R.; A087 9W32 COASTAL ENGINEERING RESEARCH CENTER FORT SELVOIR VA F/6 8/3 THE SPN ENERGY FLUX METHOD FOR PREDICTING LONGSHORE TRANSPORT R-ETC(U)

jUN 80 C SALVIN, C R SCHwEPPE

UNiCLASSIFIED CERCTP-80ONR I.uuhhuuuuuIu

-EEEIhEE iIIIIIIIIII
TP 80-4

The SPM Energy Flux Method

for Predicting Longshore Transport Rate

by

Cyril Galvin and Charles R. Schweppe

TECHNICAL PAPER NO. 80-4

JUNE 1980

Approved for public release;

!C. U.S. ARMY, CORPS OF ENGINEERS

COASTAL ENGINEERING

URESEARCH CENTER

Kingman Building

Fort Belvoir, Va. 22060
Reprint or republication of any of this material shall give appropriate

credit to the U.S. Army Coastal Engineering Research Center.

Limited free distribution within the United States of single copies of

this publication has been made by this Center. Additional copies are

available from:

National Technical Information Service

A TTN: Operations Diviiion

5285 Port Royal Road

Springfield, Virginia 22161

The findings in this report are not to be construed as an official

Department of the Army position unless so designated by other

authorized documents.

i
UNCLASSIFIED

SECURITY CLASSIFICATION OF THIS PAGE ("7on Date Entered)

REPORT DOCUMENTATION PAGE BEFORE MsL GFORM

I. REPORT NUMBER 2. GOVT ACCESSION NO. 3. RECIPIENT'S CATALOG NUMBER

TP 80-4 h_ bAo qr 4. TITLE (and Subtitle) S. TYPE OF REPORT & PERIOD COVERED

- THEE NERGY FLUX METHOD FOR PREDICTING I Technical ,Paper

ON-- TRANORT ITE.

7. AUTHOR(*) S. CONTRACT OR GRANT NUMBER(e)

/P'I ~ Cy T i7avin

Charles R./Schweppe

9. PERFORMING ORGANIZATION NAME AND ADDRESS 10. PROGRAM ELEMENT. PROJECT. TASK

AREA & WORK UNIT NUMBERS

Department of the Army/

Coastal Engineering Research Center (CERRE-CP) D31196

Kingman Building, Fort Belvoir, Virginia 22060

II. CONTROLLING OFFICE NAME AND ADDRESS 12. REPORT DATE

Department of the Army June 1980

Coastal Engineering Research Center 13. NUMBER OF PAGES

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16. DISTRIBUTION STATEMENT (of thle Report)

Approved for public release; distribution unlimited.

17. DISTRIBUTION STATEMENT (of the abstract emtered In Block 20, it different from Report)

II. SUPPLEMENTARY NOTES

IS. KEY WORDS (Continue on reverse side It necessary and Identify by block number)

Coastal engineering Longshore transport rate

Energy flux Wave height

2& 4I01TW A C F( itosl an reveres, 804b If ad ida dsti fy by block number)

This report explains in detail the energy flux method in Section 4.532 of

the Shore Protection Manual (SPM). Appendix A describes the derivation of four

energy flux factors. Appendix B explains how the significant wave height enters

these equations. Appendix C identifies the data that led to the prediction of

longshore transport rate from the energy flux factor. The importance of the

correct formulation of breaker speed, and its effect on estimates of breaker

angle are demonstrated. The report describes the steps used to arrive at the

energy flux method, but it does not critically analyze those steps.

W, ,oA" 47 3 rmo o OF ,, I NOSIS s OBSOLETE DOO .* " 03 ETUNCLASSIFIED

SECUmhTY CLASSIFICATION OF THIS PAGE (10a Des Entered)
PREFACE

This report is published to provide coastal engineers with detailed

explanations of three frequently asked questions concerning the energy flux

method described in Section 4.532 of the Shore Protection Manual (SPM). The

questions involve the derivaton of equations for the energy flux factor,

the use of significant wave height in these equations, and the data used to

relate energy flux factors to longshore transport rate. These questions

require more elaborate explanations than is possible in the SPM. The work

was carried out under the coastal processes research program of the U.S. Army,

Coastal Engineering Research Center (CERC).

The report was written by Dr. Cyril Galvin, formerly Chief, Coastal

Processes Branch, and Charles R. Schweppe, formerly a student trainee in

Coastal Processes Br-ahch- under the geieral supervision of R.P. Savage, Chief,

Research Division, CERC. The authors wish to thank C.M. McClennan for advice

on the derivation, D.L. Harris for help with the significant wave height,

T.L. Walton for pointing out the importance of the breaker speed estimate, and

P. Vitale, principal investigator of the longshore transport prediction work

unit.

Comments on this publication are invited.

Approved for publication in accordance with Public Law 166, 79th Congress,

approved 31 July 1945, as supplemented by Public Law 172, 88th Congress,

approved 7 November 1963.

TEDE E.SHOP -

Colonel, Corps of Engineers

Commander and Director

Accession ba.

WDC TAB

Juntification.

By_

-AY--A.- ~abilt a

Dist A:aI tI Po

special
CONTENTS

Page

CONVERSION FACTORS, U.S. CUSTOMARY TO METRIC (SI) ... ....... 5

SYMBOLS AND DEFINITIONS .......... .................... 6

I INTRODUCTION ............ ......................... 9

II DISCUSSION OF THE THREE APPENDIXES ....... .............. 9

1. The Equations (App. A) ......... .................. 9

2. Wave Height (App. B) ...... ................... .... 10

3. The Data (App. C) ........ .................... ... 10

III WAVE SPEED AND BREAKER ANGLE ........ ................. 10

IV SUMMARY ........... ............................ ... 12

LITERATURE CITED ........ ....................... .... 13

APPENDIX

A DERIVATION OF LONGSHORE ENERGY FLUX FACTOR ... .......... .. 15

B DISTINCTION BETWEEN SIGNIFICANT AND ROOT-MEAN-SQUARE WAVE

HEIGHTS IN PREDICTING LONGSHORE TRANSPORT RATES .......... ... 25

c FIELD AND LABORATORY DATA IN THE SPM ENERGY FLUX DISCUSSION. 30

4
CONVERSION FACTORS, U.S. CUSTOMARY TO METRIC (SI) UNITS OF MEASUREMENT

U.S. customary units of measurement used in this report can be converted to

metric (SI) units as follows:

Multiply by To obtain

inches 25.4 millimeters

2.54 centimeters

square inches 6.452 square centimeters

cubic inches 16.39 cubic centimeters

feet 30.48 centimeters

0.3048 meters

square feet 0.0929 square meters

cubic feet 0.0283 cubic meters

yards 0.9144 meters

square yards 0.836 square meters

cubic yards 0.7646 cubic meters

miles 1.6093 kilometers

square miles 259.0 hectares

knots 1.852 kilometers per hour

acres 0.4047 hectares

foot-pounds 1.3558 newton meters

millibars 1.0197 x 10- 3 kilograms per square centimeter

ounces 28.35 grams

pounds 453.6 grams

0.4536 kilograms

ton, long 1.0160 metric tons

ton, short 0.9072 metric tons

degrees (angle) 0.01745 radians

Fahrenheit degrees 5/9 Celsius degrees or Kelvins1

1To obtain Celsius (C) temperature readings from Fahrenheit (F) readings, use

formula: C = (5/9) (F -32).

