienclosed basins (bays and estuaries) also may be sub-
stantial, but the fall in water level is influenced by the coupling to the
sea. There are some detrimental effects as a result of setdown, such as
making water-pumping facilities inoperable due to exposure of the intake,
increasing the pumping heads of such facilities, and causing navigational
hazards because of decreased depths.
However, rises in water levels (setup rather than setdown) are of
most concern. Abnormal rises in water level in nearshore regions will not
only flood low-lying terrain, but provide a base on which high waves can
attack the upper part of a beach and penetrate farther inland. Flooding
of this type combined with the action of surface waves can cause severe
damage to low-lying land and backshore improvements.
Wind-induced surge, accompanied by wave action, accounts for most of
the damage to coastal engineering works and beach areas. Displacement of
stone armor units of jetties, groins and breakwaters, scouring around
structures, accretion and erosion of beach materials, cutting of new in-
lets through barrier beaches, and shoaling of navigational channels can
often be attributed to storm surge and surface waves. Moreover, surge can
increase hazards to navigation, impede vessel traffic, and hamper harbor
operations. A knowledge of the increase and decrease in water levels
that can be expected during the life of a coastal structure or project is
necessary to design structures that will remain functional.
3.862 Storms. A storm is an atmospheric disturbance characterized by
high winds which may or may not be accompanied by precipitation. Two
distinctions are made in classifying storms: a storm originating in the
tropics is called a troptcal storm; a storm resulting from a cold and
warm front is called an extratropical storm. Both of these storms can
produce abnormal rises in water level in shallow water near the edge of
water bodies. The highest water levels produced along the entire gulf
coast and from Cape Cod to the south tip of Florida on the east coast
generally result from tropical storms. High water levels are rarely
caused by tropical storms on the lower coast of California. Extreme
water levels in some enclosed bodies, such as Lake Okeechobee, Florida
can also be caused by a tropical storm. Highest water levels at other
coastal locations and most enclosed bodies of water result from extra-
tropical storms.
A severe tropical storm is called a hurricane when the maximum
sustained wind speeds reach 75 miles per hour (65 knots). Hurricane
winds may reach sustained speeds of more than 150 miles per hour (130
knots). Hurricanes, unlike less severe tropical storms, generally are
well organized and have a circular wind pattern with winds revolving
around a center or eye (not necessarily the geometric center). The eye
is an area of low atmospheric pressure and light winds. Atmospheric
pressure and wind speed increase rapidly with distance outward from the
eye to a zone of maximum wind speed which may be anywhere from 4 to 60
3-83
nautical miles from the center. From the zone of maximum wind to the
periphery of the hurricane, the pressure continues to increase; however,
the wind speed decreases, The atmospheric pressure within the eye is the
best single index for estimating the surge potential of a hurricane. This
pressure is referred to as the central pressure index (CPI). Generally
for hurricanes of fixed size, the lower the CPI, the higher the wind speeds.
Hurricanes may also be characterized by other important elements, such as
the radius of maximum winds (R) which is an index of the size of the storm,
and the speed of forward motion of the storm system (Vp). A discussion of
the formation, development and general characteristics of hurricanes is
given by Dunn and Miller (1964).
Extratropical storms that occur along the northern part of the east
coast of the United States accompanied by strong winds blowing from the
northeast quadrant are called northeasters. Nearly all destructive north-
easters have occurred in the period from November to April; the hurricane
season is from about June to November. A typical northeaster consists of
a single center of low pressure and the winds revolve about this center,
but wind patterns are less symmetrical than those associated with hurri-
canes.
3.863 Factors of Storm Surge Generation. The extent to which water
levels will depart from normal during a storm depends on several factors.
The factors are related to the:
(a) characteristics and behavior of the storm;
(b) hydrography of the basin;
(c) initial state of the system; and
(d) other effects that can be considered external to the system.
Several distinct factors that may be responsible for changing water levels
during the passage of a storm may be identified as:
(a) astronomical tides
(b) direct winds
(c) atmospheric pressure differences
(d) earth's rotation
(e) rainfall
(f) surface waves and associated wave setup
(g) storm motion effects.
The elevation of setup or setdown in a basin depends on storm inten-
sity, path or track, overwater duration, atmospheric pressure variation,
speed of translation, storm size, and associated rainfall. Basin charac-
teristics that influence water-level changes are basin size and shape,
and bottom configuration and roughness, The size of the storm relative
to the size of the basin is also important. The magnitude of storm
surges is shown in Figures 3-41 and 3-42. Figure 3-41 shows the differ-
ence between observed water levels and predicted astronomical tide levels
during Hurricane Carla (1961) at several Texas and Louisiana coast tide
stations. Figure 3-42 shows high water marks obtained from a storm survey
made after Hurricane Carla. Harris (1963b) gives similar data from other
hurricanes.
3-84
3.864 Initial Water Level. Water surfaces on the open coast or in en-
closed or semienclosed basins are not always at their normal level prior
to the arrival of a storm. This departure of the water surface from its
normal position in the absence of astronomical tides, referred to as an
intttal water level, is a contributing factor to the water level reached
during the passage of a storm system. This level may be 2 feet above
normal for some locations along the U.S. Gulf coast. Some writers refer
to this difference in water level as a forerunner in advance of the storm
due to initial circulation and water transport by waves particularly when
the water level is above normal. Harris (1963b) on the other hand, indi-
cates that this general rise may be due to short-period anomalies in the
mean sea level not related to hurricanes. Whatever the cause, the initial
water level should be considered when evaluating the components of open-
coast storm surge. The existence of an initial water level preceeding the
approach of Hurricane Carla is shown in Figure 3-41 and in a study of the
synoptic weather charts for this storm. (Harris, 1963b.) At 0700 hours
(Eastern Standard Time) 9 September 1961, the winds at Galveston, Texas
were about 10 mph, but the open coast tide station (Pleasure Pier) shows
the difference between the observed water level and astronomical tide to
be above 2 feet. Rises of this nature on the open coast can also affect
levels in bays and estuaries.
There are other causes for departures of the water levels from normal
in semienclosed and enclosed basins, such as the effects of evaporation
and rainfall. Generally, rainfall plays a more dominant role since these
basins are affected by direct rainfall and can be greatly affected by
rainfall runoff from rivers. The initial rise caused by rainfall is due
to rains preceding the storm; rains during the passage of a storm have a
time-dependent effect on the change in water level.
3.865 Storm Surge Prediction. The design of coastal engineering works is
usually based on a life expectancy for the project and on the degree of
protection the project is expected to provide. This design requires that
the destgn storm have a specified frequency of occurrence for the partic-
ular area. An estimate of the frequency of occurrence of a particular
storm surge is required. One method of making this estimate is to use
frequency curves developed from statistical analyses of historical water
level data. Table 3-8, based on National Ocean Survey tide gage records,
indicates observed extreme storm surge water levels including wave setup
The water levels are those actually recorded at the various tide stations,
and do not necessarily reflect the extreme water levels that may have
occurred near the gages. Values in this table may differ from gage-station
values because of corrections for seasonal and secular anaomalies. The
frequency of occurrence for the highest and lowest water levels may be
estimated by noting the length of time over which observations were made.
The average yearly highest water level is an average of the highest water
level from each year during the period of observation. Extreme water
levels are rarely recorded by water level gages, partly because the gages
tend to become inoperative with extremely high waves, and partly because
the peak storm surge often occurs between tide gage stations. Post-storm
surveys showed water levels, as the result of Hurricane Camille, August
3-85
Pelican Island
r=)
ow Chemical Plant "B"
w Chemical Plant "A"
Freeport
Brazoria Co. Nav. Dist.
13 ie
i
i
?
r
r
Brakes Bayou(Beaumont)
Port Arthur Channel
Naval Base Pier (Orange)
poie oes
Galveston
5
Mud Bayou pas
: Bridge
Colorado River Locks (Matagorda)
Port O'Connor
Turning Basin CE Field Office(Corpus Christi)
vetties Gage (Port Aransas)
10
Hurricane Stage
Tropical Stage meeoos
Frontal Stage — camo mone
© Position at 0700 EST
@ Position at 1900 EST.
>
(from Harris, 1963 b)
Storm Surge and Observed Tide Chart. Hurricane Carla,
7-12 September 1961. Insert Maps for Freeport and
Galveston, Texas, Areas
Figure 3-41.
DEINE
LEGEND
Time Eye of Hurricane
Nearest to Station
September, |96!
September, |961
RANARRNe
COLINAS
Missing data
Mud Bayou Brid
2
5
\
Jetties Gage
Port Aransas)
8F-Observed Tide
4r(
Ca)
-
°
o
wu
ae
9
V
Bet ie Sia [Real aL Shey
12
—
|
N
W ‘ai ‘
seuss
September, |96/
é
==
September, 1961
September, I96I
2 Bayou Rigoud
fecal
jae |
&
|
a
|
A
Z
te)
RSS
EEIRISGS
Te
BEES
Lt teV ss)
[| lets
a ee
oootnnun un nu
4004 4004 4004
(Beaumont)
2} Naval bose bier
(Orange, Texas)
Dow Chemical A
Brakes Bayou
Port Arthur
(Channel)
Van a al
VTL |
1O0f-—-Observed Tide
12; Observed Tide
Hurricane Carla,
Insert Maps for Freeport and
September, 1961
Texas, Areas -- Continued
-41. Storm Surge and Observed Tide Chart.
7-12 September 1961.
Galveston,
o-6y
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Table 3-8. Highest and Lowest Water Levels
Location
ATLANTIC COAST
Eastport, Maine
Portland, Maine
Bar Harbor, Maine
Portsmouth, N.H.
Boston, Mass.
Woods Hole, Mass.
Providence, R.I.
Newport, R.I.
New London, Conn,
Willets Point, N.Y.
Battery, N.Y.
Montauk, N.Y.
Sandy Hook, N.J.
Atlantic City, N.J.
Philadelphia, Pa.
Lewes, Del.
Breakwater Harbor
Baltimore, Md.
Annapolis, Md.
Solomons Island, Md.
Washington, D.C.
Portsmouth, Va.
Observation | Mean | Average | Extreme
Period
1930-69
1912-69
1947-70
1927-70
1922-70
1933-70
1938-47
1957-70
1931-70
1938-70
1932-70
1920-70
1948-70
1933-70
1912-20
1923-70
1900-20
1922-70
1936-70
1902-70
1929-70
1938-70
1931-70
1935-70
Highest Water Levels
Above Mean High Water
(feet)
Date of
Record
Lowest Water Levels
Below Low Water
10.5
5.9
4.1
11
0.9
1.2
2.9
2.8
3.9 4.9 21 Dec 68
2.8 4.0 30 Nov 44
20 Nov 45
3.1 4.3 29 Dec 59
2.8 3.7 30 Nov 44
29 Dec 59
3.0 4.4 29 Dec 59
2.9 9.2 21 Sep 38
3.7 13.1 21 Sep 38
Deh 10.0 21 Sep 38
3.2 8.1 21Sep 38
3.8 9.6 21 Sep 38
2.9 a7 12 Sep 60
3.2 6.6 31 Aug 54
3a 5.7 12 Sep 60
2.8 5.2 14Sep 44
2.2 4.7 25 Nov 50
3.0 5.4 6 Mar 62
2.3 6.7 23 Aug 33
2.2 5.4 23 Aug 33
2.0 3.4 13 Aug 55
2.8 8.4 17 Oct 42
3.0 5.7 18 Sep 36
(feet)
Average | Extreme Date of
Yearly Low Record
Lowest
3.9 4.4 7Jan 43
23 May 59
30 Dec 63
2.8 Sh 7/ 30 Nov 55
2.8 3.6 30 Dec 53
2.7 3.4 30 Nov 55
3.1 3.8 25 Jan 28
24 Mar 40
2.0 2.7 8 Jan 68
2.5 3.4 5 Jan 59
2.1 2.9 25 Jan 36
2.3 3.4 11 Dec 43
3) 4.1 24 Mar 40
3.0 4.0 8 Mar 32
5 Jan 59
1.9 2.6 8 Feb 51
2.9 4.1 31 Dec 62
2.7 3.7 8 Mar 32
hil 6.7 31 Dec 62
2.5 3.0 28 Mar 55
3.4 5.0 24 Jan 08
2.8 3.8 31 Dec 62
2.1 3.4 31 Dec 62
2.9 3.9 31 Dec 62
2.0 2.4 25 Jan 36
4 Dec 42
8 Feb 59
Table 3-8. Highest and Lowest Water Levels — Continued
Location
ATLANTIC COAST
Norfolk, Va.
Sewells Point
Morehead City, N.C.
Wilmington, N.C.
Southport, N.C.
Charleston, S.C.
Fort Pulaski, Ga.
Fernandina, Fla.
Mayport, Fla.
Miami Beach, Fla.
GULF COAST
Key West, Fla.
St. Petersburg, Fla.
Cedar Key, Fla.
Pensacola, Fla.
Grand Isle, La.
Humble Platform
Bayou Rigaud
Eugene Island, La.
Galveston, Tex,
Port Isabel, Tex.
* Diurnal Range
Highest Water Levels
Above Mean High Water
(feet)
Observation | Mean | Average | Extreme
1928-70
1953-57
1935-70
1933-53
1922-70
1936-70
1897-1924
1939-1970
1928-70
1931-51
1955-70
1926-70
1947-70
1914-25
1939-70
1923-70
1949-70
1947-70
1939-70
1908-70
1944-70
Dey *
13%
1.4*
Tides
oe
1.4*
13%
ins)
1.6
De
1.6
its)
1.9
2.5
2.2
1.5
2.5
3.0
3.2
7.6
2.2
2)
6.0
10.1
3.8
Date of
Record
23 Aug 33
15 Oct 54
19 Sep 55
15 Oct 54
2 Nov 47
11 Aug 40
15 Oct 47
2 Oct 1898
9 Sep 64
8Sep 65
8 Sep 65
5 Sep 50
15 Feb 53
11 Sep 64
20 Sep 26
25 Sep 53
22 Sep 56
24 Sep 56
27 Jun 57
16,17 Aug 15
11 Sep 61
Lowest Water Levels
Below Low Water
(feet)
Average | Extreme
Yearly Low
Lowest
1.2 1.6
tcf 2.1
2.8 BS
1.4 2.2
1.4 1.7
11 1.5
1.6 2.4
2.6 oes
1.4 1.7
Date of
Record
23 Jan 28
26 Jan 28
11 Dec 54
3 Feb 40
28 Jan 34
30 Nov 63
20 Mar 36
24 Jan 1940
24Jan 40
24 Mar 36
19 Feb 28
3Jan 58
27 Aug 49
21 Oct 52
4 Feb 63
6 Jan 24
22 Nov 49
3 Feb 51
3 Feb 51
13 Jan 64
25 Jan 40
11Jan 08
31 Dec 56
7 Jan 62
Table 3-8. Highest and Lowest Water Levels — Continued
Location
PACIFIC COASTt
San Diego, Calif.
La Jolla, Calif.
Los Angeles, Calif.
Santa Monica, Calif,
San Francisco, Calif.
Crescent City, Calif.
Astoria, Oreg,
Neah Bay, Wash.
Seattle, Wash.
Friday Harbor, Wash.
Ketchikan, Alaska
Juneau, Alaska
Skagway, Alaska
Sitka, Alaska
Yakutat, Alaska
Seward, Alaska
Kodiak Island, Alaska
Womens Bay
Unalaska Island, Alaska
Dutch Harbor
Adak Island, Alaska
Sweeper Cove
Attu Island, Alaska
Massacre Bay
Period
1906-70
1925-53
1956-70
1924-70
1933-70
1898-1970
1933-46
1950-70
1925-70
1935-70
1899-1970
1934-70
1919-70
1936-41
1944-70
1945-62
1938-70
1940-70
1925-38
1949-63
1934-39
1946-70
1943-70
1943-69
t Tsunami levels not included.
Highest Water Levels
Above Mean High Water
(feet)
High
Date of
Record
Lowest Water Levels
Below Low Water
(feet)
Average | Extreme
5.7
5.2
5.4
5.4
5.7
6.9
8.3
8.2
11.3
7.7
15.3
16.4
16.7
9.9
10.1
10.6
8.8
3.7
3.7
3.3
1.8
1.8
1.8
1.8
1.5
2.4
2.6
2)
2.2
2.3
4.4
5.0
4.9
3.4
Sa
3.4
2.8
2.0
1.8
1.6
Od
1
9|
2.6
2.4
2.3,
Papal
2.3
3e1
3.8
4.1
3.3
352
5.9
6.3
6.7
4.7
4.8
4.1
3.7
2.9
aed,
2.4
20 Dec
29 Dec
20 Dec
19,20 Dec
27 Dec
17,18 Jul
24 Dec
19 Jan
9 Jan
4 Feb
17 Dec
30 Nov
6 Feb
5 Dec
30 Dec
2 Dec
2 Nov
22 Oct
2 Nov
2 Nov
13 Oct
21 Nov
14,15 Jan
28 Dec
6 Jan
68
59
68
68
40
51
52
Date of
Record
17 Dec
17 Dec
17 Dec
26 Dec
17 Dec
17 Dec
26 Dec
17 Dec
22 May
16 Jan
29 Nov
4 Jan
20 Jun
7 Jan
8 Dec
16 Jan
30 Dec
16 Jan
16 Jan
19 Jun
16 Jan
29 Dec
16 Jan
14 Jan
15,16 Jan
13 Nov
11 Nov
12,13 Nov
33
37
33
32
33
33
32
33
55
30
36
16
51
47
19
57
59
57
57
51
57
50
50
50
1969, in excess of 20 feet MSL over many miles of the open Gulf Coast,
with a peak value of 24 feet MSL near Pass Christian, Mississippi. High
water levels in excess of 12 feet MSL on the open coast and 20 feet within
bays were recorded along the Texas coast as the result of Hurricane Carla,
September, 1961. Water levels above 13 feet MSL were recorded in the
Florida Keys during Hurricane Donna, 1960.
Accumulation of data over many years in some areas, such as regions
near the North Sea, has led to relatively accurate empirical techniques
of storm surge prediction for some locations. However, these empirical
methods are not applicable to other locations. In general, not enough
storm surge observations are available in the United States to make accu-
rate predictions of storm surge. Therefore, it has been general practice
to use hypothetical design storms, and to estimate the storm-induced surge
by physical or mathematical models. Mathematical models are usually used
for predicting storm surge, since it is difficult to represent some of the
storm surge generating processes (such as the direct wind effects and
Coriolis effects) in physical laboratory models.
a. Hydrodynamic Equations. Equations that describe the storm surge
generation processes are the continuity equation expressing conservation
of mass and the equations of motion expressing Newton's second law. The
derivations are not presented here; references are cited below. The equa-
tions of motion and continuity given here represent a simplification of
the more complete equations. A more simplified form is obtained by verti-
cally integrating all governing equations and then expressing everything
in terms of either the mean horizontal current velocity or volume trans-
port. Vertically integrated equations are generally preferred in storm-
surge calculations since interest is centered in the free surface motion
and mean horizontal flow. Integration of the equations for the storm
surge problem are given by Haurwitz (1951), Welander (1961), Fortak (1962),
Platzman (1963), Reid (1964), and Harris (1967).
The equations given here are obtained by assuming:
(1) vertical accelerations are negligible,
(2) curvature of the earth and effects of surface
waves can be ignored,
(3) the fluid is inviscid, and
(4) the bottom is fixed and impermeable.
The notation and the coordinate scheme employed are shown schematic-
ally in Figure 3-43. D is the total water depth at time t, and is
given by D=d+S, where d is the undisturbed water depth and S is
the height of the free surface above or below the undisturbed depth re-
sulting from the surge. The Cartesian coordinate axes, x and y, are in
the horizontal plane at the undisturbed water level and the z axis is
directed positively upward. The x axis is taken normal to the shoreline
(positive in the shoreward direction), and the y axis is taken alongshore
(positive to the left when facing the shoreline from the sea).
3-92
Seaward
PROFILE
SE RN BEXAR S78 | Egy.
SN LAND Se Ne
a - OOO SHORELINE ——
W Wy
x
Peron Le! > ww uw
y iseclet SEA hey
PLAN VIEW
Figure 3-43. Notation and Reference Frame
The differential equations appropriate for tropical or extratropical
storm surge problems on the open coast and in enclosed and semienclosed
basins are as follows;
aM aM as a a i i
UPC ar pea eee mee Cat CUTE ny LLL Seo
at ax dy C) t) dx p p
Oe — — —_— —_ ———
Ge o “a o
aq, 5) 4 o =
Se 4 Se PE Ue a Pps eee
one 3 re ces a 2 =
sg g 3) o §& 5 5 S i)
ac 6 FI z 2 g2 Re 3 =
208 5 s =| tS ao & 2 BS
<2 A eS UO geeiee Reatoe ses
aM 3M as a ag G TD
av :
ONE, RY Deh tg Spee? ah Chere i Ok te 2) tra eee
ot oy Ox dy dy dy p p y
as aU aV
ee | —e —P ee (3-52)
S S S
M.. = ‘) u? dz; Myy = i v? dz; M, = it uvdz;
d d a
d
S S
oe i udz; V= | vdz .
d —d
The symbols are defined as:
U, V = x and y components, respectively, of the volume
transport per unit width;
t = time;
Moe» Myy» Mey = momentum transport quantities;
f = 2w sin » = Coriolis parameter;
o504
w = angular velocity of earth
(7.29 x 107° radians/second) ;
o = geographical latitude;
Tg» Tgy = X and y components of surface wind stress;
The» Thy = X and y components of bottom stress;
p = mass density of water;
W,, = x and y components of wind speed;
& = atmospheric pressure deficit in head of water;
t = astronomical tide potential in head of water;
u, v = X and y components, respectively, of current
velocity;
P = precipitation rate (depth/time) ;
g = gravitational acceleration; and
8 = angle of wind measured counterclockwise from
the x axis.
Equations 3-50 and 3-51 are approximate expressions for the equations
of motion and Equation 3-52 is the continuity relation for a fluid of
constant density. These basic equations provide, for all practical pur-.
poses, a complete description of the water motions associated with nearly
horizontal flows such as the storm surge problem. Since these equations
satisfactorily describe the phenomenon involved, a nearly exact solution
can only be obtained by using these relations in complete form.
It is possible to obtain useful approximations by ignoring some terms
in the basic equations when they are either equivalent to zero or are
negligible, but accurate solutions can be achieved only by retaining the
full two-dimensional characteristics of the surge problem. Various sim-
plifications (discussed later) can be made by ignoring some of the physi-
cal processes. These simplifications may provide a satisfactory estimate,
but they must always be considered as only an approximation.
In the past, simplified methods were used extensively to evaluate
storm surge because it was necessary to make all computations manually.
Manual solutions of the complete basic equations in two dimensions were
prohibitively expensive because of the enormous computational effort.
With high speed computers, it is possible to resolve the basic hydro-
dynamic relations efficiently and economically. As a result of computers,
several workers have recently developed useful mathematical models for
computing storm surge. These models have substantially improved accuracy,
and provide a means for evaluating the surge in the two horizontal dimen-
sions. These more accurate methods are not covered here, but are highly
recommended for resolving storm-surge problems where more exactness is
3-95
warranted by the size or importance of the problem. These methods are
recommended only if a computer is available. A brief description of
these methods and references to them follows.
Solutions to the basic equations given can be obtained by the tech-
niques of numerical integration. The differential equations are approxi-
mated by finite differences resulting in a set of equations referred to
as the numerical analogs. The finite-difference analogs, together with
known input data and properly specified boundary conditions, allow evalua-
tion at discrete points in space of both the fields of transport and water
level elevations. Because the equations involve a transient problem,
steps in time are necessary; the time interval required for these steps is
restricted to a value between a few seconds and a few minutes depending on
the resolution desired and the maximum total water depth. Thus solutions
are obtained by a repetitive process where transport values and water-level
elevations are evaluated at all prescribed spatial positions for each time
level throughout the temporal range.
These techniques have been applied to the study of long-wave propa-
gation in various water bodies by numerous investigators. Some investi-
gations of this type are listed below. Mungall and Matthews (1970) devel-
oped a variable-boundary, numerical tidal model for a fjord inlet. The
problem of surge on the open coast has been treated by Miyazaki (1963),
Leendertse (1967), and Jelesnianski (1966, 1967, and 1970). Platzman
(1958) developed a model for computing the surge on Lake Michigan result-
ing from a moving pressure front, and also developed a dynamical wind tide
model for Lake Erie. (Platzman, 1963.) Reid and Bodine (1968) developed
a numerical model for computing surges in a bay system taking into account
flooding of adjacent low lying terrain and overtopping of low barrier
islands.
b. Simplified Techniques for Determining Storm Surge. The tech-
niques described here for the determination of storm surge are simple, and
it is possible to carry out all storm surge calculations manually, using a
desk calculator or slide rule. In most cases, however, it is desirable to
employ a digital computer for the computations to reduce the effort and to
improve accuracy. It is sometimes possible to estimate surge with satis-
factory accuracy using a set of simplified equations, if the particular
problem is not too complex, and if the simplified technique can be verified
from actual prototype field data. Simpler schemes for computing storm
surge are obtained by including only those phenomena that appear signifi-
cant to the investigation; thus some of the less important terms are
omitted from Equations 3-50, 3-51 and 3-52.
(1) Storm Surge on the Open Coast. Ocean basins are large and
deep beyond the shallow waters of the Continental Shelf. The expanse of
ocean basins permits large tropical or extratropical storms to be situated
entirely over water areas allowing tremendous energy to be transferred
from the atmosphere to the water. Wind-induced surface currents, when
moving from the deep ocean to the coast, are impeded by the shoaling
bottom, causing an increase in water level over the Continental Shelf.
3-96
Onshore winds cause the water level to begin to rise at the edge of the
Continental Shelf. The amount of rise increases shoreward to a maximum
level at the shoreline. Storm surge at the shoreline can occur over long
distances along the coast. The breadth and width of the surge will depend
on the storm's size, intensity, track and speed of forward motion as well '
as the shape of the coastline, and the offshore bathymetry. The highest
water level reached at a location along the coast during the passage of a
storm is called the maximum surge for that location; the highest maximum
surge is called the peak surge. Maximum water levels along a coast do not
necessarily occur at the same time. The time of the maximum surge at one
location may differ by several hours from the maximum surge at another
location. The variation of maximum surge values and their time for many
locations along the east coast during Hurricane Carol, 1954, are shown in
Figure 3-44, This hurricane moved a long distance along the coast before
making landfall, and altered the water levels along the entire east coast.
The location of the peak surge relative to the location of the landfall
where the eye crosses the shoreline depends on the seabed bathymetry, wind-
field, configuration of the coastline, and the path the storm takes over
the shelf. For hurricanes moving more or less perpendicular to a coast
with relatively straight bottom contours, the peak surge will occur close
to the point where the region of maximum winds intersects the shoreline,
approximately at a distance R, to the right of the storm center. Peak
surge is generally used by coastal engineers to establish design water
levels at a site.
Attempts to evaluate storm surge on the open coast, and also in bays
and estuaries, when obtained entirely from theoretical considerations,
require verification particularly when simplified models are used. The
surge is verified by comparing the theoretical system response and computed
water levels with those observed during an actual storm. The comparison is
not always simple, because of the lack of field data. Most water-level
data obtained from past hurricanes were taken from high water marks in low-
lying areas some distance inland from the open coast. The few water-level
recording stations along the open coast are too widely separated for satis-
factory verification. Estimates of the water level on the open coast from
levels observed at inland locations are unreliable, since corrective adjust-
ments must be made to the data, and the transformation is difficult.
Systematic acquisition of hurricane data by a number of organizations
and individuals began during the first quarter of this century. Harris
(1963b) presented water-level data and synoptic weather charts for 28
hurricanes occurring from 1926 to 1961. Such data are useful for verifying
surge prediction techniques.
Because of the limited availability of observed hurricane surge data
for the open coast, design analysis for coastal structures is not always
based on observed water levels. Consequently a statistical approach has
evolved that takes into account the expected probability of the occurrence
of a hurricane with a specific CPI at any particular coastal location.
Statistical evaluation of hurricane parameters based on detailed analysis
of many hurricanes, have been compiled for coastal zones along the Atlantic
and Gulf coasts of the U.S. The parameters evaluated were the radius of
acon
attery
Caven Point
Fort Hamilton "Woods Hole
Newport
Montauk
Development Stage ceccceccccce
Tropical Stage eee ee ea
Hurricane Stage rs
Frontal Stage =o a> camo a
O Position at 0700 EST
@ Position at 1900 EST
(from Harris, 1963 b)
Figure 3-44. Storm Surge Chart. Hurricane Carol, 30, 31 August 1951.
Insert Map for New York Harbor
=
3
MS
Biel
Ba
pe
|
Old Point Comfort
——OEEa OD
Hampton Roads
YY
A Kal Ih
\\
/
Ch
RIPAPAPE PSN eee
HGS ts
Aug. 29 Aug. 30 Aug. 3! Sept. |
Se
| | be
Lae
HH
|
Pt
i
LEGEND
he [gee ean ek OLS 690
Part Oh erie is New coheed 1 PANS Missing Dats
Nearest to Station ————— Lh ee PRS
* Offset Due to Datum Lares | mm
Ri taulensd eat Plane Difference Monta ms NY j
p @ - Y a
mA R | pap poste o he Fe Teaien aly Shay
SP ARERR
a ; Ree mae ee
a fet alt ape othe
; ged OE
4 Lt {| | Poel iad a
aoe REE Pry.
Sb Rae 2 Swe
Scale s ° et
‘ pe:
cite tete tt hh = Missing Dota EE
pelos PEIN TTT Lo
sae ers AAR SEA
oft bed
PEEP ERE
pasauecececseeee
Aug.28 Aug.29 Aug.30 Aug. 3! Sept.!
Figure 3-44. Storm Surge Chart. Hurricane Carol, 30, 31 August 1951.
Insert Map for New York Harbor -- Continued
maximum wind R; the minimum central pressure of the hurricanes Po;
the forward speed of the hurricane Vie while approaching or crossing
the coast; and the maximum sustained wind speed W, 30 feet above the
mean water level.
Based on this analysis, the U.S. Weather Bureau (now the National
Weather Service) and U.S. Army Corps of Engineers jointly established
specific storm characteristics for use in the design of coastal struc-
tures. Because the parameters characterizing these storms are specified
from statistical considerations and not from observations, the storms
are termed hypothetical hurricanes or hypo-hurricanes. The parameters
for such storms are assumed constant during the entire surge generation
period. Graham and Nunn (1959) have developed criteria for a design storm
where the central pressure has an occurrence probability of once in 100
years. This storm is referred to as the standard project hurricane (SPH).
The mathematical model used for predicting the wind and pressure fields in
the SPH is discussed in Section 3.72, Model Wind and Pressure Fields for
Hurricanes. The SPH is defined as a "hypo-hurricane that is intended to
represent the most severe combination of hurricane parameters that is
reasonably characteristic of a region excluding extremely rare combina-
tions.'' Most coastal structures built by the U.S. Army Corps of Engineers
that are designed to withstand or protect against hurricanes are based on
design water associated with the SPH.
The construction of nuclear-powered electric generating stations in
the coastal zone made necessary the definition of an extreme hurricane
called the probable maximum hurricane (PMH). The PMH has been adopted by
the Atomic Energy Commission for design purposes to ensure public safety
and the safety of nuclear-power facilities. Procedures for developing a
PMH for a specific geographical location are given in U.S. Weather Bureau
Interim Report HUR 7-97 (1968). The PMH was defined as "A hypothetical
hurricane having that combination of characteristics which will make the
most severe storm that is reasonably possible in the region involved, if
the hurricane should approach the point under study along a critical path
and at an optimum rate of movement."
Selection of hurricane parameters and the methods used for developing
overwater wind speeds and directions for various coastal zones of the
United States are discussed in detail by Graham and Nunn (1959) and in
HUR 7-97 (1968) for the SPH and PMH. The basic design storm data should
be carefully determined, since errors may Significantly affect the final
results.
Two simple methods are presented here for estimating storm surge on
the open coast: one a quasi-static numerical scheme and the other a nomo-
graph method. These methods should never be used for estimating the surge
in bays, estuaries, rivers or in low-lying regions landward of the normal
open-coast shoreline. Neither method is entirely satisfactory for all
cases, but for many problems both appear to give reasonable solutions.
The use of each method is illustrated by estimating the peak open-coast
storm surge for an actual hurricane. The peak surges thus calculated are
compared to the surge computed by a complete two-dimensional numerical
model for the same storm.
3-100
(a) si-Static Method for Prediction of Hurricane Surge.
\This method for determining open-coast storm surge is based on theoretical
approximations of the governing hydrodynamic equations originally proposed
by Freeman, Baer, and Jung (1957). The term quast-statie method is used
here to emphasize that this method should be restricted to slow-moving,
large-scale storm systems. This method is called the Bathystrophte Storm
Tide Theory and, unlike earlier one-dimensional theories, some of the
effects of longshore flow and the apparent Coriolis force are considered.
Such an approximation of the theory can be described as a quasi-static
method in which a numerical solution is obtained by successively integrat-
ing wind stresses over the Continental Shelf from its seaward edge to the
shore for a predetermined interval of time.
This simplified method assumes that storm surge responds instanta-
neously to the onshore wind stresses, advection of momentum can be ignored,
longshore sea surface is uniform, and no flow is assumed normal to the
shore which is treated as a seawall. Barometric effects and precipita-
tion also can be neglected. Setup due to atmospheric pressure difference
can be estimated from another source, and added to the final design water
level. Based on the preceding assumptions, Equations 3-50 and 3-51 reduce
to
T
po fy = (3-53)
ox p
i eee
aye er thy (3-54)
ot p
Conservation of mass is not considered because, (1) there is no flow
perpendicular to shore, (2) the longshore flow is assumed independent of
y and, (3) the water level is assumed slowly changing. The forces (ex-
pressed in mass times acceleration per unit area) involved and the corres-
ponding response of the sea for the bathystrophic approximation are shown
in Figure 3-45. As indicated in the figure, the surface shear force act-
ing in the x-direction t,, and the apparent Coriolis force is balanced
by the hydrostatic pressure force pgA(dS/9x). Moreover, the surface shear
force acting in the y-direction Tt is balanced by the bottom shear
force Thy and the inertial force ~p(dV/dt).
Bretschneider and Collins (1963) developed a computational model
based on this Bathystrophic Theory and applied it to open-coast surge
problems for the region around Corpus Christi, Texas. Marinos and
Woodward (1968) modified the Bretschneider and Collins model and calibra-
ted it for various reaches along the Texas coast by using three hurricanes
of record, and also made parametric studies of hypo-hurricane surges for
the entire coast of Texas.
In some cases the underlying assumptions made in the development of
this theory are not satisfied. Thus, as a consequence of assuming that
the onshore wind stresses cause an instantaneous change of the water level,
i.e., U = 0, the traverse line (the line over the Continental Shelf along
which computations are carried out) must always be taken perpendicular io
3-10!
uotzeutxorddy otydozyskyzeg Loy sasuodsoy pue sedt0q FO ITJeWEYIS “Sp-¢g eINBTYy
awl}
Ayianab
\\ 404DM 40 Ajisuap
Jazawosinod sljol409d
S$Sd4js WOJZOog 40 UauodWoOd-K
YY peeds puim 40 Syuauodwod A‘x="M
So peeds Pulm
(xX 40) Y4PIM 4lun
N \ jad jsodsud4s4, 4a,0mM }O yuauodwod-A
> S$SO14S PUIM 4O Sjuauodwod A‘x=
N
SS ———(] yydap |D4sOL
MS Mojaqg yidep
dnyas
| ai0us 40 dnyas |O4OL
= ]@Aa7 J9sOMIINES
|
(
‘uolpOpuasaidas [o1404did
49109)9 0 BAIB Of payso4sip
uaeq SADY $e@|D2S SNOI4IDA :31LON
As, «xs
'
6
a = 'd
+
bi
Ky, eX
"
> 3
= A
ot at
" "
vu a
2 1s
= IMS
7QN3937
3-102
the bottom contours for valid computations. The above assumption also
implies that there is no flooding landward of the shore; thus, there is
a deficiency in the method when substantial flooding occurs.
The bottom contours of the actual seabed are rarely straight and para-
llel; however, the traverse line can often be oriented so that it is nearly
perpendicular to the contours in an average way. For complex offshore
bathymetry, such an approximation would be invalid. For storms moving
more or less perpendicular to a coastline, the traverse line can be taken
through, or anywhere to the right of, the region of maximum winds, but
never to the left of this region. Many other factors such as the angle
of approach of a storm, the coastline configuration, and inertial effects,
limit the use of such a simple approach.
The computation model given here is based on Bathystrophic Storm Tide
Theory as described by Bodine (1971). Although Bodine applied both manual
and digital computer calculation methods to the open coast storm surge
problem, only the manual method is presented.
The bottom and surface shear stresses are assumed to vary according
to:
7b KVIVI
OE
pas kay (bottom shear stress) (3-55)
T
— = kW? cos @
(wind shear stress) (3-56)
in which K is a dimensionless bottom friction coefficient, k isa
dimensionless surface friction coefficient, W is the wind speed, and
6 is the angle between the x-axis and the local wind vector. The bottom
friction coefficient K is related to the coefficient of Chezy C and
the Darcy-Weisbach friction factor fr as follows:
=2=- (3-57)
Typical bottom conditions result in a value of K that lies in the
range between 2 x 1073 and 5 x 1073. For a first estimate, a value
of K = 2.5 x 1073 may be assumed. This coefficient is used in cali-
brating the model. It not only accounts for energy dissipation at
the bed, but may be used to adjust for inexact modeling and deficien-
cies caused by ignoring some of the hydrodynamic processes involved.
3-103
The wind stress coefficient is based on one given by Van Dorn (1953) and
others which is assumed to be a function of wind speed; thus,
keep een for W < W, (3-58)
1
~k
W 2
(e
Kaw ces Ke i for W > Ww. (3-59)
where the constants kK, and Ky are usually taken to be 1.1 x 10-© and
2.5 x 10°, respectively. W, is a critical wind speed taken as 14 knots
(16 miles per hour).
Introducing the bottom and free-surface stress relations into
Equations 3-53 and 3-54 gives
as = au [fV + kW? cos 0] (3-60)
Ox gD
av ?
oN Eevee (3-61)
ot D
an estimate of the amount of setup that can be attributed to the onshore
effects. The setup attributed to longshore effects can be obtained by
rewriting Equation 3-60 in the following two-component form:
aS, aa kW? cos 0 (3-62)
Ox gD
OS yaar te (3-63)
ox gD
The total setup along the x-axis is the sum of the two components or,
bd + ar». (3-64)
In the finite-difference numerical solution of the reduced equations,
values of S, and S, are evaluated at points spaced Ax apart along a
Single Cartesian axis (the traverse line). The values of S, the total
setup for the increment, can be regarded as being the water level midway
between two points along the traverse line, midway between x and x + Ax
at a time t. The longshore volume transport of water V is also evaluate
between points. Wind stress data and the Coriolis parameter are supplied
at the points x and x + Ax. The subscripts and superscripts i and n
3-104
are used to denote discrete points in space and time, respectively. The
quantities Ax and At are allowed to be nonuniform spacings along the
traverse line and time increments, respectively. The finite difference
forms of Equations 3-61, 3-62, and 3-63 are then given as follows:
nt1 Ax n+1
(AS. ie Pe (A; + A;4 1)
2gDi 44, (3-65)
met ae n+1
AST eg ree WE Ee): Vise (3-66)
2gDi 4%
1
= +1 n
(5) [e+ Beal + (B+ Bo)" At + Vix
ee (3-67)
oe 1 + Kivi \e
it, it
where A and B are the kinematic forms of the wind stress given by:
A = kW’ cos@ (3-68)
B = kW’ sin@ (3-69)
The time step specified by the ordinal number n represents a time
level at which AS,, AS and V are known, while n+ 1 represents the
new time level for which the quantities ASy, AS, and V are to be
determined.
The total water depth at the mid-interval between two time steps is
given by:
n n+1
ee ay + pitas 3
Dim 2 43° 2 33 (S, + Spey:
(3-70)
1
+(F)( lisse) (Sap) i+1]” ‘i [Sap : (Sap)iei]"**)
where
S. = initial setup
Sq = astronomical tide
SAp = atmospheric pressure setup.
3-105
The initial setup refers to the water level at the time the storm surge
computations are started. An approximate relationship giving the atmos-
pheric pressure setup in feet when pressure is expressed in inches of
mercury is
Sap = 114 (p,—p) (1-e**) (3-71)
where p, is the pressure at the periphery of the storm, and r is the
radial distance from the storm center to the computation point on the
traverse line.
The total depth at the new time level is evaluated by the relation:
nti _ « i+ 1
Di+¥ 7
+1 n
+ §, + S074 (SoS, Ney
(3-72)
+ : [(Sap); + (Sap) ea)"
The total water level rise at the coast is the sum of all component water
level changes resulting from the meteorological storm plus those components
which are not related to the storm. Hence the total setup is given by
Sp = St Sy Sap ft Se ah we op (3-73)
where Sy is a component due to the wave setup in the region shoreward of
the breaker line and is related to the breaker height by Equation 3-49.
(See Section 3.85, Wave Setup.) The setup component Sz accounts for the
setup resulting from local conditions such as bottom configuration, coast-
line shape, or other flows influencing the system, such as flows from bay
inlets or rivers. This component can only be estimated from a full under-
standing of the influence of topographic and hydrographic features not
considered in the numerical calculations. Contributions made by the
various setup components to the total surge are shown in Figure 3-46.
_ Equations 3-65, 3-66, and 3-67 can be rewritten in a more useful form
by introducing dimensional constants for terms frequently appearing in the
equations. Hence,
(48,) Fes AP ASS gh (3-74)
Di4¥%
C, Ax
(48,) fy = as (sin), + (sine); 4] View (3-75)
ity
1
View = (| (8; + Beas)” + (B+ Bias)"*7] ae Vie (3-76)
n =4 nt+%
i + Cy | Ving lAbK Den
3-106
Elevation in Feet above MSL Datum
Bottom Depth (Feet- MSL)
(0) =
Continental Shelf
100 oe
LEGEND
200
Breaking Wave Setup
300 S, = x-Component Setup
Sy = y- Component Setup
400 Sap= Atmospheric Pressure Setup
S, = Astronomical Tide
500
Design Water Level Including
Surface- Wave Setup
a
iia
ial:
om
<a
es
=
ge |
aad
a:
eer
ee
as eee
=
ames
beagle
ee
ep |
See
Cie:
Ss
—==
Mean Sea Level (MSL)
MCE
S_ = Initial Water Level
400 200 (0) 10 20 30 40 50 60
Distance in Yards Distance from Coast (Nautical Miles )
(from Bodine,1971)
Figure 3-46. Various Setup Components Over the Continental Shelf
3-107
The values of the dimensional constants C,, Cy, and C3 in Equations
3-74, 3-75 and 3-76 depend on the units used in performing the calculations.
Table 3-9 gives the dimensions of the variables used in the numerical
scheme in three systems of units and the corresponding value of the
constants for each system. The first column of units is given in the
metric system while the other two are given in mixed units in the English
system.
Table 3-9. Systems of Units for Storm Surge Computations
Units and Constant Values
Parameters - ; ;
nm mi
ft eye
ft/sec? ft/sec?
iF IEE
(km/hr) 2 (nm/hr) 2 (mi/hr) 2
km2/hr nm2/hr mi2/hr
hr7! hr7! hr7!
hr hr
269
141
(1000) 2 (6080) 2
The assumptions for this model are reasonably good only when the
momentum of the water is increasing under the influence of the wind, i.e.,
when the right side of Equation 3-76 is positive. This condition can be
considered in the calculations by using the relation
(3-77)
For a derivation of this equation see Bodine (1971). Should V ata
particular time level, as evaluated from Equation 3-76, be less than the
value obtained from Equation 3-77, then the result from Equation 3-77 is
ignored. However, if the opposite is true, then as an estimate, V is
taken equal to the maximum possible value as given by Equation 3-77.
3-108
It is customary to assume that the system is initially in a state of
equilibrium and V = 0, and S is uniform along the traverse line at the
start of the computational scheme. Thus, computations should be initiated
for conditions prior to the arrival of strong winds over the Continental
Shelf. Although the real system would seldom, if ever, be in a complete
initial state of equilibrium, errors in assuming it to be are of little
consequence in the computational scheme after several time steps in the
calculations, because the effects of the forcing functions will eventually
predominate.
To demonstrate the computational procedures, the storm surge in the
Gulf of Mexico resulting from Hurricane Camille (1969) is calculated.
Hurricane Camille was an extremely severe storm that crossed the eastern
part of the Gulf of Mexico with the eye of the storm making landfall at
Bay St. Louis, Mississippi, at about 0500 Greenwich Mean Time (GMT) on
18 August 1969. Unusually high water levels were experienced along the
gulf coast during the passage of Camille because of the intense winds and
relatively shallow water depths which extend far offshore. The storm surge
is calculated and compared with the peak surge generated by Camille.
Information published by the U.S. Weather Bureau for Hurricane Camille
in HUR 7-113 (1969) indicates that R = 14 nautical miles and Vp = 13 knots
would be representative of these values which are assumed to be invariant
while the storm moved over the Continental Shelf. The wind data and track
for Hurricane Camille have been published by the Weather Bureau in HUR
7-113A (1970). The track, together with the traverse line used in the
present calculations, is shown in Figure 3-47. Overwater wind speeds and
directions are shown in Figure 3-48. A profile of the seabed along the
traverse line is shown in Figure 3-49.
Tide records from the region affected by the storm show the mean water
level to be about 1.2 feet above normal before being affected by the storm.
This value is taken to represent the initial water level. The range of
astronomical tides in this location is about 1.6 feet. A constant value of
0.8-foot above MSL is used in the computations; the final surge hydrograph
at the coast can be subsequently corrected to account for the predicted
variations of the tide. Variations in the initial water level and astro-
nomical tides may be added algebraically to the storm surge calculations
without seriously affecting the final results. The atmospheric pressure
difference, a is needed for evaluating the pressure setup com-
ponent, S,,. Data given in HUR 7-113 (1969) suggest that p, = 26.73
inches of mercury and p, = 29.92 inches of mercury are representative for
the hurricane. All of these values are assumed to remain constant for the
calculations.
The wind data, basin profile, and hurricane characteristics provide
the basic information needed in making an estimate of the peak storm surge
associated with Hurricane Camille. The time intervals, distance increments
along the traverse line, etc., are given in the computational steps to
follow.
3-109
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{oot
ps
=
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: 1
i}
028 | 088
== oles =~ (5)
068
3-110
69 8ny LI — LW 00ZI 2)
OTT Tue) ouvotsriny
‘(sqzouy) STOAOS] 9oeFINS OOY-0E “8P-¢F PINSTY
(OL6T ‘VETI-Z “UNH
‘uoneoyqnd neoing Joyiwany ‘S$’ wIOJ})
3-III
2100 GMT — 17 Aug 69
EB
1800 GMT — 17 Aug 69
D
S=ht2
0300 GMT — 18 Aug 69
G
0000 GMT — 18 Aug 69
F
30-Foot Surface Isovels (knots), Hurricane Camille -- Continued
Figure 3-48.
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eRe eae ee eect IL { } oH —
ag zi Hi isiziele! Le 1 lal
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3-114
This quasi-linear computational scheme can be used for manual com-
putations; however, since the calculations are repetitive, they can be
performed more efficiently by using a digital computer. When carried out
manually, the technique is laborious, tedigus and subject to error.
Bodine (1971) gives a computer program based on the numerical scheme
presented here. Other programs based on similar numerical schemes using
the quasi-static methods have been developed.
When manual computation of storm surge is necessary, a systematic,
tabular procedure must be adopted to permit stepping through all of the
discrete computational points in space for each time increment. Table
3-10 represents a recommended procedure. One table is required for each
time increment. Table 3-10 corresponds to the time of peak surge for
Hurricane Camille; preceding tables required to bring the calculations to
this point are not included since those calculations are similar. The
first table in the series must reflect the initial conditions; thus V is
taken to be zero and S is assumed uniform over the system.
Manual surge calculations for Hurricane Camille give a peak surge of
25.03 feet, say 25 feet (MLW) or 24.2 feet (MSL). The bottom friction
coefficient selected for this particular example was K = 0.003 and the
surge was found to be insensitive to small changes in the friction coeffi-
cient. Computer calculations using a friction coefficient of 0.003 result-
ed in a peak surge of 25.19 feet and a bottom friction coefficient of
0.0025 resulted in a peak surge of 25.40 feet. For some basins and storm
systems, the bottom shear stresses are more significant in determining
water levels. Therefore, it is important to select a bottom friction
coefficient by verification (i.e., by comparing calculated results with
observed water levels). After such verification, the model may be used
to estimate the storm surge from hypothetical hurricanes for the same
geographical region.
The surge hydrograph (water level as a function of time) for Hurricane
Camille is shown in Figure 3-50 for the most landward computational point
on the traverse line. This figure shows that the water level rose for
about the first 8 hours, but then began to fall gradually until about 27
hours of computational period had elapsed, then began to rise rapidly.
A study of the local wind fields during this period shows that the winds
had an onshore component in the early stages of the storm, then the winds
began blowing offshore for several hours before the principal rise at the
coast.
(b) Nomograph Method. A simplified method for obtaining
a first approximation to the peak storm surge of a hurricane can be based
on an empirical analysis of past records, an empirical analysis of a sys-
tematic set of calculations with numerical models, or a combination of the
two. Jelesnianski (1972) combined empirical data from Harris (1959) with
his theoretical calculations to produce a set of nomograms that permit the
rapid estimation of peak surge for any geographical location when a few
parameters characterizing a storm are known.
The first nomogram (Fig. 3-51) permits an estimate of the peak surge
S7, generated by an idealized hurricane with specified CPI and radius of
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ge a abs Bde mbs.
Figure 3-5I. Preliminary Estimate of Peak Surge
2-ts
maximum wind that is moving perpendicular to a shoreline with a speed of
15 MPH. This nomogram indicates there is a critical storm size as reflec-
ted by the radius of maximum winds, R. For a given pressure drop greater
than zero, the highest peak surge is produced for a critical value of R
equal to 30 miles and any value of R_ greater or less than this value
results in a lesser value of the peak surge.
A second factor Fg given in Figures 3-52 and 3-53 adjusts for the
effects of variations in bathymetric characteristics along the gulf and
Atlantic coasts. A third factor Fy given in Figure 3-54, adjusts for
the effects of storm speed and the angle with which the storm track
intercepts the coast.
The predicted peak storm surge Sp is then given by
55 esr Ss Eng (3-78)
Jelesnianski (1972) applied the scheme to the 43 storms given by Harris
(1959) that entered land south of New England during the period from 1893
to 1957. The peak surges reported by Harris are plotted against the peak
surges predicted by the nomograph method in Figure 3-55. The two-
dimensional hurricane model and storm surge prediction model described by
Jelesnianski (1967) was used for all calcualtions without adjustment for
local variations in friction coefficient or other efforts to calibrate
the model for individual storms. For many of the hurricanes, the post-
storm surveys conducted were of limited scope and probably did not disclose
the true peak surge. Thus, at least a part of the spread between observed
and computed values must be due to the observed data. In addition to the
peak surge, other nomograms for computing other storm surge parameters are
given by Jelesnianski (1967).
An example problem illustrating the use of the nomogram method follows:
FTE HE KE TEI IR RES KR PKS Fe. EXAMPLE PROBLEM Fe Re eR Te OR I A eR ea ee
GIVEN: Parameters for Hurricane Camille are:
Ap = 3.19 inches of mercury (in. Hg.)
Vr = 13 knots
R = 14 nautical miles (n.m.)
FIND: Estimate peak open-coast surge produced by Hurricane Camille.
3-20
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(Jelesnianski , 1972)
Shoaling Factors on East Coast
Figure 3-53.
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S123
(from Jelesnianski, 1972)
Figure 3-54. Correction Factor for Storm Motion
(TSW aAogp 4904) °s ‘abung yDaq pansasqg
16
(Feet above MSL)
Computed Peak Surge, Sp
. Comparison of Observed and Computed Peak Surges
3-55
Figure
(for 43 storms with a landfall south of New England
from 1893-1957)
3-124
SOLUTION: Because of the units used in the various nomograms, the units
of the parameters are converted to
_ 3.19 (1000)
Ap = 3.19iin: He. 39.53
= 108 millibars .
Note: 29.53 inches of Hg. ~ 1000 millibars
Vp
13 knots © 14.5 mph
R 14n.m. © 15.6 miles.
Using Figure 3-51 with values of Ap and R,
S; © 22 feet (MSL)
Based on the approximate landfall location (west of Biloxi) of
Hurricane Camille (Fig. 3-47), the shoaling factor ys is determined
from Figure 3-52.
F, © 1.23 .
To evaluate the correction for storm motion Fy the angle of storm
approach to the coast wp must. be first determined. The definition of
the angle w is shown schematically in the insert in Figure 3-54.
From Figure 3-47 wp is estimated to be 102°. From Figure 3-54 for
y = 102° and Vp = 14.5 mph,
Fy © 0.97 .
The peak surge given by Equation 3-78
S, = S;Fs Fy
S, = (22) (1.23) (0.97)
S, = 26.2 feet, say 26 feet (MSL)
Seo
Jelesnianski (1972) has calculated the open coast surge for
Hurricane Camille with a full two-dimensional mathematical model. The
maximum surge envelope along the coast, based on computations from this
model, is shown in Figure 3-56 where the zero distance corresponds to
the point of landfall of the hurricane eye. This figure shows that
the peak surge estimated by this method is about 25+ feet MSL.
Thus, by three independent estimates, it has been found that the peak
surge is about 25 feet MSL which corresponds approximately to that
observed (U.S. Geological Survey) at Pass Christian, Mississippi of
24.2 feet MSL. It would be expected that a slightly higher peak water
elevation occurred because Pass Christian is located a few miles left of
the position where the maximum winds made landfall.
It is rare that such a close agreement is found when estimating the
peak surge with these dissimilar models. Normally, because of the
difference in these predictive schemes, it can be expected that peak
surge estimates may deviate by as much as 25 percent. For well-
formulated schemes properly applied, there is usually a trade-off
between reliability of the estimate and the computational effort.
Ct Ee ee I Se See i Se Ee Te Na Te ee ae Pt hy GY SSeS
(c) Predicting Surge for Storms other than Hurricanes.
Although the basic equations for water motion in response to atmospheric
stresses are equally valid for nonhurricane tropical and extratropical
storms, the structures of these storms are not nearly so simple as that
of a hurricane. Because the storms display much greater variability in
structure, it is difficult to derive a proper wind field. Moreover, no
system of classifying these storms by parameters has been developed
similar to hurricane classification by such parameters as radius to
maximum winds, forward motion of the storm center, and central pressure.
Criteria however have been established for a Standard Project North-
easter for the New England coast north of Cape Cod as given by Peterson
and Goodyear (1964). Specific standard-project storms other than hurri-
canes are not presently available for other coastal locations. Estimates
of design-storm wind fields can be made by meteorologists working directly
with climatological weather maps, and by use of statistical wind records
on land and assuming that they blow toward shore for a significant dura-
tion over a long, straight line fetch.
Once the wind field is determined, estimation of the storm surge may
be determined by methods based on the complete basic formulas or the quasi-
static method given. The nomogram method cannot be used, since this scheme
is based on the hurricane parameters.
(2) Storm Surge in Enclosed Basins. An example of an inclined
water surface caused by wind shearing stresses over an enclosed body of
water occurred during passage of the hurricane of 26-27 August 1949 over
the northern part of Lake Okeechobee, Florida. After the lake level was
inclined by the wind, the wind direction shifted 180° in 3 hours, resulting
3-126
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ef
(4984) Syy61aH abing w40ys pajoipaid
= Ira
3
in a turning of the height contours of the lake surface. However, the
turning of the contours lagged behind the wind so that for a time the
wind blew parallel to the water level contours instead of perpendicular
to them. Contour lines of the lake surface from 1800 hours on 26 August
to 0600 hours on 27 August 1949 are shown in Figure 3-57. The map con-
tours for 2300 hours on 26 August show the wind blowing parallel to the
highest contours at two locations. (Haurwitz, 1951), (Saville, 1952),
(Sibul, 1955), (Tickner, 1957), and (U.S. Army, Corps of Engineers, 1955) .)
Recorded examples of wind setup on the Great Lakes are available from
the U.S. Lake Survey, National Ocean Survey, and NOAA. These observations
have been used for the development of theoretical methods for forecasting
water levels during approaching storms and for the planning and design of
engineering works. As a result of the need to predict unusually high
stages on the Great Lakes, numerous theoretical investigations have been
made of wind setup for that area. (Harris, 1953), (Harris and Angelo,
1962), (Platzman and Rao, 1963), (Jelesnianski, 1958), (Irish and
Platzman, 1962), and (Platzman, 1958, 1963, 1965, and 1967).)
Water level variations in an enclosed basin cannot be estimated satis-
factorily if a basin is irregularly shaped, or if natural barriers such as
islands affect the horizontal water motions. However, if the basin is
simple in shape and long compared to width, then water level elevations
may be reasonably calculated using the hydrodynamic equations in one
dimension. Thus if the motion is considered only along the x-axis (major
axis), and advection of momentum, pressure deficit, astronomical effects
and precipitation effects are neglected, then Equations 3-50 and 3-52
reduce to
ateees tse
aes g ae (75 Tp) (3-79)
aS- ey, SOU (3-80)
ot Ox
If it is further assumed that steady state exists, then Equation 3-79 becomes
dSiyvai fel
AS = eat (7, + 7) (3-81)
The bottom stress is taken in the same direction as the wind stress, since
for equilibrium conditions the flow near the bottom is opposite to flow
induced by winds in the upper layers. Theoretical development of this
wind setup equation was given by Hellstrom (1941), Keulegan (1951), and
others. The mechanics of the various determinations have differed some-
what, but the resultant equation has been about the same. This wind
setup equation is expressed as:
k’np ,W? F
MSS SS — OBL) 3-82
a (3-82)
3-128
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(GS61°3 30 9 Awsy sn)
ee 4z 9nvy 0090
or °
sam ginivie
VoVAVA
WovAVA vovAVA J
sod vod ‘Luod
wReY O2B0dxX] 40 A¥VONNOS
DART V3S_ mvam “1904
&} BBHOLMOD advsUNe anv
wnoK wad
BOK BHA BWO4sD!
OOlMRd BINH OF HOS ALIDOTBA
OAV MOlAIBuIO GNM JevuRAV
“VS M'Laas MI MOLLWADIa BDvaUNe
UBAVA OMIMOHE'NOLLVLE OMIOVE
ON3937
VoWAVA
iuod
92 9Nv oOoB!
VovAVA Ze =) 7 é vovAvE
Auoe
S-le9
where
k’ = a numerical constant = 0.003
Pq = air density
W = Wind velocity
F = Fetch length
8 = angle between wind and the fetch
me = et tes] (bse 1 Nore ns 1.30
AS = Wind setup. This value represents
the difference in water level
between the two ends of the fetch
D = average depth of the fetch.
An approximate expression of Equation 3-82 is given by
CW?F
S=
cos 6 (3-83)
where C is a coefficient having dimensions of time squared per unit
length. Saville (1952) in a comprehensive investigation of setup data
obtained from Lake Okeechobee found that C is approximately 1.165 x 1073
when W is given miles per hour, F in miles and D and AS in feet.
This coefficient is almost identical with that of the Netherlands Zuider
Zee formula (1.25 x 1073).
Equation 3-83 is often useful in making the first approximation of
the setup in an enclosed basin. Its advantage is that the setup can be
evaluated with fewer computations. The surge can be estimated more satis-
factorily by segmenting an enclosed basin into reaches and using a numeri-
cal integration procedure to solve Equation 3-81 for the various reaches
in the basin. Bretschneider (Ippen, 1966) presented solutions in parameter
form for Equation 3-81, and compiled these solutions in tables for different
conditions. These tables can be used to estimate the storm surge for a
rectangular channel of constant depth with either an exposed bottom or a
nonexposed bottom and a basin of regular shape. Such solutions may provide
an estimate of the water level variation in some basins.
A more complete scheme than those described above follows. This
method accounts for the time dependence of the problem; more specifically
Equations 3-79 and 3-80 are used. Although this more complete method
provides a better approximation of the surge problem, it is done at the
expense of increasing the computations.
3-130
When a basin enclosed with vertical sides has a well-defined major
axis and a gradually varying cross section, it is possible to use a
dynamic one-dimensional, computational model for evaluating fluctuations
in water level resulting from a forcing mechanism such as wind stress,
and also to account for some of the effects of a varying cross section.
The validity of such a model depends on the behavior of the storm system
and the geometric configuration of the basin.
Equations 3-79 and 3-80, when the varying width b of the basin is
introduced, can be written in terms of the volume flow rate Q(x,t) as
aQ as
ar eee (1 ') (3-84)
ey UC) (3-85)
ot b dx
where x is taken along the major axis of the basin, and for any time t,
A is the cross-section area, and D is the average depth A/b or
D=d+S.
Various schemes have been proposed for evaluating the water level
changes in an enclosed body of water by using the differential equations.
(Equations 3-84 and 3-85.) The formulation of the problem and the numer-
ical scheme given here is from Bodine, Herchenroder and Harris (1972).
The surface stress and bottom stress terms are taken identical to the
terms given for the quasi-static method for open-coast surge. (See
Equations 3-55 and 3-56, Section 3-865b(1)(a).) The stress terms, in
terms of the volume flow rate, become
T _ KQIQI (3-86)
p (Db)?
$s
x = kWW, = kW? cos 6 (3-87)
Substituting Equations 3-86 and 3-87 into Equation 3-84 gives
dQ dS __ KQIQ|
ee DW W. 9 a (Ae ee =
ot wh Ba D*b (S88)
3-13!
A finite difference representation of Equations 3-88 and 3-85 may be
expressed in the form
+1 1 At +1
Qe1 = G lot + > (biey + bj43/2) (kW" cos O)i+t
2.
(3-89)
iS yy +A a4) (Srey ~ Sin34).
2Ax itz it 3/2 it it 3/y
n+1 _ qn PACT: zs. +1
Siem = Stay Tt Tee (Q — G+)” (3-90)
1 2
where,
AKAt | Q”,, |
G=1+ ies Sa eu MDT he Bote (3-91)
(Disw +Diap)” (disw + diva)
The value of G is greater than unity for most flow conditions except
for the case when the flow vanishes (Q = 0). The subscripts and super-
scripts i and n are used to denote discrete points in space and time,
respectively. A schematic of the grid system used is shown in Figure 3-58.
It is seen that S at the new time level (t + At) is first evaluated
based on the known values of Q, D, S, A and b(x) lying on triangle (1)
and subsequently followed by an evaluation of S at the new time level
based on known values lying on triangle (2). The solutions at successive
time levels are obtained by a marching process with Q evaluated at the
new time level for all integer steps along x; S is evaluated for all
mid-integer steps along x. Width as a function of x, b(x), is taken
constant for all t and the total depth D is permitted to vary with
time. Thus the cross-section area of the basin A is a function of
distance along the major axis of the basin and a function of time.
The computational scheme is a combined boundary and initial value
problem. At each end of the basin, it is assumed that there is no flow
across the boundary, thus Q=0 at the boundary. The initial conditions
assumed are that Q=0 and S_ is uniform throughout the basin.
The scheme requires that for numerical stability, the time increment
specified for successive calculations At be taken less than Ax//gDmazx,
where Dmqax is the maximum depth (d+ S) anticipated in the basin during
passage of a storm system. (Abbott, 1966.)
The restriction imposed by the criterion for numerical stability
results in a trade-off between the resolution obtained in the solution
and the number of calculations involved. Decreasing Ax gives better
resolution of the surge, but requires a smaller At and increases the
number of computational steps. It is important to choose Ax small
enough so that reasonable resolution of the surge is obtained, but large
enough to reduce the computations. The choice of a Ax depends on the
problem involved,
Selse
we3skS PIN “gg-¢ emNsTY
(2261 ‘'S!4J0H Ppuod JeposUaYdIEH ‘aulpog wod})
XV (2%/_ +!)
XV(I+!)
1V
i Wy
IL TK
Tee
XV (2, +!)
JIONVIYL
xv! BCE 7 =!)
Sn
XV(I-!)
4V (1-¥)
$Vu
#V(1+U)
SA }ey)
For such a recursive type calculation, computer computations reduce
the effort required. However, since this is a one-dimensional formulation
of the problem, it is possible to carry out the necessary computations
manually.
The March 1955 storm on Lake Erie is used to demonstrate the scheme.
A plan view of Lake Erie is shown in Figure 3-59. The width and cross-
section area of the Lake are shown in Figure 3-60, mean bottom profile in
Figure 3-61, and wind and wind direction data for the March 1955 storm in
Figure 3-62.
In the following example problem computations are made at a single
spatial point where Q is evaluated at a distance of 3Ax from Buffalo
(Fig. 3-59) at a time when the wind speeds are approximately maximum.
Tables similar to Table 3-10 should be used in manual calculations.
C2 eS et Ch ea Me ro A? RP CP O39 EXAMPLE PROBLEM Ko FN OR OR ae ok ieee
GIVEN: Ax = 1Oemales
Me 3S Wierd hese
g = 79000 mi/hr2
K = 0.003
The wind at the new time level is 50.5 miles per hour and 6 = Lig(
The corresponding water surface widths for this section are bj+1/2 = 26.3
miles and bj43/2 = 21.0 miles. The values from preceding calculations at
the previous time level are
Oe = 0.0707 mi3/hr.
Se =i, 5e52¢ ft.
n
Mays = 0.426 mi2
n -2
A43/2 = 0827 8ami'<
n 0
Dipy/2 = 0-0162 mi.
n é
Di43/2 = 0-0132 mi.
Also required is the value of ope
is given as +0.0915 mi3/hr.
from the previous spatial step which
3-134
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9 Lud
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3NY087109 180d
Luod
3-135
Section Area ( 10® Square Feet)
26
Sosvaanaaeeee E 5 = eee) cones bocce cuuetos:
= Peete Eau a : f 5
ra EE ae ao t
a) Peet me Tt ewe 23 ae
wa t + Peet 7 t ot : 4
Ht a : : :
Hoe } -- Ett
+ i Hee
t aa i r m0 } rm ia
i tot r oe = ot i i pao! i Ht i t
Li. if im a i stot im TTI
rt tt HH T mt tt Sept feeb :
ae ! H ei street ase fouted eetoeepiaseetoeet
t i ++ i H Cet + 1 4
stitesonts EE cpistanestestartitestaat puibitei Leet
Ht : : as +f
: t - - - :
HH im gaaeah Vaal jug Gaves Gageaeq (saugsesqa (2 5 =H
im T t ct a 2 im i tT
Toy + Tey is BEE
t +. TH 1t t ioe 1
+ Ey — + tt Het + a
i T T L Be T Tt +
H Ht cet =EN : tet Sean Boar - Et i
‘men r im 1] a i r Tt at Tr TT
i aaeo i tt i T a ra ae Pee i BBs
— a a
— 3
;
HH
-
:
1
ee
: 4
+ aan
; : — rH
: : : :
Ea Ht nae +
SEE
i
CLet
ol
t =
80
100 120 140
160
Distance from Toledo (Statute Miles)
i im for ; ; (e)
180 200 220 240 260
(after Platzman and Rao, 1963)
Figure 3-60. Cross-Section Area and Surface Width—Lake Erie
3-136
Section Width (105Feet )
7 as aa fee
Ht H zi
" +E it
: : + aa _
$ T i bit
10 : fe : : : -
: : egeuaes cae: HH
t ae: SE oe: imeem ae! one
20} fri riunirrinrnsnimriindie
ce PERSE
= cH aeeS EES
30 7 SeLEiitoees HE
T a ae a5 Saunt Guana
40 : ae ape saeiotartt eaietet
i t : t san pooaaeeet H+ t
: + > cose see aH —
50 a oud aetestasesseseeinesaatasten’ ci
a t : Pat gene beees
ert aeoaena cL
Depth ( Feet)
roa)
ro)
|
N
jo)
ase : He
Ht Ht :
aS
80F =
t TH
BbecieCereseesescsrsee a booed buses teeds babes teeed (ese oop a IB SEOR ESR aS Ba:
; tHe : Eeeet Sos osone oe Gad Seo eseoe eee ee at oa a 1 : it +e eet
90-— ge avanenend stotesozes ugzs czars tazestate:seezszeztatatzzazezt ini tats mitepsaats
80
Distance from Toledo ( Statute Miles)
Figure 3-61. Mean Bottom Profile of Lake Erie
3= 34%
GG6| YOIOW ‘W4O}S — 2113 @YD7 40y UOI}DaIIG pud paads pulm “ZQg-¢ a4nbI4
(sanoH) awit
psec puce ysl2
002! O0vd 008!
TT ial 7 ’ 1] (0) — 0081! -
1
+H
= 0lsi=
asiMyI0(9
= G0E)=
=O02=
0)
(‘H'd'W) paads pulm
(s9016aq ) a4D7 40 SIXiy YyIM ajbuy
asIMyo0|919}uUN0D
|
°
°
ro)
— 0021
— 00S!
Meas DO Bg ed
09 — 008!
3-138
FIND: Wind setup at the new time level.
SOLUTION: The wind stress coefficie:t given by Equation 3-59 is
Wolfe: LieX 105° +-2.5 X 100° oule
Saeed so = = 1. A 50.5
=a2.27 ee 10-°
From Equation 3-91
4KAt | Q?, , |
Soe e aha 2 2 n
(Di4% + Diz3/.)” (diay + b+)
4 (0.003) (0.2) | 0.0707
at (0.003) (0.2) | 0.0707 | zy
(0.0162)? + (0.0132)7] (26.3 + 21.0)
The two terms needed in the evaluation of Cig (Equation 3-89) are
given by
n+1
= (bi+% + bi4 3p) (kw? cos Ore
0.2
= Zi (26.3 + 21.0) (2.27 X 107°). (50.5)? (1) = 0.0274
and
gAt
wax (Aith t Aisa) (Sie 7 Sizan)”
_ 79,000 (0.2) (0.426 + 0.278) (3.32 —3.76) _
(2) (10) (5,280) r
— 0.0463
The volume flow rate from Equation 3-89 is
‘OMe co [0.070 + 0.0274 — 0.0463] = 0.0513 mi? /hr.
“OT
3=139
and finally, the water level at this discrete point in space and time
is given by Equation 3-90 as
n+1 n At
5 = een Q =O ae
nh Fh T AS by (Q — Qes
(0.2) (0.0915 — 0.0513) (5280)
(10) (26.3)
tl
Shove ae
3.48 feet, say 3.5 feet .
The significant digits indicated in the above computations do not
reflect the accuracy of the numerical procedure, but are retained to
reduce the accumulation of round-off errors.
The wind setup hydrograph for the ends of the lake as determined by
computer is shown in Figure 3-63. The storm winds blew in the general
direction toward Buffalo at the northeastern end of the lake as indi-
cated by the setdown at Toledo and the setup at Buffalo. The spatial
steps Ax taken are quite large; smaller increments in Ax would give
a more accurate estimate. Calculations with the mathematical model
were initiated with the system in a calm state, i.e., Q=S=0.
The wind setup profile along the major axis of the lake determined
by the numerical scheme is shown in Figure 3-64 for three time periods
during the storm. The nodal point where the computed water surface
crosses the stillwater level occurs near the center of the lake, but
this nodal point can vary with time.
Although the comparison of the computed and observed wind setup is
not in complete agreement, particularly at the beginning and late stages
of the storm, the method gives reasonable results for the wind setup
amplitudes. To engineers, it is frequently the maximum departure of
the water level from its normal position that is of greatest concern.
Results of the simplified model should be interpreted with care, since
many of the physical processes which may be significant have been neg-
lected. Wind and bottom stress laws, in particular, are oversimplified
for the Lake Erie problem. Better agreement can be expected with two-
dimensional schemes such as the one developed by Platzman (1963), since
they more accurately model for the physical processes involved.
Ce a eh et ee Ot HO te Pe Te ee te er UT eS
(3) Storm Surge in Semienclosed Basins. It is generally im-
possible to make reliable estimates of storm surge in semienclosed basins
(bays, and estuaries) with less exact procedures such as those described
for specific problems within enclosed basins or on the open coast. This
is because bays and estuaries are nearly always irregular in shape, and
basin geometry is often further complicated by the presence of islands,
Navigational channels, and harbors. Moreover, many of these basins have
3-140
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extensive low-lying areas surrounding them which may be subject to exten-
sive flooding during severe storm conditions. Also, these basins are
usually shallow, and on the windward side of the basin substantial bottom
areas can become exposed. Thus, during the passage of a storm system,
many semienclosed basins have time-dependent moving boundaries. The
water present at any time in a bay or estuary is also dependent upon the
sea state because of the interrelation between these two bodies of water.
Because of the above characteristics of semienclosed basins, a simplified
approach does not usually provide a satisfactory estimate of water motions,
and methods based on full two-dimensional dynamic approaches should be
employed.
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3-154
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3-156
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3-157
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CHAPTER 4
LITTORAL
PROCESSES
NORTH CAROLINA COAST — 13 February 1973
CHAPTER 4
LITTORAL PROCESSES
4.1 INTRODUCTION
Littoral processes result from the interactions among winds, waves,
currents, tides, sediments, and other phenomena in the littoral zone. The
purpose of this chapter is to discuss those littoral processes which involve
sediment motion. Shores erode, accrete, or remain stable, dependéng on the
rates at which sediment is supplied to and removed from the shore. Excessive
erosion or accretion may endanger the structural integrity or functional
usefulness of a beach, or other coastal structures. Therefore an under-
standing of littoral processes is needed to predict erosion or accretion
rates. An aim of coastal engineering design is to maintain a stable shore-
line where sediment supplied to the shore balances that which is removed.
This chapter presents information needed for understanding the effects of
littoral processes on coastal engineering design.
4.11 DEFINITIONS
In describing littoral processes, it is necessary to use clearly
defined terms. Commonly used terms, such as "beach" and "shore," have
specific meanings in the study of littoral processes, as shown in the
Glossary. (See Appendix A.)
4.111 Beach Profile. Profiles perpendicular to the shoreline have char-
acteristic features that reflect the action of littoral processes. (See
Figure 1-1, Chapter 1, and Figures A-1 and A-2 of the Glossary for spe-
cific examples.) At any given time, a profile may exhibit only a few
specific features, but usually a dune, berm and beach face can be identi-
fied.
Profiles across a beach adapt to imposed wave conditions in ways as
illustrated in Figure 4-1 by a series of profiles taken between February
1963 and November 1964 at Westhampton Beach, N.Y. The figure shows how
the berm built up gradually from February through August 1963, then was
cut back in November through January, and was rebuilt in March through
September 1964. This process is typical of a cyclical process of storm
erosion in winter and progradation with the lower, and often longer,
waves in summer.
4.112 Areal View. Figure 4-2 shows three generalized charts of different
U.S. coastal areas, all to the same scale. Figure 4-2a shows a rocky coast,
well-indented, where sand is restricted to local pocket beaches. Figure
4-2b shows a long straight coast with an uninterrupted sand beach. Figure
4-2c shows short barrier islands interrupted by inlets. These are some of
the different coastal configurations which reflect differences in littoral
processes and the local geology.
4.12 ENVIRONMENTAL FACTORS
4,121 Waves. Water waves are the principal cause of most shoreline
changes. Without wave action on a coast, most of the coastal engineering
Datum = MTL = Mean Tide Level
Initial Survey 11 October 1962
\W - MTL-Beach Intersect
(Surveys did not always reach MTL)
—
BS VO Ol GG an
Breyey (este
5 Nov 64
——_ IO Sep 64
13 Jul 64
—— || May 64
—— [2 ihe fc:
——} I dem
=e dela. (3c
—— 118 Weir Ge
———— NOVO
Oct 63
Sep 63
S57) aes
ELZEVE | AA at qe nee reek | eleemn
|
——— lB Ae) SS)
————
———— JORVul Gs
—— 9 May 63
—— 24 Apr 63
20 9 Apr 63
2| Mar 63
12 Mar 63
2! Feb 63
al 7 Feb 63
-300-200 -1I00 O 100 200 300
Distance from Mean Water Line of Initial Survey, feet
Elevation, feet
ro)
MTL
o--
Figure 4-1. Typical Profile Changes with Time, Westhampton
Beach, New York
4-2
Rhode Island
.--- (See USC&GS Chart 1210)...
a. Rocky Coast with Limited Beaches
Northeastern Florida
(See USC & GS Chart 1246)
OCEAN
b. Straight Barrier Island Shoreline
Southern New Jersey
(See USC & GS Chart 1217)
BAY
OCEAN Scale in Miles
TTS ga
c. Short Barrier Island Shoreline
Figure 4-2. Three Types of Shoreline
4-3
problems involving littoral processes would not occur. A knowledge of
incident waves, or of surf, is essential for coastal engineering planning,
design, and construction.