To obtain Kelvin (K) readings, use formula: K = (5/9) (F -32) + 273.15.

5
SYMBOLS AND DEFINITIONS

A(w) spectrum function of the amplitude of n (see eq. B-5)

at vertical distance from breaker wave trough to MWL (see Fig. A-3)

B(t) envelope function (see eq. B-6 and Fig. B-2)

b crest length between orthogonals (see Fig. A-2)

b subscript for condition at the breaker

C wave velocity

d water depth

E wave energy density (see eq. B-i) with subscripts

g acceleration of gravity

H wave height

H' deepwater height assuming no refraction

mean value of the highest fraction of wave heights above a height

II which has a probability, p, of occurrence (see eqs. B-8 and B-10)

H, 8 root-mean-square height (eq. B-3)

1i8 significant wave height (eq. B-13)

h water depth measured from wave crest to bottom (Fig. A-3)

h* generalized water depth under a breaking wave (eq. 2)

i subscript for conditions at point i (Fig. A-2)

Kr refraction coefficient (eq. A-11)

K8 shoaling coefficient (eq. A-12)

I. wavelength

fn natural log symbol

N iiumber of waves

n ratio of group velocity to individual wave velocity (eq. A-17)

o subscript for conditions in deep water (except eq. B-6)

P) energy flux, per unit length of shoreline, defined by equation (A-6)

P(1) probability function of wave height

6
SYMBOLS AND DEFINITIONS--Continued

P* energy flux per unit length of wave crest (eq. A-i)

P, part of the energy flux in waves defined by equation (A-7)

Pk8 approximate evaluation of Pt for conditions in the surf zone

p fraction of wave heights, in a train of N waves, whose height exceeds

H (eq. B-8)

Q volume rate of longshore transport

s distance in the longshore direction between adjacent orthogonals (Fig.

A-2)

T wave period

t time, or conversion factor, seconds per year

w weight density of water

a angle between wave crest and shoreline (Fig. A-2)

8 ratio of breaker depth to breaker height (Fig. A-3)

rl departure of water surface from MWL (Fig. B-2)

p mass density of water

a ratio at/Hb (Fig. A-3)

wwo wave frequency or central wave frequency (eq. B-6)

7

.~ - ~
THE SPM ENERGY FLUX METHOD FOR PREDICTING LONGSHORE TRANSPORT RATE

by

Cyril Calvin and Charles R. Schweppe

I. INTRODUCTION

The Shore Protection Manual (SPM) (U.S. Army, Corps of Engineers, Coastal

Engineering Research Center, 1977) describes procedures for estimating quantities important in coastal engineering design. Among the most important of

these is the estimation of longshore transport rate. Aside from relying

solely on judgment or historical data, the principal way to estimate longshore

transport rate is by use of the energy flux method, described in Section 4.532

of the SPM. Since the SPM provides design guidance rather than explanation,

the derivations of the energy flux equations are only briefly indicated in

the manual.

This report supplements the SPM with a documented development of the energy

flux method as it is presented on pages 4-96 to 4-102 of the SPM. The documentation is mainly found in the three appendixes to this report: Appendix A,

Derivation of the Longshore Energy Flux Factor; Appendix B, Distinction Between

Significant and Root-Mean-Square Wave Heights in Predicting Longshore Transport

Rates; and Appendix C, Field and Laboratory Data Appearing in the SPM Energy

Flux Discussion. The subject matter of these appendixes are the three most

commonly questioned aspects of the SPM presentation on the energy flux method.

Each appendix gives an independent explanation of one topic without necessary

cross-reference to other parts of the report.

This report only describes what was done to arrive at the energy flux

method already published in the SPM. The assdmptions necessary to arrive at Nthe formulation in the SPM are described, but not reviewed, although these

assumptions were closely examined by reviewers when Chapter 4 of the SPM was

under preparation (May 1972 to August 1973). The current issue of the SPM is

the third edition (1977). Section 4.532 on the energy flux method is the same

in all three editions, with the exception of an error on the ordinate scale of

Figure 4-36 which was corrected after the first (1973) printing. Galvin and

Vitale (1977) compared the energy flux method documented in the SPM with its

predecessor, TR-4, Shore Protection, Planning and Design (U.S. Army, Corps of

Engineers, 1966).

II. DISCUSSION OF THE THREE APPENDIXES

1. The Equations (App. A).

The energy flux method relates longshore transport rate, Q, to wave

conditions by use of the longshore energy flux factor, Pk,. Two equations

are needed: an equation that converts wave conditions into P£8, and an equation that predicts Q from Pk., In the SPM, four theoretically equivalent equations for longshore energy flux, P., are developed from smallamplitude, linear theory. From these four equations, four design equations

for the longshore energy flux factor in the surf zone, P', are derived.

Although each equation for Pk is equivalent to any of the three other

equations for Pk, no two of the four Pk8 equations are exactly equivalent

because each Pts equation was derived from a different Pt equation using

a different set of approximations.

9 _1

lG,,IO P,.GE BI...-UOT,.. II'6' L .+ ...... .. .. ... , 

--
The four equations for PX, equations 4-31 to 4-34, and the four equations

for P£8, equations 4-35 to 4-38, are given in Tables 4-7 and 4-8, respectively,

in the SPM, and they are reproduced in Appendix A.

Appendix A derives the eight equations and identifies the assumptions used.

These derivations involve only simple algebra and the assumptions are standard.

However, the exact formulation of the assumptions does affect the value of

coefficients in the equations.

2. Wave Height (App. B).

The energy flux factor P depends sensitively on wave height. In one

equation, it is an H dependence; in another one, H3 ; and in two equations, H5/2. In all, the height enters Pks primarily as the energy density, F,

which depends on H2 . Since, in nature, height varies from one wave to the

next, the average energy density of a group of waves will not be determined

from the average height, but rather from a height which produces the average

value of H2 . This is the root-mean-square (rms) height, firms .

The height commonly used in coastal engineering work is neither the average

height nor the Hrns, but the significant height, H.. Appendix B explains

the relations between Hrms and H., and how the energy flux factor is calibrated for use with H ..

3. The Data (App. C).

To obtain an equation to predict Q from PZ., it is necessary to have

N sufficient data for wave conditions to compute Pk, and to measure values of

Q at the time the wave conditions are measured. Appendix C identifies the

sources of data which led to the relation between Q and Pks shown in Figure

4-37 and equation 4-40 of the SPM.

III. WAVE SPEED AND BREAKER ANGLE

This report primarily provides documentation for the energy flux method as

given in Section 4.532 of the SPM. It does not critically evaluate the method.

However, this sectiorn does examine the effect of assumptions about wave speed

and breaker angle on the computed values of Q to enable the user to form a

judgment of the overall accuracy.

The energy flux is proportional to group velocity. The group velocity for

linear waves in shallow water is equal to wave speed. For energy flux entering

the surf zone, the appropriate wave speed is the speed of the breaker. Breaker

characteristics have long been assumed to be described by solitary wave theory.