Three important aspects of a study of waves on beaches are: the
theoretical description of wave motion; the climatological data for waves
as they occur on a given segment of coast; and the description of how
waves interact with the shore to move sand.
The theoretical approach can provide a useful description of water
motion caused by waves when the limiting assumptions of the theory are
satisfied. Surprisingly, the small-amplitude theory (Section 2.23) and
aspects of solitary wave theory (Section 2.27) have proved useful beyond
the limits assumed in their derivations.
The theoretical description of water-wave motion provides estimates
of water motion, longshore force, and energy flux due to waves. These
estimates are useful in understanding the effect of waves on sediment
transport, but currently (1973) the prediction of wave-induced sediment
motion for engineering purposes relies heavily on empirical coefficients
and judgement rather than on theory.
Statistical distributions of wave characteristics along a given shore-
line provide a basis for describing the wave climate of a coastal segment.
Important wave characteristics affecting sediment transport near the beach
are height, period, and direction of breaking waves. Breaker height is
Significant in determining the quantity of sand in motion; breaker direc-
tion is a major factor in determining longshore transport direction and
rate. Waves affect sediment motion in the littoral zone in two ways:
they initiate sediment movement, and they drive current systems that trans-
port the sediment once motion is initiated.
4.122 Currents. Water waves induce an orbital motion in the fluid beneath
them. (See Section 2.23.) These are not closed orbits, and the fluid
experiences a slight wave-induced drift, or mass transport. Magnitude and
direction of mass transport are functions of elevation above bottom and
wave parameters (Equation 2-55), and are also influenced by wind and tem-
perature gradients. The action of mass transport, extended over a long
period, can be important in carrying sediment onshore or offshore, par-
ticularly seaward of the breaker position.
As waves approach breaking, wave-induced bottom motion in the water
becomes more intense, and its effect on sediment becomes more pronounced.
Breaking waves create intense local currents (turbulence) that move sedi-
ment. As waves cross the surf zone after breaking, the accompanying fluid
motion is mostly uniform horizontal motion, except during the brief pass-
age of the breaker front where significant turbulence occurs. Since wave
crests at breaking are usually at a slight angle to the shoreline, there
is usually a longshore component of momentum in the fluid composing the
breaking waves. This longshore component of momentum entering the surf
4-4
zone is the principal cause of longshore currents - currents that flow
parallel to the shoreline within the surf zone. These longshore currents
are largely responsible for the longshore sediment transport.
There is some mean exchange between the water flowing in the surf
zone and the water seaward of the breaker zone. The most easily seen of
these exchange mechanisms are the rip currents (Shepard and Inman, 1950),
which are concentrated jets of water flowing seaward through the breaker
zone.
4.123 Tides and Surges. In addition to wave-induced currents, there are
other currents affecting the shore that are caused by tides and storm
surges. Tide-induced currents can be impressed upon the prevailing wave-
induced circulations, especially near entrances to bays and lagoons and
in regions of large tidal range. (Notices to Mariners and the Coastal
Pilot often carry this information.) Tidal currents are particularly
important in transporting sand in shoals and sand waves around entrances
to bays and estuaries.
Currents induced by storm surges (Murray, 1970) are less well known
because of the difficulty in measuring them, but their effects are un-
doubtedly significant.
The change in water level by tides and surges is a significant factor
in sediment transport, since, with a higher water level, waves can then
attack a greater range of elevations on the beach profile. (See Figure
1-7.) The appropriate theory for predicting storm surge levels is dis-
cussed in Section 3.8.
4,124 Winds. Winds act directly on beaches by blowing sand off the
beaches (deflation) and by depositing sand in dunes. (Savage and Wood-
house, 1968.) Deflation usually removes the finer material, leaving
behind coarser sediment and shell fragments. Sand blown seaward from
the beach usually falls in the surf zone; thus it is not lost, but is
introduced into the littoral transport system. Sand blown landward from
the beach may form dunes, add to existing dunes, or be deposited in
lagoons behind barrier islands.
For dunes to form, a significant quantity of sand must be available
for transport by wind, as must features that act to trap the moving sand.
Topographic irregularities, the dunes themselves, and vegetation are the
principal features that trap sand.
The most important dunes in littoral processes are foredunes - the
line of dunes immediately landward of the beach. They usually form be-
cause beachgrasses growing just landward of the beach will trap sand blown
landward off the beach. Foredunes act as a barrier to prevent waves and
high water from moving inland, and provide a reservoir of sand to replen-
ish the nearshore regime during severe shore erosion.
The effect of winds in producing currents on the water surface is
well documented, both in the laboratory and in the field. (Bretschneider,
4-5
1967; Keulegan, 1951; and van Dorn, 1953.) These surface currents drift
in the direction of the wind at a speed equal to 2 to 3 percent of the
wind speed. In hurricanes, winds generate surface currents of 2 to 8
feet per second. Such wind-induced surface currents toward the shore
cause significant bottom return flows which may transport sediment sea-
ward; similarly, strong offshore winds can result in an offshore surface
current, and an onshore bottom current which can aid in transporting
sediment landward.
4.125 Geologic Factors. The geology of a coastal region affects the
supply of sediment on the beaches and the total coastal morphology. Thus,
geology determines the initial conditions for littoral processes, but geo-
logic factors are not usually active processes affecting coastal engineer-
ing.
One exception is the rate of change of sea level with respect to
land which may be great enough to influence design, and should be exam-
ined if project life is 50 years or more. On U.S. coasts, typical rates
of sea level rise average about 1 to 2 millimeters per year, but changes
range from -13 to +9 millimeters per year. (Hicks, 1972.) (Plus means
a rise in sea level with respect to local land level.)
4.126 Other Factors. Other principal factors affecting littoral processes
are the works of man and activities of organisms native to the particular
littoral zone. In engineering design, the effects on littoral processes
of construction activities, the resulting structures, and structure main-
tenance must be considered. This consideration is particularly necessary
for a project that may alter the sand budget of the area, such as jetty
or groin construction. In addition biological activity may be important
in producing carbonate sands, in reef development, or (through vegetation)
in trapping sand on dunes.
4.13 CHANGES IN THE LITTORAL ZONE
Because most of the wave energy is dissipated in the littoral zone,
this zone is where beach changes are most rapid. These changes may be
short-term due to seasonal changes in wave conditions and to occurrence
of intermittent storms separated by intervals of low waves, or long-term
due to an overall imbalance between the added and eroded sand. Short-term
changes are apparent in the temporary redistribution of sand across the
profile (Fig. 4-1); long-term changes are apparent in the more nearly per-
manent shift of the shoreline. (See Figures 4-3, 4-4, and 4-5.)
Maximum seasonal or storm-induced changes across a profile, such as
those shown in Figure 4-1, are typically on the order of a few feet verti-
cally and from 10 to 100 feet horizontally. (See Table 4-1.) Only during
extreme storms, or where the available sand supply is restricted, do un-
usual changes occur over a short period.
Typical seasonal changes on southern California beaches are shown in
Table 4-1. (Shepard, 1950.) These data show greater changes on the beach
4-6
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4-9
Table 4-1. Seasonal Profile Changes on Southern California Beaches
Locality
Marine Street
Beacon Inn
South Oceanside
San Onofre
Surf Beach
Fence Beach
Del Mar
Santa Monica Mountains
Point Mugu
Rincon Beach
Goleta Beach
Point Sur
South Side
North Side
Carmel Beach (South)
Point Reyes
Scripps Beach
Range A
Range B
Range C
Range D
Range E
Range F
Range G
Range H
Scripps Pier
Vertical * Horizontal
(ft at MWL (ft.)
5.9
8.2 iLiL7/
5.0 41
4.2 43
10.8 87
3.4 —
6.2 a
6.0 ——
Poff -——
0.5 —
1.8 —-
6.3 —
6.7 -——-
Bl ———
1.6 88
SEZ 240
5.6 260
5.3 250
3.3 ——
11,5) 67
0.7 15
1.6 44
3.8 33
3.6T 30T
4.3 38
Seah B2it
Des 28
3.41 34F
i672 12
A § ia;
2.9 30}
1.9t 16t
at
bet )onp | at MWE Ce)
27 Nov 45— 5 Apr 46
2 Nov 45—19 Mar 46
2 Nov 45—26 Apr 46
6 Nov 45—12 Apr 46
6 Nov 45—12 Apr 46
2 Nov 45—10 May 46
28 Aug 45—13 Mar 46
28 Aug 45—13 Mar 46
22 Aug 45—13 Mar 46
22 Aug 45—13 Mar 46
27 Aug 45—14 Mar 46
26 Aug 45—14 Mar 46
26 Aug 45—16 Mar 46
19 Nov 45—29 Apr 46
7 Nov 45—10 Apr 46
7 Nov 45—10 Apr 46
7 Nov 45— 9 May 46
7 Nov 45—24 Apr 46
7 Nov 45—10 May 46
19 Oct 45—10 Apr 46
7 Nov 45—24 Apr 46
13 Oct 37—26 Mar 38
26 Mar 38—30 Aug 38
30 Aug 38—13 Feb 39
13 Feb 39—22 Sep 39
22 Sep 39—24 Jan 40
24 Jan 40—18 Sep 40
18 Sep 40—16 Apr 41
16 Apr 41—17 Sep 41
17 Sep 41—29 Apr 42
29 Apr 42—30 Sep 42
(from Shepard 1950)
* Vertical erosion measured at berm for all localities except Scripps Beach and
Scripps Pier where the mean water line (MWL) was used.
+ Accretion values.
than are typical of Atlantic coast beaches. (Urban and Galvin, 1969;
Zeigler and Tuttle, 1961.) Available data indicate that the greatest
changes on the profile are in the position of the beach face and of the
longshore bar - two relatively mobile elements of the profile. Beaches
change in plan view as well. Figure 4-6 shows the change in shoreline
position at seven east coast localities as a function of time between
autumn 1962 and spring 1967.
Comparison of beach profiles before and after storms suggests ero-
sion of the beach above MSL from 10,000 to 50,000 cubic yards per mile
of shoreline during storms expected to recur about once a year. (DeWall,
et al., 1971; and Shuyskiy, 1970.) While impressive in aggregate, such
sediment transport is minor compared to longshore transport of sediment.
Longshore transport rates may be greater than 1 million cubic yards per
year.
The long-term changes shown in Figures 4-3, 4-4, and 4-5 illustrate
shorelines of erosion, accretion, and stability. Long-term erosion or
accretion rates are rarely more than a few feet per year in horizontal
motion of the shoreline, except in localities particularly exposed to
erosion, such as near inlets or capes. Figure 4-5 indicates that shore-
lines can be stable for a long time. It should be noted that the erod-
ing, accreting, and stable beaches shown in Figures 4-3, 4-4, and 4-5
are on the same barrier island within a few miles of each other.
Net longshore transport rates along ocean beaches range from near
zero to 1 million cubic yards per year, but are typically 100,000 to
500,000 cubic yards per year. Such quantities, if removed from a 10- to
20-mile stretch of beach year after year, would result in severe erosion
problems. The fact that many beaches have high rates of longshore trans-
port without unusually severe erosion suggests that an equilibrium condi-
tion exists on these beaches, in which the material eroded is balanced by
the material supplied; or in which seasonal reversals of littoral trans-
port replace material previously removed.
4.2 LITTORAL MATERIALS
Littoral materials are the solid materials (mainly sedimentary) in
the littoral zone on which the waves and currents act.
4.21 CLASSIFICATION
The characteristics of the littoral materials are a primary input
to any coastal engineering design. Median grain size is the most fre-
quently used design characteristic.
4.211 Size and Size Parameters. Littoral materials are classified by
grain size into clay, silt, sand, gravel, cobble, and boulder. Several
Size classifications exist, of which two, the Unified Soil Classification,
(based on the Casagrande Classification) and the Wentworth Classification,
are most commonly used in coastal engineering. (See Figure 4-7.) The
4-II
Position of Mean Sea Level Shoreline (+ Seaward, -Landward), in feet
Figure 4-6.
Misquamicut Beach, ne
Jones Beach, N.Y.
Ludlam Island,N.J.
Long Beach Island, N.J.
Fluctuations in Location of Mean Sealevel
Shoreline on Seven East Coast Beaches
Unified Soil Classification is the principal classification used by engi-
neers. The Wentworth classification is the basis of a classification
widely used by geologists, but is becoming more widely used by engineers
designing beach fills.
For most shore protection design problems, typical littoral mate-
rials are sands with sizes between 0.1 and 1.0 millimeters or, in phi
units, between 3.3 and 0 phi. According to the Wentworth classification,
sand size is in the range between 0.0625 and 2.0 millimeters (4 and -1
phi); according to the Unified Soil Classification, it is between 0.074
and 4.76 millimeters (3.75 and -2.25 phi). Within these sand-size ranges,
engineers commonly distinguish size classes by median grain size measured
in millimeters, based on sieve analyses.
Samples of typical beach sediment usually have a few relatively
large particles covering a wide range of diameters, and many small par-
ticles within a small range of diameters. Thus, to distinguish one
sample from another, it is necessary to consider the small differences
(in absolute magnitude) among the finer sizes more than the same differ-
ences among the larger sizes. For this reason, all sediment size classi-
fications exaggerate absolute differences in the finer sizes compared to
absolute differences in the coarser sizes.
As shown in Figure 4-7, limits of the size classes differ. The
Unified Soil Classification boundaries correspond to U.S. Standard Sieve
Sizes. The Wentworth classification varies as powers of 2 millimeters;
that is, the size classes have limits, in millimeters, determined by the
relation 2°, where n is any positive or negative whole number, including
zero. For example, the limits on sand size in the Wentworth scale are
0.0625 millimeters and 2 millimeters, which correspond to 2 * and 2}
millimeters.
This property of having class limits defined in terms of whole number
powers of 2 millimeters led Krumbein (1936) to propose a phi unit scale
based on the definition:
Phi units (6) = — log, (diameter in mm.) (4-1)
Phi unit scale is indicated by writing $ or phi after the numerical
value. The phi unit scale is shown in Figure 4-7. Advantages of phi
units are:
(a) Limits of Wentworth size classes are whole numbers in phi
units. These pas limits are the negative value of the exponent, n, in
the relation 2". For example, the sand size class ranges from +4 to -1,
in phi units.
(b) Sand size distributions typically are near lognormal, so that
a unit based on the logarithm of the size better emphasizes the small
significant differences between the finer particles in the distribution.
4-13
Wentworth Scale Phi Units Grain HS: Séuidard Unified Soil
Diameter Classification
(Size Description) Ptaay gl eye eeee (USC)
Cobble
64.0
19.0
4.76
4.0
2.0
1.0
0.5
0.42
0.25
0.125
0.074
0.0625
0.00391 Silt or Clay
0.00024.
oO ealOE. d (mm)
Figure 4-7. Grain Size Scales (Soil Classification)
(c) The normal distribution is described by its mean and standard
deviation. Since the distribution of sand size is approximately lognormal,
then individual sand size distributions can be more easily described by
units based on the logarithm of the diameter rather than the absolute diam-
eter. Comparison with the theoretical lognormal distribution is also a
convenient way of characterizing and comparing the size distribution of
different samples.
Of these three advantages, only (a) is unique to the phi units. The
other two, (b) and (c), would be valid for any unit based on the logarithm
of size.
Disadvantages of phi units are:
(a) Phi units increase as absolute size in millimeters decreases.
(b) Physical appreciation of the size involved is easier when the
units are millimeters rather than phi units.
(c) The median diameter can be easily obtained without phi units.
(d) Phi units are dimensionless, and are not usable in physically
related quantities where grain size must have units of length such as
grain-size, Reynolds number, or relative roughness.
Size distributions of samples of littoral materials vary widely.
Qualitatively, the size distribution of a sample may be characterized by
a diameter that is in some way typical of the sample, and by the way that
the sizes coarser and finer than the typical size are distributed. (Note
that size distributions are generally based on weight, rather than number
of particles.)
A size distribution is described qualitatively as well-sorted if all
particles have sizes that are close to the typical size. If all the par-
ticles have exactly the same size, then the sample is perfectly sorted.
If the particle sizes are distributed evenly over a wide range of sizes,
then the sample is said to be well-graded. A well-graded sample is poorly
sorted; a well-sorted sample is poorly graded.
The medtan (Mg) and the mean (M) define typical sizes of a sample of
littoral materials. The median size, Mg in millimeters, is the most com-
mon measure of sand size in engineering reports. It may be defined as
My = de, (4-2)
where ds5gq is the size in millimeters that divides the sample so that half
the sample, by weight, has particles coarser than the ds5g size. An equiv-
alent definition holds for the median of the phi size distribution, using
the symbol Mjg instead of Mj.
Several formulas have been proposed to compute an approximate mean
(M) from the cumulative size distribution of the sample. (Otto, 1939;
4-15
Inman, 1952; Folk and Ward, 1957, McCammon, 1962.) These formulas are
averages of 2, 3, 5, or more symmetrically selected percentiles of the
phi frequency distribution, such as the formula of Folk and Ward.
%i6 + bso + bq
Mg = ime To (4-3)
where $ is the particle size in phi units from the distribution curve
at the percentiles equivalent to the subscripts 16, 50 and 84 (Fig. 4-8);
¢, is the size in phi units that is exceeded by x percent (by dry weight)
of the total sample. These definitions of percentile (after Griffiths, 1967,
p.- 105) are known as graphic measures. A more complex method - the method
of moments - can yield more precise results when properly used.
To a good approximation, the median, Mg is interchangeable with the
mean, (M), for most beach sediment. Since the median is easier to deter-
mine it is widely used in engineering studies. For example, in one CERC
study of 465 sand samples from three New Jersey beaches, the mean computed
by the method of moments averaged only 0.01 millimeter smaller than the
median for sands whose average median was 0.30 millimeter (1.74 phi).
(Ramsey and Galvin, 1971.)
The median and the mean describe the approximate center of the sedi-
ment size distribution. In the past, most coastal engineering projects
have used only this size information. However, for more detailed design
calculations of fill quantities required for beach restoration projects
(Sections 5.3 and 6.3), it is necessary to know more about the size dis-
tribution.
Since the actual size distributions are such that the log of the size
is approximately normally distributed, the approximate distribution can be
described (in phi units) by the two parameters that describe a normal dis-
tribution - the mean and the standard deviation. In addition to these
two parameters (mean and standard deviation), skewness and kurtosis
describe how far the actual size distribution of the sample departs from
this theoretical lognormal distribution.
Standard deviation is a measure of the degree to which the sample
spreads out around the mean and may be approximated using Inman's (1952)
definition, by
iyo Sr (4-4)
where $g, is the sediment size, in phi units, that is finer than 84 per-
cent by weight, of the sample. If the sediment size in the sample actually
has a lognormal distribution, then Sy is the standard deviation of the
4-16
Cumulative Percent Coarser
MENACE EE
PERDUE OOGERDOGHRRDGOREUTERRREDEOGHRAOORERRORGREUSERRERORORERE? GUN’ |
CO PEER
EECEEEEEEEECEEEE EEA ee HEHEHE
To GRRE LEREREBERSE ’ aaa a
fe T_16te¢ 4 zs Teiste SSEESSESESSSsSeseeees”, fl Pe
ht Cotta A
GAH City
PAM |
afiiis ke Saeann
aoe ft Saeeee eee
INTUTOONONITOTAAAAUIIN Tn TACT AAMT
BUUAUNGUAGUSANOUSGSTEOUOUEGUOUNOUSAUOUUAUSUNOHGUUOUORUOUGNNY”<AVEQEOUABHOUNONOUIGUEL
aft He Her
HiT HSBEEIGBEOGEREEE Poo SRERREREEE EEE HEEL nena Mee
HH SOEUEGHGEE SEEGER 24 FHGNSGEUGE FOEHGERDGE TOE
Pa Po
Seeeee Hg A HH Po a8 @! SERREURAUEEaE Hoo
SoSG, éocee Be ci BESS Bee Pt]
Bi
HH
ag
3
[|
[|
T}
cH
a anes
Hi
it
PERE dy v di
SC) I a
FING
Le
it
HH
a
ite
aE
ert ST
all
rt
!
Hata PTET
Hn Tat as
4 y Seceeaes poo
2.5 3.0 3.5
Diameter (phi)
oe Et
IES 10 080.706 05 0.4 0.3 0.2 0.15 0.1 008 006
Diameter (mm)
Figure 4-8. Example Size Distribution
sediment in phi units. For perfectly sorted sediment, o = 0. For
typical well-sorted sediments, oy =~ 0.5.
The degree by which the phi size distribution departs from symmetry
is measured by the skewness (Inman, 1952), as
M, —M
Boats emis ad (4-5)
) op 4
where Mg is the mean, Mjp is the median, and og is the standard
deviation in phi units. For a perfectly symmetric distribution, the mean
equals the median, and the skewness is zero.
Extensive literature is available on the definition, use, and impli-
cation of o, a, and other measures of the size distribution. (Inman,
1952; Folk and Ward, 1957; McCammon, 1962; Folk, 1965, 1966; and
Griffiths, 1967), but despite a long history of investigation and a
considerable background of data, applications of size distribution infor-
mation "are still largely empirical, qualitative, and open to alternative
interpretations." (Griffiths, 1967, p. 104.)
Currently, median grain size is the most commonly reported sand size
characteristic, probably because there are only limited data to show the
usefulness of other size distribution parameters in coastal engineering
design. However, the standard deviation (Equation 4-4) must also be
given as a parameter for use in beach fill design. (See Section 5.3;
Krumbein and James, 1965; Vallianos, 1970; Berg and Duane, 1968.)
4.212 Composition. In addition to classification by size, littoral
material may be classified by composition. For shore protection pur-
poses, composition normally is not an important factor, since the domi-
nant littoral materials are quartz sands which are durable and chemically
inert for periods longer than typical project lifetimes. However, sedi-
ment composition is useful when the material departs from this expected
condition. Other than quartz, littoral material may be composed of car-
bonates (usually shell, coral, and algal material), organics (most often
peat), and clays and silts (marsh and tidal flat deposits).
4.213 Other Characteristics. In addition to size and composition, sedi-
ments have a number of other properties by which they may be classified.
Table 4-2 lists some density-related properties. Radioactive properties
of naturally occurring thorium minerals have been used as tracers in beach
sands. (Kamel, 1962; Kamel and Johnson, 1962, p. 324.) Other properties
more directly related to soil mechanics studies are found in soil mechanics
manuals.
4.22 SAND AND GRAVEL
By definition the word sand refers to a size class of material, but
sand also implies the particular composition, usually quartz (silica).
4-(8
Table 4-2. Density of Littoral Materials
Specific Gravity (dimensionless)
Quartz 2.65
Calcite Pai?)
Heavy Minerals > 2.87 (commonly 2.87 to 3.33)
Unit Weight* (Ibs./ft?)
Uniform Sand
loose
dense
Mixed Sand
loose
dense
Clay
stiff glacial
soft, very organic
*From Terzaghi and Peck, 1967.
In tropical climates, calcium carbonate, especially shell material,
is often the dominant material in beach sand. In temperate climates,
quartz and feldspar grains are the most abundant, commonly accounting for
about 90 percent of beach sand. (Krumbein and Sloss, 1963, p. 134.)
Because of its resistance to physical and chemical changes, and its
common occurrence in terrestrial rocks, quartz is the most common mineral
found in littoral materials. The relative abundance of non-quartz mate-
rials is a function of the relative importance of the sources supplying
the littoral zone and the materials available to those sources, The small
amount of heavy minerals (specific gravity greater than 2.87) usually
found in sand samples may indicate the source area of the material
(McMaster, 1954; Giles and Pilkey, 1965; Judge, 1970), and thus they
may be used as natural tracers. Such heavy minerals may form black or
reddish concentrations at the base of dune scarps, along the berm, and
around inlets. Occasionally heavy minerals occur in concentrations great
enough to justify mining them as a metal ore. (Everts, 1971; Martens, 1928.)
Table 4-3 from Pettijohn (1957, p. 117) lists the 26 most common minerals
found in beach sands.
Sand is by far the most important littoral material in coastal engi-
neering design. However, in some localities, such as New England, Oregon,
Washington, and countries bordering on the North Sea, gravel and shingle
are locally important. Gravel-sized particles are often rock fragments,
that is, a mixture of different minerals, whereas sand-sized particles
usually consist of single mineral grains.
4-19
Table 4-3. Minerals Occurring in Beach Sand
Common Dominant Constituents*
Quartz—may average about } Feldspar—typically only Calcite—includes shell, coral,
70 percent in beach sand; 10 to 20 percent in beach algal fragments, and oolites;
varies from near 0 to over sands, but may be much varies from 0 to nearly 100
99 percent. more, particularly in percent; may include signifi-
regions of eroding igneous cant quantities of aragonite.
rock.
Common Accessory Minerals (Adopted from Pettijohn, 1957)
Andalusite Epidote *Muscovite
Apatite Garnet Rutile
Aragonite Hornblende Sphene
Augite Hypersthene-enstatite Staurolite
Biotite Ilmenite Tourmaline
Chlorite Kyanite Zircon
Diopside Leucoxene Zoisite
*Dolomite Magnetite
" -* These are light minerals with specific gravity not exceeding 2.87. The remaining
minerals are heavy minerals with specific gravity greater than 2.87. Heavy minerals
make up less than 1 percent of most beach sands.
4.23 COHESIVE MATERIALS
The amount of fine-grained, cohesive materials, such as clay, silt,
and peat, in the littoral zone depends on the wave climate, contributions
of fine sediment from rivers and other sources, and recent geologic his-
tory. Fine-grain size material is common in the littoral zone wherever
the annual mean breaker height is below about 1.0 foot. Fine material is
found at or near the surface along the coasts of Georgia, western Florida
between Tampa and Cape San Blas, and in large bays such as Chesapeake Bay
and Long Island Sound. These are all areas of low mean breaker height.
In contrast, fine sediment is seldom found along the Pacific coast of
California, Oregon, and Washington, where annual mean breaker height
usually exceeds 2.5 feet.
Where rivers bring large quantities of sediment to the sea, the
amount of fine material remaining along the coast depends on the balance
between wave action acting to erode the fines and river deposition act-
ing to replenish the fines. (Wright and Coleman, 1972.) The effect of
the Mississippi River delta deposits on the coast of Louisiana is a pri-
mary example.
Along eroding, low-lying coasts, the sea moves inland over areas
formerly protected by beaches, so that the present shoreline often lies
where tidal flats, lagoons and marshes used to be. The littoral materials
on such coasts may include silt, clay and organic material at shallow
depths. As the active sand beach is pushed back, these former tidal flats
4-20
and marshes then outcrop along the shore. (e.g. Kraft, 1971.) Many barrier
islands along the Atlantic and Gulf coasts contain tidal and marsh deposits
at or near the surface of the littoral zone. The fine material is often
bound together by the roots of marsh plants to form a cohesive deposit
that may function for a time as beach protection.
4.24 CONSOLIDATED MATERIAL
Along some coasts, the principal littoral materials are consolidated
materials, such as rock, beach rock, and coral, rather than unconsolidated
sand. Such consolidated materials protect a coast and resist shoreline
changes.
4,241 Rock. Exposed rock along a shore indicates that the rate at which
sand is supplied to the coast is less than the potential rate of sand
transport by waves and currents. Reaction of a rocky shore to wave attack
is determined by the structure, degree of lithification, and ground water
characteristics of the exposed rock, and by the severity of the wave
climate. Protection of eroding cliffs is a complex problem involving
geology, rock mechanics, and coastal engineering. Two examples of these
problems are the protection of the cliffs at Newport, Rhode Island (U.S.
Army, Corps of Engineers, 1965) and at Gay Head, Martha's Vineyard,
Massachusetts. (U.S. Army, Corps of Engineers, 1970.)
Most rocky shorelines are remarkably stable, with individual rock
masses identified in photos taken 50 years apart. (Shepard and Grant,
1947.)
4.242 Beach Rock. A layer of friable to well-lithified rock often occurs
at or near the surface of beaches in tropical and subtropical climates.
This material consists of local beach sediment cemented with calcium car-
bonate, and it is commonly known as beach rock. Beach rock is important
to coastal engineers because it provides added protection to the coast,
greatly reducing the magnitude of beach changes (Tanner, 1960), and be-
cause beach rock may affect construction activities. (Gonzales, 1970.)
According to Bricker (1971), beach rock is formed when saline waters
evaporate in beach sands, depositing calcium carbonate from solution. The
present active formation of beach rock is limited to tropical coasts, such
as the Florida Keys, but rock resembling beach rock is common at shallow
depths along the east coast of Florida, on some Louisiana beaches, and re-
lated deposits have been reported as far north as the Fraser River Delta
in Canada. Comprehensive discussion of the subject is given in Bricker
(1971) and Russell (1970).
4.243 Organic Reefs. Organic reefs are wave-resistant structures reach-
ing to about mean sea level that have been formed by calcium carbonate
secreting organisms. The most common reef-building organisms are herma-
typic corals and coralline algae. Reef-forming corals are usually re-
stricted to areas having winter temperatures above about 18°C (Shepard,
1963, p. 351), but coralline algae have a wider range. On U.S.
4-2Il
coastlines, active coral reefs are restricted to southern Florida, Hawaii,
Virgin Islands and Puerto Rico. On Some of the Florida coast, reeflike
structures are produced by sabellariid worms. (Kirtley, 1971.) Organic
reefs stabilize the shoreline and sometimes affect navigation.
4.25 OCCURRENCE OF LITTORAL MATERIALS ON U.S. COASTS
Littoral materials on U.S. coasts vary from consolidated rock to
clays, but sand with median diameters between 0.1 and 1.0 millimeters
(3.3 and 0 phi) is most abundant. General information on littoral mate-
rials is in the reports of the U.S. Army Corps of Engineers' National
Shoreline Study; information on certain specific geological studies is
available in Shepard and Wanless (1971); and information on specific
engineering projects is published in Congressional documents and is
available in reports of the Corps of Engineers.
4.251 Atlantic Coast. The New England coast is generally characterized
by rock headlands separating short beaches of sand or gravel. Exceptions
to this dominant condition are the sandy beaches in northeastern Massachu-
setts, and along Cape Cod, Martha's Vineyard, and Nantucket.
From the eastern tip of Long Island, New York, to the southern tip
of Florida, the littoral materials are characteristically sand with median
diameters in the range of 0.2 to 0.6 millimeter (2.3 to 0.7 phi). This
material is mainly quartz sand. In Florida, the percentage of calcium
carbonate in the sand tends to increase going south until, south of the
Palm Beach area, the sand becomes predominantly calcium carbonate. Size
distributions for the Atlantic coast, compiled from a number of sources,
are shown in Figure 4-9. (Bash, 1972.) Fine sediments and organic sedi-
ments are common minor constituents of the littoral materials on these
coasts, especially in South Carolina and Georgia. Beach rock and coquina
are common at shallow depths along the Atlantic coast of Florida.
4.252 Gulf Coast. The Gulf of Mexico coasts of Florida, Alabama, and
Mississippi are characterized by fine white sand beaches and by stretches
of swamp. The swampy stretches are mainly in Florida, extending from
Cape Sable to Cape Romano, and from Tarpon Springs to the Ochlockonee
River. (Shepard and Wanless, 1971, p. 163.)
The Louisiana coast is dominated by the influence of the Mississippi
River which has deposited large amounts of fine sediment around the delta
from which wave action has winnowed small quantities of sand. ‘uit scna
has been deposited along barrier beaches offshore of a deeply indented
marshy coast. West of the delta is a 75-mile stretch of shelly sand
beaches and beach ridges.
The Texas coast is a continuation of the Louisiana coastal plain ex-
tending about 80 miles to Galveston Bay; from there a series of long, wide
barrier islands extends to the Mexican border. Littoral materials in this
area are dominantly fine sand, with median diameters between 0.1 and 0.2
millimeter (3.3 and 2.3 phi).
Median Diameter ,(mm.)
4 LONG ISLAND
ee ie ES x feet bsees = ets
NORTH CAROLINA
\ cas
DELMARVA
ial easy ad $8 song gwah 2.) sei oly when not a ted
0.6 0.64
Peak
“NORTH CAROLINA | SOUTH CAROLINA (
Figure 4-9.
FLORIDA.
100 200 Miles
Sand Size Distribution Along the U.S. Atlantic Coast
4-23
Phi Units (0)
4.253 Pacific Coast. Sands on the southern California coast range in
size from 0.1 to 0.6 millimeter (3.3 to 0.7 phi). (Emery, 1960, p. 190.)
The northern California coast becomes increasingly rocky, and coarser
material becomes more abundant. The Oregon and Washington coasts include
considerable sand (Bascom, 1964), with many rock outcrops. Sand-sized
sediment is contributed by the Columbia River and other smaller rivers.
4.254 Alaska. Alaska has a long coastline (47,300 miles), and is corres-
pondingly variable in littoral materials. However, beaches are generally
narrow, steep, and coarse-grained; they commonly lie at the base of sea-
cliffs. (Sellman, et al., 1971, p. D-10.) Quartz sand is less common and
gravel more common here than on many other U.S. coasts.
4,255 Hawaii. Much of the Hawaiian islands is bounded by steep cliffs,
but there are extensive beaches. Littoral materials consist primarily of
bed rock, and white sand formed from calcium carbonate produced by marine
invertebrates. Dark colored basaltic and olivine sands are common where
river mouths reach the sea. (Shepard and Wanless, 1971, p. 497, U.S. Army,
Corps of Engineers, 1971.)
4.256 Great Lakes. The U.S. coasts of the Great Lakes vary from high
bluffs of clay, shale, and rock, through lower rocky shores and sandy
beaches, to low marshy clay flats. (U.S. Army Corps of Engineers, National
Shoreline Study, August 1971, North Central Division, p. 13.) The litto-
ral materials are quite variable. Specific features are discussed, for
example, by Bowman (1951); Hulsey (1962); Davis (1964-65); Bajorunas and
Duane (1967); Berg and Duane (1968); Saylor and Upchurch (1970); Hands
(1970); and Corps of Engineers (1953, 1971).
4.26 SAMPLING LITTORAL MATERIALS
Sampling programs are designed to provide information about littoral
materials on one or more of the following characteristics:
(a) typical grain size (usually median size),
(b) size distribution,
(c) composition of the littoral materials,
(d) variation of (a), (b), and (c), with horizontal and vertical
position on the site, and
(e) possible variation in (a), (b), (c), and (d) with time.
A sampling program will depend on the intended purpose of the samples,
the time and money availablé for sampling, and an inspection of the site
to be sampled. A brief inspection will often identify the principal vari-
ations in the sediment and suggest the best ways to sample these variations.
Sampling programs usually involve beach and nearshore sands and potential
borrow sources.
4-24
The extent of sampling depends on the importance of littoral materials
as related to the total engineering problem. The sampling program should
specify:
(a) horizontal location of sample,
(b) spacing between samples,
(c) volume of sample,
(d) vertical location and type of sampled volume (e.g. surface layer
or vertical core),
(e) technique for sampling,
(f) method of storing and documenting the sample.
Beaches typically show more variation across the profile than along
the shore, so sampling to determine variation in the littoral zone should
usually be along a line perpendicular to the shoreline.
For reconnaissance sampling, a sample from both the wetted beach face
and from the dunes is recommended. More extensive samples could be ob-
tained at constant spacings across the beach or at different locations on
the beach profile. Spacings between sampling lines are determined by the
variation visible along the beach or by statistical techniques.
Many beaches have subsurface layers of peat or other fine material.
If this material will affect the engineering problem, vertical holes or
borings should be made to obtain samples at depth.
Sample volume should be adequate for analysis. For sieve analysis,
about 50 grams are required; for settling tube analysis, smaller quanti-
ties will suffice, but at least 50 grams are needed if other studies are
required later. A quarter of a cup is more than adequate for most uses.
Sand often occurs in fine laminae on beaches. However, for engineer-
ing applications it is rarely necessary to sample individual laminae of
sand. It is easier and more representative to take an equidimensional
sample that cuts across many laminae. Experience at CERC suggests that
any method of obtaining an adequate volume of sample covering a few inches
in depth usually gives satisfactory results. Cores should be taken where
pile foundations are planned.
The sample is only as good as the information identifying it. The
following minimum information should be recorded at the time of sampling:
locality, date and time, position on beach, remarks, and initials of col-
lector. This information must stay with the sample, which is best ensured
by fixing it to the sample container or placing it inside the container.
Unless precautions are taken, the sample label may deteriorate due to
moisture, abrasion, or other causes. Improved labels result by using
4-25
ballpoint ink on plastic strip (plastic orange flagging commonly used by
surveyors). Some information may be preprinted by rubber stamp on the
plastic strip using indelible laundry ink. The advantage is that the
label can be stored in the bag with the wet sample without the label
deteriorating or the information washing or wearing off.
4.27 SIZE ANALYSES
Three common methods of analyzing a beach sediment for size are:
visual comparison with a standard, sieve analysis, and settling tube
analysis.
The mean size of a sand sample can be estimated qualitatively by
visually comparing the sample with sands of known sizes. Standards can
be easily prepared by sieving out selected diameters, or by selecting
samples whose sizes are already known. The standards may be kept in
labeled transparent vials, or glued on cards. If glued, care is neces-
sary to ensure that the particles retained by the glue are truly repre-
sentative of the standard.
Good, qualitative, visual estimates of mean size are possible with
little previous experience. With experience, such visual estimates
become semiquantitative. Visual comparison with a standard is a useful
tool in reconnaisance, and in obtaining interim results pending a more
complete laboratory size analysis.
4.271 Sieve Analysis. Sieves are graduated in size of opening according
to the U.S. Standard series. These standard sieve openings vary by a fac-
tor of 1.19 from one opening to the next larger (by the fourth root of 2,
or quarter phi intervals), e.g., 0.25, 0.30, 0.35, 0.42, and 0.50 milli-
meters (2.00, 1.75, 1.50, 1.25, 1.00 phi). The range of sieve sizes used,
and the size interval between sieves selected can be varied as required.
Typical beach sand can be analyzed adequately using sieves with openings
ranging from 0.062 to 2.0 millimeters (4.0 to -1.0 phi), in size incre-
ments increasing by a factor of 1.41 (half-phi intervals).
Sediment is usually sieved dry. However, for field analysis or for
size analysis of sediment with a high content of fine material, it may be
useful to wet-sieve the sediment. Such wet-sieve analyses are described
by) (e.g., Lee, Yancy, and Wilde, 1970, p. 4).
Size analysis by sieves is relatively slow, but provides a widely
accepted standard of reference.
4,272 Settling Tube. Spherical sedimentary particles settle through
water at a speed that increases as the particle weight increases. Since
most sand is approximately spherical quartz, or calcium carbonate with a
specific gravity near quartz, particle size is proportional to particle
weight. Thus fall velocity can be used to measure size. (e.g., Colby and
Christensen, 1956; Zeigler and Gill, 1959; and Gibbs, 1972.) Figure 4-31
shows fall velocity for quartz spheres as a function of temperature.
4-26
There are numerous types of settling tubes; the most common is the
visual accumulation tube (Colby and Christensen, 1956), of which there
are also several types. The type now used at CERC (the rapid sediment
analyzer or RSA) works in the following way:
A 3- to 6-gram sample of sand is dropped through a tube filled with
distilled water at constant temperature. A pressure sensor near the
bottom of the tube senses the added weight of the sediment supported by
the column of water above the sensor. As the sediment falls past the
sensor, the pressure decreases. The record of pressure versus time is
empirically calibrated to give size distribution based on fall velocity.
(Zeigler and Gill, 1959.)
The advantage of settling tube analysis is its speed. With modern
settling tubes, average time for size analyses of bulk lots can be about
one-fifth the time required by sieves.
It is often claimed that a settling tube also provides a physically
more realistic size analysis than a sieve, since the fall velocity takes
into account the hydrodynamic effects of shape and density. However,
this claim has not been documented, and may be questioned in view of the
limited knowledge concerning the fluid mechanics of a sand sample falling
in a settling tube - the lead particles encounter effectively laminar flow,
the trailing particles encounter turbulent flow, and all particles inter-
act with each other.
Because of lack of an accepted standard settling tube, rapidly chang-
ing technology, possible changes in tube calibration, and the uncertainty
about fluid mechanics in settling tubes, it is recommended that all set-
tling tubes be carefully calibrated by running a range of samples through
both the settling tube and ASTM standard sieves. After thorough initial
calibration, the calibration should be spot-checked periodically by running
replicate sand samples of known size distribution through the tube.
4.3 LITTORAL WAVE CONDITIONS
4.31 EFFECT OF WAVE CONDITIONS ON SEDIMENT TRANSPORT
Waves arriving at the shore are the primary cause of sediment trans-
port in the littoral zone. Higher waves break further offshore, widening
the surf zone and setting more sand in motion. Changes in wave period or
height result in moving sand onshore or offshore. The angle between the
crest of the breaking wave and the shoreline determines the direction of
the longshore component of water motion in the surf zone, and usually the
lonyshore transport direction. For these reasons, knowledge about the
wave climate - the combined distribution of height, period, and direction
through the seasons - is required for an adequate understanding of the
littoral processes of any specific area.
4-27
4.32 FACTORS DETERMINING LITTORAL WAVE CLIMATE
The wave climate at a shoreline depends on the offshore wave climate,
caused by prevailing winds and storms, and on the bottom topography that
modifies the waves as they travel shoreward.
4.321 Offshore Wave Climate. Wave climate is the distribution of wave
conditions averaged over the years. A wave condition is the particular
combination of wave heights, wave periods, and wave directions at a given
time. A specific wave condition offshore is the result of local winds
blowing at the time of the observation and the recent history of winds
in the more distant parts of the same water body. For local winds, wave
conditions offshore depend on the wind velocity, duration, and fetch.
For waves reaching an observation point from distant parts of the sea,
a decay factor is added which preferentially filters out the higher,
shorter period waves with increasing distances. (Chapter 3 discusses
wave generation and decay.)
4.322 Effect of Bottom Topography. As waves travel from deep water, they
change height and direction because of refraction, shoaling, bottom fric-
tion, and percolation. Laboratory experiments indicate that height and
apparent period are also changed by nonlinear deformation of the waves in
shallow water.
Refraction is the bending of wave crests due to the slowing down of
that part of the wave crest which is in shallower water. (See Section
2.32.) As a result, refraction tends to decrease the angle between the
wave crest and the bottom contour. Thus, for most coasts, refraction re-
duces the breaker angle and spreads the wave energy over a longer crest
length.
Shoaling is the change in wave height due to conservation of energy
flux. (See Section 2.32). As a wave moves into shallow water the wave
height first decreases slightly, and then increases continuously to the
breaker position, assuming friction and refraction effects are negligible.
Bottom friction is important in reducing wave height where waves
mist travel long distances in shallow water. (Bretschneider, 1954.)
There has been only limited field study of nonlinear deformation in
shallow water waves (Byrne, 1969), but because such deformation is common
in laboratory experiments (Galvin, 1972), it is expected that such phe-
nomena are also common in the field. An effect of nonlinear deformation
is to split the incoming wave crest into two or more crests affecting
both the resulting wave height and the apparent period.
Offshore islands, shoals, and other variations in hydrography also
shelter parts of the shore. In general, bottom hydrography has the
greatest influence on waves traveling long distances in shallow water.
Because of the effects of bottom hydrography, nearshore waves generally
have different characteristics than they had in deep water offshore.
4-28
Such differences are often visible on aerial photos. Photos may
show two or more distinct wave trains in the nearshore area, with the
wave train most apparent offshore decreasing in importance as the surf
zone is approached. (e.g., Harris, 1972.) The difference appears to be
caused by the effects of refraction and shoaling on waves of different
periods. Longer period waves, which may be only slightly visible off-
shore, may become the most prominent waves at breaking, because shoaling
increases their height relative to the shorter period waves. Thus, the
wave period measured from the dominant wave offshore may be less than the
wave period measured from the dominant wave entering the surf zone when
two wave trains of unequal period reach the shore at the same time.
4.323 Winds and Storms. The relation of a shoreline locality to the
seasonal distribution of winds and to storm tracks is a major factor
in determining the wave energy available for littoral transport. For
example, strong winter winds in the northeastern United States usually
are from the northwest, but because they blow from land to sea, they do
not produce large waves at the shore. These northwest winds often imme-
diately follow a northeaster - a low pressure system with strong north-
east winds that generate high waves offshore.
A storm near the coastline will influence wave climate with storm
surge and high seas; a storm offshore will influence wave climate only
by swell. The relation between the meteorological severity of a storm
and the resulting beach change is complicated. (See Section 4.35.)
Storms are not uniformly distributed in time or space: storms vary
seasonally and from year to year; storms originate more frequently in
some areas than in others; and storms follow characteristic tracks deter-
mined by prevailing global circulation and weather patterns.
An investigation of 170 damaging storms affecting the east coast of
the U.S. from 1921-1962 (Mather, et al., 1964), classified the storms into
eight types based on origin, structure, and path of movement. Of these
eight types, although 33 percent were hurricanes, two types, comprising
only 19 percent of the total, characterized by weather fronts east and
south of the U.S. coasts produced more damage per storm because of long
fetches. (Damage is defined by Mather, et al., as "at best some water
damage,'' and includes "wave damage, coastal flooding, and tidal inunda-
tion,'' but specificially excludes wind damage.)
The probability that a given section of coast will experience storm
waves depends on its ocean exposure, its location in relation to storm
tracks, and the shelf bathymetry.
4.33 NEARSHORE WAVE CLIMATE
4.331 Mean Value Data on U.S. Littoral Wave Climates. Wave height and
period data for some localities of the U.S. are becoming increasingly
available (e.g., Thompson and Harris, 1972), but most localities still
lack such data. However, wave direction is difficult to measure, and
consequently direction data are rarely available.
4-29
The quality and quantity of available data often do not justify
elaborate statistical analysis. Even where adequate data are available,
a simple characterization of wave climate meets many engineering needs.
While mean values of height and, to a lesser degree, period are useful,
data on wave direction are generally of insufficient quality for even
mean value use. Table 4-4 compiles mean annual wave heights collected
from a number of wave gages and by visual observers along the coasts of
the United States. Visual observations were made from the beach of waves
near breaking. The gages were fixed in depths of 10 to 28 feet.
Wave data treated in this section are limited to nearshore observa-
tions and measurements. Consequently waves were fully refracted and had
been fully affected by bottom friction, percolation, and nonlinear changes
in wave form by shoaling. Thus, these data differ from data that would
be obtained by simple shoaling calculations based on the deepwater wave
statistics. In addition, data are normally lacking for the rarer, high-
wave events. For these reasons, the data should not be used for struc-
tural design, since they are only applicable to the particular site
where they were obtained. Normal design practice is based on deepwater
wave statistics which are then adjusted to the shallow-site conditions.
However, the nearshore data are useful in littoral transport calculations.
Mean wave height and period from a number of visual observations by
coastguardmen at shore stations are plotted by month in Figures 4-10 and
4-11, using the average values of stations within each of five coastal
segments. Table 4-4 and Figures 4-10 and 4-11 show average values char-
acteristic of the wave climate in exposed coastal localities. Visual
height data represent an average value of the higher waves just before
they first break. The data provide only approximate indications of the
height distributions, but mean values of these distributions are useful.
In Figure 4-10 the mintmum monthly mean littoral zone wave height
averaged for the California, Oregon, and Washington coasts exceeds the
maximum mean littoral zone wave height averaged for the other coasts.
This difference greatly affects the potential for sediment transport in
the respective littoral zones, and should be considered by engineers
when applying experience gained in a locality with one nearshore wave
climate to a problem at a locality with another wave climate.
4.332 Mean vs Extreme Conditions. Section 3.22 contains a discussion
of wave height distributions and the relations between various wave
height statistics, such as the mean, significant, and RMS heights, and
extreme values. In general, a group of waves from the same record can be
approximately described by a Rayleigh distribution. (See Section 3.22.)
However, a different distribution appears necessary to describe the dis-
tribution of significant wave heights, where each significant wave height
is from a different wave record at the locality. (See Figure 4-12.)
Visual analysis of waves recorded on chart paper is discussed in
Section 3.22 and by Draper (1967), Tucker (1963), Harris (1970), and
Magoon (1970). Spectrum analysis of wave records is discussed in
4-30
(43) 44619H ADM asOYSJOaN AjysUoW UDaW
Oct
Mean Monthly Nearshore Wave Heights for Five Coastal Segments
Figure 4-10.
SATY LOZ (SWTeD Butpnytouy) spotzeg eAeM otOYsIeeN ATYQUOW UeOW
Ya]
Sjususes [e1SeOD
AON 490 des Bny jnp une Aow $ ady
be
1B
-
i
aid
Gi
co
4OW 84
u
ie)
“II-p ean3sty
t
(99S) pOldag AADM asOYSsDaN AjysuOW UDA;
4-32
Table 4-4. Mean Wave Height at Coastal Localities of Conterminous United States
Mean Annual
Location Mean Annual
iS. Wave Height (ft.) Wave Height (ft.)
Atlantic Coast
New Jersey (cont.)
Location
Maine
Moose Peak + Ludlam Island
New Hampshire Maryland
Hampton Beach Ocean City
Massachusetts Virginia
Nauset iF Assateaque
+ Cape Cod *Virginia Beach
Rhode Island Virginia Beach
North Carolina
*Nags Head
Nags Head
+ Wrightsville
Oak Island
+ Holden Beach
Georgia
St. Simon Island
Point Judith
+ Misquamicut
New York
+ Southampton
+ Westhampton
+ Jones Beach
Short Beach
New Jersey
Monmouth Florida
+ Deal Daytona Beach
Toms River Ponce deLeon
+ Brigantine *Lake Worth
*Palm Beach
+ Boca Raton
Hillsboro
Gulf Coast
*Atlantic City
Atlantic City (BEP
+ Atlantic City (CG)
| Florida
*Naples
Florida (cont.)
+ Navarre Beach
Cape San Blas Santa Rosa
+ Panama City Louisania
+Greyton Beach Grand Island
+Crystal Beach Texas
*Galveston
+ Beasley Park
Pacific Coast
California California (cont.)
Point Loma 2.1 Point Arguello
#South Carlsbad oy, + Natural Bridges
+ Carlsbad 2.9 + Thornton
*Huntington Beach Le + Goat Rock
+Huntington 2.6 Point Arena
+ Bolsa Chica 2.2 + Prairie Creek
+Leo Carrillo aS. Oregon
+PEG at Point Mugu 3.0 Umpqua River
*Point Mugu 2.7 Yaquina Bay
#McGrath 3.5 Washington
+Carpinteria 1.8 Willapa Bay
Point Conception Cape Flattery
+El Capitan
CERC wave gage records.
The following mean wave heights are from visual (nearbreaker) observations.
t CERC Beach Evaluation Program,
t CERC Littoral Environment Observation Program.
Unmarked Coast Guard Observations.
Sp109ay JDaA-| 10} SaBDg aADM |D}S00D WOdy SyYyBIay aADM juodIyIUBIS yo UOINGI4}sIG “Zi-b a4NbI4
(226) ‘81410} puo uosdwoy, wo4s) Pe}DDIpU| UDY} 49}Da19 jUadded
09 0S -0 O02 OlS6. 18 = Z SH9
}ylaH aADM fuDIIjIUbIS
jaa}
34
Section 3.23 and by Kinsman (1965), National Academy of Sciences (1963),
and Neumann and Pierson (1966).
For the distribution of significant wave heights as defined by the
data reduction procedures at CERC (Thompson and Harris, 1972), the data
fit a modified exponential distribution of form
LA 2 ~ H, as
F (H,>H,) =e 0 (4-6)
where Hg is the significant height, Hs the significant height of in-
terest, Hg min is the approximate "minimum significant height," and o
is the significant wave height standard deviation. This equation depends
on two parameters, Hg min and o which are related to the mean height,
H =H
s s min
=P hc (4-7)
he If Hg min Or oO are not available, but the mean significant height,
Hs is known, then an approximation to the distribution of (4-6) can be
obtained from the data of Thompson and Harris (1972, Table 1), which
suggest
H
s min
~ 0.38 H, . (4-8)
This approximation reduces Equation 4-6 to a one-parameter distribution
depending only on mean significant wave height
1.61 H, —0.61 i
F (H, >Hi.) e H, (4-9)
Equation 4-9 is not a substitute for the complete distribution function,
but when used with the wave-gage data on Figure 4-12, it provides an
estimate of higher waves with agreement within 20 percent. Greater
scatter would be expected with visual observations.
4.34 OFFICE STUDY OF WAVE CLIMATE
Information on wave climate is necessary for understanding local
littoral processes. Usually, time does not permit obtaining data from
the field, and it is necessary to compile information in an office study.
The primary variables of engineering interest for such a compilation are
wave height and direction.
Shipboard observations covering conterminous U.S. coasts and other
ocean areas are available as summaries (Summary of Synoptic Meterological
Observations, SSMO) through the National Technical Information Service,
Springfield, Va. 22151. See Harris (1972) for a preliminary evaluation
of this data for coastal engineering use.
q-39
When data are not available for a specific beach, the wave climate
can be estimated by extrapolating from another location, after correcting
for differences in coastal exposure, winds, and storms.
On the east, gulf, and Great Lakes coasts, local winds are often
highly correlated with the direction of longshore currents. Such wind
data are available in "Local Climatological Data" sheets published monthly
by the National Weather Service, National Oceanographic and Atmospheric
Agency (NOAA) for about 300 U.S. weather stations. Other NOAA wind-data
sources include annual summaries of the Local Climatological Data by sta-
tion (Local Climatological Data with Comparative Data), and weekly sum-
maries of the observed weather (Daily Weather Maps), all of which can be
ordered from the Superintendent of Documents, U.S. Government Printing
Office, Washington, D.C. 20402.
Local weather data are often affected by conditions in the neighbor-
hood of the weather station, so care should be used in extrapolating
weather records from inland stations to a coastal locality. However,
statistics on frequency and severity of storm conditions do not change
appreciably for long reaches of the coast. For example, in a study of
Texas hurricanes, Bodine (1969) felt justified in assuming no difference
in hurricane frequency along the Texas coast. In developing information
on the Standard Project Hurricane, Graham and Nunn (1959) divided the
Atlantic Coast into zones 200 miles long and the gulf coast into zones
400 miles long. Variation of most hurricane parameters within zones is
not great along straight open stretches of coast.
The use of weather charts for wave hindcasting is discussed in Sec-
tion 3.4. Computer methods for generating offshore wave climate are now
(1973) under test and development. However, development of nearshore
wave climate from hindcasting is usually a time-consuming job, and the
estimate obtained may suffer in quality because of the inaccuracy of
hindcast data, and the difficulty of assessing the effect of nearshore
topography on wave statistics. At the present time, if available at the
specific location, statistics based on wave-gage records are preferable
to hindcast statistics when wave data for the shallow-water conditions
are required.
Other possible sources of wave climate information for office stud-
ies include aerial photography, newspaper records, and comments from
local residents.
Data of greater detail and reliability than that obtained in an
office study can be obtained by recording the wave conditions at the
shoreline locality for at least 1 year by the use of visual observers
or wave gages. A study of year-to-year variation in wave height statis-
tics collected at CERC wave gages (Thompson and Harris, 1972), indicates
that six observations per day for 1 year gives a reliable wave height
distribution function to the 1 percent level of occurrence. At the gage
at Atlantic City, one observation a day for 1 year provided a useful
height-distribution function.
4.35 EFFECT OF EXTREME EVENTS
Infrequent events of great magnitude, such as hurricanes, cause sig-
nificant modification of the littoral zone, particularly to the profile
of a beach. An extreme event could be defined as an event, great in
terms of total energy expended or work done, that is not expected to occur
at a particular location, on the average, more than once every 50 to 100
years. Hurricane Camille in 1969 and the East Coast Storm of March 1962
can be considered extreme events. Because large storms are infrequent,
and because it does not necessarily follow that the magnitude of a storm
determines the amount of geomorphic change, the relative importance of
extreme events is difficult to establish.
Wolman and Miller (1960) suggested that the equilibrium profile of
a beach is more related to moderately strong winds that generate moderate
storm waves, rather than to winds that accompany infrequent catastrophic
events. Saville (1950) showed that for laboratory tests with constant
wave energy and angle of attack there is a particular critical wave steep-
ness at which littoral transport is a maximum. Under field conditions,
there is probably a similar critical value that produces transport out of
proportion to its frequency of occurrence. The winds associated with this
critical wave steepness may be winds generated by smaller storms, rather
than the winds associated with extreme events.
A review of studies of beach changes caused by major storms indi-
cates that no general conclusion that can be made concerning the signifi-
cance of extreme events. Many variables affect the amount of damage a
beach will sustain in a given storm.
Most storms move large amounts of sand from the beach offshore, but
after the storm, the lower waves that follow tend to restore this sand to
the face of the beach. Depending on the extent of restoration, the storm
may result in little permanent change. Depending on the path of the storm
and the angle of the waves, a significant amount of material can also be
moved alongshore. If the direction of longshore transport caused by the
storm is opposite to the net direction of transport, the sand will prob-
ably be returned in the months after the storm and permanent beach changes
effected by the storm will be small. If the direction of transport before,
during, and after the storm is the same, then large amounts of material
could be moved by the storm with little possibility of restoration. Suc-
cessive storms on the same beach may cause significant transport in oppo-
site directions. (e.g. Everts, 1973.)
There are some unique events that are only accomplished by catastro-
phic storms. The combination of storm surge and high waves allows water
to reach some areas not ordinarily attacked by waves. These extreme con-
ditions may result in the overtopping of dunes and in the formation of
washover fans and inlets. (Morgan, et al., 1958; Nichols and Marston,
1939.) Some inlets are periodically reopened by storms and then sealed
by littoral drift transported by normal wave action.
4-37
The wave climate at a particular beach also determines the effect
a storm will have. In a high-energy climate, storm waves are not much
larger than ordinary waves, and their effects may not be significant. An
example of this might be northeasters occurring at Cape Cod. In a low-
energy wave climate, where transport volumes are usually low, storm waves
can move significant amounts of sand, as do hurricanes on the gulf coast.
The type of beach sediment is also important in storm-induced changes.
A storm can uncover sediments not ordinarily exposed to wave action, and
thus alter the processes that follow the storm. (Morgan, et al., 1958.)
In sand-deficient areas where the beach is underlain by mud, the effects
of a storm can be severe and permanent.
The effects of particular storms on certain bedches are described in
the following paragraphs. These examples illustrate how an extreme event
may affect the beach.
In October 1963, the worst storm in the memory of the Eskimo people
occurred over an ice-free part of the Arctic Ocean, and attacked the coast
near Barrow, Alaska. (Hume and Schalk, 1967.) Detailed measurements of
some of the key coastal areas had been made just before the storm. Freeze-
up just after the storm preserved the changes to the beach until surveys
could be made the following July. Most of the beaches accreted 1 to 2
feet, although Point Barrow was turned into an island. According to Hume
and Schalk, "The storm of 1963 would appear to have added to the Point the
sediment of at least 20 years of normal longshore transport."" Because of
the low-energy wave climate and the short season in which littoral pro-
cesses can occur at Barrow, this storm significantly modified the beach.
A study of two hurricanes, Carla (1961) and Cindy (1963), was made by
Hayes (1967). He concluded that "the importance of catastrophic storms
as sediment movers cannot be over-emphasized,"' and observed that, in low-
energy wave climates, most of the total energy is expended in the near-
shore zone as a series of catastrophes. In this region, however, the rare
"extreme" hurricane is probably not as significant in making net changes
as the more frequent moderate hurricanes.
Surprisingly, Hurricane Camille, with maximum winds of 200 mph, did
not cause significant changes to the beaches of Mississippi and Louisiana.
Tanner (1970) estimated that the sand transport along the beach appeared
to have been an amount equal to less than a year's amount under ordinary
conditions, and theorized that "the particular configuration of beach, sea
wall, and coastal ridge tended to suppress large scale transport."
Hurricane Audrey struck the western coast of Louisiana in June, 1957.
The changes to the beach during the storm were not extreme nor permanent.
However, the storm exposed marsh sediments in areas where sand was defi-
cient, and ''set the stage for a period of rapid shoreline retreat follow-
ing the storm." (Morgan, et al., 1958.) Indirectly, then, the storm was
responsible for significant geomorphic change.
4-38
A hurricane (unnamed) coincided with spring tide on the New England
coast on 21 September 1938. Property damage and loss of life were both
high. A storm of this magnitude was estimated to occur about once every
150 years. A study of the beach changes along a 12-mile section of the
Rhode Island coast (Nichol and Marsten, 1939) showed that most of the
changes in the beach profile were temporary. The net result was some
cliff erosion and a slight retrogression of the beaches.
Beach changes from Hurricane Donna which hit Florida in September
1960 were more severe and permanent. In a study of the southwestem
coast of Florida before and after the storm, Tanner (1961), concluded
that "Hurricane Donna appears to have done 100 year's work, considering
the typical energy level thought to prevail in the area."
On 1 April 1946, a tsunami struck the Hawaiian Islands with runup in
places as high as 55 feet above sea level. (Shepard, et al., 1950.) The
beach changes were similar to those inflicted by storm waves although "in
only a few places were the changes greater than those produced during nor-
mal storm seasons or even by single severe storms.'' Because a tsunami is
of short duration, extensive beach changes do not occur, although property
damage can be quite high.
Several conclusions can be drawn from the above examples. If a beach
has a sufficient sand supply and fairly high dunes, and if the dunes are
not breached, little permanent modification will result from storms, except
for a brief acceleration of the normal littoral processes. This accelera-
tion will be more pronounced on a shore with low-energy wave conditions.
4.4 NEARSHORE CURRENTS
Nearshore currents in the littoral zone are predominantly wave-induced
motions superimposed on the wave-induced oscillatory motion of the water.
The net motions generally have low velocities, but because they transport
whatever sand is set in motion by the wave-induced water motions, they are
important in determining littoral transport.
There is only slight exchange of fluid between the offshore and the
surf zone. Onshore-offshore flows take place in a number of ways, which
at present are not fully understood.
4.41 WAVE-INDUCED WATER MOTION
In idealized deepwater waves, water particles have a circular motion
in a vertical plane perpendicular to the wave crest (Fig. 2-4, Section
2.235), but this motion does not reach deep enough to affect sediment on
the bottom. In depths where waves are affected by the bottom, the circu-
lar motion becomes elliptical, and the water at the bottom begins to move.
In shallow water, the ellipses elongate into nearly straight lines. At
breaking, particle motion becomes more complicated, but even in the surf
zone, the water moves forward and backward in paths that are mostly hori-
zontal, with brief, but intense, vertical motions produced by the passage
4-39
of the breaker crest. Since it is this wave-induced water particle motion
that causes the sediment to move, it is useful to know the length of the
elliptical path traveled by the water particles and the maximum velocity
and acceleration attained during this orbit.
The basic equations for water-wave motion before breaking are dis-
cussed in Chapter 2. Quantitative estimates of water motion are possible
from small-amplitude wave theory (Section 2.23), even near breaking where
assumptions of the theory are not valid. (Dean, 1970; Eagleson, 1956.)
Equations 2-13 and 2-14, in Section 2.234 give the fluid-particle velocity
components u, w in a wave where small-amplitude theory is applicable.
(See Figure 2-3 for relation to wave phase and water particle accelera-
tion.)
For sediment transport, the conditions of most interest are those
when the wave is in shallow water. For this condition, and making the
small-amplitude assumption, the horizontal length 2A, of the path moved
by the water particle as a wave passes in shallow water is approximately
H T./gd
2A = aoe (4-10)
and the maximum horizontal water velocity is
H/gd
Unathie Waar (4-11)
The term under the radical is the wave speed in shallow water.
x Re eK kK KK kK K kK kK kK * * * EXAMPLE PROBLEM * * * * * * * * * * *® * * * *
GIVEN: A wave 1 foot high with a period of 5 seconds is progressing shore-
ward in a depth of 2 feet.
FIND:
(a) Calculate the maximum horizontal distance 2A the water particle
moves during the passing of a wave.
(b) Determine the maximum horizontal velocity Ungy Of a water
particle.
(c) Compare the maximum horizontal distance 2A with the wavelength
in the 2-foot depth.
(d) Compare the maximum horizontal velocity Unag» with the wave
speed, C.
SOLUTION:
(a) Using Equation 4-10, the maximum horizontal distance is
HT ./gd
‘Spas 2nd
1) 322212)
2A = ——————_ =_ 3.2 feet .
2m (2)
(b) Using Equation 4-11, the maximum horizontal velocity is
H./gd
max 2d
1/32.2 (2)
Umax = a Eee = 2.0 feet per second .
(c) Using the relation L = TYgd to determine the shallow-water
wavelength,
L = 5,/32.2 (2) = 40.1 feet .
From (a) above, the maximum horizontal distance 2A is 3.2 feet
therefore the ratio 2A/L is
2S ae
L 40.1 nae
(d) Using the relation C = Ved (Equation 2-9) to determine the
shallow-water wave speed
C = 32.2 (2) = 8.0 feet per second .
From (b) above the maximum horizontal velocity Ugg,, is 2.0
feet per second. Therefore the ratio Ug,/C is
u 2.0
aX = — = 0.25.
C 8.0
eee) Meee: erie eee Ve aise Cea) eK) Re de en ie Cae) el eo ie) Se ice Se ae es ee a donde, ans) See
Although small-amplitude theory gives a fair understanding of many
wave-related phenomena, there are important phenomena that it does not
predict. Observation and a more complete analysis of wave motion show
that particle orbits are not closed. Instead, the water particles
advance a little in the direction of the wave motion each time the wave
4-4!
passes. The rate of this advance is the mass transport velocity; (Equa-
tion 2-55, Section 2.253). This velocity becomes important for sediment
transport, especially for sediment suspended above ripples seaward of the
breaker.
For conditions evaluated at the bottom (z = -d), the maximum bottom
velocity, Umasc( dq) » given by Equation 2-13 determines the average bottom
mass transport velocity, U(-q) obtained from Equation 2-55, according to
the equation
2
w—g)= (Vmera)) (4-12)
(— 4) 2C 5
where C is the wave speed given by Equation 2-3. Equation 2-55, and
thus Equation 4-12, does not include allowance for return flow which
must be present to balance the mass transported in the direction of wave
travel. In addition, the actual distribution of the time-averaged net
velocity depends sensitively on such external factors as bottom character-
istics, temperature distribution, and wind velocity. (Mei, Liu, and Carter,
1972.) Most observations show the time-averaged net velocity near the
bottom is directed toward the breaker region from both sides. (See Inman
and Quinn, (1952), for field measurements in surf zone; Galvin and Eagle-
son, (1965) for laboratory observations; and Mei, Liu and Carter (1972,
p. 220), for comprehensive discussion.) However, both field and labora-
tory observations have shown that wind-induced bottom currents may be
great enough to reverse the direction of the shoreward time-averaged
wave-induced velocity at the bottom when there are strong onshore winds.
(Cook and Gorsline, 1972; and Kraai, 1969.)
4.42 FLUID MOTION IN BREAKING WAVES.
During most of the wave cycle in shallow water, the particle velo-
city is approximately horizontal and constant over the depth, although
right at breaking there are significant vertical velocities as the water
is drawn up into the crest of the breaker. The maximum particle velocity
under a breaking wave is approximated by solitary wave theory (Equation
2-66) to be
Up max — C= J/g (H+d) , (4-13)
where (H+d) is the distance measured from crest of the breaker to the
bottom.
Fluid motions at breaking cause most of the sediment transport in
the littoral zone, because the bottom velocities and turbulence at break-
ing suspend more bottom sediment. This suspended sediment can then be
transported by currents in the surf zone whose velocities are normally
too low to move sediment at rest on the bottom.
The mode of breaking may vary significantly from spilling to plung-
ing to collapsing to surging, as the beach slope increases or the wave
4-42
steepness (height-to-length ratio) decreases. (Galvin, 1967.) Of the
four breaker types, spilling breakers most closely resemble the solitary
waves whose speed is described by Equation 4-13. (Galvin, 1972.) Spill-
ing breakers differ little in fluid motion from unbroken waves (Divoky,
LeMehaute, and Lin, 1970), and thus tend to be less effective in trans-
porting sediment than plunging or collapsing breakers.
The most intense local fluid motions are produced by plunging break-
ers. As the wave moves into shallower depths, the front face begins to
steepen. When the wave reaches a mean depth about equal to its height,
it breaks by curling over at the crest. The crest of the wave acts as a
free-falling jet that scours a trough into the bottom. At the same time,
just seaward of the trough, the longshore bar is formed, in part by sedi-
ment scoured from the trough and in part by sediment transported in rip-
ples moving from the offshore.
The effect of the tide on nearshore currents is not discussed, but
tide-generated currents may be superimposed on wave-generated nearshore
currents, especially near estuaries. In addition, the changing elevation
of the water level as the tide rises and falls may change the area and
the shape of the profile through the surf zone, and thus alter the near-
shore currents.
4.43 ONSHORE-OFFSHORE CURRENTS
4.431 Onshore-Offshore Exchange. Field and laboratory data indicate
that water in the nearshore zone is divided by the breaker line into
two distinct water masses between which there is only a limited exchange
of water.
The mechanisms for the exchange are: mass transport velocity in
shoaling waves, wind-induced surface drift, wave-induced setup, currents
induced by irregularities on the bottom, rip currents, and density cur-
rents. The resulting flows are significantly influenced by, and act on,
the hydrography of the surf and nearshore zones. Figure 4-13 shows the
nearshore current system measured for particular wave conditions on the
southern California coast.
At first observation, there appears to be extensive exchange of
water between the nearshore and the surf zone. However, the breaking
wave itself is formed largely of water that has been withdrawn from the
surf zone after breaking. (Galvin, 1967.) This water then reenters the
surf zone as part of the new breaking wave, so that only a limited amount
of water is actually transferred offshore. This inference is supported
by the calculations of Longuet-Higgins (1970, p. 6788) which show that
little mixing is needed to account for observed velocity distributions.
Most of the exchange mechanisms indicated act with speeds much slower
than the breaking-wave speed, which may be taken as an estimate of the
maximum water particle speed in the littoral zone indicated by Equation
ANUS
Scripps Bed
2 December 1948
°
Wave Period I5 Seconds teu
Wave from WNW 469 §
Le)
e265 oy
= > .25-.50KN °
== 50- 1.OKN
a= >| Knot
—~—% Observed Current (not measured) °
© Starting Position of Surface Float ee \t A Scripps
H, = Breaker Height ‘4 U/ institution
GS Float Recovery Area bod
¥ SNH, =3.5'
H,=5.5)
H,=6
SCALE IN FEET
° 500 1000
[== Se =e=S=o)
(from Shepard and Inman, 1950)
Figure 4-13. Nearshore Current System Near La Jolla Canyon, California
4.432 Diffuse Return Flow. Wind- and wave-induced water drift, pres-
sure gradients at the bottom due to setup, density differences due to
suspended sediment and temperature, and other mechanisms produce patterns
of motion in the surf zone that vary from highly organized rip currents
to broad diffuse flows that require continued observation to detect. Dif-
fuse return flows may be visible in aerial photos as fronts of turbid
water moving seaward from the surf zone. Such flows may be seen in the
photos reproduced in Sonu (1972, p. 3239).
4.433 Rip Currents. Most noticeable of the exchange mechanisms between
offshore and surf zone are rip currents. (See Figure 4-14, and Figure
A-7, Appendix A.) Rip currents are concentrated jets that carry water
seaward through the breaker zone. They appear most noticeable when long,
high waves produce wave setup on the beach. In addition to the classi-
cal rip currents, there are other localized currents directed seaward
from the shore. Some are due to concentrated flows down gullies in the
beach face, and others can be attributed to interacting waves and edge
wave phenomena. (Inman, Tait, and Nordstrom, 1971, p. 3493.) The origin
of rip currents is discussed by Arthur (1962), and Sonu (1972).
Three-dimensional circulation in the surf is documented by Shepard and
Inman (1950), and this complex flow needs to be considered, especially in
evaluating the results of laboratory tests for coastal engineering purposes.
However, at present, there is no proven way to predict the conditions that
produce rip currents or the spacing between rips. In addition, data are
lacking that would indicate quantitatively how important rip currents are
as sediment transporting agents.
4.44 LONGSHORE CURRENTS
4.441 Velocity and Flow Rate. Longshore currents flow parallel to the
shoreline, and are restricted mainly between the zone of breaking waves
and the shoreline. Most longshore currents are generated by the long-
shore component of motion in waves that obliquely approach the shoreline.
Longshore currents typically have mean values of 1 foot per second
or less. Figure 4-15 shows a histogram of 5,591 longshore current veloc-
ities measured at 36 sites in California during 1968. Despite frequent
reports of exceptional longshore current speeds, most data agree with
Figure 4-15 in showing that speeds above 3 feet per second are unusual.
A compilation of 352 longshore current observations, most of which appear
to be biased toward conditions producing high speed, showed that the maxi-
mum observed speed was 5.5 feet per second, and that the highest observa-
tions were reported to have been wind-aided. (Galvin and Nelson, 1967.)
Although longshore currents generally have low speeds, they are important
in littoral processes because they flow along the shore for extended peri-
ods of time, transporting sediment set in motion by the breaking waves.
The most important variable in determing the longshore current veloc-
ity is the angle between the wave crest and the shoreline. However, the
volume rate of flow of the current and the longshore transport rate depend
4-45
Figure 4-14. Typical Rip Currents, Ludlam
Island, New Jersey
mostly on breaker height. The outer edge of the surf zone is determined
by the breaker position. Since waves break in water depths approximately
proportional to wave height, the width of the surf zone on a beach in-
creases with wave height. This increase in width increases the cross
section of the surf zone.
2400
2000
Total of 5591 Observations
March-December 1968
1600
1200
800
Number of Observations
400
=5 -4 = 2) ae =| (0) | 2 3 4 5
Longshore Current Velocity, (feet per sec)
Figure 4-15. Distribution of Longshore Current Velocities. Data taken
from CERC California LEO Study (See Szuwalski 1970).
If the surf zone cross section is approximated by a triangle, then
an increase in height increases the area (and thus the volume of the flow)
as the square of the height, which nearly offsets the increase in energy
flux (which increases as the 5/2 power of height). Thus, the height is
important in determining the width and volume rate of longshore current
flow in the surf zone. (Galvin, 1972.)
Longshore current velocity varies both across the surf zone (Longuet-
Higgins, 1970b) and in the longshore direction (Galvin and Eagleson, 1965).
Where an obstacle to the flow, such as a groin, extends through the surf
zone, the longshore current speed downdrift of the obstacle is low, but
it increases with distance downdrift. Laboratory data suggest that the
current takes a longshore distance of about 10 surf widths to become fully
developed. These same experiments (Galvin and Eagleson, 1965) suggest that
the velocity profile varies more across the surf zone at the start of the
flow than it does downdrift where the flow has fully developed. The ratio
of longshore current speed at the breaker position to longshore current
speed averaged across the surf zone varied from about 0.4 where the flow
started to about 0.8 or 1.0 where the flow was fully developed.
4-47
4.442 Velocity Prediction. The variation in longshore current velocity
across the surf zone and along the shore, and the uncertainties in vari-
ables such as the surf zone hydrography, make prediction of longshore
current velocity uncertain. There are three equations of possible use
in predicting longshore currents: Longuet-Higgins (1970); an adaptation
from Bruun (1963); and Galvin (1963). All three equations require co-
efficients identified by comparing measured and computed velocities, and
all three show about the same degree of agreement with data. Two sets of
data (Putnam, et al., 1949, field data; Galvin and Eagleson, 1965, labora-
tory data) appear to be the most appropriate for checking predictions.
The radiation stress theory of Longuet-Higgins (1970a, Equation 62),
as modified by fitting it to the data, is the one recommended for use
based on its theoretical foundation. The other two semiempirical equa-
tions may provide a check on the Longuet-Higgins prediction. Written
in common symbols (m is beach slope; g is acceleration of gravity; Hp
is breaker height; T is wave period; and a, is angle between breaker
crest and shoreline), these equations are:
a. Longuet-Higgins.
vp. Mo om (gH, )” sin 2ap) , (4-14)
where
_ 0.694 1(26)-”
(4-15)
ff
According to Longuet-Higgins (1970a, p. 6788), vp is the longshore cur-
rent speed at the breaker position, JT is a mixing coefficient which
ranges between 0.17 (little mixing) and 0.5 (complete mixing), but is
commonly about 0.2; 8 is the depth-to-height ratio of breaking waves
in shallow water taken to be 1.2 and fr is the friction coefficient,
taken to be 0.01. Using these values, “M,= 9.0.