Solitary theory can be used to locate the breaker point using the rule-of-thumb

db = 1.28 Hb (1)

and to estimate the breaker speed using an equation of the form,

C =Vgh* (2)

I0
where h* is a depth. If h* is set equal to the mean depth, then the breaker

speed is that of linear theory

or using equation (1)

= 6.42 Vb (feet per second) (3)

If h* is set equal to the depth under the crest of the breaking wave, and

the depth is properly corrected for the depression of the trough below mean

water level, the result is

C 8.02 Ab (feet per second) (4)

If h* is set equal to + db without correcting for depression of trough

below mean water level, the speed will be

Cb = 8.57 VHib (feet per second) (5)

Since wave speed enters as a linear multiplier in the equation for energy

flux, the estimated energy flux will vary directly as the estimated wave speed.

The field data plotted on Figure 4-37, from which the relation between Q and

Pk, is obtained, include estimates of Pk, using equations (3) and (5). The

nine data points from Watts (1953) and Caldwell (1956) have energy flux values

computed with equation (3). The 14 data points from Komar (1969) include

energy flux estimates based on both equations (3) and (5). Equation (4) is

used in the derivations for all P., equations, as shown in Appendix A.

Since the Pis equations used in the SPM depend on equation (4) for wave

Nspeed, it is evident that wave speed has not been treated consistently throughout the analysis. The Pk8 computed with the equations in the SPM (depending

on eq. 4 for wave speed) will be 25 percent too high for the plotted relation

between Q and Pk. for the points of Watts (1953) and Caldwell (1956) on

Figure 4-37.

The error between the SPM P., equations and the 14 data points of Komar

(1969) is more difficult to evaluate. The depth used by Komar is the 20-minute

time average at the wave gage (P.D. Komar, Oregon State University, personal

communication, 1978). This depth was often significantly deeper than the

breaker depth, as can be judged from the fact that the average crest angle at

the wave gage was about 57 percent greater than the average breaker angle in

Komar's data (App. IV of Komar, 1969). Thus, the use of the depth at the gage

raises the speed above the linear theory estimate of equation (3) toward the

value of equation (4). Since it is not easy to evaluate this effect, and since

the relation between PZ8 and Q is heavily dependent on the 14 plotted points

of Komar, the error due to using equation (4) for breaker speed in calculating

Pis is concluded to be less than 25 percent and may be negligible.

Moreover, the Komar (1969) data for energy flux probably include an overestimate of the breaker angle. This is because breaker angle, ctb' was

computed from the angle at the sensor, a, by

Cb sin b -- s in
where Cb was effectively equation (5) and C was equation (3). Since

equation (5) probably overestimates wave speed (by about 7 percent) and

equation (3) underestimates it (by about 10 percent), the resulting breaker

angle probably should be multiplied by (1 - 0.10)/(1 + 0.07) or 0.84 to be

theoretically correct.

The net result of the variable estimates of wave speed and breaker angle

is to suggest that equation (4) is a logical compromise, and this is what is

used in the SPM equations.

IV. SUMMARY

This report describes the energy flux method of estimating longshore

transport rate and provides detailed explanations of the three most frequently

asked questions about this method (see Apps. A, B, and C). The following general conclusions result from this study.

1. Energy flux may be estimated by four separate methods, depending on

the available field data. The results show that the energy flux factor, Ps,

is proportional to any one of the following groups of wave variables (App. A):

(a) Hb51 2 sin 2ab

(b) H0 5/2 (cos C,0)114 sin 2co

(c) H2 T sin ab cos ao

(d) sin ao

2. The wave height used in these equations is the significant wave height

(App. B).

3. Longshore transport rate, Q, is directly proportional to the energy

flux factor, Pz." The indicated proportionality constant is 7,500 for traditional units (Q in cubic yards per year and Pk, in foot-pounds per second

per foot). Appendix C describes the data used to derive this constant.

4. There is uncertainty in the proportionality constant because of variations in field data, in the equation for breaker speed, and in measurement of

breaker angle. An intuitive estimate of the uncertainty in the constant is

±40 percent.

S. The energy flux method is expected to be improved with further

knowledge.

2
LITERATURE CITED

BRETSCHNEIDER, C.L., and REID, R.O., "Modification of Wave Height Due to

Bottom Friction, Percolation, and Refraction," TM-45, U.S. Army, Corps of

Engineers, Beach Erosion Board, Washington, D.C., Oct. 1954.

CALDWELL, J.M., "Wave Action and Sand Movement Near Anaheim Bay, California,"

TM-68, U.S. Army, Corps of Engineers, Beach Erosion Board, Washington, D.C.,

Feb. 1956.

DAS, M.M., "Longshore Sediment Transport Rates: A Compilation of Data," MP

1-71, U.S. Army, Corps of Engineers, Coastal Engineering Research Center,

Washington, D.C., Sept. 1971.

DAS, M.M., "Suspended Sediment and Longshore Sediment Transport Data Review,"

Proceedings of the 13th International Conference on Coastal Engineering,

American Society of Civil Engineers, Vol. II, July 1972, pp. 1027-1048

(also Reprint 13-73, U.S. Army, Corps of Engineers, Coastal Engineering

Research Center, Fort Belvoir, Va., NTIS 770 179).

FAIRCHILD, J.C., "Laboratory Tests of Longshore Transport," Proceedings of

the 12th Conference on Coastal Engineering, American Society of Civil

Engineers, Vol. II, Sept. 1970, pp. 867-889.

GALVIN, C.J., Jr., "Breaker Travel and Choice of Design Wave Height," Journal

of the Waterways and Harbors Division, Vol. 95, No. WW2, May 1969, pp. 175-

200 (also Reprint 4-70, U.S. Army, Corps of Engineers, Coastal Engineering

Research Center, Fort Belvoir, Va., NTIS 712 652).

GALVIN, C.J., Jr., and VITALE, P., "Longshore Transport Prediction - SPM 1973

Equation," Proceedings of the 15th Conference on Coastal Engineering,

American Society of Civil Engineers, Vol. I, July 1976, pp. 1133-1148.

HARRIS, D.L., "Characteristics of Wave Records in the Coastal Zone," Waves on

Beaches and Resulting Sediment Transport, R.E. Meyer, ed., Academic Press,

New York, 1972, pp. 1-51.

HARRIS, D.L., "Wave Energy-Estimates," U.S. Army, Corps of Engineers, Coastal

Engineering Research Center, Fort Belvoir, Va., unpublished, Oct. 1973.

IVERSEN, H.W., "Waves and Breakers in Shoaling Water," Proceedings of the

Third Conference on Coastal Engineering, American Society of Civil Engineers,

1952, pp. 1-12.

JOHNSON, J.W., "Sand Transport by Littoral Currents," Technical Report Series

3, Issue 338, Institute of Engineering Research, University of California,

Berkeley, Calif., June 1952.

KINSMAN, B., Wind Waves, Their Generation and Propagation on the Ocean

Surface, 1st ed., Prentice-Hall, Englewood Cliffs, N.J., 1965.

KOMR, P.D., "The Longshore Transport of Sand on Beaches," unpublished Ph.D.

'1.esis, University of California, San Diego, Calif., 1969.

KRUMBEIN, W.C., "Shore Currents and Sand Movement on a Model Beach," TM-7,

U.S. Army, Corps of Engineers, Beach Erosion Board, Washington, D.C.,

Sept. 1944.