Applying equation 4-14 to the two sets of data yields predictions
that average about 0.43 of the measured values. In part, these predicted
speeds are lower because vp, as given in Equation 4-14 is for the speed
at the breaker line, whereas the measured velocities are mostly from the
faster zone of flow shoreward of the breaker line. (Galvin and Eagleson,
1965.) Therefore, Equation 4-14 multiplied by 2.3 leads to the modified
Longuet-Higgins equation for longshore current velocity:
v = 20.7 m(gH,)” sin 2a, , (4-16)
used in Figure 4-16. Further developments in the Longuet-Higgins' (1970b
and 1971) theory permit calculation of velocity distribution, but there
is no experience with these predictions for longshore currents flowing
on erodible sand beds.
Measured Longshore Current Speed (ft/sec)
S) OEE DDE e DO
DD OSBER EEE RES aES Shek SEOs
aa
AOC | |
BREORE ERA SESS
na fog
es
SUGEEVUOESESEGEE SeBBSE
Ee BESRR HHrralor tt
sie
0 eTaRauanerfatavavaet PEEEE EEE
0 | 2 3
aa
wane
BSESEE BEBSaea
PEO Ot ae aE etapa et tee mafia
pa pL EE BLY ARES Y Ueto aaa
SESS EREE TRAPS SAREE BRERA PA ERPS
ee eT
BB ait C real
anae \eé
eae eee eee
ei a ae Gaia
Co Feo ai,
had ners My auncieaee +
144 agate EL to
BHabSaSuas
EaeeB etait taluiclatetal sie
BSR aso
leson | | | |
Seabees Wehiisiat te
Putnom 25188
£
5 6
Computed Longshore Current Speed (ft/sec)
Figure 4-16. Measured Versus Predicted Longshore Current Speed
b. Bruun (1963 as Modified).
v, = M, (gH,)” [ mH, (sin 2ay) /t\* , (4-17)
where vp is the mean velocity in the surf zone where the flow is fully
developed, and My involves a friction factor of the Chezy kind (see
Galvin, 1967, p. 297.)
c. Galvin (1963).
vp = KgmTsin 2a, , (4-18)
where vp is the mean velocity in the surf zone where the flow is fully
developed, and K is a coefficient depending on breaker height-to-depth
ratio and the ratio of trough depression on breaker height. To a good
approximation, K may be taken as 1.0. (Galvin and Eagleson, 1965.)
4.45 SUMMARY
The major currents in the littoral zone are wave-induced motions
superimposed on the wave-induced oscillatory motion of the water. The
net motions generally have low velocities, but because they transport
whatever sand is set in motion by the wave-induced water motions, they
are important in determining littoral transport.
Evidence indicates that there is only slight exchange of fluid
between the offshore and the surf zone.
Longshore current velocities are most sensitive to changes in breaker
angle, and to a lesser degree, to changes in breaker height. However, the
volume rate of flow of the longshore current is most sensitive to breaker
height, probably proportional to H*. The modified Longuet-Higgins equation
(4-16) is recommended for predicting mean longshore current velocity of
fully developed flows, and the two semiempirical equations (4-17 and 4-18)
are available as checks on the Longuet-Higgins equation.
4.5 LITTORAL TRANSPORT
4.51 INTRODUCTION
4.511 Importance of Littoral Transport. Sediment motions indicated by
the shoreline configuration in Figure 4-17 are aspects of littoral trans-
port. If the coast is examined on satellite imagery as shown in Figure
4-17, only its general characteristics are visible. At this elevation,
the shore consists of bright segments that are straight or slightly curved.
The brightness is evidence of sand, the most common material along the
shore. The straightness often is evidence of sediment transport.
In places, the straight segments of shoreline cut across preexisting
topography. Elsewhere, the shoreline segments are separated by wide 1la-
goons from the irregular mainland. The fact that the shore is nearly
straight across both mainland and irregular bays is evidence of headland
4-50
ATLANTIC
OCEAN
: a ~*, * ay 7
‘Jones. Beach” |
Long, Tsigtd® |
New Mock |
wre (New York Bight)
pre
: “ Se ee a ?
Po ae eG, 3
+i, -
3%
Sandy Hook,
New Jerseys ©~
e
New Jersey. oe)
Figure 4-17. Coasts in Vicinity of New York Bight
erosion, accompanied by longshore transport which has carried sand along
the coast to supply the barriers and spits extending across the bays.
The primary agent producing this erosion and transport is the action of
waves impinging on the shore.
Littoral transport is the movement of sedimentary material in the
littoral zone by waves and currents. The littoral zone extends from the
shoreline to just beyond the most seaward breakers.
Littoral transport is classified as onshore-offshore transport or as
longshore transport. Onshore-offshore transport has an average net direc-
tion perpendicular to the shoreline; longshore transport has an average
net direction parallel to the shoreline. The instantaneous motion of
sedimentary particles has both an onshore-offshore and a longshore com-
ponent. Onshore-offshore transport is usually the most significant type
of transport in the offshore zone, except in regions of strong tidal
currents. Both longshore and onshore-offshore transport are significant
in the surf zone.
Engineering problems involving littoral transport generally require
answers to one or more of the following questions:
(a) What are the longshore transport conditions at the site?
(Needed for the design of groins, jetties, navigation channels, and
inlets.)
(b) What is the trend of shoreline migration over short and long
time intervals? (Needed for design of coastal structures, including
navigation channels.)
(c) How far seaward is sand actively moving? (Needed in the design
of sewage outfalls and water intakes.)
(d) What is the direction and rate of onshore-offshore sediment
motion? (Needed for sediment budget studies and beach fill design.)
(e) What is the average shape, and the expected range of shapes,
for a given beach profile? (Needed for design of groins, beach fills,
Navigation structures and flood protection.)
(f) What effect will a postulated structure or project have on
adjacent beaches and on littoral transport? (Needed for design of all
coastal works.)
This section presents recommended methods for answering these and
related questions. The section indicates accepted practice based on
field observations and research results. Section 4.52 deals with onshore-
offshore transport, presenting material pertinent to answering questions
(b) through (f). Section 4.53 deals with longshore transport, presenting
material pertinent to questions (a), (b), and (f).
4-52
4.512 Zones of Transport. Littoral transport occurs in two modes: bed-
load transport, the motion of grains rolled over the bottom by the shear
of water moving above the sediment bed; and suspended-load transport, the
transport of grains by currents after the grains have been lifted from
the bed by turbulence.
Both modes of transport are usually present at the same time, but it
is hard to distinguish where bedload transport ends and suspended-load
transport begins. It is more useful to identify two zones of transport
based on the type of fluid motion initiating sediment motion: the off-
shore zone where transport is initiated by wave-induced motion over rip-
ples, and the surf zone where transport is initiated by the passing break-
er. In either zone, net sediment transport is the product of two pro-
cesses: the periodic wave-induced fluid motion that initiates sediment
motion, and the superimposed currents (usually weak) which transport the
sediment set in motion.
a. Offshore Zone. Waves traveling toward shallow water eventually
reach a depth where the water motion near the bottom begins to affect the
sediment on the bottom. At first, only low-density material (such as sea-
weed and other organic matter) moves. This material oscillates back and
forth with the waves, often in ripple-like ridges parallel to the wave
crests. For a given wave condition, as the depth decreases, water motion
immediately above the sediment bed increases until it exerts enough shear
to move sand particles. The sand then forms ripples with crests parallel
to the wave crests. These ripples are typically uniform and periodic,
and sand moves from one side of the crest to the other with the passage
of each wave.
As depth decreases to a value several times the wave height, the veloc-
ity distribution with time changes from approximately sinusoidal to a dis-
tribution that has a high shoreward component associated with the brief
passage of the wave crest, and lower seaward velocities associated with
the longer time interval occupied by the passage of the trough. As the
shoreward water velocity associated with the passing crest decreases and
begins to reverse direction over a ripple, a cloud of sand erupts upward
from the lee (landward) side of the ripple crest. This cloud of sand
drifts seaward with the seaward flow under the trough. At these shallow
depths, the distance traveled by the cloud of suspended sediment is two
or more ripple wavelengths, so that the sand concentration at a point
above the ripples usually exhibits at least two maximums during the pass-
age of the wave trough. These maximums are the suspension clouds shed by
the two nearest upstream ripples. The approach of the next wave crest
reverses the direction of the sand remaining suspended in the cloud. The
landward flow also drags material shoreward as bedload.
For the nearshore profile to be in equilibrium with no net erosion or
accretion, the average rate at which sand is carried away from a point on
the bottom must be balanced by the average rate at which sand is added.
Any net change will be determined by the net residual currents near the
bottom which transport sediment set in motion by the waves. These currents,
4-53
the subject of Section 4.4, include longshore currents and mass-transport
currents in the onshore-offshore direction. It is possible to have ripple
forms moving shoreward while residual currents above the ripples carry
suspended sediment clouds in a net offshore direction. Information on the
transport of sediment above ripples is given in Bijker (1970), Kennedy and
Locher (1972), and Mogridge and Kamphuis (1972).
b. Surf Zone. The stress of the water on the bottom due to turbu-
lence and wave-induced velocity gradients moves sediment in the surf zone
with each passing breaker crest. This sediment motion is both bedload
and suspended-load transport. Sediment in motion oscillates back and
forth with each passing wave, and moves alongshore with the longshore
current. On the beach face, the landward termination of the surf zone,
the broken wave advances up the slope as a bore of gradually decreasing
height, and then drains seaward in a gradually thinning sheet of water.
Frequently, the draining return flows in gullies and carries sediment to
the base of the beach face.
In the surf zone, ripples cause significant sediment suspension, but
here there are additional eddies caused by the breaking wave. These eddies
have more energy and are larger than the ripple eddies. The greater energy
suspends more sand in the surf zone than offshore. The greater eddy size
mixes the suspended sand over a larger vertical distance. Since the size
is about equal to the local depth, significant quantities of sand are sus-
pended over most of the depth in the surf zone.
Since breaking waves suspend the sediment, the amount suspended is
partly determined by breaker type. Data from Fairchild (1972, Figure 5),
show that spilling breakers usually produce noticeably lower suspended
sediment concentrations than do plunging breakers. See Fairchild (1972)
and Watts (1953) for field data; Fairchild (1956 and 1959) for lab data.
Typical suspended concentrations of fine sand range between 20 parts per
million and 2 parts per thousand by weight in the surf zone, and are about
the same near the ripple crests in the offshore zone.
Studies of suspended sediment concentrations in the surf zone by
Watts (1953) and Fairchild (1972) indicate that sediment in suspension in
the surf zone may form a significant portion of the material in longshore
transport. However, present understanding of sediment suspension, and
the practical difficulty of obtaining and processing sufficient suspended
sediment samples have limited this approach to predicting longshore trans-
port.
4.513 Profiles. Profiles are two-dimensional vertical sections showing
how elevation varies with distance. Coastal profiles (See Figs. 4-1 and
4-18) are usually measured perpendicular to the shoreline, and may be
shelf profiles, nearshore profiles, or beach profiles. Changes on near-
shore and beach profiles are interrelated, and are highly important in
the interpretation of littoral processes. The measurement and analysis
of combined beach and nearshore profiles is a major part of most engineer-
ing studies of littoral processes.
Vertical Distance (Arbitrary Datum), feet
Beach Profile
(horizontal scale divided by 100)
(vertical exageration = 2)
Nearshore Profile
(horizontal scale divided by 10)
(vertical exageration = 10)
Inner Continental Shelf Profile
(vertical exageration = 50)
Horizontal Distance (Arbitrary Origin), feet
Figure 4-18. Three Scales of Profiles, Westhampton, Long Island
4-55
a. Shelf Profiles. The shelf profile is typically a smooth, concave-
up curve showing depth to increase seaward at a rate that decreases with
distance from shore. (bottom profile in Figure 4-18.) The smoothness of
the profile may be interrupted by other superposed geomorphic features,
such as linear shoals. (Duane, et al., 1972.) Data for shelf profiles
are usually obtained from charts of the National Ocean Survey (formerly,
U.S. Coast and Geodetic Survey).
The measurable influence of the shelf profile on littoral processes
is largely its effect on waves. To an unknown degree, the shelf may also
serve as a source or sink for beach'sand. Geologic studies show that
much of the outer edge of a typical shelf profile is underlain by rela-
tively coarse sediment, indicating a winnowing of fine sizes. (Dietz,
1963; Milliman, 1972; and Duane, et al., 1972.) Landward from this resi-
dual sediment, sediment often becomes finer before grading into the rela-
tively coarser beach sands.
b. Nearshore Profiles. The nearshore profile extends seaward from
the beach to depths of about 30 feet. Prominent features of most near-
shore profiles are longshore bars; see middle profile of Figure 4-18 and
Section 4.525. In combination with beach profiles, repetitive nearshore
profiles are used in coastal engineering to estimate erosion and accre-
tion along the shore, particularly the behavior of beach fill, groins,
and other coastal engineering structures. Data from nearshore profiles
must be used cautiously. (see Section 4.514.) Under favorable condi-
tions nearshore profiles have been used in measuring longshore transport
rates. (Caldwell, 1956.)
c. Beach Profiles. Beach profiles extend from the foredunes, cliffs,
or mainland out to mean low water. Terminology applicable to features of
the beach profile is in Appendix A (especially Figures A-1 and A-2). The
backshore extends seaward to the foreshore, and consists of one or more
berms at elevations above the reach of all but storm waves. Berm sur-
faces are nearly flat and often slope landward at a slight downward angle.
(See Figure 4-1.) Berms are often bounded on the seaward side by a break
in slope known as the berm crest.
The foreshore is that part of the beach extending from the highest
elevation reached by waves at normal high tide seaward to the ordinary
low water line. The foreshore is usually the steepest part of the beach
profile. The boundary between the backshore and the foreshore may be
the crest of the most seaward berm, if a berm is well developed. The
seaward edge of the foreshore is often marked by an abrupt step at low
tide level.
Seaward from the foreshore, there is usually a low-tide terrace which
is a nearly horizontal surface at about mean low tide level. (Shepard,
1950; and Hayes, 1971.) The low-tide terrace is commonly covered with sand
ripples and other minor bed forms, and may contain a large bar-and-trough
system, which is a landward-migrating sandbar (generally parallel to the
shore) common in the nearshore following storms. Seaward from the low-tide
4-56
terrace (seaward from the foreshore, if the low-tide terrace is absent)
are the longshore troughs and longshore bars.
4.514 Profile Accuracy. Beach and nearshore profiles are the major
source of data for engineering studies of beach changes; sometimes lit-
toral transport can be estimated from these profiles. Usually, beach
and nearshore profiles are measured at about the same time, but differ-
ent techniques are needed for their measurement. The nearshore profile
is usually measured from a boat or amphibious craft, using an echo sounder
or leadline. or from a sea sled. (Kolessar and Reynolds, 1965-66; and
Reimnitz and Ross, 1971.) Beach profiles are usually surveyed by standard
leveling and taping techniques.
The accuracy of profile data is affected by four types of error:
sounding error, spacing error, closure error, and error due to temporal
fluctuations in the sea bottom. These errors are more significant for
nearshore profiles than for beach profiles.
Saville and Caldwell (1953) discuss sounding and spacing errors.
Sounding error is the difference between the measured depth and the
actual depth. Under ideal conditions, average sonic sounding error may
be as little as 0.1 foot, and average leadline sounding error may be
about twice the sonic sounding error. (Saville and Caldwell, 1953.) (This
suggests that sonic sounding error may actually be less than elevation
changes caused by transient features like ripples. Experience with suc-
cessive soundings in the nearshore zone indicates that errors in practice
may approach 0.5 foot.) Sounding errors are usually random and tend to
average out when used in volume computations, unless a systematic error
due to the echo sounder or tide correction is involved. Long-period
water level fluctuations affect sounding accuracy by changing the water
level during the survey. At Santa Cruz, California, the accuracy of
hydrographic surveys was +1.5 feet due to this effect. (Magoon, 1970.)
Spacing error is the difference between the actual volume of a seg-
ment of shore and the volume estimated from a single profile across that
segment. Spacing error is potentially more important than sounding error,
since survey costs of long reaches usually dictate spacings between near-
shore profiles of thousands of feet. For example, if a 2-mile segment of
shore 4,000 feet wide is surveyed by profiles on 1,000-foot spacings, then
the spacing error is about 9 cubic yards per foot of beach front per survey,
according to the data of Saville and Caldwell (1953, Figure 5). This error
equals a major part of the littoral budget in many localities.
Closure error arises from the assumption that the outer ends of
nearshore profiles have experienced no change in elevation between two
successive surveys. Such an assumption is often made in practice, and
may result in significant error. An uncompensated closure error of 0.1
foot, spread over 1,000 feet at the seaward end of a profile, implies a
change of 3.7 cubic yards per time interval per foot of beach front where
4-57
the time interval is the time between successive surveys. Such a volume
change may be an important quantity in the sediment budget of the litto-=
ral zone.
A fourth source of error comes from assuming that the measured beach
profiles (which are only an instantaneous picture), represent a long-term
condition. Actually, beach and nearshore profiles change rapidly in re-
sponse to changing wave conditions, so that differences between succes-
sive surveys of a profile may merely reflect temporary differences in
bottom elevation caused by storms and seasonal changes in wave climate.
Such fluctuations obliterate long-term trends during the relatively short
time available to most engineering studies. This fact is illustrated for
nearshore profiles by the work of Taney (196la, Appendix B) who identified
and tabulated 128 profile lines on the south shore of Long Island that had
been surveyed more than once from 1927 to 1956. Of these, 47 are on
straight shorelines away from apparent influence by inlets, and extend
from Mean Low Water (MLW) to about -30 feet MLW. Most of these 47 pro-
files were surveyed three or more times, so that 86 separate volume
changes are available. These data lead to the following conclusions:
(a) The net volume change appears to be independent of the time
between surveys, even though the interval ranged from 2 months to 16
years. (See Figure 4-19.)
(b) Gross volume changes (the absolute sums of the 86 volume changes)
are far greater than net volume changes (the algebraic sums of the 86 vol-
ume changes). The gross volume change for all 86 measured changes is 8,113
cubic yards per foot; the net change is -559 cubic yards per foot (loss in
volume).
(c) The mean net change between surveys, averaged over all pairs of
surveys, is -559/86 or -6.5 cubic yards per foot of beach. The median
time between surveys is 7 years, giving a nominal rate of volume change
of about -1 cubic yard per year per foot.
These results point out that temporary changes in successive surveys
of nearshore profiles are usually much larger than net changes, even when
the interval between surveys is several years. These data show that care
is needed in measuring nearshore profiles if results are to be used in
engineering studies. The data also suggest the need for caution in inter-
preting differences obtained in two surveys of the same profiles.
The positions of beach profiles must be marked so that they can be
recovered during the life of the project. The profile monuments should
be tied in by survey to local permanent references. If there is a long-
term use for data at the profile positions, the monuments should be ref-
erenced by survey to a state coordinate system or other reference system,
so that the exact position of the profile may be recovered in the future.
Even if there is no anticipated long-term need, future studies in any _
coastal region are likely, and will benefit greatly from accurately sur-
veyed, retrievable benchmarks.
500
EY: eek)
OYCDOGDO OOO
Unit Volume Change Between Surveys, yd>/yr/ft
ro)
oO [@)
09 o0G0O
Oo} OO
100
(VOX 20)
0 |jo®
OO@ COcaD
@/@
NM
ro)
fo)
oO
-500
Time Between Surveys, months
( based on data from Taney, 196! a)
Figure 4-19. Unit Volume Change Versus Time Between Surveys for Profiles
on South Shore of Long Island. Data from Profiles Extending
from MSL to about the -30 depth Contour.
For coastal engineering, the accuracy of shelf profiles is usually
less critical than the accuracy of beach and nearshore profiles. Gener-
ally, observed depth changes between successive surveys of the shelf do
not exceed the error inherent in the measurement. However, soundings
separated by decades suggest that the linear shoals superposed on the
profile do show small but real shifts in position. (Moody, 1964, p. 143.)
Charts giving depths on the continental shelves may include soundings
that differ by decades in date.
Plotted profiles usually use vertical exaggeration or distorted
scales to bring out characteristic features. This exaggeration may lead
to a false impression of the actual slopes. As plotted, the three pro-
files in Figure 4-18 have roughly the same shape, but this sameness has
been obtained by vertical exaggerations of 2x, 10x, and 50x.
Sand level changes in the beach and nearshore zone may be measured
quite accurately from pipes imbedded in the sand. (Inman and Rusnak, 1956;
Urban and Galvin, 1969; and Gonzales, 1970.)
4.52 ONSHORE-OFFSHORE TRANSPORT
4.521 Sediment Effects. Properties of individual particles which have
been considered important in littoral transport include: size, shape,
immersed specific gravity, and durability. Collections of particles
have the additional properties of size distribution, permeability, and
porosity. These properties determine the forces necessary to initiate
and maintain sediment motion.
For typical beach sediment, size is the only property that varies
greatly. However, quantitative evaluation of the size effect is usually
lacking. A gross indication of a size effect is the accumulation of
coarse sediment in zones of maximum wave energy dissipation, and deposi-
tion of fine sediment in areas sheltered from wave action. (e.g. King,
1972, pp. 302, 307, 426.) Sorting by size is common over ripples (Inman,
1957) and large longshore bars (Saylor and Hands, 1970). Field work on
Size effects in littoral transport does not permit definite conclusions.
(King, 1972, p. 483; Inman, Komar, and Bowen, 1969; Castanho, 1970; Ingle,
1966, Figure 112; Yasso, 1962; and Zenkovich, 1967a.)
The shape of most littoral materials is approximately spherical;
departures from spherical are usually too slight to affect littoral
transport.
Immersed specific gravity (specific gravity of sediment minus spec-
ific gravity of fluid) is theoretically an important physical property of
the sediment particle. (Bagnold, 1963.) However, the variation in immersed
specific gravity for typical littoral materials in water is small since
most beach sediments are quartz (immersed specific gravity = 1.65), and
most of the remainder are calcium carbonate (immersed specific gravity
= 1.9). Thus, little variation in littoral transport is expected from
variation in immersed specific gravity.
4-60
Durability (resistance to abrasion, crushing, and solution) is usu-
ally not a factor within the lifetime of an engineering project. (Kuenen,
1956; Rusnak, Stockman, and Hofmann, 1966; and Thiel, 1940.) Possible
exceptions may include basaltic sands on Hawaiian beaches (Moberly, 1968),
some fragile carbonate sands which may be crushed to finer sizes when sub-
ject to traffic, (Duane and Meisburger, 1969, p. 44), and carbonate sands
which may be soluble under some conditions. (Bricker, 1971.) In general,
recent information lends further support to the conclusion of Mason (1942)
that, "On sandy beaches the loss of material ascribable to abrasion...
occurs at rates so low as to be of no practical importance in shore pro-
tection problems."
Size distribution and its relation to sediment sorting may be impor-
tant for design of beach fills. (See Sections 5.3 and 6.3.) Permeability
and porosity affect energy dissipation (Bretschneider and Reid, 1954; Bret-
schneider, 1954) and wave runup. (See Section 7.21; and Savage, 1958.)
Sediment properties are physically most important in determining fall
velocity and the hydraulic roughness of the sediment boundary. Fall velo-
city effects are important in onshore-offshore transport. Hydraulic rough-
ness effects have been insufficiently studied, but they appear to affect
initiation of sediment transport and energy dissipation. This may be par-
ticularly true for flow of the swash on the plane surface of the foreshore.
(Everts, 1972.)
4.522 Initiation of Sediment Motion. Considerable hydraulic and coastal
engineering research has been devoted to the initiation of sediment motion
under moving water. From this research has come general agreement (Graf,
1971, Chapter 6; Hjulstrom, 1939; and Everts, 1972) that the initiation of
motion on a level bed of fine or medium sand requires less shear (lower
velocities) than the initiation of motion on a level bed of silt or gravel;
(Figure 4-7 for size classes). It is also generally agreed that critical
entraining velocities for sand are usually less than 1 foot per second.
Velocities of wave-induced water motion in the offshore zone can be
estimated fairly well from the equation of small-amplitude theory. (See
Chapter 2.) This theory leads to Equation 2-13 which can be transformed
into a dimensionless expression for velocity at the sand surface (z = -d)
Umax — T T
Hsieh 3 ee pees se a (4-19)
H sinh and/
which is plotted in dimensionless form in Figure 4-20 and for common values
of wave period in Figure 4-21. Figures 4-20 and 4-21 give maximum bottom
particle velocity, umax(_j), as a function of depth, wave period, and
local wave height. This prediction by linear theory for the offshore zone,
and the related results from solitary-wave theory for the zone near break-
ing, agree fairly well with measurement in the field. (Inman and Nasu,
1956; and Cook and Gorsline, 1972, Figures 5 and 6.).
4-6l
“yydag anijojay
(dimensionless )
ini
H
Umax (
Figure 4-20. Maximum Wave Induced Bottom Velocity as a Function
of Relative Depth
62
Depth ( feet )
+
|
|!) DCD Di!!! Died
4D ap Wap ap Wap tap
feet,
|
:
3
Depth
Ty) > CI) CDICD
| 4 { +4
| 1} ] | | |
} EUDUTED PEGS TA PT u
| ana f i} Hitt |
Vil ] PTT | AnH) | i iii |
HEH \} me uaaayy | | | | WUE
| | i} | | if | | i i
beet INDI AURSUT NTH HTH RADHA FRNET FANT
i | | ; b b ) | | | | \)
MS Cn | hud | | LH |
Lt} | |
| | | | | iH
i | t | |
| | | }
| | | t
| |
Umox (-g)!
H
(dimensionless )
Figure 4-21. Maximum Bottom Velocity from Small Amplitude Theory
4-63
ke ek kK kK KK RK RK KK K F * EXAMPLE PROBLEM * * * * * * * * ¥ ®¥ ¥ ® FE KK
GIVEN: A wave in depth d = 200 feet, with period T = 9 seconds, and a
maximum bottom velocity Umax (-d) 2 1.0 foot per second.
FIND: The minimum wave height that will create the given bottom velocity.
SOLUTION: Calculate
gt?
ic 20
5.12 (9)?
415 feet .
and
200
&
iy pac
0.482
Entering Figure 4-20 with d/Lo = 0.482, determine
Umax; ay! A,
ae = 0.30,
T
= os “max (_q)
0.30 ,
H aap ACES 30 feet.
0.30
Thus a 30-foot minimum wave height with a 9-second wave period is needed
to create a bottom velocity equal to or greater than 1 foot per second
in 200 feet of water. Alternatively, a curve for a 9-second period can
be interpolated in Figure 4-21 and Sead can be read from the
curve's intersection with the 200-foot depth.
La a Pa PS Pe Te en a Jee Sea er el eR a MN a, PR eet oak tee eR SD
As a wave moves into shallower water, both bottom velocity and size
of water-particle orbit increase. For some low velocities where the hori-
zontal orbit is small, sand moves as individual grains rolling across the
surface, but for most conditions, once quartz sand begins to move, ripples
form (Kennedy and Falcon, 1965; Carstens, et al., 1969; Cook, 1970). Thus,
the initiation of sediment motion is usually indicated by the formation
of sediment ripples.
Figure 4-22, from Inman (1957) and including data from two earlier
studies, shows the velocity needed to start motion in a sediment bed of
a given grain size. These results are in general agreement with other
studies relating critical velocity to grain size. Also shown in this
figure are maximum velocities above which ripples tend to be smoothed
off, in qualitative agreement with conditions for bed forms in unidirec-
tional flows. (Southard, 1972.)
From Figure 4-22, it appears that maximum wave-induced bottom veloc-
ities between 0.4 and 1.0 foot per second are sufficient to initiate sand
motion under waves. In field studies, Inman (1957) found that ripples are
always present whenever computed maximum velocities exceed 0.33 foot per
second, and Cook and Gorsline (1972) report ripples above a velocity range
of 0.5 to 0.6 foot per second. Equation 4-19 can be used to determine the
combination of wave conditions and depth that produces any given critical
value of uma at the bottom. Figure 4-23 shows the relation between depth
and wave height, for given wave periods, for a critical velocity, Umax =
0.5 foot per second.
4.523 Seaward Limit of Significant Transport. Figures 4-20 through 4-23
and a knowledge of offshore wave climate suggest that waves can move bot-
tom sediments over most of the Continental Shelf (to depths of 100 to 400
feet or more) during some time of the year. Geologic studies indicate
that fine material has been winnowed from the surficial sediments over
much of the shelf. (Shepard, 1963; and Dietz, 1963.) The question is,
what is the maximum depth to which the rate of sand movement is signifi-
cant in coastal engineering? This section discusses field data that
supply partial answers to this question.
a. Bathymetry. Dietz (1963) and others point out that waves rework
nearshore sands, smoothing out irregularities by longshore and onshore-
offshore transport. This smoothing produces a quasi-equilibrium surface
in the nearshore zone which forms relatively straight contours, nearly
parallel to the shoreline.
Most bathymetric charts with closely spaced contours as illustrated
by Figures 4-24 and 4-25 show that isobaths near the shore run parallel
to the shoreline; further offshore, the contours may indicate linear
shoals (Duane, et al., 1972), or other irregular submarine features.
Following the idea of Dietz (1963), the depth below which shore-
parallel contours give way to irregular contours is assumed to mark the
local transition between the nearshore zone where sands are moved by the
waves in significant quantities and the offshore zone where sand is moved
in lesser quantities. Possible exceptions to this shore-parallel contour
rule are the contours around river deltas and inlets, or where reefs and
ledges occur in the nearshore zone.
4-65
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of 0.5 Ft/Sec ( Based on Linear Theory )
4-67
30°10’ X\
DXN
Contour Interval 5 feet (from Dietz, 1963, p. 984)
up to 40, 2 feet thereafter Feet
—_
(0) 2000 4000
Figure 4-24. Nearshore Bathymetry with Shore-Parallel
Contours off Panama City, Florida
Bathymetry, such as that in Figures 4-24 and 4-25, suggests that the
depth to the deepest shore-parallel contour is usually constant along the
shore for distances of several miles, but that this depth may vary with
longshore distances of about 10 miles. (See Figure 4-25.) The depth to
the deepest shore-parallel contour may depend on the contour spacing, but
this is not important if contour intervals are small relative to the total
depths involved. In general, the deepest shore-parallel contour is between
15 and 60 feet. In most localities, this depth is somewhat deeper than
that at which nearshore profiles are presumed to close-out, These near-
shore contours probably reflect longer term adjustment to extreme storms
that occur rarely during the typical time interval between repetitive
nearshore profile surveys.
b. Size Distribution. Geologic studies (Milliman, 1972; and Curray,
1965) suggest that littoral sands grade seaward into finer materials before
the relatively coarse sands of the shelf are reached. In some places the
boundary between the coarser shelf sediment and this finer material is quite
sharp. (Pilkey and Frankenberg, 1964.) The finer material is currently
4-68
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4-69
interpreted as bounding the seaward edge of sediment moved by waves in sig-
nificant quantities. This band of finer material suggests that there is
little exchange between littoral and shelf sands in most places.
c. Sand Budget Balancing. Onshore sand transport has been suggested
as a source of sand for several coastal localities that lack other obvious
sand sources. (Shepard, 1963, pp. 176-177; and Pierce, 1969.) For example,
Pierce suggests that the offshore must supply 440,000 cubic yards per year
of sand to the southern segment of the Outer Banks of North Carolina be-
cause other known sources do not balance the budget.
d. Transport in Nearshore Zone. Although theoretical, experimental,
and field data show that waves move sand some of the time over most of the
Continental Shelf, most of the data suggest that sand from the shelf is not
a significant contributor to the sediment budget of the littoral zone.
Sand transport from the nearshore zone is more likely. Surveys show
that sand in the nearshore zone does move, although it is difficult to meas-
ure direction of motion. The presence of shore-parallel contours along most
open shores is evidence that the waves actively mold the sand bottom in the
nearshore zone. However, the time scale of transport in the nearshore zone
may be relatively slow.
In tests at Santa Barbara, California, and at Atlantic City and Long
Branch, New Jersey, dredged sands were dumped offshore in depths ranging
from 15 to 40 feet, but no measurable onshore migration of the sand re-
sulted for times of about 1 year. (Hall and Herron, 1950.) Radioactive
tracers have shown that gravel moves slowly landward in 30 feet of water
at a rate of about 0.5 cubic yard per year per foot of beach. (Crickmore
and Waters, 1972.)
At shallower depths in the nearshore zone, onshore sand transport
following storms is well documented. Transport of sediment suspended
over ripples by the mass transport velocity is more than adequate to
return sand eroded from the beach or to transport sand eroded from the
nearshore bottom to the beach.
e. Summary on Seaward Limit. The deepest shore-parallel contour
appears to be a usable estimate of the maximum depth where significant
sand transport can be expected. This depth varies from 15 feet to per-
haps 60 feet or more along U.S. coasts. This choice may be modified for
specific wave conditions using Figure 4-23 to find the depth where maxi-
mum wave-induced velocity first exceeds 0.5 foot per second. If this
depth is less than the depth of the deepest shore-parallel contour, it
should be used as the seaward limit for the given wave conditions.
4.524 Beach Erosion and Recovery.
a. Beach Erosion. Beach profiles change frequently in response to
winds, waves, and tides. Profiles are also affected by events in the long-
shore direction that influence the longshore transport of sand. The most
4-70
notable rapid rearrangement of a profile is by storm waves, especially
during storm surge (Section 3.8) which enables the waves to attack at
higher elevations on the beach. (see Figure 1-7.)
The part of the beach washed by runup and runback is the beach face.
Under normal conditions, the beach face is contained within the fore-
shore, but during storms, the beach face is moved shoreward by the cut-
ting action of the waves on the profile. The waves during storms are
steeper, and the runback of each wave on the beach face carries away more
sand than is brought to the beach by the runup of the next wave. Thus
the beach face migrates landward, cutting a scarp into the berm. (See
Figure 1-7.)
In moderate storms, the storm surge and accompanying steep waves will
subside before the berm has been significantly eroded. In severe storms, or
after a series of moderate storms, the backshore may be completely eroded,
after which the waves will begin to erode the coastal dunes, cliffs, or main-
land behind the beach.
The extent of storm erosion depends on wave conditions, storm surge,
the stage of the tide and storm duration.
Potential damage to property behind the beach depends on all these
factors and on the volume of sand stored in the beach-dune system when a
storm occurs.
For planning and design purposes, it is useful to know the magnitude
of beach erosion to be expected during severe storms. Table 4-5 tabulates
the effect of four notable extratropical storms along the Atlantic coast
of the U.S. This table provides information on typical observed order-of-
Magnitude values for beach erosion above mean sea level (MSL) from single ~
storms.
For the storm of 17 December 1970, information is available from
seven localities (Column 2 of Table 4-5). (DeWall, et al., 1971.) The
three other storms include two closely spaced storms affecting Jones Beach,
New York, in February 1972 (Everts, 1972), and a storm that affected the
northern New Jersey coast in November 1953. (Caldwell, 1959.) Character-
istics that distinguish one storm from another are duration and storm
surge. (See Colums 9 and 10, Table 4-5.) Storm waves lasted about 1 day
for the 17 December 1970 storm, and about 2 days for the other three
storms. (See Colum 9.) Storm surge elevation varied from a low of 2.8
feet to a high of 6 feet in the New York Bight area. The November 1953
storm combined longer duration and high storm surge; the 17 December 1970
storm had short duration and moderate storm surge; and the February 1972
storms both had longer duration, one with moderate storm surge and the
other with low storm surge.
Duration and storm surge (Columns 9 and 10) are consistent with storm
damage data (Columns 11 through 16, although the effect is influenced by
the choice of profiles included in each study. The December 1970 storm
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includes all profiles surveyed. The two February 1972 storms at Jones
Beach include all profiles away from the influence of inlets, reducing
the number of profiles from 15 in the December 1970 storm to 10 in the
February 1972 storms. The November 1953 storm gives relatively high
storm damage which may partly result from the long time interval between
pre-storm survey and the storm. (See Columns 1 and 3.) These results
are also affected by the fact that they omit some profiles ''believed to
be influenced by the presence of a seawall or a bulkhead." (Caldwell,
19595 pe 42)
Although the data in Table 4-5 are not completely comparable, the
results do suggest that the average volume of sand eroded above mean sea
level from beaches about 5 or more miles long has a certain range of val-
ues. A moderate storm may remove 4 to 10 cubic yards per foot of beach
front above MSL; an extreme storm (or a moderate storm that persists for
a long time) may remove 10 to 20 cubic yards per foot; rare storms that
are most destructive in beach erosion due to a combination of intensity,
duration, and orientation may remove 20 to 50 cubic yards per foot. These
values are average for beaches 5 to 10 miles or more long, and they are
compatible with other, less complete, data for notable storms. (Caldwell,
1959; Shuyskiy, 1970; and Harrison and Wagner, 1964.) For comparative
purposes, a berm 100 feet wide at an elevation of 10 feet MSL contains 37
cubic yards per foot of beach front, a quantity that would be adequate
except for extreme storms.
In terms of horizontal changes rather than the volume changes in
Table 4-5, a moderate storm can erode a typical beach 75 to 100 feet or
more, and leave it exposed to greater erosion if a second storm follows
before the beach has recovered. This possibility should be considered
in design and placement of beach fills and other protective measures.
Extreme values of erosion may be more useful than mean values for
design. Column 17 of Table 4-5 suggests that the ratio of the most ero-
ded above-MSL-profile to the average profile for east coast beaches ranges
from about 1.5 to 6. If the average erosion per profile is based only on
those profiles showing net erosion, then this ratio is probably between
1.5 and 3.
Although the dominant result of storms on the above MSL part of
beaches is erosion, most post-storm surveys show that the storm produces
local accretion as well. Of the 90 profiles from Cape Cod, Massachusetts,
to Cape May, New Jersey, surveyed immediately after the December 1970
storm, 16 showed net accretion above mean sea level. (Compare Columns 4,
11, and 12 in Table 4-5.) Similar results are indicated for a number of
more severe storms. (Caldwell, 1959.)
The storm surveys also show that the shoreline on many beaches may
prograde seaward even though the profile as a whole loses volume, or
vice versa. This possibility suggests caution in interpreting aerial
photos of storm damage. (Everts, 1973.)