13
LONGUET-HIGGINS, M.S., "On the Statistical Distribution of the Height of Sea

Waves," Journal of Marine Research, Vol. 11, No. 3, 1952, pp. 245-266.

LONGUET-HIGGINS, M.S., "Longshore Currents Generated by Obliquely Incident

Sea Waves, 1," Journal of Geophysical Research, Vol. 75, No. 33, Nov. 1970,

pp. 6788-6801.

MOORE, G.W., and COLE, A.Y., "Coastal Processes, Vicinity of Cape Thompson,

Alaska," Trace Element Investigation Report 753, U.S. Geological Survey,

Washington, D.C., Jan. 1960.

PRICE, W.A., and TOMLINSON, K.W., "The Effect of Groynes on Stable Beaches,"

Proceedings of the 11th Conference on Coastal Engineering, American Society

of Civil Engineers, Vol. 1, 1969, pp. 518-525.

SAUVAGE, M.G., and VINCENT, M.G., "Transport Littoral Formation de Fleches et

de Tombolos," Proceedings of the Fifth Conference on Coastal Engineering,

American Society of Civil Engineers, 1954, pp. 296-328.

SAVILLE, T., Jr., "Model Study of Sand Transport Along an Infinitely Long

Straight Beach," Transactions of the American Geophysical Union, Vol. 31,

1950, pp. 555-565.

SAVILLE, T., Jr., "Discussion of Laboratory Determination of Littoral Transport Rates," Journal of Waterways and Harbors Division, Vol. 88, No. WW4,

Nov. 1962, pp. 141-143.

SHAY, E.A., and JOHNSON, J.W., "Model Studies on the Movement on Sand Transported by Wave Action Along a Straight Beach," Issue 7, Series 14, Institute

of Engineering Research, University of California, Berkeley, Calif., 1951.

SVERDRUP, H.U., and MUNK, W.H., "Wind, Sea, and Swell: Theory of Relations

for Forecasting," Publication No. 601, U.S. Navy Hydrographic Office,

Washington, D.C., Mar. 1947.

THORNTON, E.B., "Longshore Current and Sediment Transport," TR-5, Department

of Coastal and Oceanographic Engineering, University of Florida, Gainesville,

Fla., Dec. 1969.

U.S. ARMY, CORPS OF ENGINEERS, "Shore Protection Planning and Design," TR-4,

3d ed., Coastal Engineering Research Center, Washington, D.C., 1966.

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

Protection Manual, 3d ed., Vols. I, II, and III, Stock No. 008-022-00113-1,

U.S. Government Printing Office, Washington, D.C., 1977, 1,262 pp.

WALTON, T.L., Jr., "Littoral Drift Computations Along the Coast of Florida by

Use of Ship Wave Observations," unpublished Thesis, University of Florida,

Gainesville, Fla., 1972.

WATTS, G.M., "A Study of Sand Movement at South Lake Worth Inlet, Florida,"

TM-42, U.S. Army, Corps of Engineers, Beach Erosion Board, Washington, D.C.,

Oct. 1953.

WIEGEL, R.L., Oceanographic Engineering, 1st ed., Prentice-Hall, Englewood

Cliffs, N.J., 1964.

14
APPENDIX A

DERIVATION OF LONGSHORE ENERGY FLUX FACTOR

This appendix derives four formulas for longshore energy flux, Pk, and

four formulas for corresponding approximations, the longshore energy flux

factors, P.. These eight formulas are given in the SPM in Tables 4-7 and

4-8, based on assumptions summarized in Table 4-9. For convenience, Tables

4-7 and 4-8 are reproduced here as Figure A-1 (p. 4-97 of the SPM).

The basic longshore energy flux derivation is given in Galvin and Vitale

(1977), who made use of earlier work by Walton (1972). The derivation also

benefits from Longuet-Higgins' (1970) work on conservation of momentum flux

in the surf zone. Equations well known from linear wave theory are presented

without derivation but are keyed to other chapters in the SPM, or to the development in Wiegel (1964) where more detailed derivation is provided.

1. Equations for Pk.

The derivation for Pk proceeds as follows: Assume a coast with contours

that are parallel to a straight shoreline (Fig. A-2). Waves approaching this

coast are assumed to be described by linear small-amplitude theory. In general,

a wave crest that makes an angle ao with the shoreline when in deep water

will refract to make an angle ai at some shallower depth (Fig. A-2), where ai

is related to a. by Snell's law. In what follows, the subscript o refers

to deepwater conditions, and the symbols are those used in the SPM.

The path of a wave passing through point i is shown on Figure A-2 as the

dashed orthogonal labeled "wave path." The flux of energy in the direction of

wave travel per unit Zength of wave crest at point i is given by

P*=C E

Sn C E (A-I)

where C, is the wave group velocity, C is the wave phase velocity,

n = C9 /C, and E is the energy density, the total average energy per unit

area of sea surface, and is defined as (eq. 2-39 in the SPM)

w 82 (A-2)

where w is the weight density of water, 64.0 pounds per cubic foot for

seawater, and H is the wave height. (This wave height is the height of a

uniform periodic wave. How it relates to the significant wave height and to

wave heights characterizing wave height distributions is discussed in App. B.)

From equation (A-l), the energy flux in the direction of wave travel, at

point i, for crest length, bi , is (see eq. 7 in Galvin and Vitale, 1976)

P* bi Ei C9 bi (A-3)

15
Table 4.7. Longshore Energy Flux,Pg, for a Single Periodic Wave in

Any Specified Depth. (Four Equivalent Expressions from

Small-Amplitudc Theory)

Equation Pe Data Required

(energy/time/distance) (any consistent units)

4-31 2C (Esin 2cr) d, T, H,

4-32 C ( T 0 sin 2%) d, T, H0 , ao

4-33 K'Co ( 410 sin 2c0 T, H0 ,cc

4-34 (2C) (K" C )-T C (Y4 E sin 2 ) d,T, H, aoof

no subscript indicates a variable at the specified depth

where small-amplitude theory is valid

Cg = group velocity (see assumption lb. Table 4-9)

Co = decpwater

d = water depth

H = significant wave height

T = wave period

a= angle between wave crest and shoreline

KR = refraction coefficient ICV-s a

Table 4-8. Approximate Formulas for Computing Longshore

Energy Flux Factor, PC., Entering the Surf Zone

Equation Pis Data Required

(ft.-lbs./sec./ft. of beach front) (ft.-sec. units)

4-35 32.1 Hb512 sin 2ab Hb,ab

4-36 18.3 H0 

5 2 (cos o,)114 sin 2*0 Ho, o

4-37 20.5 TH 2 sin b cos Oo T, Ho, %,b

4-38 100.6 (H3/T) sin ao T. Hb , %_o

H. = deepwater

Hb = breaker position

H = significant wave height

T = wave period

a = angle between wave crest and shoreline

See Table 4-7 for equivalent small amplitude equations and

Table 4-9 for assumptions uscd in deriving Pis from PV.