4-74
b. Beach Recovery. The typical beach profile left by a severe storm
is a simple, concave-upward curve extending seaward to low tide level or
below. (See top of Figure 4-26.) The sand that has been eroded from the
beach is deposited mostly as a ramp or bar in the surf zone that exists at
the time of the storm. Immediately after the storm, beach repair begins
by a process that has been documented in detail. (e.g., Hayes, 1971; Davis,
et al., 1972; Davis and Fox, 1972; and Sonu and van Beek, 1971.) Sand that
has been deposited seaward of the shoreline during the storm begins moving
landward as a sandbar with a gently sloping seaward face and a steeper land-
ward face. (See Figure 4-26.) These bars have associated lows (runnels)
on the landward side and occasional drainage gullies across them. (King,
1972, p. 339.) These systems are characteristic of post-storm beach accre-
tion under a wide range of wave, tide, and sediment conditions. (Davis,
et al., 1972.) They are sometimes called ridge-and-runnel systems.
The processes of accretion occur as follows. Sand is transported
landward over the nearly flat seaward face of the bar by the waves. At
the bar crest, the sand avalanches down the landward slip face. If the
process continues long enough, the bar reaches the landward limit of
storm erosion where it is "welded" onto the beach. (e.g. Davis, et al.,
1972.) Further accretion continues by adding layers of sand to the top
of the bar which, by then, is a part of the beach. (See Figure 4-27.)
Berms may form immediately on a post-storm profile without an inter-
vening bar-and-trough, but the mode of berm accretion is quite similar
to the mode of bar-and-trough growth. Accretion occurs both by addition
of sand laminae to the beach face (analogous to accretion on the seaward-
dipping top of the bar in the bar-and-trough) and by addition of sand on
the slight landward slope of the berm surface when waves carrying sedi-
ment overtop the berm crest (analogous to accretion on the landward dip-
ping slip face of the bar). This process of berm accretion is also illus-
trated in Figure 4-1.
The rate at which the berm builds up or the bar migrates landward to
weld onto the beach varies greatly, apparently in response to: wave con-
ditions, beach slope, grain size, and the length of time the waves work
on the bars. (Hayes, 1971.) Compare the slow rate of accretion at Crane
Beach in Figure 4-26 (mean tidal range 9 feet, spring range 13 feet) with
the rapid accretion on the Lake Michigan shore in Figure 4-27 (tidal range
esse than 0525) foot).
Post-storm studies by CERC show that the rate of post-storm replenish-
ment by bar migration and berm building is usually rapid immediately after
the storm. This rapid buildup is important in evaluating the effect of
severe storms because (unless surveys are made within hours after the
storm) the true extent of erosion during the storm is likely to be ob-
scured by the post-storm recovery. Lack of surveys before the start of
post-storm recovery may explain some survey data that show MSL accretion
on profiles that have lost volume.
6) 100
Figure 4-26.
Summer Accretion 29 May —7 September 1967
Station CBA, Crane Beach
Ipswich, Massachusetts
ee
ae dt ae
o-— .
WS Sesh aw
~ ore. —
=—-——_——. Bene
ae orn
_—_— — = — a
_
Seite =— —
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—
~ Se eee Toa
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—
~ - =~ —
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-_
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200 300
Feet
29 May
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sat seus
aS 24 July
SS =o
oe 8 Aug
Se ee ae eee
Para Se 14 Aug
= egal 22 Aug
Cle
sts i 7 Sept wi w
400 500
(from Hayes, 1972)
Slow Accretion of Ridge-and-Runnel at Crane
Beach, Massachusetts
(ueSTYyoTW ‘PUeTIOH)
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| | | | | |
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4-7
The ideal result of post-storm beach recovery is a wide backshore
that will protect the shore from the next storm. Beach recovery may be
prevented when the period between successive storms is too short. Main-
tenance of coastal protection requires knowledge of the necessary width
and elevation of the backshore appropriate to local conditions, and
adequate surveillance to determine when this natural sand reservoir is
diminished to a point where it may not protect the backshore during the
next storm.
4.525 Bar-Berm Prediction. High, steep waves scour the beach,eroding the
foreshore into a simple concave-upwards profile. The material eroded from
the beach is deposited offshore as a longshore bar. Waves of low steepness
tend to push sand onto the beach, usually as migrating longshore-bar sys-
tems which eventually become part of the beach. In contrast to the concave-
up eroded profile discussed previously, the accreted profile is concave-
downward. Idealized eroded and accreted profiles (measured in a prototype-
scale wave tank) across the beach and nearshore zone are shown in Figure
4-28.
To design a beach that contains a reservoir of sand in the backshore
sufficient to survive a design storm, a minimum requirement is the ability
to distinguish wave conditions that cause eroded profiles from those that
cause accretion. Usually, it is assumed that a berm characterizes an ac-
creted profile and that a bar characterizes an eroded profile. (See Figure
4-28.) This picture is somewhat idealized. A sharp berm crest between
backshore and foreshore is often lacking, and on some beaches the berm is
absent, so that the top of the foreshore reaches the dune or cliff line.
Berms are illustrated in Figures 4-1 and 4-27.
Similarly, the idealized longshore bar seaward of an eroded beach,
(middle profile of Figure 4-18) is often absent, and in its place there
may be several subdued bars or a platform extending to the breaker line
at nearly constant depth.
a. Longshore Bars. The term bar has been applied to a number of
quite different coastal features, including barrier islands (the "off-
shore bar'' of Johnson, 1919), ridge-and-runnel systems, and linear shoals.
(Duane, et al., 1972.) Longshore bars are unrelated to any of these fea-
tures. They appear to most nearly resemble a ridge-and-runnel system,
but differ in that longshore bars are located at the breaker position and,
at least in part, are eroded out of the bottom by the falling breaker,
whereas the ridge-and-runnel system is an accretionary feature migrating
landward across the surf zone.
The typical longshore bar, as described from the observations of
Shepard (1950) and the experiments of Keulegan (1948), is a ridge of sand
parallel to the shore and formed at the breaking position of large plung-
ing breakers. Longshore bars seem most directly related to the height of
larger breakers (not necessarily of maximum height). The depth to the sea-
ward bar increases with the height of the larger breakers along the Pacific
coast. (Shepard, 1950.) Bars form readily in tidal seas, but seem better
4-78
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4-79
developed in tideless seas such as Lake Michigan (Fig. 4-27). (Saylor
and Hands, 1970.) Keulegan (1948) found that the ratio of depth of long-
shore trough to depth of bar was approximately 1.69 in laboratory experi-
ments, but most field measurements showed less depth difference, averag-
ing about 1.23 (based on MLLW) for 276 measurements from the Scripps pier.
(Shepard, 1950.) According to Shepard, bars are not significant on slopes
steeper than 4° (1 on 14).
There is evidence that longshore bars, as described, are formed by a
transient condition when waves of a given height and period plunge on a
relatively plane sand slope. Shepard found that steep storm waves elimi-
nate the bars rather than build them, and that bars form after the largest
storm wave subsided. Such a relation is consistent with plunging breaker
conditions predicted by the breaker type parameter (Galvin, 1972). Be-
cause bars are formed by high waves, they may persist through long inter-
vals of low waves. Once formed, bars may trigger the breaking of higher
waves, dissipating the wave energy and thus reducing beach erosion. (Davis
and Fox, 1972; and Zwamborn, Fromme, and Fitzpatrick, 1970.)
Laboratory observations show that longshore bars form when waves
plunge, and that these bars are absent when waves spill. With constant
wave conditions, a wave may plunge initially on a steep sand slope and
form a bar. . The beach then erodes forming a flatter slope, which changes
the breaker type to spilling, which eliminates the pronounced longshore
bar. For other constant wave conditions, a wave may spill initially on
a gentle sand slope, and no pronounced bar forms. Later in the test, the
breaker position migrates closer to the steeper foreshore, the breaker
begins to plunge, and a longshore bar forms.
b. Steepness Effect. The distinction between profiles with pro-
nounced berms (usually without bars) and profiles with eroded foreshores
(often with longshore bars) is well known, (Nayak, 1970, and Johnson,
1956.) Early laboratory results suggested that the shape of the profile
depends on deepwater steepness, Ho/Lo, of the waves reaching the beach.
Between 1936 and 1956, laboratory experiments were made which led to the
assumption that beach profiles generally eroded if H)/Lo exceeded 0.025
and accreted if Hj/Lo was less than about 0.02.
However, neither field data nor prototype-size laboratory experiments
support this widely used criterion. (Saville, 1957.) Field and prototype-
size laboratory data of Saville showed that beaches eroded at signifi-
cantly lower deepwater steepness than the value of 0.025 derived from
model laboratory experiments. Saville (1957) concluded that the absolute
size of wave height was probably as important as steepness in determining
the profile.
c. Dimensionless Fall Time. Prediction of accreted (berm) or erod-
ed (bar) profiles is possible using a dimensionless fall-time parameter
F = 4 (4-20)
Oo
V,T
where H, is deepwater wave height, T is wave period, and Vr is the
fall velocity of the beach sediment. (Dean, 1973.)
The fall-time parameter Fp, is plotted against deepwater steep-
ness, Ho/Lo, in Figure 4-29 using the profile data of Rector (1954, Table
1, Column 25) and Saville (1957, unpublished). These data include wave
heights ranging from about 0.05 foot to 5.0 feet, a range compatible with
field conditions. They also include a range of initial slopes.
In Figure 4-29, the line of demarcation between deposition offshore
and deposition onshore is approximately at the value Fo = 1. More com-
plete separation is possible when F, is plotted against H,/Mg. (See
Figure 4-30.)
Values of Fy, > 1 indicate that the time required by the particle
to fall a distance about equal to the maximum depth in the surf zone is
greater than the time available between arriving wave crests. Thus,
values of Fy significantly greater than 1 suggest significant concen-
trations of suspended sediment, which are expected to diffuse seaward
and deposit offshore.
Since values of Vr range from about 0.066 foot per second (2 centi-
meters per second) for fine sand to 0.49 foot per second (15 centimeters
per second) for coarse sand (Fig. 4-31), Fy ranges from about 0.25 for
low swell on coarse sand beaches to 10 or more for storm waves on fine
sand beaches. Such values suggest that typical field and laboratory con-
ditions define a range of conditions within which the importance of sus-
pended sediment in the surf zone may vary from significant to negligible.
The effect of temperature on fall velocity (Fig. 4-31) is important
enough to be critical under some conditions. It appears that temperature
by itself, in its effect on fall velocity, can change a profile from erod-
ing to accreting.
To summarize results on berm-bar criteria, the dimensionless fall
time, F, = H,/(V-T), Equation 4-20 provides an estimate of the separa-
tion between berm and bar-type profiles. A value of F, between 1 and 2
appears to be the critical value, with F, = 2 being more appropriate for
prototype-size waves. For F, less than the critical value, the beach
accretes above MLW. The effect of fall velocity is important. Deepwater
wave steepness by itself is an unreliable criterion for prototype condi-
tions.
4-8|
Dimensionless Fall Time, Ho /(V,T)
osition Onshore
Oz 0.42 Orgaal
(Kohler and Galvin, 1973)
00! +—+—- 4—+-+-4 > + pees 0ee eee Se: peaay Fs + - pRE ERS HY LAPP hs) uk 4
0.00! 0.002 0.004 0.0! 002 004 00701
Deepwater Steepness, Ho/Lo
Figure 4-29. Berm-Bar Criterion Based on Dimensionless Fall
Time and Deep Water Steepness
Demensionless Fall Time, Ho/(V¢T)
(Eee Bae
oe
10 20 3040 60 100 200 400 1000 2000 5000 10,000
(Kohler and Galvin, 1973)
Wave Height-to-Grain Size Ratio, Ho/Mg (ft/ft)
Figure 4-30. Berm-Bar Criterion Based on Dimensionless Fall
Time and Height-to-Grain Size Ratio
@|D9S AVIS PsOpuUDIS 4sajAy
TNORrROAMoON TOO TONNYONDWO
--- TT 0
—-NNNMM-
50 70 100
(from Schulz, et al.,1954)
20 30
.
AN
.
XA |
Sk
lil fed REI Case Eg WW Ges em eet aE
|G Ee mar SEEEEESST UG) ENN NNT ne nj eeESE ES oa
0.5 0.7
Fall Velocity (cm/sec )
0.2 0.3
.05 0.07 0.1
| peomase
ity e
ieee fate ee
epee! casei
il sees 4
i ED POE |
ue separ seman:
0.1
(WW) JajawoIG ajdI440q
Fall Velocity of Quartz Spheres in Water as a Function
of Diameter and Temperature
Figure 4-31.
4.526 Slope of the Foreshore. The foreshore is the steepest part of the
beach profile. The equilibrium slope of the foreshore is a useful design
parameter, since this slope, along with the berm elevation, determines
minimum beach width.
The slope of the foreshore tends to increase as the grain size in-
creases. (U.S. Army, Beach Erosion Board, 1933; Bascom, 1951; and King,
1972, p. 324.) This relationship between size and slope is modified by
exposure to different wave conditions (Bascom, 1951; and Johnson, 1956);
by specific gravity of beach materials (Nayak, 1970; and Dubois, 1972);
by porosity and permeability of beach material (Savage, 1958); and prob-
ably by the tidal range at the beach. Analysis by King (1972, p. 330)
suggests that slope depends dominantly on sand size, and also signifi-
cantly on an unspecified measure of wave energy.
Figure 4-32 shows trends relating slope of the foreshore to grain
size along the Florida Panhandle, New Jersey-North Carolina, and the
U.S. Pacific coasts. Trends shown on the figure are simplifications of
actual data, which are plotted in Figure 4-33. The trends show that,
for constant sand size, slope of the foreshore usually has a low value
on Pacific beaches, intermediate value on Atlantic beaches, and high
value on Gulf beaches,
This variation in foreshore slope from one region to another appears
to be related to the mean nearshore wave heights. (See Figures 4-10, 4-11,
and Table 4-4.) The gentler slopes occur on coasts with higher waves. An
increase in slope with decrease in wave activity is illustrated by data
from Half Moon Bay (Bascom, 1951), and is indicated by the results of King
E972, ps4 552i
The inverse relation between slope and wave height is partly caused
by the relative frequency of steep or high eroding waves which produce
gentle foreshore slopes and low accretionary post-storm waves which pro-
duce steeper beaches. (See Figures 4-1, 4-26, and 4-27.)
The relation between foreshore slope and grain size shows greater
scatter in the laboratory than in the field. However, the tendency for
slope of the foreshore to increase with decreasing mean wave height is
supported by laboratory data of Rector (1954, Table 1). In this labora-
tory data, there is an even stronger inverse relation between deepwater
steepness, Ho/Lo, and slope of the foreshore than between Hg and the
slope.
To summarize the results on foreshore slope for design purposes, the
following statements are supported by available data:
(a) Slope of the foreshore on open sand beaches depends principally
on grain size, and (to a lesser extend) on nearshore wave height.
(b) Slope of the foreshore tends to increase with increasing median
grain size, but there is significant scatter in the data.
4-85
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height, again with scatter.
(d) For design of beach profiles on ocean or gulf beaches, use
Figure 4-32, keeping in mind the large scatter in the basic data on Fig-
ure 4-33, mich of which is caused by the need to adjust the data to
account for differences in nearshore wave climate.
4.53 LONGSHORE TRANSPORT RATE
4.531 Definitions and Methods. Ltttoral drift is the sediment (usually
sand) moved in the littoral zone under action of waves and currents. The
rate, Q, at which littoral drift is moved parallel to the shoreline is
the longshore transport rate. Since this movement is parallel to the
shoreline, there are two possible directions of motion, right and left,
relative to an observer standing on the shore looking out to sea. Move-
ment from the observer's right to his left is motion toward the left, in-
dicated by the subscript %t. Movement toward the observer's right is
indicated by the subscript rt.
Gross longshore transport rate, Qg> is the sum of the amounts of
littoral drift transported to the right’ and to the left, past a point on
the shoreline in a given time period.
Os On * One (4-21)
Similarly, net longshore transport rate, Q,, is defined as the
difference between the amounts of littoral drift transported to the right
and to the left past a point on the shoreline in a given time period.
Raz Oe Qe - ; (4-22)
The quantities Qnt, Qet, Q® and have engineering uses: for
example, is used to predict shoaling rates in uncontrolled inlets;
Q is used for design of protected inlets and for predicting beach ero-
sion on an open coast; Q z¢ and Q+ are used for design of jetties and
impoundment basins behind weir jetties. In addition, Qg provides an
upper limit on other quantities.
Occasionally, the ratio
Ege
; 4-23
Q, (4-23)
a
is known, rather than the separate values Qy; and Q,¢. Then & is
related to Q, in terms of y by
City)
= ———— , 4-24
Q On (as) ( )
This equation is not very useful when y approaches 1.
4-88
Longshore transport rates are usually given in units of volume per
time (cubic yards per year in the U.S.). Typical rates for oceanfront
beaches range from 10° to 10© cubic yards per year. (See Table 4-6.)
These volume rates typically include about 40 percent voids and 60 per-
cent solids.
At present, there are four basic methods to use for the prediction
of longshore transport rate:
1. The best way to predict longshore transport at a site is to
adopt the best known rate from a nearby site, with modifictions based
on local conditions.
2. If rates from nearby sites are unknown, the next best way to
predict transport rates at a site is to compute them from data showing
historical changes in the topography of the littoral zone (charts, sur-
veys, and dredging records are primary sources).
3. If neither Method 1 nor Method 2 is practical, then it is
accepted practice to use either measured or calculated wave conditions
to compute a longshore component of "wave energy flux" which is related
through an empirical curve to longshore transport rate. (Das, 1972.)
4. A recently developed empirical method (Galvin, 1972) is avail-
able to estimate gross longshore transport rate from mean annual near-
shore breaker height. The gross rate, so obtained, can be used as an
upper limit on net longshore transport rate.
Method 1 depends largely on engineering judgement and local data.
Method 2 is an application of historical data, which gives usable answers
if the basic data are reliable and available at reasonable cost, and the
interpretation is based on a thorough knowledge of the locality. By
choosing only a few representative wave conditions, Method 3 can usually
supply an answer with less work than Method 2, but with correspondingly
less certainty. Because calculation of wave statistics in Method 3 follows
an established routine, it is often easier to use than researching the
hydrographic records and computing the changes necessary for Method 2.
Method 4 requires mean nearshore breaker height data. Section 4.532
utilizes Methods 3 and 4. Methods 1 and 2 are discussed in Section 4.8.
4.532 Energy Flux Method. Method 3 is based on the assumption that long-
shore transport rate, Q, depends on the longshore component of energy flux
in the surf zone. The longshore energy flux in the surf zone is approxi-
mated by assuming conservation of energy flux in shoaling waves, using
small-amplitude theory, and then evaluating the energy flux relation at
the breaker position. The energy flux per unit length of wave crest, or,
equivalently, the rate at which wave energy is transmitted across a plane
of unit width perpendicular to the direction of wave advance is (from
Section 2.238, combining Equations 2-39 and 2-40):
P= pelsiee is
BS eae (4-25)
Table 4-6. Longshore Transport Rates from U.S. Coasts
Predominant | Longshore*
: Ren Date of
Location Direction of Transport Reference
Record
Transport (cu.yd./yr.)
Atlantic Coast
200,000 | 1946-55 \S. Army (1955a)
493,000 1885-1933 .S. Army (1954b)
436,000 1933-51 .S. Army (1954b)
200,000 | 1922-25 .S. Army (1954b)
300,000 1947-53 .S. Army (1954b)
360,000 | 1930-31 \S. Army (1954b)
250,000 | 1939-41 \S. Army — (1954b)
400,000 1935—46 .S. Army (1954b)
400,000 1935-46 .S. Congress (1953a)
200,000 .S. Congress (1953b)
150,000 | 1934—36 .S. Army (1948a)
29,500 1850—1908 -S. Congress (1948)
75,000 | 1850-1908 | U.S. Army (1955b)
150,000 | 1925—30 \S. Army (1947)
to
225,000
Suffolk County, N.Y.
Sandy Hook, N.J.
Sandy Hook, N.J.
Asbury Park, N.J.
Shark River, N.J.
Manasquan, N.jJ.
Barneget Inlet, N.J.
Absecon Inlet, N.J. t
Ocean City, N.J. f
Cold Spring Inlet, N.J.
Ocean City, Md.
Atlantic Beach, N.C.
Hillsboro Inlet, Fla.
Palm Beach, Fla.
ANMnNnNNNNAAZAAZZAZZS
Gulf of Mexico
Pinellas County, Fla. 50,000 1922-50 U.S. Congress (1954a)
Perdido Pass, Ala. 200,000 1934—53 U.S. Army = (1954c)
Pacific Coast
Santa Barbara, Calif.
Oxnard Plain Shore, Calif.
Port Hueneme, Calif.
Santa Monica, Calif.
El Segundo, Calif.
Redondo Beach, Calif.
Anaheim Bay, Calif. t
Camp Pendleton, Calif.
280,000 | 1932-51 Johnson _—(1953)
1,000,000 | 1938-48 U.S. Congress (1953d)
500,000 U.S. Congress (1954b)
270,000 | 1936—40 U.S. Army (1948)
162,000 1936—40 U.S. Army (1948b)
30,000 U.S. Army (1948)
150,000 1937-48 U.S. Congress (1954c)
100,000 1950—52 U.S. Army (1953a)
AMnNNNN
Great Lakes
Milwaukee County, Wis. 8,000 1894-1912 .S. Congress (1946)
Racine County, Wis. 40,000 1912-49 .S. Congress (1953e)
Kenosha, Wis. 15,000 1872—1909 .S. Army (1953b)
Ill. State Line to Waukegan 90,000 .S. Congress (19536)
Waukegan to Evanston, Ill. 57,000 .S. Congress (1953f)
South of Evanston, Ill. 40,000 .S. Congress (1953f)
Waikiki Beach t a a ee eee ace,
(from Wiegel, 1964; Johnson, 1957)
* Transport rates are estimated net transport rates, Qy. In some cases, these approximate the gross transport rates, Qs.
t Method of measurement is by Accretion except for Absecon Inlet, and Ocean City, New Jersey, and Anaheim Bay,
California, by Erosion, and Waikiki Beach, Hawaii, by Suspended Load Samples.
4-90
If the wave crests make an angle, a with the shoreline, the energy flux
in the direction of wave advance per untt length of beach is
€. cosia™, (4-26)
and the longshore component is given by
Po) = P cosa sina = Pe HC cosa sina ,
8 g&
or since cos a sina = 1/2 sin 2a
ig] eye :
Po 46 H C, sin 2a . (4-27)
The surf-zone approximation of Pg is written as Pgg.
= P68
Pos = 76 HBC, sin 2ay « (4-28)
Usable formulations of this surf-zone approximation can be obtained by
several methods. The principal approximation is in evaluating Cg and H
at the breaker position. It is standard practice to approximate the group
velocity, Cg, by the phase velocity, C, at breaking. The phase velo-
city may then be approximated by either linear wave theory (Equation 2-3)
or by solitary wave theory (Equation 4-13).
Figure 4-34 presents the longshore component of wave energy flux in
a dimensionless form, P,/pg* HE T, as a function of breaker steepness,
H,/eT’, and the angle the wave crest makes with the shoreline in either
deep water a,» or at the breaker line, ap Figure 4-34 is based on
Equation 4-28 using linear wave theory to determine C, and assuming that
refraction is by straight parallel bottom contours. Figure 4-35 can be
used to determine the longshore component of wave energy flux when breaker
height, “Hp, period, 7, and, angle; %p, are known--for example, for
surf observation data. The use of Figure 4-34 is illustrated by an
example problem below.
For linear theory, in shallow water, Cg ~ C and
Poe =, ap © smZepe, (4-29)
where Hp and ap are the wave height and direction at breaking and C
is the wave speed from Equation 2-3, evaluated in a depth equal to 1.28
Hp.
Brae
Hy
iat
ieee
tf i
geuanuge
aus
Hao
it
+
iae
1
iit
sStsz
+t
i
i tH
Tots
+t
Judi
uit
tot
Att
80° 100°
60°
Deepwater Wave Angle, Qo, degrees
less Form
imension
Longshore Component of Wave Energy Flux in D
as a Function of Breaker Conditions
4-34
Figure
= 2
kok * & kk kK KX kk *& * * EXAMPLE PROBLEM * * * * * * * * & FE RFR FER
GIVEN: A breaking wave with height, Hp = 4 feet; period T = 7 seconds.
Surf observations indicate that the wave crest at breaking makes an
angle, ap = 6° with the shoreline.
FIND:
(a) The longshore component of wave energy flux.
(b) The angle the wave made with the shoreline when it was in deep
water, do.
SOLUTION: Calculate
DNA a
— = ———— = 0.00254 .
gT? 32.2 (7.0)?
Enter Figure 4-34 to the point where the line aes ap = 6° crosses
Hp/gT? = 0.00254 and read Py/og* He T = 6.7 x 10°* from ae left axis
and dg = 18° from the bottom axis.
The longshore energy flux can then be calculated as,
La)
ll
= 0.00067 pg? HZ T
Pp = 0.00067 (2) (32:2)? G20)7 17:0) «
P, = 155.6, say 160 ft.-lb./ft.-sec. ,
and
a./= 18? .
Meek Kes ie ie ae Re ee Be Biden ae) eA ae oe ie Ee oe | oe eee oe eK OR RSE RR eh HK ok OR
For offshore conditions, the group velocity is equal to one-half the
deepwater wave speed C,, where Cy is given by Equation 2-7, and the
refraction coefficient, Kp, can be determined by the methods of Section
Dec ce HENGE),
Py = — T(H, Kp)? sin 2a, . (4-30)
Ss
Figure 4-35 also presents en longshore component of wave energy flux
in a einens eae form, Pe/ (pg He T), as a function of deepwater wave steep-
ness, H)/gT*, and the angle the ‘wave crest makes with the shoreline in
either deep water, Wp, Or at the breaker line, op. Refraction by
straight parallel bottom contours is again assumed. As illustrated by
4-93
N
g
23
ne = (=|
Seater
Lie é &
La a3
IC ar
| G8
ES |
|
|
2 cs cD 2.
ie) S) ° °
o 7 = = =
vt Mm N =
(12H 50)
Deepwater Wave Angle, Qo, degrees
Figure 4-35 Longshore Component of Wave Energy Flux as a Function of Deepwater Wave Conditions
the example problem below, Figure 4-35 can be used to determine the
longshore component of wave energy flux when deepwater wave height, Ho,
period, T, and deepwater wave angle, 0 , are known.
Ree eek ee gee eee eee ANID IE IP ROBIEEM. cp cous ace Sets eee ee
GIVEN: A wave in deep water has a height, Hp = 5 feet and a period,
T = 7 seconds. While in deep water, the wave crest makes an angle,
a, = 25° with the shoreline.
(a) The longshore component of wave energy flux.
(b) The angle the wave makes with the shoreline when it breaks (assum-
ing refraction is by straight, parallel bottom contours).
SOLUTION: Calculate,
LS 5.0
— = ——__ = 0.0032 .
eT 32.2 (7.0)
Enter Figure 4-35 with ag = 25° and H,/gT* = 0.0032, read Pp/pg* H2 T=
1.5 x 10°3. This corresponds with a breaker angle ap = 9.5° which is
obtained by interpolation between the dashed curves of constant ap.
Therefore,
Poa ss (0.0015) pg? H? ibe
Pp = (0.0015) (2) (32.2)? (5)? (7.0) ,
Py = 544.3, say 540 ft.-lb./ft.-sec. ,
and
kee EK BR RR ek OR BOR RR KR KH KR KK RB OK RRR ce oe) OR) Re es ee) eee =
Equations 4-25 and 4-28 are valid only if there is a single wave
train with one period and one height. However, most ocean wave condi-
tions are characterized by a variety of heights with a distribution usu-
ally described by a Rayleigh distribution. (See Section 3.22.) For a
Rayleigh distribution, the correct height to use in Equation 4-28 or in
the formulas shown in Table 4-8 is the root-mean-square amplitude. How-
ever, most wave data are available as significant heights, and coastal
engineers are used to dealing with significant heights.
Significant height is implied in all equations for Ppy3. The value
of Pps, computed using significant height is approximately twice the
4-95
value of the exact energy flux for sinusoidal wave heights with a Rayleigh
distribution. Since this means that Py, is proportional to energy flux
and not equal to it, Pgs is referred to as the Longshore energy flux
factor in Table 4-8 and the following sections.
Longshore energy flux in this general case (Pg) is given by equa-
tion 4-27. This is an exact equation for the longshore component of
energy flux in a single small-amplitude, periodic wave. This equation
is good for any specified depth, but because the wave refracts, Pg will
have different values as the wave moves into shallower water. The value
of Pg in Equation 4-27 can be manipulated through use of small-amplitude
wave theory to obtain the four equivalent formulas for Pg shown in Table
4-7.
In order to use Pg for longshore transport computations in the surf
zone, it is necessary to approximate Pg for conditions at the breaker
position. These approximations are shown as Pegg in Table 4-8, evaluated
in foot-pound per second units. The bases for these approximations are
shown in Table 4-9. Measurements show that the longshore transport rate
depends on Pgg. (See Figures 4-36 and 4-37.)
As implied by the definition of Pgs, the energy flux factors in
Figures 4-36 and 4-37 are based on significant wave heights. The plotted
Peg values were obtained in the following manner. For the field data of
Watts (1953) and Caldwell (1956), the original references give energy flux
factors based on significant height, and these original data (after unit
conversion) are plotted as Pp, in Figures 4-36 and 4-37, Similarly, the
one field point of Moore and Cole (1960), as adopted by Saville (1962),
is assumed to be based on significant height. (See Figure 4-37.) Finally,
the field data of Komar (1969), are given in terms of root-mean-square
energy flux. This energy flux is multiplied by a factor of 2 (Das, 1972),
converted to consistent units, and then plotted in Figure 4-36 and 4-37.
For laboratory conditions (Fig. 4-36 only), waves of constant height
are assumed. When these heights are used in the equations of Table 4-8,
the result is an approximation of the exact longshore energy flux. In
order to plot the laboratory data in terms of an energy-flux factor con-
sistent with the plotted field data, this energy flux is multiplied by 2
before plotting in Figure 4-36.
For the purpose of this section, it is assumed that the shoaling co-
efficient, K,, for nearshore breaking waves is equivalent to the breaker
height index, Hp/Ho', found from observation. (See Figure 2-65.)
The choice of equations to determine Pgg depends on the data avail-
able. The right hand columns of Tables 4-7 and 4-8 tabulate the data
required to use each of the formulas. An example using the second Pgg
formula is given in Section 4.533.
Possible changes in wave height due to energy losses as waves travel
over the Continental Shelf are not considered in these equations. Such
4-96
Table 4-7. Longshore Energy Flux,Pp, for a Single Periodic Wave in
Any Specified Depth. (Four Equivalent Expressions from
Small-Amplitude Theory)
Equation Po Data Required
(energy/time/distance) (any consistent units)
2C, (4 E sin 2a)
C (4E, sin 2a,)
KRC, (4 E, sin 2a)
(2C) (KR C,J" C, (AE sin 2a,)
no subscript indicates a variable at the specified depth
where small-amplitude theory is valid
= group velocity (see assumption 1b, Table 4-9)
= deepwater
water depth
= significant wave height
= wave period
ae dr a?
i
= angle between wave crest and shoreline
: Soe COS Go
Kp= refraction coefficient , SSS
cos @
Table 4-8. Approximate Formulas for Computing Longshore
Energy Flux Factor, P»., Entering the Surf Zone
P Data Required
g
(ft.-Ibs./sec./ft. = beach front) | (ft.-sec. units)
B2al H,si2 sin 2a
18:3 >” (cos'a,)!* sin 2a,
20.5 T He sin a, COs a,
100.6 (Hj /T) sin oe
H, = deepwater
H, = breaker position
H = significant wave height
T = wave period
@ = angle between wave crest and shoreline
See Table 4-7 for equivalent small amplitude equations and
Table 4-9 for assumptions used in deriving Py, from Pp».
4-97
Table 4-9. Assumptions for Pg, Formulas in Table 4-8.
1. Formula 1 — Equation 4-35
a. Energy density at breaking is given by linear theory,
E = (pg H})/8 ~ 8H}
b. Group velocity equals wave speed at breaking, and breaking speed is given by
solitary wave theory according to the approximation. (Galvin, 1967, Equation 11.)
C, = C ® (2gH,)” = 8.02 (H,)”
c. a can be replaced by ap
2. Formula 2 — Equation 4-36
a. Same as 1b above
b. Hy is related to H, by refraction and shoaling coefficients, where the coefficients
are evaluated at the breaker position
Hy = Kp K, H,
c. Refraction coefficient Kp given by small-amplitude theory; shoaling coefficient
K, assumed constant, so that
d. (H,)* = 1.14 (cosa,)* H,”
if (cos a,) = 1.0
and (IK 72 = 1.14
3. Formula 3 — Equation 4-37
a. Refraction coefficient at breaking is given by small-amplitude theory.
4. Formula 4 — Equation 4-38
Same as 1a above
Same as 1b above
Same as 3a above
aoe ©
Cos = 1.0
NOTE: Constants evaluated for foot-pound-second units. Small-amplitude theory is assumed valid in
deep water. Nearshore contours are assumed to be straight and parallel to the shoreline.
4-98
10?
°
om
ro)
oO
104
For Design, use Figure 4-37
rs)
w
Longshore Transport Rate, Q, yd 3/yr
10?
@ Field Data, Significant Height
© Laboratory Data, Quartz Sand, Periodic Waves
fox? 1o7! | 10 102 105 loz
Longshore Energy Flux Factor , Poss ft.-lbs /sec/ft. of beach front
Figure 4-36. Longshore Transport Rate Versus Energy Flux Factor for Field
and Lab Conditions
EH EE
jaae
aug
(44 sad ph) ‘OQ ayoy jsodsuody aJoys6u07
Ft-Ibs/sec/ft of beach front
Posy
?
Longshore Energy Flux Factor
Design Curve for Longshore Transport Rate versus Energy
Only field data are included.
Flux Factor.
4-37.
igure
B
4-100
changes may reduce the value of Pp», ‘when deepwater wave height statis-
tics are used as a starting point for computing Py,. (Walton, 1972;
Bretschneider and Reid, 1954; and Bretschneider, 1954.)
The Equations 4-35 through 4-38 in Table 4-8 are related to the Equa-
tion for E, previously recommended for use with this method (Caldwell,
1956, Equations 5 and 6; or the equations in Figure 2-22, page 175, CERC
Technical Report No. 4, 1966 edition) by a constant
E, = (8.64 X 104) Pp. (4-39)
where E, is in units of foot-pounds per foot per day and Pg, is in
units of foot-pounds per foot per second.
The term in parenthesis for Equation 4-32 in Table 4-7 is identical
to the longshore force of Longuet-Higgins (1970a). This longshore force
also correlates well with the longshore transport rate.
The relation between Q and Pg, in Figures 4-36 and 4-37 can be
approximated by
Q = (7.5 X 103) B, (4-40)
Equation 4-40 tends to overestimate Q at the higher values of Po,
for the plotted field data, but it falls below the estimated rates com-
puted from the data of Johnson (1952). (See Das, 1972, Figure 6.) The
value of 7.5 x 10% in Equation 4-40 is approximately twice the equiva-
lent value from the design curve of CERC Technical Report No. 4, 1966
edition, and is about 5 percent greater than the value estimated by Komar
and Inman (1970).
Judgement is required in applying Equation 4-40. Although the data
in Figures 4-36 and 4-37 appear to follow a smooth trend, the log-log
scale compresses the data scatter. For example, the average difference
between the plotted points from field data and the prediction given by
Equation 4-40 is at least 28 percent of the value of prediction (average
difference derived by Das (1972) is 42 percent). In addition, some in-
complete measurements suggest transport rates ranging from two orders of
Magnitude below the line (Thornton, 1969) to one order of magnitude above
the line. (Johnson, 1952.) These additional data are plotted by Das
(1972).
As an aid to computation, Figures 4-38 and 4-39 give lines of con-
stant Q based on Equation 4-40 and Equations 4-35 and 4-36 for Pgs
given in Table 4-8. To use Figures 4-38 and 4-39 to obtain the longshore
4-101
transport rate, only the (Hp, op) data and Figure 4-38, or the (Hy, 4)
data and Figure 4-39 are needed. If the shoaling coefficient is signifi-
cantly different from 1.3, multiply the Q obtained from Figure 4-39 by
the factor 0.88 VKs. (See Table 4-9, Assumption 2d.)
Figure 4-39 applies accurately only if ao, is a point value. If ap
is a range of values, for example a 45° sector implied by the direction
NE, then the transport evaluated from Figure 4-39 using a single value
of oa, for NE may be 12 percent higher than the value obtained by aver-
aging over the 45° sector implied by NE. The more accurate approach is
given in the example problem of the next section.
The unit for Q is a volume of deposited quartz sand (including
voids in the volume) per year. Bagnold (1963) suggests using immersed
weight instead of volume in the unit for longshore transport rate (Section
4.521), since immersed weight is the pertinent physical variable related
to the wave action causing the sediment transport. Use of an immersed
weight unit does eliminate the difference between lightweight material
and quartz that occurs if volume units are used. (Das, 1972.) However,
in coastal engineering design, it is the volume and not the immersed
weight of eroded or deposited sand that is important, and since beach
sand is predominantly quartz (specific gravity 2.65), volume is directly
proportional to immersed weight. On some beaches the sand may be calcium
carbonate which has a specific gravity ranging from 2.87 (calcite) to
2.98 (aragonite) in pure form. Naturally occurring oolitic aragonite
sand with a specific gravity of 2.88 (Monroe, 1969), has an immersed
weight 14 percent greater than pure quartz sand. Since the longshore
transport Equation 4-40, with one exception (Watts, 1953; and Das, 1971,
p. 14), is based on quartz sand, then oolitic sand beaches may have
slightly lower longshore transport rates than is suggested by comparison
with data from quartz sand beaches. However, the scatter in the data
(Fig. 4-36) makes such a specific gravity effect difficult to detect.
4,533 Energy Flux Example (Method 3). Assume that an estimate of the
longshore transport rate is required for a locality on the north-south
coastline along the west edge of an inland sea. The locality is in an
area where stronger winds blow out of the northwest and north, resulting
in a deepwater distribution of height and direction as listed in Table
4-10. Assume the statistics were obtained from visual observations
collected over a 2-year interval at a point 2 miles offshore by seamen
aboard vessels entering and leaving a port in the vicinity. This type
of problem, based on SSMO wave statistics (Section 4.34), is discussed
in detail by Walton, (1972), and Walton and Dean (1973). Shipboard
data are subject to uncertainty in their applicability to littoral trans-
port, but often they are the only data available. It is assumed that
shipboard visual observations are equivalent to significant heights.
(Cartwright, 1972; and Walton, 1972.)
4-102
SUSSEEER
Bes
See oe: =! é
a
(4284) ‘Gy ‘}yBiaH 4ayDa1g aADM jUuddIZIUBIS
4-103
Breaker Angle, @, (degrees )
Figure 4-38. Longshore Transport Rate as a Function of Breaker Height and Breaker Angle
Significant Deepwater Height, Ho, ( feet )
9.0 : :
8.0 : : : :
7.0 R= gereusteseer tac Spiess
6.O FH Seeeea AtadsseEEene teen ta aseeonEe PEERY Haze
5. OREN SrHarE seasieieraanelnestanee es sceta feeeeenee’ rf
Prt
EEE EE EEE EEE HERE t ea Casoeeeees ea HEE
lO?) 202 150re, 402 50° 60707
Deepwater Angle, Q,, (degrees)
( Do not use averaged angle )
PSaees
°
Figure 4-39. Longshore Transport Rate as a Function of Deepwater
Height and Deepwater Angle
4-104
Table 4-10. Deepwater Wave Heights, in Percent by Direction, off East-Facing
Coast of Inland Sea
Compass Direction N
ae 90°
H,t (ft.)
il
2
3
4
5
8
*Calm conditions, or waves from SW, W, or NW.
t Shipboard visual observations assumed equivalent to significant height
(See Walton, 1972.)
This problem could be solved using Figure 4-39, but for illustration,
and because of a slightly higher degree of accuracy from the direction
data given, the problem is illustrated in detail.
In this example, the available data are the joint frequency distri-
bution of H, and oy. For each combination of ag and H,, the
corresponding Qa, Ho is calculated for Table 4-11 in the following
manner. The basic equation is a form of Equation 4-40 written
= E/E P 4-41
Qo. Hy | 2s} oy» Hy ( )
where f is the decimal frequency, which is the percent frequency in
Table 4-10, divided by 100. The constant, A, is of the type used in
Equation 4-40.
Since the available data are ag and Hg, the appropriate equation
for Pp, is given in Table 4-8, If A= 7.5 x 10° as in Equation 4-40
and Equation 4-36 in Table 4-8 are used,
Oye) = 1373x2108 Pere F (ag) + (4-42)
where
F(a.) = {(cosa,)% sin 2a,} (4-43)
This direction term, F(a,), requires careful consideration. A
compass point direction for the given data (Table 4-10) represents a 45°
4-105
sector of wave directions. If F(a) is evaluated at a, = 45° (NE or
SE in the example problem), it will have a value 12 percent higher than
the average value for F(a,) over 45° sector bisected by the NE or SE
directions. Thus, if the data warrant a high degree of accuracy, Equa-
tion 4-43 should be averaged by integrating over the sector of directions
involved.
Table 4-11. Computed Longshore Transport for East-Facing Coast of Inland Sea
Qa, > Ho in cu.yd./yr. from Equation 4-42
1261102 49] TDD OO TO P52 X10 fp — 563% 102 10 BO lO
5105 X1108% |S nsO 4 KO Fh 2.876108) 1 2740" | ete
11.13) 6 1020S 52 AAO? | 3960K 10" i eS Ocal Deen ato
11.42)56907 4) 936103 0¢ 1102
9.98 X 103
32131 X08
(8.35 X 103)
71.50 X 10? | 131.78 X 103 | 16.70 X 103 | —35.92 X 103 | — 5.69 X 103 —
= (71.50 + 131.78 + 8.35) X 103 = 212 X 103 or 212,000 cu.yd./yr.
= (8.35 + 35.92 + 5.69) X 10? = 50 X 10? or 50,000 cu.yd./yr.
Q, BS"Qh = Qo = 212° 103° = "50: K-10? ="162"X 10° (or 162/000" cu.ydeye
= Q, + Qg = 212 X 103 + 50 X 10? = 262 X 10° or 262,000 cu.yd./yr.
ae
|
*Coast runs N-S so frequencies of waves from N and S are halved.
tCalculation of this number is shown in detail in the text.
If F(a,) is evaluated at a, = 0 (waves from the E in the example
problem), then F(aj) = 0. Actually, a, = 0° is only the center of a
45° sector which can be expected to produce transport in both directions.
Therefore, F(ag) should be averaged over 0° to 22.5° and 0° to -22.5°,
giving F(a.) = + 0.370 rather than 0. The + or - sign comes out of the
sin 209 term in F(ag) (Equation 4-43), which is defined such that trans-
port to the right is positive, as implied by Equation 4-22.
A further complication in direction data is that waves from the north
and south sectors include waves traveling in the offshore direction. It
is assumed that, for such sectors, frequency must be multiplied by the
fraction of the sector including landward traveling waves. For example
the frequencies from N and S in Table 4-10 are multiplied by 0.5 to
obtain the transport values listed in Table 4-11.
4-106
To illustrate how values of Qy,, q, listed in Table 4-11 were
calculated, the value of Qy, is here calculated for Hp, = 1 and
the north direction, the top value in the first column on Table 4-11.
The direction term, F(aj9), is averaged over the sector from a = G75
to a= 90°, i.e., from NNE to N in the example. The average value of
F(a9) is found to be 0.261. Hy to the 5/2 power is simply 1 for this
case. The frequency given in Table 4-10 for H, = 1 and direction =
north (NW to NE) is 9 percent or in decimal terms, 0.09. This is mul-
tiplied by 0.5 to obtain the part of shoreward directed waves from the
north sector (i.e., N to NE) resulting in f = 0.09 (0.5) = 0.045. Put-
ting all these values into Equation 4-42 gives
Qy,z = 1.373 X 105 (0.045) (1)/2 (0.261)
= 1,610 yd3/yr. (See Table 4-11)
Table 4-11 indicates the importance of rare high waves in determin-
ing the longshore transport rate. In the example, shoreward moving 8-foot
waves occur only 0.5 percent of the time, but they account for 12 percent
of the gross longshore transport rate. (See Table 4-11.)
Any calculation of longshore transport rate is an estimate of poten-
ttal longshore transport rate. If sand on the beach is limited in quan-
tity, then calculated rates may indicate more sand transport than there
is sand available. Similarly, if sand is abundant, but the shore is
covered with ice for 2 months of the year, then calculated transport
rates must be adjusted accordingly.
The procedure used in this example problem is approximate and
limited by the data available. Equation 4-42, and the other approxi-
mations listed in Table 4-11, can be refined if better data are avail-
able. An extensive discussion of this type of calculation is given by
Walton (1972).
Although this example is based on shipboard visual observations of
the SSMO type (Section 4.34), the same approach can be followed with
deepwater data from other sources, if the joint distribution of height
and direction is known, At this level of approximation, the wave period
has little effect on the calculation, and the need for it is bypassed as
long as the shoaling coefficient (or breaker height index) reasonably
satisfies the relation (K,) 2/2 = 1.14. (See Assumption 2d, Table 4-9.)
For waves on sandy coasts, this relation is reasonably satisfied. (e.g.,
Bigelow and Edmondson, 1947, Table 33; and Goda, 1970, Figure 7.)
4.534 Empirical Prediction of Gross Longshore Transport Rate (Method 4).
Longshore transport rate depends partly on breaker height, since as
breaker height increases, more energy is delivered to the surf zone. At
the same time, as breaker height increases, breaker position moves off-
shore widening the surf zone and increasing the cross-section area through
which sediment moves.
4-l07
Galvin (1972) showed that when field values of longshore transport
rate are plotted against mean annual breaker height from the same locality,
a curve
Q=2X 10°H, (4-44)
forms an envelope above almost all known pairs of (Q, Hp), as shown in
Figure 4-40. Here, as before, Q is given in units of cubic yards per
year; Shp rane feet.
Figure 4-40 includes all known (Q, Hp) pairs for which both Q and
Hp are based on at least 1 year of data, and for which Q is considered
to be the gross longshore transport rate, » defined by Equation 4-21.
Since all other known (Q, Hp) pairs plot below the line given by Equation
4-41, the line provides an upper limit on the estimate of longshore trans-
port rate. From the defining equations for and Q,, any line that
forms an upper limit to longshore transport rate must be the gross trans-
port rate, since the quantities Qpz, Qgz, and Q,, as defined in Section
4.531, are always less than or equal to Q®>
In Equation 4-44, wave height is the only independent variable, and
the physical explanation assumes that waves are the predominant cause of
transport. (Galvin, 1972.) Therefore, where tide-induced currents or
other processes contribute significantly to longshore transport, Equation
4-44 would not be the appropriate approximation. The corrections due to
currents may either add or subtract from the estimate of Equation 4-44,
depending on whether currents act with or against prevailing wave-induced
transport.
4.535 Method 4 Example (Empirical Prediction of Gross Longshore Transport
Rate. Near the site of the problem outlined in Section 4.533, it is de-
sired to build a small craft harbor. The plans call for an unprotected
harbor entrance, and it is required to estimate costs of maintenance
dredging in the harbor entrance. The gross transport rate is a first
estimate of the maintenance dredging required, since transport from
either direction could be trapped in the dredged channel. Wave height
statistics were obtained from a wave gage in 12 feet of water at the end
of a pier. (See Columns (1) and (2) of Table 4-12.) Heights are avail-
able as empirically determined significant heights. (Thompson and Harris,
1972.) (To facilitate comparison, the frequencies are identical to the
deepwater frequencies of onshore waves in Table 4-10 for the problem of
Section 4.533. That is, the frequency associated with each H, in Table
4-12 is the sum of the frequencies of the shoreward Hg on the correspond-
ing line of Table 4-10.)
The breaker height, Hp, in the empirical Equation 4-44 is related
to the gage height, Hg, by a shoaling-coefficient ratio, (Ks)p/ (Kg) g>
where (K,)p, is the shoaling coefficient (Equation 2-44), (H/H, in
4-108
Longshore Transport Rate, Q, (yd° per yr)
i i]
t Pe jig Wh 1
1 Tt | T
al on
| [ | {
f SE - 3 +
T i
rit panna
jy 1
1 ia
i
i
t
T 1
T
1
!
|
19 1 coe
i
1
T
=
}
= S 1
I
1 ONT NONIS NS
: cm
SS eed cE
3 +
+ T
8 =
S :
. NNN
N SS N uu
SS SAVIN il
WV Ss
SEV i |
: =
3 <3 S
NESS if
D Si
MAYQ
S SS S
N WH . i
= —
( i"
5
r i
i oon a
t i
S S Hit rH
a £ ~ ~ at
9 S 1 |
E PHS i H
SS ne ond oo SO 8 8G OU pe Oe fel me so = + E
+t on bat ft mma Semen een es oes am ing eeu
Fa a5 = ttt iS +
Haag Ht t
ii + 1a Dag a
“T a’ 18 T
] T T7 rT
: Poo
| et
meno ee
co i
| Ri . |
? |
|
104 —
0.1 O2 04 06 081.0 2 4 627-8710
Mean Breaker Height ,Hp,( feet ) (Galvin,1972)
Figure 4-40. Upper Limit on Longshore Transport Rates
4-109
Table 4-12. Example Estimate of Gross Longshore Transport Rate for Shore of Inland Sea
(1) H, = significant height reduced from gage records, assumed to correspond
to the height obtained by visual observers
(2) f = decimal frequency of wave heights
(K,)
(4) aye assumed shoaling-coefficient ratio.
i)
ae poe f
(5) Hy = B ey By = 1.062 6
g&
On — EX 10> Hz = 2.26 X 10° cu.yd./yr. from Equation 4-44
Note that shoreward-moving waves exist only 51 percent of the time.
4-110
Table C-1 Appendix C) evaluated at the breaker position and ({Ks)g is
the shoaling coefficient evaluated at the wave gage:
K
eu (4-45)
K, or H/H4 can be evaluated from small-amplitude theory, if wave-period
information is available from the wave gage statistics. For simplicity,
assume shoaling-coefficient ratios as listed in Colum 4 of Table 4-12.
Such shoaling coefficient ratios are consistent with the shoaling co-
efficientof K,=1.3 (between deepwater and breaker conditions) assumed
in deriving Pps (Table 4-9), and with the fact that waves on the inland
sea of the problem would usually be steep, locally generated waves.
Column 5 of the table is the product fH, (Ks) p/ (Kg) q- The sum
(1.06 feet) of entries in this column is assumed equivalent to the aver-
age of visually observed breaker heights. Substituting this value in
Equation 4-44, the estimated gross longshore transport rate is 226,000
cubic yards per year. It is instructive to compare this value with the
value of 262,000 cubic yards per year obtained from the deepwater example.
(See Table 4-11.) The two estimates are not expected to be the same,
since the same wave statistics have been used for deep water in the first
problem and for a 12-foot depth in the second problem. However, the numer-
ical values do not differ greatly. It should be noted that the empirical
estimate just obtained is completely independent of the longshore energy
flux estimate of the deepwater example.
In this example, wave gage statistics have been used for illustrative
purposes. However, visual observations of breakers, such as those listed
in Table 4-4, would be even more appropriate since Equation 4-44 has been
"calibrated" for such observations. On the other hand, hindcast statis-
tics would be less satisfactory than gage statistics due to the uncertain
effect of nearshore topography on the transformation of deepwater statis-
tics to breaker conditions.
4.6 ROLE OF FOREDUNES IN SHORE PROCESSES
4.61 BACKGROUND
The cross section of a barrier island shaped solely by marine hydrau-
lic forces has three distinct subaerial features: beach, crest of island,
and deflation plain. (See Figure 4-41.) The dimensions and shape of the
beach change in response to varying wave and tidal conditions (Section
4.524), but usually the beach face slopes upward to the island crest - the
highest point on the barrier island cross section. From the island crest,
the back of the island slopes gently across the deflation plain to the
edge of the lagoon separating the barrier island from the mainland. These
three features are usually present on duneless barrier island cross sec-
tions; however, their dimensions may vary.
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Island crest elevation is determined by the nature of the sand form-
ing the beach, and by the waves and water levels of the ocean. The beach
and waves interact to determine the elevation of the limit of wave runup -
the primary factor in determining island crest elevation. Normally the
island crest elevation is almost constant over long sections of beach.
However, duneless barrier island crest elevations vary with geographical
area. For example, the crest elevation typical of Core Banks, North
Carolina, is about +6 feet MSL; +4 feet MSL is typical for Padre Island,
Texas; +11 feet MSL is typical for Nauset Beach, Massachusetts.
Landward of the upper limit of wave uprush or berm crest are the back-
shore and the deflation plain. This area is shaped by the wind, and in-
frequently by the flow of water down the plain when the island crest is
overtopped by waves. (e.g., Godfrey, 1972.) Obstructions which trap
wind-transported sand cause the formation of dunes in this area. (See
discussion in Section 6.4 Sand Dunes.) Beachgrasses which trap wind-
transported sand from the beach and the deflation plain are the major
agent in creating and maintaining foredunes.
4.62 ROLE OF FOREDUNES
Foredunes, the line of dunes just behind a beach, have two primary
functions in shore processes. First, they prevent overtopping of the
island during some abnormal sea conditions. Second, they serve as a
reservoir for beach sand.
4.621 Prevention of Overtopping. By preventing water from overtopping,
foredunes prevent wave and water damage to installations landward of the
dune. They also block the water transport of sand from the beach area to
the back of the island and the flow (overwash) of overtopping sea water.
Large reductions in water overtopping are effected by small increases
in foredune crest elevations. For example, the hypothetical 4-foot dune
shown in Figure 4-41 raises the maximum island elevation about 3 feet to
an elevation of 6 feet. On this beach of Padre Island, Texas, the water
levels and wave runup maintain an island crest elevation of +4 feet MSL
(about 2 feet above MHW). This would imply that the limit of wave runup
in this area is 2 feet (the island crest elevation of 4 feet minus the
MHW of 2 feet). Assuming the wave runup to be the same for all water
levels, the 4-foot dune would prevent significant overtopping at water
levels up to 4 ft MSL (the 6-foot effective island height at the dune
crest minus 2 feet for wave runup). This water level occurs on the aver-
age once each 5 years along this section of coast. (See Figure 4-42.)
Thus, even a low dune, one which can be built with vegetation and sand
fences in this area in 1 year (Woodard et al., 1971) provides consider-
able protection against wave overtopping. (See Section 5.3 and 6.3.)
Foredunes or other continuous obstructions on barrier islands may
cause unacceptable ponding from the land side of the island when the la-
goon between the island and mainland is large enough to support the needed
wind setup. (See Section 3.8.) There is little danger of flooding from
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4-114
this source if the lagoon is less than 5 miles wide. Where the lagoon
is wider (especially 10 miles or greater) flooding from the lagoon side by
wind setup should be investigated before large dune construction projects
are undertaken.
4.622 Reservoir of Beach Sand. During storms, erosion of the beach occurs
and the shoreline recedes. If the storm is severe, waves attack and erode
the foredunes and supply sand to the beach; in later erosion stages, sand
is supplied to the back of the island by overwash. (Godfrey, 1972.)
Volumes of sand eroded from beaches during storms have been estimated
in recent beach investigations. Everts (1973) reported on two storms dur-
ing February 1972 which affected Jones Beach, New York. The first storm
eroded an average of 27,000 cubic yards per mile above mean sea level for
the 9-mile study area; the second storm (2 weeks later) eroded an average
of 35,000 cubic yards of sand per mile above mean sea level at the same
site. Losses at individual profiles ranged up to 120,000 cubic yards per
mile. Davis (1972) reported a beach erosion rate on Mustang Island, Texas,
following Hurricane Fern (September 1971), of 12.3 cubic yards per linear
foot of beach for a 1,500-foot stretch of beach (about 65,000 cubic yards
per mile of beach). On Lake Michigan in July 1969, a storm eroded an aver-
age of 3.6 cubic yards per linear foot of beach (about 29,000 cubic yards
per mile) from an 800-foot beach near Stevensville, Michigan. (Fox, 1970.)
Because much of the eroded sand is usually returned to the beach by wave
action soon after the storm, these volumes are probably representative of
temporary storm losses.
Volumes equivalent to those eroded during storms have been trapped
and stored in foredunes adjacent to the beach. Foredunes constructed
along Padre Island, Texas, and Core Banks, North Carolina, (Section 6.43
and 6.447) contain from 30,000 to 80,000 cubic yards of sand per mile of
beach. Assuming the present rate of entrapment of sand continues for the
next 3 years at these sites, sand volumes ranging from 50,000 to 160,000
cubic yards per mile of beach will be available to nourish eroding beaches
during a major storm. Sand volumes trapped during a 30-year period by
European beachgrass at Clatsop Spit, Oregon, averaged about 800,000 cubic
yards per mile of beach. Thus, within a few years, foredunes can trap and
store a volume of sand equivalent to the volumes eroded from beaches dur-
ing storms of moderate intensity.
4.623 Long-Term Effects. Dolan, (1972-73) advances the concept that a
massive, unbroken foredune line restricts the landward edge of the surf
zone during storms causing narrower beaches and thus increased turbulence
in the surf zone. The increased turbulence causes higher sand grain attri-
tion and winnowing rates and leads to accelerated losses of fine sand, an
erosive process that may be detrimental to the long-range stability of bar-
rier islands. However, as discussed in Section 4.521, the effects of sedi-
ment size are usually of secondary importance in littoral transport pro-
cesses - processes which are important in barrier island stability. In
addition, geographical location is probably more important in determining
beach sand size than dune effects, since both fine and coarse sand beaches
4 hI'S
front major foredune systems in different geographical locations. For
example, fine sand beaches front a massive foredune system on Mustang
Island, Texas, and coarse sand beaches front dunes on the Cape Cod spits.
Godfrey (1972) discusses the effect of a foredune system on the
long term stability of the barrier islands of the Cape Hatteras and Cape
Lookout National Seashores, North Carolina. Important implicit assump-
tions of the discussion are that no new supply or inadequate new supplies
of sand are available to the barrier island system, and that rising sea
level is, in effect, creating a sand deficit by drowning some of the
available island volume. The point of the geomorphic discussion is that
under such conditions the islands must migrate landward to survive. A
process called "oceanic overwash" (the washing of sand from low foredunes
or from the beach over the island crest onto the deflation plain by over-
topping waves) is described as an important process in the landward migra-
tion of the islands. Since a foredune system blocks overtopping and pre-
vents oceanic overwash, foredunes are viewed as a threat to barrier island
stability.
Granted the implicit assumptions and a geologic time frame, the
geomorphic concept presented has convincing logic and probably has merit.
However, the assumptions are not valid on all barrier islands or at all
locations in most barrier islands or at all locations in most barrier
island systems. Too, most coastal engineering projects are based on
a useful life of 100 years or less. In such a short period, geologic
processes, such as sea-level rise, have a minor effect in comparison with
the rapid changes caused by wind and waves. Therefore, the island crest
elevation and foredune system will maintain their elevation relative to
the mean water level on stable or accreting shores over the life of most
projects. On eroding shores, the foredunes will eventually be eroded
and overwash will result in shoreward migration of the island profile;
sand burial and wave and water damage will occur behind the original
duneline. Therefore, planning for and evaluation of the probable suc-
cess of a foredune system must consider the general level of the area of
the deflation plain to be protected, the rate of sea level rise, and the
rate of beach recession.
4.7 SEDIMENT BUDGET
4.71 INTRODUCTION
4.711 Sediment Budget. A sediment budget is based on sediment removal,
transportation and deposition, and the resulting excesses or deficiencies
of material quantities. Usually, the sediment quantities are listed
according to the sources, sinks, and processes causing the additions and
subtractions. In this chapter, the sediment is usually sand, and the
processes are either littoral processes or the changes made by man.
The purpose of a sediment budget is to assist the coastal engineer
by: identifying relevant processes; estimating volume rates required for
design purposes; singling out significant processes for special attention;
and, on occasion, through balancing sand gains against losses, checking
the accuracy and completeness of the design budget.
4-16
Sediment budget studies have been presented by Johnson (1959), Bowen
and Inman (1966), Vallianos (1970), Pierce (1969), and Caldwell (1966).
4.712 Elements of Sediment Budget. Any process that increases the
quantity of sand in a defined control volume is called a source. Any
process that decreases the quantity of sand in the control volume is
called a stink. Usually, sources are identified as positive and sinks
as negative. Some processes (longshore transport is the most important)
function both as source and sink for the control volume, and these are
called convecting processes.
Potnt sources or point stinks are sources or sinks that add or sub-
tract sand across a limited part of a control volume boundary. A tidal
inlet often functions as a point sink. Point sources or sinks are gener-
ally measured in units of volume per year.
Line sources or line sinks are sources or sinks that add or subtract
sand across an extended segment of a control volume boundary. Wind trans-
port landward from the beaches of a low barrier island is a line sink for
the ocean beach. Line sources or sinks are generally measured in units of
volume per year per unit length of shoreline. To compute the total effect
of a line source or sink, it is necessary to multiply this quantity by the
total length of shoreline over which the line source or sink operates.
The following conventions are used for elements of the sediment
budget:
Q; is a point source
Q; is a point sink
q; is a line source
Gz; is a line sink
These subscripted elements of the sediment budget are identified by name
in Table 4-13 according to whether the element makes a point or line con-
tribution to the littoral zone, and according to the boundary across which
the contribution enters or leaves. Each of the elements is discussed in
following sections.
The length of shoreline over which a line source is active is in-
dicated by, ,bz and the total contribution of the line source or line
Sankey? sQ7 "or V0.2") so that “in general
Q* = bq; . (4-46)
4-(1%
Table 4-13. Classification of Elements in the Littoral Zone Sediment Budget
Longshore Ends
of
Littoral Zone
Point Source Qg ro) QQ G
(cu.yd./yr.) Offshore shoal or Rivers, streams* Replenishment Longshore
island transport in*
Location of Offshore Side of Onshore Side of Within
Source or Sink Littoral Zone Littoral Zone Littoral Zone
Point Sink or @ Q on
(cu.yd./yr.) Submarine canyon Inlets* Mining, extractive Longshore
dredging transport out*
Line Source qd; q) Gs
(cu.yd./yr./ft. of beach) Sand transport Coastal erosion Beach erosion*
from the offshore including erosion CaCO, production
of dunes and cliffs*
Line Sink q, q5 q3
(cu.yd./yr./ft. of beach) | Sand transport to Overwash Beach storage*
the offshore Coastal land and CaCO, losses
dune storage
*Naturally occurring sources and sinks that usually are major elements in the sediment budget.
It is often useful to specify a source or sink as a fraction, kj,
of the gross longshore transport rate:
Q, = by. (4-47)
In a complete sediment budget, the difference between the sand added
by all sources and the sand removed by all sinks should be zero. In the
usual case, a sand budget calculation is made to estimate an unknown ero-
Sion or deposition rate. This estimated rate will be the difference result-
ing from equating known sources and sinks. The total budget is shown sche-
matically as follows:
Sum of Sources - Sum of Sinks = 0, or
Sum of Known Sources - Sum of Known Sinks = Unknown (Sought)
Source or Sink
4 4 SOLER 3 £
Z SQre WENO? =| KL OT-t s Dee le) —10,. (4-48)
Sil pS 3 = Gl LS 1
The GF are obtained using Equation 4-46 and the appropriate q,; and
bz. The subscript, 7t, equals 1, 2, 3, or 4 and corresponds to the sub-
scripts in Table 4-13.
4-118
4.713 Sediment Budget Boundaries. A sediment budget is used to identify
and quantify the sources and sinks that are active in a specified area.
By so doing, erosion or deposition rates are determined as the balance
of known sinks and sources. Boundaries for the sediment budget are deter-
mined by the area under study, the time scale of interest, and study pur-
poses. In a given study area, adjacent sand budget compartments (control
volumes) may be needed with shore-perpendicular boundaries at significant
changes in the littoral system. For example, compartment boundaries may
be needed at inlets, between eroding and stable beach segments, and between
stable and accreting beach segments. Shore-parallel boundaries are needed
on both the seaward and landward sides of the control volumes. They may be
established wherever needed, but the seaward boundary is usually established
at or beyond the limit of active sediment movement, and the landward bound-
ary beyond the erosion limit anticipated for the life of the study. The
bottom surface of a control volume should pass below the sediment layer
that is actively moving, and the top boundary should include the highest
surface elevation in the control volume. Thus, the budget of a particular
beach and nearshore zone would have shore parallel boundaries landward of
the line of expected erosion and at or beyond the seaward limit of signifi-
cant transport. A budget for barrier island sand dunes might have a bound-
ary at the bay side of the island and the landward edge of the backshore.
A sediment budget example and analysis are shown in Figure 4-43.
This example considers a shoreline segment along which the incident wave
climate can transport more material entering from updrift. Therefore
the longshore transport in the segment is being fed by a continuously
eroding sea cliff. The cliff is composed of 50 percent sand and 50 per-
cent clay. The clay fraction is assumed to be lost offshore while the
sand fraction feeds into the longshore transport. The budget balances
the sources and sinks using the following continuity equation:
Sum of Known Sources - Sum of Known Sinks = Difference
An example calculation is shown in Figure 4-43.
4.72 SOURCES OF LITTORAL MATERIAL
4.721 Rivers. It is estimated that rivers of the world bring about 3.4
cubic miles or 18.5 billion cubic yards of sediment to the coast each year
(volume of solids without voids). (Stoddard, 1969; from Strakhov, 1967.)
Only a small percentage of this sediment is in the sand size range that is
common on beaches. The large rivers which account for most of the volume
of sediment carry relatively little sand. For example, it is estimated
(Scruton, 1960) that the sediment load brought to the Gulf of Mexico each
year by the Mississippi River consists of 50 percent clay, 48 percent silt,
and only 2 percent sand. Even lower percentages of sand seem probable for
other large river discharges. (See Gibbs, 1967, p. 1218, for information
on the Amazon River.) But smaller rivers flowing through sandy drainage
areas may carry 50 percent or more of sand. (Chow, 1964, p. 17-20.) In
southern California, sand brought to the coast by the floods of small
rivers is a significant source of littoral material. (Handin, 1951; and
Norris, 1964.)
4-l19
qb
Offshore Sink
WATER
|
|
|
|
|
225
ERODING SHORELINE - PLAN VIEW
(Not to Scale )
|i Budget Boundary ———————=|
if
i] -
| kG Recession
_as—First Survey Profile
Second Survey Profile
Seaword Limit of
Active Erosion
SECTION A-A
(Not to Scale)
Assumptions Budget Calculations
G = 100,000 yd?/yr. Sum of Sources — Sum of Sinks = Difference
i 6)
qa 2 yd?/yr. /tt. Find Q;
- _ 3 Se - Sq IE
q, = 0.5 yd°/yr./tt (Qi qs bli (Q, + q,b) =o
Eoinone (108 + 1.0 x 10*) —(Q) + 0.5 X 104) = 0
QZ = 110,000—5,000
Q = 105,000 yd?/yr.
Figure 4-43. Basic Example of Sediment Budget
4-120
Most of the sediment carried to the coast by rivers is deposited
in comparatively small areas, often in estuaries where the sediment is
trapped before it reaches the coast. (Strakhov, 1967.) The small frac-
tion of sand in the total material brought to the coast and the local
estuarine and deltaic depositional sites of this sediment suggest that
rivers are not the immediate source of sediment on beaches for much of
the world's coastline. Many sources of evidence indicate that sand-
sized sediment is not supplied to the coasts by rivers on most segments
of the U.S. Atlantic and Gulf coasts. Therefore, other sediment sources
must be important.
4.722 Erosion of Shores and Cliffs. Erosion of the nearshore bottom,
the beach, and the seaward edge of dunes, cliffs, and mainland (Fig. 4-44)
results in a sand loss. In many areas, erosion from cliffs of one area is
the principal source of sand for downdrift beaches. Kuenen (1950) esti-
mates that beach and cliff erosion along all coasts of the world totals
about 0.03 cubic mile or 160 million cubic yards per year. Although this
amount is only about 1 percent of the total solid material carried by
rivers, it is a major source in terms of sand delivered to the beaches,
Since the sand fraction in the river sediments is usually small, and is
usually trapped before it reaches the littoral zone. Shore erosion is an
especially significant source where older coastal deposits are being ero-
ded, since these usually contain a large fraction of sand.
If an eroding shore maintains approximately the same profile above
the seaward limit of significant transport while it erodes, then the ero-
sion volume per foot of beach front is the vertical distance from dune
base or berm crest to the depth of the seaward limit (h), multiplied by
the horizontal retreat of the profile, Ax. (See Figure 4-44.)
Figure 4-44 shows three equivalent volumes, all indicating a net ero-
sion of hAx. To the right in Figure 4-44 is a typical beach profile.
The dashed line profile below it is the same as the solid line profile.
The horizontal distance between solid and dashed profiles is Ax, the
horizontal retreat of the profile due to (assumed) uniform erosion, The
unit volume loss, hAx, between dune base and depth to seaward limit is
equivalent to the unit volume indicated by the slanted parallelogram on
the middle of Figure 4-44. The unit volume of this parallelogram, hAx,
is equivalent to the shaded rectangle on the left of Figure 4-44. If the
vertical distance, h is 40 feet, and Ax = 1 foot of horizontal erosion,
then the unit volume lost is 40/27, or 1.5 cubic yards per foot of beach
front.
4.723 Transport from Offshore Slope. An uncertain and potentially signifi-
cant source in the sediment budget is the contribution from the offshore
slope. However, hydrography, sediment size distribution, and related evi-
dence discussed in Section 4.523 indicate that contributions from the con-
tinental shelf to the littoral zone are probably negligible in many areas.
Most shoreward moving sediment appears to originate in areas fairly close
to shore. Significant onshore-offshore transport takes place within the
littoral zone due to seasonal and storm-induced profile changes and to
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4-122
erosion of the nearshore bottom and beaches, but in the control volume
defined, this transport takes place within the control volume. Transport
from the offshore has been treated as a line source.
In some places, offshore islands or shoals may act as point sources
of material for the littoral zone. For example, the drumlin islands and
shoals in Boston Harbor and vicinity may be point sources for the nearby
mainland.
4,724 Windblown Sediment Sources. To make a net contribution to the
littoral zone in the time frame being considered, windblown sand must
come from a land source whose sand is not derived by intermediate steps
from the same littoral zone. On U.S. ocean coasts, such windblown sand
is not a significant source of littoral materials. Where wind is impor-
tant in the sediment budget of the ocean shore, wind acts to take away
sand rather than to add it, although local exceptions probably occur.
However, windblown sand can be an important source, if the control
volume being considered is a beach on the lagoon side of a barrier island.
Such shores may receive large amounts of windblown sand.
4.725 Carbonate Production. Dissolved calcium carbonate concentration
in the ocean is near saturation, and it may be precipitated under favor-
able conditions. In tropical areas, many beaches consist of calcium car-
bonate sands; in temperate zones, calcium carbonate may be a significant
part of the littoral material. These calcium carbonate materials are gen-
erally fragments of shell material whose rate of production appears to in-
crease with high temperature and with excessive evaporation. (See Hayes,
1967.) Oolitic sands are a nonbiogenic chemical precipitate of calcium
carbonate on many low latitude beaches.
Quantitative estimates of the production of calcium carbonate sedi-
ment are lacking, but maximum rates might be calculated from the density
and rate of growth of the principal carbonate-producing organisms in an
area. For example, following northeasters along the Atlantic coast of
the U.S., the foreshore is occasionally covered with living clams thrown
up by the storm from the nearshore zone. One estimate of the annual con-
tribution to the littoral zone from such a source would assume an average
shell thickness of about 0.04 foot completely covering a strip of beach
100 feet wide all along the coast. On an annual basis, this would be
about 0.15-cubic yard per year per foot of beach front. Such a quantity
is negligible under almost all conditions. However, the dominance of
carbonate sands in tropical littoral zones suggests that the rate of pro-
duction can be much higher.
4.726 Beach Replenishment. Beach protection projects often require
placing sand on beaches. The quantity of sand placed on the beach in
such beach-fill operations may be a major element in the local sediment
budget. Data on beach-fill quantities may be available in Corps of Engi-
neer District offices, in records of local government engineers, and in
dredging company records. The exact computation of the quantity of a
4-123
beach fill is subject to uncertainties: the source of the dredged sand
often contains significant but variable quantities of finer materials that
are soon lost to the littoral zone; the surveys of both the borrow area
and the replenished area are subject to uncertainty because sediment trans-
port occurs during the dredging activities; and in practice only limited
efforts are made to obtain estimates of the size distribution of fill
placed on the beach. Thus, the resulting estimate of the quantity of
suitable fill placed on the beach is uncertain. More frequent sampling
and surveys could help identify this significant element in many sedi-
ment budgets.
4.73 SINKS FOR LITTORAL MATERIALS
4.731 Inlets and Lagoons. Barrier islands are interrupted locally by
inlets which may be kept open by tidal flow. A part of the sediment moved
alongshore by wave action is moved into these inlets by tidal flow. Once
inside the inlet, the sediment may deposit where it cannot be moved sea-
ward by the ebb flow. (Brown, 1928.) The middleground shoals common to
many inlets are such depositional features. Such deposition may be reduced
when the ebb currents are stronger than the flood currents. (Johnson,
1956.)
It is evident from aerial photography (e.g., of Drum Inlet, N.C.,
Fig. 4-45) that inlets do trap significant quantities of sand. Caldwell's
(1966) estimate of the sand budget for New Jersey, calculates that 23
percent of the local gross longshore transport is trapped by the seven
inlets in southern New Jersey, or about 250,000 cubic yards per year for
each inlet. In a study of the south shore of Long Island, McCormick (1971)
estimated from the growth of the floodtide delta of Shinnecock Inlet (shown
by aerial photos taken in 1955 and 1969) that this inlet trapped 60,000
cubic yards per year. This amounts to about 20 percent of the net long-
shore transport (Taney, 196la, p. 46), and probably less than 10 percent
of the gross transport. (Shinnecock Inlet is a relatively small inlet.)
It appears that the rate at which an inlet traps sediment is higher imme-
diately after the inlet opens than it is later in its history.
4.732 Overwash. On low barrier islands, sand may be removed from the
beach and dune area by overwashing during storms. Such rates may average
locally up to 1 cubic yard per year per foot. Data presented by Pierce
(1969) suggest that for over half of the shoreline between Cape Hatteras
and Cape Lookout, North Carolina, the short term loss due to overwash was
0.6 cubic yard per year per foot of beach front. Figure 4-46 is an aerial
view of overwash in the region studied by Pierce (1969). Overwash does
not occur on all barrier islands, but if it does, it may function as a
source for the beach on the lagoon side.
4.733 Backshore and Dune Storage. Sand can be temporarily withdrawn
from transport in the littoral zone as backshore deposits and dune areas
along the shore. Depending on the frequency of severe storms, such sand
may remain in storage for intervals ranging from months to years. Back-
shore deposition can occur in hours or days by the action of waves after
4-124
S
me
nN
=
Brat
~s
NK
x
Figure 4-45. Sediment Trapped Inside Old Drum Inlet,
North Carolina
4-125
Figure 4-46.
a/INY TLV
a. ae
jI0
.&
$=
‘
(1 November 1971)
Overwash on Portsmouth Island, North
Carolina
4-126
storms. Dune deposits require longer to form - months or years - because
wind transport usually moves material at a lesser rate than wave transport.
If the immediate beach area is the control volume of interest, and budget
calculations are made based on data taken just after a severe storm, allow-
ance should be made in budget calculations for sand that will be stored in
berms through natural wave action. (See Table 4-5.)
4.734 Offshore Slopes. The offshore area is potentially an important
sink for littoral material. Transport to the offshore is favored by:
storm waves which stir up sand, particularly when onshore winds create
a seaward return flow; turbulent mixing along the sediment concentration
gradient which exists between the sediment-water mixture of the surf zone
and the clear water offshore; and the slight offshore component of gravity
which acts on both the individual sediment particles and on the sediment-
water mixture.
It is often assumed that the sediment sorting loss that commonly
reduces the volume of newly placed beach fill is lost to the offshore
slopes. (Corps of Engineers, Wilmington District, 1970; and Watts,
1956.) A major loss to the offshore zone occurs where spits build into
deep water in the longshore direction. Sandy Hook, New Jersey, is an
example. (See Figure 4-47.) It has been suggested (Bruun and Gerritsen,
1959) that ebb flows from inlets may sometimes cause a loss of sand by
jetting sediment seaward into the offshore zone.
The calculation of quantities lost to the offshore zone is difficult,
since it requires extensive, accurate, and costly surveys. Some data on
offshore changes can be obtained by studies of sand level changes on rods
imbedded in the sea floor (Inman and Rusnak, 1956), but without extending
the survey beyond the boundary of the moving sand bed, it is difficult
to determine net changes.