4-97

Figure fk-I. Page 4-97 of the SPM, showing four

formulas for longshore energy flux,

P., and four formulas for longshore

energy flux factors, Pts'

16
Orthogonal 1 Orthogonal 2

Wove Crest

Tangent to Crest

Gi at

-s1-

Bottom

Contours

Wave

/ I Path

Breaker LineI

Surf

Zone

Beach

Figure A-2. Definitions for conservation of energy flux for

shoaling wave (after Walton, 1972).

17
In the small-amplitude theory assumed, energy is not dissipated and does

not cross wave orthogonals. Therefore, the energy flux in the direction of

wave travel must remain constant between orthogonals, i.e., between deep water

and point i.

P~bo = l1tb. = total wave power between orthogonals (A-4)

From the geometry of Figure A-2, it is obvious that bi changes with

position on the wave path. However, the distance between adjacent orthogonals,

s, measured in the longshore direction, does not change because orthogonal 2

must be identical to orthogonal I in all respects except being displaced along

the coast by a distance s. Therefore, at any point, i, in the wave path

bi = s cos aj (A-5)

For the straight parallel contours assumed, the distance s is arbitrary.

Therefore, divide both sides of equation (A-4) by s, and set s equal to the

unit of distance. In the SPM, this unit is 1 foot; in metric units, it would

be 1 meter. This procedure defines a total wave power, per unit length of

shoreline, indicated by P

P = P* cos ai (A-6)

In the Appendix, i indicates any point on the path where equations (A-i)

to (A-5) are valid. With this understanding, the subscript i is dropped in

the derivation helow. Since P has both a magnitude and a direction at any

point, a longshore component, Pk, can be defined. Using equations (A-l) and / (A-6), this longshore component of energy flux can be written

Pz = (gCq cos a) sin a (A-7)

By use of the identity

sin 2 a = 2 cos a sin a (A-8)

equation (A-7) can he written as the first of the four alternate forms for

P. given in Table 4-7 of the SPM, i.e., equation (4-31) in the SPM:

S2 C ( sin 2a) (A-9)

The equation is given in this form so that the term in parentheses corresponds

to the longshore force term used in Longuet-Higgins (1970). The energy density,

T, is proportional to the local height squared, H2 , (eq. A-2). The local

height can be related to the deepwater height, HO , by equations (7.8) and

(7.9) in Wiegel (1964)

I = KI K, I0 (A-10)

18
where KR is the refraction coefficient

/ cos a ) 1 1 2

Cos C1

and K is the shoaling coefficient

K8 = (Cu°) 1 2 (A-12)

K. for later use may also be approximated by the breaker height index

K 1ib (A-13)

0

where Hb is the wave height at the point of breaking. Thus,

K:2 K2 E

A' S o

/ \co-- \/C \

so P9, becomes

)= K2 K2 C F sin ax cos a (A-is)

P s g o

By substituting equations (A-li) and (A-12) into equation (A-15) and canceling

like terms,

I= Cgo Eo cos Co sin a (A-16)

Equation 2.68 of Wiegel (1964) gives that

s - = ) = (A-17)

s inh 47

where d is the water depth. To a good approximation, n is equal to 0.5 in

deep water, i.e., d/l, > 0.5, and n is equal to 1.0 in shallow water, i.e.,

d/L < 0.04. (Exact values of n at these limits are 0.5117 at d/L equal t

0.5, and 0.9795 at d/L equal to 0.04.) Equation (A-16) can be further modified using Snell's law (where C is the local wave speed given by equation

(2-3) in the SPM),

Cs sin co (A-18)

0

19
and equations (A-8) and (A-17) to get the second of the equivalent forms for

Pt given in Table 4-7 of the SPM, i.e., equation (4-32) in the SPM:

pi = C G E sin 2ao) (A-19)

The third equivalent form for Pt is equation (4-33) in Table 4-7 of the SPM.

It is obtained from equation (A-is) by using equations (A-12) and (A-8),

PX = K2 C (1 Eo sin 2a) (A-20)

The fourth and last of the equivalent forms of Pk is equation (4-34) in Table

4-7 of the SPM. It is obtained by substituting equations (A-8), (A-i), and

(A-18) into equation (A-9),

p, = ( ) (9-)Cg (L _Esin 2%o (A-21)

2. Equations for P9's"

Up to this point, all results are for small-amplitude linear theory. However, the assumed relation between longshore transport and energy flux in the

surf zone requires that P, be evaluated at the breaker line where smallamplitude theory is less valid. To indicate approximations for waves entering

the surf zone, the symbol Pk. will be used in place of Pk. This approximation is called the energy flux factor, P 8., in the SPM, and like Pk, it is

measured in units of energy per second per unit length of shoreline; e.g., footpounds per second per foot. One expression for Pts will be derived from each

of the four equivalent expressions for Pk (eqs. A-9, A-19, A-20, and A-21)

to obtain the four equations in Table 4-8 of the SPM.

The energy density appears in all four equations for PX. In foot-poundsecond units, and for saltwater (w = 64 pounds per cubic foot), the energy

density is, from equation (A-2),

- w1l 2 _

8

8 1l2 (foot-pounds per square foot) (A-22)

In shallow water, group velocity equals wave speed, and near breaking, wave

speed depends on depth measured from the crest elevation, as in solitary wave

theory (Section 2.27 in the SPM).

The equation for wave speed near breaking, Cb , is an approximation.

Several approximations are possible, but the solitary wave approximation

used in obtaining the SPM equations for Pts is as follows (symbols defined

in Fig. A-3). The first approximation for the speed of the solitary wave is

20
Hb

att

db

Figure A-3. Definition of terms for breaker

speed equation.

Cb = /gh (A-23)

where

h = fib + db a t (A-24)

By dividing with ifb, the wave speed becomes

Cb 7 = KAHiT (A-25)

where K is a dimensional term depending on two ratios and g (see eq. 11 in

Galvin, 1969)

K = /g(l + - a) (A-26)

The depth-to-height ratio, , was assumed equivalent to the usual coastal

engineering rule-of-thumb (8 = 1.28). The relative depression of the trough,

a, was known to vary from 0.15 to 0.40, and a standard value of 0.28 was used

for a giving a wave speed of (eq. 4)

C Cb = 8.024W (feet per second)

It should be noted that there is recent evidence that both and u

should have somewhat lower values than those used to obtain equation (4) but

the net result of the reductions does not significantly change the coefficient

in equation (4).

A third approximation is the replacement

= (A-27)

where the subscript b refers to breaker zone conditions.

21
I

Substituting the approximations represented by equations (4), (A-22), and

(A-27) into equation (A-9) yields Pk, approximations for each of the four

equations for Pk in Table 4-7.

Equation (4-31) for P. in Table 4-7 of the SPM is approximated by

P 2 2(8.02 i

1112 ) -L (8 H2) sin 2a b

b 4 bb

32.1 5/2 sin 2a (foot-pounds per second per foot) (A-28)

which involves only height and direction at the breaker. This is equation

(4-35) of Table 4-8 in the SPM.

In the same way, equation (4-32) for P. (eq. A-19) can be reduced to

P9 = 16.0 1/2 Ho sin 2ao (foot-pounds per second per foot) (A-29)

Hb is put in terms of Ho using equations (A-10) and (A-11) to get

/ (cos a01

1

/

4 (KsI°i) 1/2 (A-30) b \COScab/ s

In this equation, cos ab equals 1.0 to a good approximation. For example,

even if ab has a high value of 200 (exceeded less than 5 percent of the time

on straight beaches), (cos 200)1/4 = 0.98.

So, using this approximation and substituting equation (A-30) into equation

(A-29),

P = 16.0 115/2 (cos 0o)1i/4 K1/2 sin 2a (foot-pounds per second

per foot) (A-31)

The shoaling coefficient, KS, is approximated by the breaker height index.

Hb/T , obtained from the experimental work of Iversen (1952). Related data are

in Figure 2-65 of the SPM. I- is the unrefracted deepwater water height. If

the slope and period are known, a steepness can be computed, and Llb/H1 obtained

from these experimental results. However, the period in offshore wave statistics

correlates poorly with the littoral wave period (Harris, 1972). A reasonable

approximation without using the period is obtained by observing that Hb/H,

ranges from 0.95 to 1.7 for plunging and spilling waves, a center range of about

1.3 for expectable slopes and moderately steep waves. Therefore, K. is assumed

here to be 1.3. Since Q depends on the square root of K., this assumed

value will usually give Q to within 10 percent of the value from lversen's

data when steepness and slope information is available.

This approximation reduces equation (A-31) to a convenient approximation

involving only deepwater height and direction, which is equation (4-36) in the

SPM:

I' = 18.3 115/2 (cos Co )1/4 sin 2a (foot-pounds per second

per foot) (A-32)

22
The deepwater wave velocity, Co, can be approximated by (see eq. 2.37

in Wiegel, 1964)

CO = T 5 12 T (feet per second) (A-33) 2 T

Using this along with equations (A-8), (A-11), and (A-22), the third expression

for PZ. can be obtained from equation (A-20):

/Cos a11

P -0I (5.12 T) 1 (8 H')(2 c , il 9's co( 4 0 b b

= 20.5 T I2 cos OL sin (foot-pounds per second per foot) (A-34)

0 0

This is equation (4-37) in the SPM.

The final equation for P,, equation (A-21), reduces to

I Cos ac0-1 112 1

2 (8.02 11c/s2) \ 5.12 T (8.02 b (8 b Cos a0 sin a

= 100.5 (b) cos ab sin ao (foot-pounds per second per foot) (A-35)

by use of equations (4), (A-8), (A-11), (A-22), and (A-33). The cos a- term

in many cases, is close enough to 1.0 to be ignored. Therefore, equation (A-35)

can be further reduced to:

= 100 sin a (foot-pounds per second per foot) (A-36)

This is equation (4-38) in the SPM.

The four approximate equations for PZs (eqs. A-28, A-32, A-34, and A-36)

are given in Table 4-8 of the SPM. Unlike the four exact equations (Pk in

Table 4-7), these approximations are not equivalent, due to the different

assumptions made in deriving them. Trial solutions of all four approximations

suggest that they give values of P,, that agree within a factor of 2. Figures

4-36 and 4-37 in the SPM show that variation of P , by a factor of 2 is within

the scatter of the data defining the dependence of Q on P1%.

All results in this appendix, including the Pg. approximations, are for

a coast with straight, parallel (but not necessarily evenly spaced) contours.

All results assume no friction loss, which may become an important omission

for waves traveling long distances over shallow, flat slopes (Bretschneider

and Reid, 1954; Walton, 1972). Additional assumptions used in the derivations

are summarized in Table 4-9 of the SPM. The essentially approximate basis

for the longshore transport prediction must be recognized when the energy flux

method is used. The more that the prototype conditions agree with the conditions

23
assumed in these derivations, the greater the confidence in the resulting

estimate.

3. Summary.

This appendix provides the derivation for eight equations given in the

SPM. These eight equations are the four equivalent equations for the longshore energy flux, Pk (Table 4-7), and the four nonequivalent equations for

the longshore energy flux factor, PkS (Table 4-8) given in the SPM. The

correspondence between equation numbers in this appendix and the SPM is as

follows:

P. in SPM Table 4-7 PkS in SPM Table 4-8

SPM This Appendix SPM This Appendix

4-31 A-9 4-35 A-28

4-32 A-19 4-36 A-32

4-33 A-20 4-37 A-34

4-34 A-21 4-38 A-36

Tables 4-7 and 4-8 of the SPM are reproduced as Figure A-1 on page 16 of this

Appendix.

24
APPENDIX B

DISTINCTION BETWEEN SIGNIFICANT AND ROOT-MEAN-SQUARE WAVE HEIGHTS

IN PREDICTING LONGSHORE TRANSPORT RATES

The four equivalent expressions for longshore energy flux, Pk, given

in Table 4-7 of the SPM are developed from small-amplitude linear theory (see

App. A), and each expression for Pz contains the wave energy density, E,

as a linear factor. The energy density per wave is

8 H2 (B-i)

where w is the weight density of seawater, and H is the wave height measured

from crest to trough (see Fig. B-i). For a group of waves, N in number, each

with wave height Hi, for i = 1 to N, the average energy density is givei by

N

(E) =1 . avg N l-=l 8

= w (HrMS)2 

(13-2)

where H., is the root-mean-square (rms) wave height given by

N

(H s ) 2 1 2 (B-3)

rMS) N l.

StilWater

Level

Figure B-1. Definition of wave height and amplitude for

simple sinusoidal wave function.

Thus, the rms wave height is the proper wave height to use in evaluating

and hence Pk. In conditions where a uniform train of periodic waves exists,

approximated in some laboratory experiments, all of the tj are equal. Thus,

by equation (B-3), Hi is identical to lrms. In such cases Ili is interchangeable with Hrms in equation (B-2). In natural conditions, however, such

as ocean waves approaching a shore, the wave heights are usually not uniform.

In such cases, the entire distribution of wave heights to determine HrMs must

be considered or some other type of average wave height computed. It has been

the general practice of coastal engineers to use an average wave height called

the significant wave height, II., assumed equivalent to the mean of the highest

25
one-third of the wave heights in the distribution. This value was selected

since it appeared to approximate the results of wave heights as reported by

experienced observers making visual estimates of wave heights in the ocean

(Sverdrup and Munk, 1947). In addition, since from equation (B-I), E is

proportional to 12, most of the energy is carried by the higher waves. So

[i is a better representative wave height than the mean wave height in most

coastal engineering problems.

However, the significant wave height, H, does not equal the rms wave

height, Hirms, so I-Is cannot be used to evaluate E or PP directly. Thus,

the relation between H. and arms must be determined. This can be done for

restrictive conditions following Longuet-Higgens' (1952) work (see also Kinsman,

1965 (Sec. 3.4) and Harris, 1973).

To develop a relation between Ii and hrms, it is assumed that the total

wave energy density, E, as given in equation (B-l), depends on the potential

and kinetic energy densities, Ep and E, respectively, both given by

E= K= -6 112 (B-4)

where 11 = H... For this to hold, the following conditions must occur:

(a) The waves are linear and have small amplitudes.

(b) All waves of a single frequency arrive from the same direction.

(Multidirectional waves will not seriously affect the result as long as waves

at a given frequency do not come from more than one direction.)

The wave heights will form a Rayleigh distribution if the following conditions are also assumed:

(a) The wave spectrum contains a single, narrow band of frequencies.

(b) The wave energy comes from a large number of different sources

of random phase.