4.735 Submarine Canyons. Probably the most frequently mentioned sinks
for littoral materials are submarine canyons. Shepard (1963) and Shepard
and Dill (1966) provide extensive description and discussion of the origin
of submarine canyons. The relative importance of submarine canyons in
sediment budgets is still largely unknown.
Of 93 canyons tabulated by Shepard and Dill (1966), 34 appear to be
receiving sediment from the coast, either by longshore transport or by
transport from river mouths. Submarine canyons are thought to be espe-
cially important as sinks off southern California. Herron and Harris
(1966, p. 654) suggest that Mugu Canyon, California, traps about 1 mil-
lion cubic yards per year of the local littoral drift.
The exact mechanism of transport into these canyons is not clear,
even for the La Jolla Canyon (California) which is stated to be the most
extensively studied submarine feature in the world. (Shepard and Buff-
ington, 1968.) Once inside the canyons, the sediment travels down the
floors of the heads of the canyons, and is permanently lost to the litto-
ral zone.
4-127
Sandy Hook,
New Jersey
OCEAN
S
ce
K
=
a
~
~
=
(14 September 1969)
Growth of a Spit into Deep Water, Sandy
Figure 4-47.
Hook, New Jersey
4-128
4.736 Deflation. The loose sand that forms beaches is available to be
transported by wind. After a storm, shells and other objects are often
found perched on pedestals of sand left standing after the wind eroded
less protected sand in the neighborhood. Such erosion over the total
beach surface can amount to significant quantities. Unstabilized dunes
may form and migrate landward, resulting in an important net loss to the
littoral zone. Examples include some dunes along the Oregon coast (Coo-
per, 1958), between Pismo Beach and Point Arguello, California (Bowen
and Inman, 1966); central Padre Island (Watson, 1971); and near Cape Hen-
lopen, Delaware (Kraft, 1971). Typical rates of transport due to wind
range from 1 to 10 cubic yards per year per foot of beach front where
wind transport is noticeable. (Cooper, 1958; Bowen and Inman, 1966;
Savage and Woodhouse, 1968; and Gage, 1970.) However average rates prob-
ably range from 1 to 3 cubic yards per year per foot.
The largest wind-transported losses are usually associated with
accreting beaches that provide a broad area of loose sand over a period
of years. Sand migrating inland from Ten Mile River Beach in the vicin-
ity of Laguna Point, California, is shown in Figure 4-48.
Study of aerial photographs and field reconnaissance can easily
establish whether or not important losses or gains from wind transport
occur in a study area. However, detailed studies are usually required
to establish the importance of wind transport in the sediment budget.
4.737 Carbonate Loss. The abrasion resistance of carbonate materials
is much lower than quartz, and the solubility of carbonate materials is
usually much greater than quartz. However, there is insufficient evidence
to show that significant quantities of carbonate sands are lost from the
littoral zone in the time scale of engineering interest through either
abrasion or solution.
4.738 Mining and Dredging. From ancient times, sand and gravel have
been mined along coasts. In some countries, for example Denmark and
England, mining has occasionally had undesirable effects on coastal settle-
ments in the vicinity. Sand mining in most places has been discouraged by
legislation and the rising cost of coastal land, but it still is locally
important. (Magoon, et al., 1972.) It is expected that mining will become
more important in the offshore area in the future. (Duane, 1968, and Fisher,
1969. )
Such mining must be conducted far enough offshore so the mined pit
will not act as a sink for littoral materials, or refract waves adversely,
or substantially reduce the wave damping by bottom friction and percolation.
Material is also lost to the littoral zone when dredged from naviga-
ble waters (channels and entrances) within the littoral zone, and the
dredged material is dumped in some area outside of the littoral zone.
These dump areas can be for land fill, or in deep water offshore. This
action has been a common practice, because the first costs for some
dredging operations are cheaper when done this way.
4-129
PA OFLC
(24 May 1972)
Figure 4-48. Dunes Migrating Inland Near Laguna Point, California
4-130
4.74 CONVECTION OF LITTORAL MATERIALS
Sources and sinks of littoral materials are those processes that
result in net additions or net subtractions of material to the selected
control volume. However, some processes may subtract at the same rate
that they add material, resulting in no net change in the volume of lit-
toral material of the control volume.
The most important convecting process is longshore sediment transport.
It is possible for straight exposed coastlines to have gross longshore
transport rates of more than 1 million cubic yards per year. On a coast
without structures, such a large can occur, and yet not be apparent
because it causes no obvious beach changes. Other convecting processes
that may produce large rates of sediment transport with little noticeable
change include tidal flows, especially around inlets, wind transport in
the longshore direction, and wave-induced currents in the offshore zone.
Since any structure that interrupts the equilibrium convection of
littoral materials will normally result in erosion or accretion, it is
necessary that the sediment budget quantitatively identify all processes
convecting sediment through the study area. This is most important on
shores with high waves.
4,75 RELATIVE CHANGE IN SEA LEVEL
Relative changes in sea level may be caused by changes in sea level
and changes in land level. Sea levels of the world are now generally ris-
ing. The level of inland seas may either rise or fall, generally depend-
ing on hydrologic influences. Land level may rise or fall due to tectonic
forces, and land level may fall due to subsidence. It is often difficult
to distinguish whether apparent changes in sea level are due to change in
sea level, change in land level, or both. For this reason, the general
process is referred to as relative change in sea level.
While relative changes in sea level do not directly enter the sedi-
ment budget process, the net effect of these elevation changes is to move
the shoreline either landward (relative rise in sea level) or seaward
(relative fall in sea level). It thus can result in the appearance of a
gain or loss of sediment volume.
The importance of relative change in sea level on coastal engineer-
ing design depends on the time scale and the locality involved. Its
effect should be determined on a case-by-case basis.
4.76 SUMMARY OF SEDIMENT BUDGET
Sources, sinks, and convective processes are summarized diagrammati-
cally in Figure 4-49 and listed in Table 4-14. The range of contributions
or losses from each of these elements is described in Table 4-14 measured
as a fraction of the gross longshore transport rate, or as a rate given
in cubic yards per year per foot of beach front. The relative importance
4-|31
of elements in the sand budget varies with locality and with the bound-
aries of the particular littoral control volume. (These elements are
classified as point or line sources or sinks in Table 4-13, and the budget
is summarized in Equation 4-48.)
LAND OCEAN
Cliff, Dune and
Backshore Erosion Offshore Slope
(source or sink?)
Beach Replenishment Suibpenine Cane
Rivers
Dredoi
Dune and Backshore rece nS
Storage Littoral Zone
Winds
isepbyasd eatin) Calcium Carbonate
(production and loss)
Inlets and Lagoons
Including Overwash
Mining
Longshore Currents
Tidal Currents
Longshore Winds
Figure 4-49, Materials Budget for the Littoral Zone
In most localities, the gross longshore transport rate significantly
exceeds other volume rates in the sediment budget, but if the beach is
approximately in equilibrium, this may not be easily noticed.
The erosion of beaches and cliffs and river contributions are the
principal known natural sources of beach sand in most localities. Inlets,
4-132
Table 4-14. Sand Budget of the Littoral Zone
Sources
Rivers and streams The major source in the limited areas where rivers carry sand to the littoral
zone. In affected areas notable floods may contribute several times Q,.
Cliff, dune and backshore erosion | Generally the major source where rivers are absent. 1 to 4 cu.yd./yr./ft.
Transport from offshore Quantity uncertain.
Wind transport Not generally important as a source.
CaCO3 production Significant in tropical climate. The value of 0.25 cu.yd./yr./ft. seems
reasonable upper limit on temperate beach.
Beach replenishment Varies from 0 to greater than Q,.
Sinks
Inlets and lagoons May remove from 5 to 25 percent of Q, per inlet. Depends on number of
inlets, inlet size, tidal flow characteristics, and inlet age.
Overwash Less than 1 cu.yd./yr./ft. at most, and limited to low barrier islands.
Beach storage Temporary, but possibly large, depending on beach condition when budget
is made. (See Table 4-5, pages 4-72, 4-73.)
Offshore slopes
Uncertain quantity. May receive much fine material, some coarse material.
Submarine canyons Where present, may intercept up to 80 percent of Q,.
Deflation Usually less than 2 cu.yd./yr./ft. of beach front, but may range up to 10
cu.yd./yr./ft.
CaCO3 loss Not known to be important.
Mining and dredging
May equal or exceed Qg in some localities.
Convective Processes
Longshore transport (waves) May result in accretion of Qg, erosion of Qn, or no change depending on
conditions of equilibrium.
Tidal Currents May be important at mouth of inlet and vicinity, and on irregular coasts
with high tidal range.
Winds Longshore winds are probably not important, except in limited regions.
4-133
lagoons and deep water in the longshore direction comprise the principal
known natural sinks for beach sand. Of potential, but usually unknown,
importance as either a source or a sink is the offshore zone seaward of
the beach.
The works of man in beach replenishment and in mining or dredging
may provide major sources or sinks in local areas. In a few U.S. locali-
ties, submarine canyons or wind may provide major sinks, and calcium car-
bonate production by organisms may be a major source.
* kok kk ok ok koe ok * & * * BXYAMPLE PROBLEM * * * * * *% * * K ® HO FO * *
GIVEN:
(a) An eroding beach 4.4 miles long at root of spit that is 10 miles
long. Beaches on the remainder of the spit are stable. (See
Figure 4-50a.)
(b) A uniform recession rate of 3 feet per year along the eroding
4.4 miles.
(c) Depth of lowest shore parallel contour is -30 feet MSL, and
average dune crest elevation is 15 feet MSL.
(d) Sand is accumulating at the tip of the spit at an average rate of
400,000 cubic yards per year.
(e) The variation of y along the beaches of the spit is shown in
Figure 4-51. (y = Qg+/Qp¢3 Equation 4-23.)
(f) No sand accumulates to the right of the erosion area; no sand is
lost to the offshore.
(g) A medium-width jettied inlet is proposed which will breach the
spit as shown in Figure 4-50a.
(h) The proposed inlet is assumed to trap about 15 percent of the
gross transport, Q&:
(i) The 1.3-mile long beach to the right of the jettied inlet will
stabilize (no erosion) and realign with y changing to 3.5.
(j) The accumulation at the end of the spit will continue to grow
at an average annual rate of 400,000 cubic yards per year after
the proposed inlet is constructed.
4-134
(a) Site Sketch
Reach 4 Reach
Bay ey Proposed Inlet
(b) Before Inlet
Reach 4 Reach 3 Reach 2 Reach 1
Q(4) = 400,000 | Qi(3,4) = 490,000 | Q(2.3) = 530,000 | Q,(,.2) = 474,000
Qn(4) = 400,000 | Q,(3 4) = 400,000 | Q,(. 3) = 318,000 | Q,(4 2) = 284,000
Q#(2,3) = 105,000}] Q.(..2) = 95,000
Q4(3,4) = 45,000
QFe(a,4) _ 149:000 = 425,000|{ Q44(,,2) = 379,000
Qyi(4) = 400,000 D2 3)
b(3)93(3) = 82,000 bia )93(2) = 34,000
(c) After Inlet
Q(4) = 400,000 | Q(3.4) = 490,000 | Q.5) = 512,000 2) = 474,000
Q.(4) = 400,000 | Q,(3 4) = 400,000 | Q,(. 4) = 284,000 3) = 284,000
Q#(3, a = Qre(2,3) = 114,000 | Q(z 2) = 95,000
CO Qyr(2,3) = 398,000 |Qy,1.2) = 379,000
LEGEND
b(3)93(3) = 193,000 @& = 77,000
Q Gross Volume
Q, Net Volume
Q,, Volume to Right
Qo Volume to Left
Q, Inlet Sink Volume
bq; Erosion Source Volume
Figure 4-50. Summary of Example Problem Conditions and Results
4-99
Reach 4
(average y =9)
Reaches |-3
(average y =4)
5
Distance from Spit Tip -Miles
Figure 4-51. Variation of y with Distance Along the Spit,
before Inlet Condition
(a) Annual littoral drift trapped by inlet.
(b) After-inlet erosion rate of the beach to the left of the inlet.
(c) After-inlet nourishment needed to maintain the historic erosion
rate on the beach to the left of the inlet.
(d) After-inlet nourishment needed to eliminate erosion left of the
inlet.
SOLUTION:
Divide the beach under study into four sand budget compart-
ments (control volumes called reaches) as shown in Figure 4-50a. Shore
perpendicular boundaries are established where important changes in the
littoral system occur. To identify and quantify the before-inlet sys-
tem, the continuity of the net transport rate along the spit must be
established. The terminology of Figure 4-43 and Table 4-13 is used
4-136
for the sand budget calculation. The average annual volume of material
contributed to the littoral system per foot of eroding beach reaches
2eand Ses:
x mg ie (15 + 30) (3)
(2) ia %(3) = ESS 27
5.0 yd3/yr/ft.
Then, from Equation 4-46 the total annual contribution of the eroding
beaches to the system can be determined as:
Qa) + Qi) = (1.3 mi. + 3.1 mi.) (5280 ft./mi.) (5 yd?/yr//ft.)
= 1.16 X 105 yd3/yr.
Since there is no evidence of sand accumulation or erosion to the right
of the eroding area, the eroding beach material effectively moves to the
left becoming a component of the net transport volume (Q,) toward the
end of the spit. Contiuity requires the erosion volume and Reach 1 Q,
must combine to equal the acretion at the end of the spit (400,000 cubic
yards per year). Thus, Q, at the root of the spit is
Q,(1,2) ~ 400,000 yd*/yr. — 116,000 yd?/yr.
Q
= 3
tt) 284,000 yd?/yr.
Q, across the boundary between Reaches 2 and 3 (Qi(s 3)) sLES |S
>
+
Q.(2,3) Q,.(1,2) 4 ba) (45(2)) fe
i
= 284,000 yd3/yr. + (1.3 mi.) (5,280 £) (5 yd3/yr./ft.)
= 318,000 yd3/yr.
Q, across the boundary between Reaches 3 and 4 is:
ul +
2.63,4) ~ Qn(2,3) + Gy (Ge)
= 318,000 yd?/yr. ae ((Sall soe) (5.280 S| (5 yd?/yr/ft.)
= 4000,000 yd3/yr.
This Q,(3,4) moves left across reach 4 with no additions or subtrac-
tions, and since the accretion rate at the end of the spit is 400,000
cubic yards per year, the budget balances. Knowing Q, and y for
each reach, gross transport, Q&» transport to the right Q,; and
transport to the left Q,; can be computed using the following equa-
tions:
4-137
From Equation 4-24
a (Ua
ana (2
then
Q.= Se
J 1+¥7 i
and
Qoe = Qy 7 Qe:
Q» Qnt> Qo¢, and Q, for each reach are shown in Figure 4-50b.
Now the after-inlet condition can be analyzed.
Q,(1,2) = 284,000 yd?/yr. (same as “before inlet”) ,
Q1(2,3) = Qu(1,2) = 284,000 yd?/yr. (reach 2 is stable) .
The gross transport rate across the inlet with the new y = 3.5 using
Equation 4-24, is:
5 Q,(2,3) ty)
sy Na rf Sa i
4.5
Q,(2,3) = (284,000 yd3/yr.) a
Q
(2,3) = 512,000 yd2/yr. .
The inlet sink (Q5) = 15 percent of Q (253)
Q, = 512,000 ydi/yr. X 0:15 7
Q, = 77,000 yd2/yr.
The erosion value from Reach 3 now becomes:
Reach 3 erosion = spit accretion + inlet sink - net littoral drift
right of inlet.
bG) (GG) — SG4* @- Ves>
E = 3/yr, + 3/yr, — 284 /yr. ,
bes) (45(g)) = 400,000 yd?/yr. + 77,000 yd2/yr. — 284,000 yd?/yr
+
bi3) (453) = 193,000 yd3/yr. .
4-138
Nourishment needed to maintain historic erosion rate on Reach 3 beach
Ss
Reach 3 nourishment = Reach 3 erosion "after inlet'' - Reach 3 erosion
"before inlet".
3(3) = bi) (45(5)) after inlet — bs) (45(3)) before inlet .
QGcz) = 193,000 yd2/yr. — 82,000 yd?/yr. ,
G3) 111,000 yd3/yr. .
If reach 3 erosion is to be eliminated, it will be necessary to provide
nourishment of 193,000 cubic yards per year.
» Qz, and Q,, for the after-inlet condition are computed using
Equation 4-24 and related equations. The after-inlet sand budget is
shown in Figure 4-50c.
Mee ARSE I ee GEA dese Re LR GRR i) ke UE Be ie) ee en OI dey RL ee) el ele ie ee
4.8 ENGINEERING STUDY OF LITTORAL PROCESSES
This section demonstrates the use of Chapter 4 in the engineering
study of littoral processes.
4.81 OFFICE STUDY
The first step in the office phase of an engineering study of litto-
ral processes is to define the problem in terms of littoral processes.
The problem may consist of several parts, especially if the interests of
local groups are in conflict. An ordering of the relative importance of
the different parts may be necessary, and a complete solution may not be
feasible. Usually, the problem will be stated in terms of the require-
ments of the owner or local interests. For example, local interests may
require a recreational beach in an area of limited sand, making it neces-
sary to estimate the potential rates of longshore and onshore-offshore
sand transport. Or a fishing community may desire a deeper channel in an
inlet through a barrier island, making it necessary to study those litto-
ral processes that will affect the stability and long-term navigability
of the inlet, as well as the effect of the improved inlet on neighboring
shores and the lagoon.
4.811 Sources of Data. The next step is to collect pertinent data. If
the problem area is located on a U.S. coastline, the National Shoreline
Study may be consulted. This study can provide a general description of
the area, and may give some indication of the littoral processes occur-
ring in the vicinity of the problem area.
4-139
Historical records of shoreline changes are usually in the form of
charts, surveyed profiles, dredging reports, beach replenishment reports
and aerial photos. As an example of such historical data, Figure 4-52
shows the positions of the shoreline at Sandy Hook, New Jersey, during
six surveys from 1835 to 1932. Such shoreline changes are useful for
computing longshore transport rates. The Corps of Engineers maintains,
in its District and Division offices, survey, dredging, and other reports
relating to Corps projects. Charts may be obtained from various federal
agencies including the Defense Mapping Agency Hydrographic Center, Geolog-
ical Survey, National Ocean Survey, and Defense Mapping Agency Topographic
Center. A map called "Status of Aerial Photography,"' which may be obtained
from the Map Information Office, Geological Survey, Washington, D.C. 20242,
shows the locations and types of aerial photos available for the U.S., and
lists the sources from which the photos may be requested. A description of
a coastal imagery data bank can be found in the interim report by Szuwalski
(1972).
Other kinds of data usually available are wave, tide, and meteoro-
logical data. Chapter 3 discusses wave and water level predictions; Sec-
tion 4.3 discusses the effects of waves on the littoral zone; and Sub-
section 4.34 presents methods of estimating wave climate and gives pos-
sible sources of data. These referenced sections indicate the wave, tide,
and storm data necessary to evaluate coastal engineering problems.
Additional information can be obtained from local newspapers, court-
house records, and from area residents. Local people can often identify
factors that outsiders may not be aware of, and can also provide quali-
titive information on previous coastal engineering efforts in the area
and their effects.
4.812 Interpretation of Shoreline Position. Preliminary interpretation
of littoral processes is possible from the position of the shoreline on
aerial photos and charts. Stafford (1971) describes a procedure for uti-
lizing periodic aerial photographs to estimate coastal erosion. Used in
conjunction with charts and topographic maps, this technique may provide
quick and fairly accurate estimates of shoreline movement, although the
results can be biased by the short-term effects of storms.
Charts show the coastal exposure of a study site, and since exposure
determines the possible directions from which waves reach the coast, expo-
sure also determines the most likely direction of longshore transport.
Direction of longshore transport may also be indicated by the posi-
tion of sand accumulation and beach erosion around littoral barriers. A
coastal structure in the surf zone may limit or prevent the movement of
sand, and the buildup of sediment on one side of the littoral barrier serv-
es as an indicator of the net direction of transport. This buildup can be
determined from dredging or sand bypassing records or aerial photos. Fig-
ure 4-53 shows the accumulation of sand on one side of a jetty. But wave
direction and nearshore currents at the time of the photo indicate that
transport then was in the opposite direction. Thus, an erroneous conclusion
4-140
Figure 4-52. Growth of Sandy Hook, New Jersey, 1835-1932
4-141
about the net transport might be made, if only wave patterns of this photo
are analyzed. The possibility of seasonal or storm-induced reversals in
sediment transport direction should be investigated by periodic inspections
or aerial photos of the sand accumulation at groins and jetties.
LAKE MICHIGAN.
“(4 December 1967)
Figure 4-53. Transport Directions at New Buffalo Harbor Jetty on
Lake Michigan
The accumulation of sand on the updrift side of a headland is illus-
trated by the beach north of Point Mugu in Figure 4-54. The tombolo in
Figure 4-55 was created by deposition behind an offshore barrier (Grey-
hound Rock, California). Where a beach is fixed at one end by a struc-
ture or natural rock formation, the updrift shore tends to align perpen-
dicular to the direction of dominant wave approach. (See Figures 4-55,
and 4-56.) This alignment is less complete along shores with signifi-
cant rates of longshore transport.
Sand accumulation at barriers to longshore transport may also be
used to identify nodal zones. There are two types of nodal zones: diver-
gent and convergent. A divergent nodal zone is a segment of shore char-
acterized by net longshore transport directed away from both ends of the
zone. A convergent nodal zone is a segment of shore characterized by net
longshore transport directed into both ends of the zone.
Figure 4-56 shows a nodal zone of divergence centered around the
fourth groin from the bridge on the south coast of Staten Island, Outer
New York Harbor. Central Padre Island, Texas, is thought to be an example
4-142
ME RACE AC
OCEAN
-
“ - =
fs :
(21 May 1972)
Figure 4-54. Sand Accumulation at Point Mugu, California
4-143
PACIFIC
OCEAN
=
%
.
.
*
.
.
(29 August 1972)
Figure 4-55. Tombolo and Pocket Beach at Greyhound
Rock, California
4-144
(14 September 1969)
F: gure 4-56. A Nodal Zone of Divergence Illustrated
by Sand Accumulation at Groins, South
Shore, Staten Island, New York
4-145
of a convergent nodal zone. (Watson, 1971.) Nodal zones of divergence
are more common than nodal zones of convergence, because longshore trans-
port commonly diverges at exposed shores and converges toward major gaps
in the ocean shore, such as the openings of New York Harbor, Delaware Bay,
and Chesapeake Bay.
Nodal zones are usually defined by long-term average transport direc-
tions, but because of insufficient data, the location of the mid-point of
the nodal zone may be uncertain by up to 10's of miles. In addition, the
short-term nodal zone most probably shifts along the coast with changes
in wave climate.
The existence, location, and planform of inlets can be used to inter-
pret the littoral processes of the region. Inlets occur where tidal flow
is sufficient to maintain the openings against longshore transport which
acts to close them. (e.g., Bruun and Gerritsen, 1959.) The size of the
inlet opening depends on the tidal prism available to maintain it. (O'Brien,
1969.) The dependence of inlet size on tidal prism is illustrated by Fig-
ure 4-57, which shows three bodies of water bordering the beach on the south
shore of Long Island, New York. The smallest of these (Sagaponack Pond)
is sealed off by longshore transport; the middle one (Mecox Bay) is partly
open; and the largest (Shinnecock Bay) is connected to the sea by Shinne-
cock Inlet, which is navigable.
ASLANT IOC: OCEAN.
(14 September 1969)
Figure 4-57. South Shore of Long Island, N.Y., Showing Closed,
Partially Closed, and Open Inlets
4-146
Detailed study of inlets through barrier islands on the U.S. Atlan-
tic and gulf coasts shows that the shape of the shoreline at an inlet can
be classified in one of four characteristic planforms. (See Figure 4-58,
adapted from Galvin, 1971.) Inlets with overlapping offset (Fig. 4-59)
occur only where waves from the updrift side dominate longshore transport.
Where waves from the updrift side are less dominant, the updrift offset
(Fig. 4-59) is common. Where waves approach equally from both sides, in-
lets typically have negligible offset. (See Figure 4-60.) Where the sup-
ply of littoral drift on the updrift side is limited, and the coast is
fairly well exposed, a noticeable downdrift offset is common as, for exam-
ple, in southern New Jersey and southern Delmarva. (See Hayes, et al.,
(1970.) These planform relations to littoral processes have been found
for inlets through sandy barrier islands, but they do not necessarily hold
at inlets with rocky boundaries. The relations hold regionally, but tem-
porary local departures due to inlet migration may occur.
4.82 FIELD STUDY
A field study of the problem area is usually necessary to obtain
types of data not found in the office study, to supplement incomplete
data, and to serve as a check on the preliminary interpretation and corre-
lations made from the office data. Information on coastal processes may
be obtained from wave gage data and visual observations, sediment sam-
pling, topographic and bathymetric surveys, tracer programs, and effects
of natural and manmade structures.
4.821 Wave Data Collection. A wave-gaging program yields height and
period data. However, visual observations may currently be the best
source of breaker direction data. Thompson and Harris (1972) deter-
mined that 1 year of wave-gage records provides a reliable estimate of
the wave height frequency distribution. It is reasonable to assume that
the same is true for wave direction.
A visual observation program is inexpensive, and may be used for
breaker direction and for regional coverage when few wave-gage records
are available. The observer should be provided with instructions, so
that all data collected will be uniform, and contact between observer
and engineer should be maintained.
4,822 Sediment Sampling. Sediment sampling programs are described in
Section 4.26. Samples are usually surface samples taken along a line
perpendicular to the shoreline. These are supplemented by borings or
cores as necessary. Complete and permanent identification of the sam-
ple is important.
4.823 Surveys. Most engineering studies of littoral processes require
surveying the beach and nearshore slope. Successive surveys provide
data on changes in the beach due to storms, or long-term erosion or
accretion. If beach length is also considered, an approximate volume
of sand eroded or accreted can be obtained which provides information
for the sediment budget of the beach. The envelope of a profile defines
4-147
Overlapping Offset
Length of arrows indicates
relative magnitude of
Longshore Transport Rate
Adequate
updrift source #
Updrift Offset
Downdrift Offset
Inadequate
updrift source
j Negligible Offset
(Galvin, 1971 )
Figure 4-58. Four Types of Barrier Island Offset
4-148
ATLANTIC OCEAN
(14 September 1969)
Figure 4-59. Fire Island Inlet, New York - Overlapping Offset
ATLANTIC OCEAN
(13 March 1963)
Figure 4-60. Old Drum Inlet, North Carolina - Negligible Offset
4-149
fluctuations of sand level at a site (Everts, 1973), and thus provides
data useful in beach fill and groin design.
Methods for obtaining beach and nearshore profiles, and the accu-
racy of the resulting profiles are discussed in Section 4.514.
4.824 Tracers. It is often possible to obtain evidence on the direction
of sediment movement and the origins of sediment deposits by the use of
tracer materials which move with the sediment. Fluorescent tracers were
used to study sand migration in and around South Lake Worth Inlet, Florida.
(Stuiver and Purpura, 1968.) Radioactive sediment tracer tests were con-
ducted to determine whether potential shoaling material passes through or
around the north and south jetties of Galveston Harbor. (Ingram, Cummins,
and Simmons, 1965.)
Tracers are particles which react to fluid forces in the same manner
as particles in the sediment whose motion is being traced, yet which are
physically identifiable when mixed with this sediment. Ideally, tracers
must have the same size distribution, density, shape, surface chemistry,
and strength as the surrounding sediment, and in addition have a physical
property that easily distinguishes them from their neighbors.
Three physical properties have been used to distinguish tracers:
radioactivity, color, and composition. Tracers may be either naturally
present or introduced by man. There is considerable literature on recent
investigations using or evaluating tracers including reviews and biblio-
graphy: (Duane and Judge, 1969; Bruun, 1966; Galvin, 1964; and Huston,
1963); models of tracer motion: (James, 1970; Galvin, 1964; Hubbell and
Sayre, 1965; and Duane, 1970); and use in engineering problems: (Hart,
1969; Cherry, 1965; Cummins, 1964; and Duane, 1970).
a. Natural Tracers. Natural tracers are used primarily for back-
ground information about sediment origin and transport directions, i.e.,
for studies which involve an understanding of sediment patterns over a
long period of time.
Studies using stable, nonradioactive natural tracers may be based
on the presence or absence of a unique mineral species, the relative
abundance of a particular group of minerals within a series of samples,
or the relative abundance and ratios of many mineral types in a series
of samples. Although the last technique is the most complex, it is often
used, because of the large variety of mineral types normally present in
sediments and the usual absence of singularly unique grains. The most
suitable natural tracers are grains of a specific rock type originating
from a localized specific area.
Occasionally, characteristics other than mineralogy are useful for
deducing source and movement patterns. Krinsley, et al., (1964) devel-
oped a technique for the study of surface textures of sand grains with
electron microscopy, and applied the technique to the study of sand trans-
port along the Atlantic shore of Long Island. Naturally occurring radio-
active materials in beach sands have also been used as tracers. (Kamel,
1962.)
4-150
One advantage of natural tracers is their tendency to "average" out
short-term trends and provide qualitatively accurate historical background
information on transport. Their use requires a minimum amount of field
work and a minimum number of technical personnel. Disadvantages include
the irregularity of their occurrence, the difficulty in distinguishing the
tracer from the sediment itself, and a lack of quantitative control on
rates of injection. In addition, natural tracers are unable to reveal
short-term changes in the direction of transport and changes in material
sources,
Judge (1970) found that heavy mineral studies were unsatisfactory as
indicators of the direction of longshore transport for beaches between
Point Conception and Ventura, California, because of the lack of unique
mineral species and the lack of distinct longshore trends which could be
used to identify source areas. North of Point Conception, grain size and
heavy mineral distribution indicated a net southward movement. Cherry
(1965) concluded that the use of heavy minerals as an indicator of the
direction of coastal sand movement north of Drakes Bay, California was
generally successful.
b. Artificial Tracers. Artificial tracers may be grouped into two
general categories: radioactive or nonradioactive. In either case, the
tracers represent particles that are placed in an environment selected
for study, and are used for relatively short-term studies of sediment
dispersion.
While particular experiments employ specific sampling methods and
operational characteristics, there are basic elements in all tracing
studies. These are: selection of a suitable tracer material, tagging
the particle, placing the particle in the environment, and detection of
the particle.
Colored glass, brick fragments and oolitic grains are a few examples
of nonradioactive particles that have been used as tracers. The most com-
monly used stable tracer is made by coating indigenous grains with bright
colored paint or flourescent dye. (Yasso, 1962; Ingle, 1966; Stuvier and
Purpura, 1968; Kidson and Carr, 1962; and Teleki, 1966.) The dyes make
the grains readily distinguishable among large sample quantities, but do
not significantly alter the physical properties of the grains. The dyes
must be durable enough to withstand short-term abrasion. The use of paints
and dyes as tracer materials offers advantages over radioactive methods in
that they require less sophisticated equipment to tag and detect the grains,
and do not require licensing or the same degree of safety precautions. How-
ever, less information is obtained for the same costs, and generally in a
less timely matter.
When using nonradioactive tracers, samples must be collected and
removed from the environment to be analyzed later by physically counting
the grains. For fluorescent dyes and paints, the collected samples are
viewed under an ultraviolet lamp and the coated grains counted.
For radioactive tracer methods, the tracer may be radioactive at the
time of injection or it may be a stable isotope capable of being detected
by activation after sampling. The tracer in the grains may be introduced
4-151
by a number of methods. Radioactive material has been placed in holes
drilled in a large pebble. It has been incorporated in molten glass
which, when hardened, is crushed and resized (Sato, et al., 1962; and
Taney, 1963). Radioactive material has been plated on the surface of
natural sediments. (Stephens, et al., 1968.) Radioactive gas (krypton
85 and xenon 133) has been absorbed into quartz sand. (Chleck, et al.,
1963; and Acree, et al., 1969.)
In 1966, the Coastal Engineering Research Center, in cooperation
with the Atomic Energy Commission, initiated a multiagency program to
create a workable radioisotopic sand tracing (RIST) program, for use in
the littoral zone. (Duane and Judge, 1969.) Tagging procedures (by
surface plating with gold 198-199), instrumentation, field surveys and
data handling techniques were developed which permit collection and
analysis of over 12,000 bits of information per hour over a survey track
about 18,000 feet long.
These recent developments in radioactive tracing permit in situ
observations and faster data collection over much larger areas (Duane,
1970) than has been possible using fluorescent or stable isotope tracers.
However, operational and equipment costs of radionuclide tracer programs
can be high.
Accurate determination of long-term sediment transport volume is not
yet possible from a tracer study, but qualitative data on sediment move-
ment useful for engineering purposes can be obtained.
Experience has shown that tracer tests can give information on
direction of movement, dispersion, shoaling sources, relative velocity
and movement in various areas of the littoral zone, means of natural by-
passing, and structure efficiency. Reasonably quantitative data on move-
ment or shoaling rates can be obtained for short-time intervals. It
should be emphasized that this type of information must be interpreted
with care, since the data are generally determined by short-term littoral
transport phenomena. However, tracer studies conducted repeatedly over
several years at the same location could result in estimates of longer
term littoral transport.
4.83 SEDIMENT TRANSPORT CALCULATIONS
4.831 Longshore Transport Rate. The example calculation of a sediment
budget in Section 4.76 is typical in that the magnitude of the longshore
transport rate exceeds by a considerable margin any other element in the
budget. For this reason, it is essential to have a good estimate of the
longshore transport rate in an engineering study of littoral processes.
A complete description of the longshore transport rate requires
knowledge of two of the five variables
(Qt > Qep QQ xg) 2
defined by Equations 4-21, 4-22, and 4-23. If any two are known, the
remaining three can be obtained from the three equations.
4-152
Section 4.531 describes four methods for estimating longshore trans-
port rate, and Sections 4.532 through 4.535 describe in detail how to use
two of these four methods. (See Methods 3 and 4.)
One approach to estimating longshore transport rate is to adopt a
proven estimate from a nearby locality, after making allowances for local
conditions. (See Method 1.) It requires considerable engineering judge-
ment to determine whether the rate given for the nearby locality is a
reliable estimate, and, if reliable, how the rate needs to be adjusted
to meet the changed conditions at the new locality.
Method 2 is an analysis of historical data. Such data may be found
in charts, maps, aerial photography, dredging records, beach fill records,
and related information. Section 4.811 describes some of these sources.
To apply Method 2, it is necessary to know or assume the transport
rate across one end of the littoral zone being considered. The most suc-
cessful applications of Method 2 have been where the littoral zone is
bounded on one end by a littoral barrier which is assumed to completely
block all longshore transport. The existence of such a complete littoral
barrier implies that the longshore transport rate is zero across the
barrier, and this satisfies the requirement that the rate be known across
the end of the littoral zone being considered. Examples of complete lit-
toral barriers include large jetties immediately after construction, or
spits building into deep quiet water.
Data on shoreline changes permit estimates of rates of erosion and
accretion that may give limits to the longshore transport rate. Figure
4-50 is a shoreline change map which was used to obtain the rate of
transport at Sandy Hook, New Jersey. (Caldwell, 1966.)
Method 3 (the energy flux method) is described in Section 4.532
with a worked example in Section 4.533. Method 4 (the empirical pre-
diction of gross longshore transport rate) is described in Section 4.534
with a worked example in Section 4.535. The essential factor in Methods
3 and 4, and often in Method 1, is the availability of wave data. Wave
data applicable to studies of littoral processes are discussed in detail
in Section 4.3.
4.832 Onshore-Offshore Motion. Typical problems requiring knowledge of
onshore-offshore sediment transport are described in Section 4.511. Four
classes of problems are treated:
(1) The seaward limit of significant sediment transport. Avail-
able field data and theory suggest that waves are able to move sand dur-
ing some days of the year over most of the Continental Shelf. However,
field evidence from bathymetry and sediment size distribution suggest
that the zone of significant sediment transport is confined close to
shore where bathymetric contours approximately parallel the shoreline.
The depth to the deepest shore-parallel contour tends to increase with
average wave height, and typically varies from 15 to 60 feet.
4-153
(2) Sediment transport in the nearshore zone. Seaward of the
breakers, sand is set in motion by waves moving over ripples, either
rolling the sand as bed load, or carrying it up in vortices as suspended
load. The sand, once in motion, is transported by mean tidal and wind-
induced currents and by the mass transport velocity due to waves. The
magnitude and direction of the resulting sediment transport are uncer-
tain under normal circumstances, although mass transport due to waves
is more than adequate to return sand lost from the beach during storms.
It appears that bottom mass transport acts to keep the sand close to the
shore, but that some material, probably finer sand, escapes offshore as
the result of the combined wind- and wave-induced bottom currents.
(3) The shape and expectable changes in shape of nearshore and
beach profiles. Storms erode beaches to produce a simple concave-up
beach profile with deposition of the eroded material offshore. Rates of
erosion due to individual storms vary from a few cubic yards per foot to
10's of cubic yards per foot of beach front. The destructiveness of the
storm in producing erosion depends on its intensity, duration, and orienta-
tion, especially as these factors affect the elevation of storm surge and
the wave height and direction. Immediately after a storm, waves begin to
return sediment to the eroded beach, either through the motion of bar-and-
trough (ridge-and-runnel) systems, or by berm building. The parameter,
Fo = Ho/ (VFfT) given by Equation 4-20 determines whether the beach erodes
or accretes under given conditions. If F, is above critical value
between 1 and 2 the beach erodes. (See Figures 4-29, and 4-30.)
(4) The slope of the foreshore. There is a tendency for the fore-
shore to become steeper as grain size increases, and to become flatter
as mean wave height increases. Data for this relation exhibit much scat-
ter and quantitative relationships are difficult to predict.
4.833 Sediment Budget. Section 4.76 summarizes material on the sediment
budget. Table 4-14 tabulates the elements of the sediment budget and in-
dicates the importance of each element. Table 4-13 classifies the ele-
ments of the sediment budget.
A sediment budget carefully defines the littoral control volume,
identifies all elements transferring sediment to or from the littoral
control volume, ranks the elements by their magnitude, and provides an
estimate of unknown rates by the balancing of additions against losses
(Equation 4-46).
If prepared with sufficient data and experience, the budget permits
an estimate of how proposed improvements will affect neighboring segments
of the littoral zone.
4-154
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