Let n(t), the departure of the water surface from the mean sea level with

respect to time, t, be expressed as the Fourier integral

n(t) = f A(w) e t dw (!-5)

where A(w) is the spectrum function (possibly complex though only the real

part of the integral in eq. B-S is taken) of the amplitude of n(t) with

respect to the frequency w. At any time, twice the amplitude equals the wave

height, Hl-, measured from trough to crest (see Fig. B-i).

26
If the spectrum function, A(w), is appreciable only within a single,

narrow frequency band of wavelength 27/w. around frequency wo, then

(t) = e i'ot f A(w) i (

W-uO ) t dw

= eiwot B(t) (B-6)

Here, the factor eiw o t represents a carrier wave of wavelength 21T/w o and

the integral B(t) represents an envelope function around the carrier wave

(see Fig. B-2). The values of the amplitudes at the maximums and minimums of

n(t) are approximately equal to the values of the envelope function at those

points. Thus, the probability distribution of the amplitudes, and hence of

the wave heights, is the same as the Rayleigh probability distribution of the

values of B(t). Therefore, the probability, P(H), that the wave height II

is between II and Ii + dll is given by

P(ll) dtl = -d le-( fli~ ns)2

= 1 e- dlI (B-7)

Envelope Function, 8 (t

Water . Time, 

Water Surface Level Function 1()

Figure B-2. Definition of envelope wave function for n(t) with

single narrow frequency band.

For N waves the fraction, p, of the wave heights which are larger than

a given wave height, II, is equal to the probability that a wave height will

exceed Ii and is given by

p = fit '(Ii) dl

= , ( B -8)

2?
Thus, H H 8 [ (1)l112 (-9)

The mean value, H , of the wave heights larger than H is, using

equations (B-7) and (B-8),

HP = p 1 f fH HP(H) dH

= e (H/Hrms)2 f Hd[e -(HIHrms)2] (B-10)

Integrating by parts, substituting equation (B-9), and dividing through by

Hrms give

HP H +((/HrMS)2 (H/Hrs)2

-\H,-,',/ f e( dHl

[Y 1p)11/2 1- x2

= FL + - /a e dx

pp a

= ( / 2 + - f e dx] (B-11) p2p 0 eox

where a = [tn(1/p)] 1/ 2 and x = 1/Hrns. The ratio of the significant wave

height, Hs, to the rms height is found by letting p = 1/3 (from the definition of H.), which gives

H8

- = 1.416

r2--- (B-12)

since 2 = 1.414. Thus,

H 2 2 If2 (B-13)

From equation (B-2), E H2 and from equation (A-9) of Appendix A,

P, E. This means that if the significant wave height is used to compute

E or P., the result will be approximately twice what it should be using

the rms wave height.

Since coastal engineers are more familiar with 118 than HrmS , all longshore transport predictions in the SPM are designed so that H8 is used when

the equation requires a wave height. (To emphasize that the resulting PZ, is

approximately twice the theoretical longshore energy flux, PL, is called the

"longshore energy flux factor" in the SPM.) The design equations in the SPM

28
have been "calibrated" for use with H. by plotting the field data in Figure

4-37 of the SPM using H. to calculate P£." The resulting line in Figure

4-37 yields equation 4-40 of the SPM:

Q = (7.5 x 103) P 8 (B-14)

The 23 field data points used to determine equation (B-14) are shown in

Figure 4-37 of the SPM. Nine data points of Watts (1953) and Caldwell (1956)

(one of Caldwell's data points is missing from Fig. 4-37 since Q and Pt8

had opposite signs; see App. C) were reported in terms of significant wave

height. Thus, the values for Pt8 are taken directly from these reports.

Similarly, the one data point of Moore and Cole (1960) (not shown in Fig. 4-37

since it plots off the scale) is assumed to be in terms of significant wave

height, although this is not stated in their report. The 14 data points of

Komar (1969), however, are reported in terms of rms wave height and so his

values for Pk8 are multiplied by a factor of 2 before being used in Figure

4-37.

In Figure 4-36 of the SPM, 161 data points from S laboratory studies

(listed in App. C) are shown in addition to the 25 field data points. In the

laboratory studies a train of (relatively) uniform waves was used in each test,

so that the wave height measured is approximately equal by definition to the

rms wave height. Since the use of the laboratory wave heights gives the

theoretical longshore energy flux based on Hrs, they were multiplied by a

factor of 2 before being used in Figure 4-36 to make them consistent with the

energy flux factors plotted for the field data.

Because it is doubtful that the numerous assumptions relating Hs to Hrms

are all valid at the same time, the basic relation between Q and PZ, (eq.

B-14) might be considered an empirical relation, calibrated for field use with

significant wave height data. The engineering application depends on how well

the equations for Pk, predict Q when used in equation (B-14).

29
APPENDIX C

FIELD AND LABORATORY DATA IN THE SPM ENERGY FLUX DISCUSSION

Figure 4-36 in the SPM is a plot of longshore transport rate, Q, versus

longshore energy flux factor, P.., for both field and laboratory data. A

similar plot of Q versus P1. (Fig. 4-37 of the SPM) presents only the field

data. The empirical relation,

Q = (7.5 x l03 ) Pz8 (C-l)

where Q is measured in cubic yards per year and Pts is measured in footpounds per second per foot of beach front, is a visual fit of the field data

in Figure 4-37. It is given as equation (4-40) in the SPM.

The field data are taken from Watts (1953), Caldwell (1956), Moore and Cole

(1960), and Komar (1969). The laboratory data are taken from Krumbein (1944),

Saville (1950), Shay and Johnson (1951), Sauvage and Vincent (1954), and

Fairchild (1970). These data are described and listed in Das (1971) and the

derivation of empirical relations of the form of equation (C-i) is summarized

in Das (1972).

The field data include 25 data points, although only 23 of them are shown

on Figure 4-37 of the SPM. Four data points come from Watts (1953). The longshore transport rate, Q, was measured by surveying the amount of sand pumped

into a detention basin by a bypassing plant on the jetty at south Lake Worth

inlet in Florida. Surveys were made after every 6 hours of pumping or daily,

whichever was more often. Four monthly totals are used. The equivalent longshore energy flux factor, Pi. (denoted as ET in Watts' paper), was computed

from linear theory

) = 112 1 - M Hyl n t sin a cos a (C-2J

where

w = weight density of seawater

1, = wavelength

H = wave height

M = a function of d/L where d is the water depth

T = wave period

U = the angle between the wave crests and the shoreline

t = conversion of seconds to days

n = solution given by equation (2.68) of Wiegel (1964),

nil + - d\ (C-3) n = 1 sinh ((:-3)

30
Substituting the value for w, taking the shallow-water approximation

that the factor [I - M(H/L)2 ] i, and evaluating L from linear theory by

L 1T 2 tanh L

L 5.12 T2 tanh (2j;d) (C-4)

equation (C-2) becomes

P = 41 TH2 n tanh (2-d) t sin a cos a (foot-pounds per day per foot) (C-5) tsL

Note that Watts (1953) computes PZs in units of foot-pounds per day per foot

of beach front, which is converted in the SPM to units of foot-pounds per second per foot of beach front.

Significant wave height and period were taken from the analysis of the wave

records (a 12-minute record every 4 hours) of a pressure gage installed at the

seaward end of the Palm Beach pier, located 17.7 kilometers (11 miles) north of

south Lake Worth inlet, in approximately 5.2 meters (17 feet) of water. Wave

direction was obtained from twice daily observations using an engineer's transit

with sighting bar and auxiliary sights from an elevated point 5.6 kilometers

(3.5 miles) north of the inlet. No mention is made of how the quantities d

and L are evaluated.

There are six data points listed in Caldwell (1956). One of these six data

points is not plotted in Figure 4-37 of the SPM. That point represents a condition where the measured longshore transport and the computed longshore energy

flux are in opposite directions. The longshore transport rate, Q, was measured by comparing successive sets of surveys of the beach out to the 6.1-meter

(20 feet) depth contour at 152.4-meter (500 feet) intervals along the 3.4-

kilometer (11,000 feet) study area immediately south of the jetties at Anaheim

Bay, California. The longshore component of wave energy flux, Pts, was computed from equation (C-5) and, as before, converted to units of foot-pounds per

second per foot of beach front for use in the SPM. Significant wave height and

period were taken from analysis of the wave records of a step-resistance wave

gage, supplemented by hindcasting when necessary, and checked by a float-type

wave gage. The gages were installed on the seaward end of the Huntington Beach

pier, located about 10 kilometers (6 miles) south of Anaheim Bay. Wave direction was obtained from wave hindcasting and wave refraction analysis using

synoptic weather charts. Again, no mention is made of how d and L are

evaluated.

One data point is taken from an observation in Moore and Cole (1960). The

longshore transport rate, Q, was measured by comparing two surveys marking

the growth of a spit across the outlet to Tasaychek Lagoon, Alaska. The longshore energy flux, Pt, was computed by Saville (1962) using the following

equation (T. Saville, CERC, personal communication, 1974)

wL1 2 t

P 8 n T sin a cos a (C-6)

31
No mention is made in Moore and Cole (1960) of how or where the wave height,

period, and angle between wave crest and beach are measured. Saville assumes

that they are deepwater values and the SPM assumes that the wave height is

significant wave height. The computed value for PZS is larger by nearly an

order of mrgnitude than in the other field data and hence does not plot on the

scale of Figure 4-37 of the SPM. It would plot below the line in Figure 4-37

given by equation (C-I).

Fourteen data points are taken from Komar (1969), 10 from work at El Moreno

Beach, Mexico, and 4 from Silver Strand Beach, California. The longshore transport rate, Q, -was calculated from the movement of fluorescent tracer sand,

determined from the amount and location of the tracer sand in regularly spaced

core samples taken on the beach about 3 to 4 hours after injection of the tracer.

Wave energy flux and wave direction were measured by an array of digital wave

sensors (wave staffs and pressure transducers) in the nearshore region. Integrating the energy densities under the spectra peak of the records of these

wave sensors gives the mean square elevation of the water surface, <n2 >. The

energy of the wave train is then

E= pg <n2> (C-7)

The wave period of the wave train is at the point of maximum energy density

for the particular spectra peak being analyzed. By knowing the water depth,

group velocity can be found. The wave energy flux per unit crest length, EC

may now be calculated. Comparing wave records of various gages of the array

produces the wave angle, a, which gives

wave energy flux Cos a

unit beach length g

Assuming no energy dissipation until breaking, the energy flux per unit beach

length at the breaker zone is

(ECq cos a 'ECg cosa (C-8)

Equation (C-8) is then multiplied by the sin ab, where ab is measured in

the surf zone or calculated from a using Snell's law

C

sin ab sin a (C-9)

to give the energy flux factor

P = (Eg cos sin ab (C-10)

In (C-9), Cb is evaluated from phase speed near breaking assumed to be (Komar,

1969)

Cb = (2.28 gi(C-1l)

Note that Komar uses rms wave height to compute values for Pt8 . To make his

values consistent with the other field data points in Figure 4-37 of the SPM,

32
which are computed from significant wave height, Komai's values for Pk.

are multiplied by a factor of 2 before being used in the SPM (see App. B).

Two additional sources of field data are included in Figure 6 of Das (1972),

a plot of Q versus Pts similar to Figure 4-36 of the SPM. These sources

are 5 data points from Johnson (1952) and 14 data points from Thornton (1969).

Neither source is included in Figures 4-36 and 4-37 of the SPM nor are they

used in the derivation of the empirical relation between Q and Ps., equation

(C-1). Johnson's data do not include wave direction, but the corresponding

points must plot above the line given by equation (C-1) in Figure 4-37 for

reasons given in Galvin and Vitale (1977). Thornton's data are from bedload

traps between the inner and outer bar, and represent minimum values of Q.

The design prediction (eq. C-l) is based solely on field data, but Figure

4-36 does show laboratory data for comparison. Das (1971) lists 177 laboratory

data points, and of these, 161 are shown in Figure 4-36 of the SPM. In the two

data points from Price and Tomlinson (1968), crushed coal with a specific gravity of 1.35 (as opposed to 2.65 for the quartz that makes up most beach sand)

was used. In 14 of the 17 data points from Sauvage and Vincent (1954), sediments with specific gravities of 1.1 and 1.4 were used. Thus, these 16 lightweight data points were not included in Figure 4-36.

The- longshore energy flux, Pps, for the laboratory data was computed,

using equation (32) of Das (1972)

pt -P9H 2 L Ksn±cs g2 8 o o R ab COS ab (C-12)

Ho is the deepwater wave height, Lo is the deepwater wavelength, and N is

the number of waves per day, given by

N = (86,400 seconds per day) T (C-13)

where the period, T, is measured in seconds, and KR is the refraction

coefficient. The leading factor of 1/2 in equation (C-12) is the deepwater

approximation for n as given in equation (C-3).

The value for HO comes from

H'

o

R

H

where H' is the deepwater wave height unaffected by refraction, the ratio

.0 H/H' is obtained from Wiegel's tables (SPM, App. C, Table C-1), and the

0

33

L ......... ..
refraction coefficient, KR, is given by equation (7-12) of Wiegel (1964)

K (Cos cao1/2 (-5 ICO(C-is)

R \Cos a

The depth, d, the period, T, and the angle a are measured in the

laboratory studies. The deepwater approximation for the wavelength, LO =

5.12 T2 , obtains d/L, and hence L, from d/L0 and Wiegel's tables. Then,

ao can be calculated from

sin a Lo O = F-sin (C-16)

and ab can be calculated from

Lb

sin a = - sin a0 (C-17)

so that equation (C-15) can be evaluated to give the refraction coefficient,

KR, at the breaker position.

To summarize, 23 field data points were used to derive equation (C-i), the

relation between Q and PkS. The values for Pt8 for the nine data points

taken from Watts (1953) and Caldwell (1956) were computed from equation (C-2),

using values of significant wave height, period, and direction, as estimated

by Watts (1953) and Caldwell (1956). The values of PgB for the 14 data

points taken from Komar (1969) were computed from equation (C-10) whose input

comes from the mean square water elevation, <n2>, obtained from the wave

sensor records, the use of linear theory, and equations (C-l1) and (C-9) to

get the ab. It is again emphasized that only field data were used to compute

equation (C-i), and the equations for Pt. require the use of the observed

significant height.

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