The SPM Energy Flux Method
for Predicting Longshore Transport Rate

by
Cyril Galvin and Charles R. Schweppe

TECHNICAL PAPER NO. 80-4
JUNE 1980

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THE SPM ENERGY FLUX METHOD FOR PREDICTING
LONGSHORE TRANSPORT RATE

8.

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Cyril Galvin
Charles R. Schweppe

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Department of the Army
Coastal Engineering Research Center (CERRE-CP)
Kingman Building, Fort Belvoir, Virginia 22060
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Department of the Army
Coastal Engineering Research Center
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Coastal engineering Longshore transport rate

Energy flux Wave height

2a.

ABSTRACT (Continue an reverse sides if neceasary and identify by block number) i z
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.

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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 Branch, under the general 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.

ZAG,
TEDPES
Colonel, Corps of Engineers
Commander and Director

IV

APPENDIX

@)

CONTENTS

CONVERSION FACTORS, U.S. CUSTOMARY TO METRIC (SI).
SYMBOLS AND DEFINITIONS.
INTRODUCTION .
DISCUSSION OF THE THREE APPENDIXES
1. The Equations (App. A).
2. Wave Height (App. B).
3. The Data (App. C)
WAVE SPEED AND BREAKER ANGLE .
SUMMARY .

LITERATURE CITED .

DERIVATION OF LONGSHORE ENERGY FLUX FACTOR .

DISTINCTION BETWEEN SIGNIFICANT AND ROOT-MEAN-SQUARE WAVE
HEIGHTS IN PREDICTING LONGSHORE TRANSPORT RATES. B/G

FIELD AND’ LABORATORY DATA IN THE SPM ENERGY FLUX DISCUSSION.

Page

25

30

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 Zo9n0 hectares
knots 1.852 kilometers per hour
acres 0.4047 hectares
foot-pounds 125558 newton meters
millibars 1019725 ORS kilograms per square centimeter
ounces Zo 555 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 Kelvins!

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

LOrmulass |G o—

@/9) P(E 32)"

To obtain Kelvin (K) readings, use formula:

Ka= (G/9)GE =32) 827505.

P(H)

SYMBOLS AND DEFINITIONS
spectrum function of the amplitude of n (see eq. B-5)
vertical distance from breaker wave trough to MWL (see Fig. A-3)
envelope function (see eq. B-6 and Fig. B-2)
crest length between orthogonals (see Fig. A-2)
subscript for condition at the breaker
wave velocity
water depth
wave energy density (see eq. B-1) with subscripts
acceleration of gravity
wave height
deepwater height assuming no refraction

mean value of the highest fraction of wave heights above a height
H which has a probability, p, of occurrence (see eqs. B-8 and B-10)

root-mean-square height (eq. B-3)

Significant wave height (eq. B-13)

water depth measured from wave crest to bottom (Fig. A-3)
generalized water depth under a breaking wave (eq. 2)
subscript for conditions at point i (Fig. A-2)

refraction coefficient (eq. A-11)

shoaling coefficient (eq. A-12)

wavelength

natural log symbol

number of waves

ratio of group velocity to individual wave velocity (eq. A-17)
subscript for conditions in deep water (except eq. B-6)

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

probability function of wave height

SYMBOLS AND DEFINITIONS--Continued
energy flux per unit length of wave crest (eq. A-1)
part of the energy flux in waves defined by equation (A-7)
approximate evaluation of Pp for conditions in the surf zone

fraction of wave heights, in a train of N waves, whose height exceeds
H (eq. B-8)

volume rate of longshore transport

distance in the longshore direction between adjacent orthogonals (Fig.
A-2)

wave period

time, or conversion factor, seconds per year
weight density of water

angle between wave crest and shoreline (Fig. A-2)
ratio of breaker depth to breaker height (Fig. A-3)
departure of water surface from MWL (Fig. B-2)

mass density of water

ratio a+/Hp (Fig. A-3)

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

fis

rol)

mi

aie

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; LBSigey

Tha
ys

THE SPM ENERGY FLUX METHOD FOR PREDICTING LONGSHORE TRANSPORT RATE

by
Cyrtl Galvin 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 quanti-
ties 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 documen-
tation 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 assumptions necessary to arrive at
the 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, Py,. Two equations
are needed: an equation that converts wave conditions into Pg,, and an
equation that predicts Q from P,y,. In the SPM, four theoretically equiva-
lent equations for longshore energy flux, P,, are developed from small-
amplitude, 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 Pp, is equivalent to any of the three other
equations for P,, no two of the four Pp, equations are exactly equivalent
because each Py, equation was derived from a different Py) equation using
a different set of approximations.

The four equations for Pp», equations 4-31 to 4-34, and the four equations
for Pp,, 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, Ppo., depends sensitively on wave height. In one
equation, it is an H2 dependence; in another one, H?; and in two equations,
Ho/ 2 ) flint ali’) the height enters Pp»), primarily as the energy density, E,
which depends on H*. 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 H*. This is the root-mean-square (rms) height, Hyg.

The height commonly used in coastal engineering work is neither the average
height nor the Hymg, but the significant height, H,. Appendix B explains
the relations between Hymg and H,, and how the energy flux factor is cali-
brated for use with H,.

3.) the Data (App. iC).

To obtain an equation to predict Q from Ppy,, it is necessary to have
sufficient data for wave conditions to compute Py, 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 Pgs _ 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 sectior 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

d, = 1.28 Hp (1)

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

C= Vgh* (2)

where h* is a depth. If h* is set equal to the mean depth, then the breaker
speed is that of linear theory

Ch = Yad,

or using equation (1)

6.42 Vi, (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

Cy = 8.02 Vip (feet per second) (4)

If h* is set equal to Hp + dp without correcting for depression of trough
below mean water level, the speed will be

Cie S657 Vp, (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
oe is obtained, include estimates of Py, 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 Pos equations, as shown in Appendix A.

Since the Pgs equations used in the SPM depend on equation (4) for wave
speed, it is evident that wave speed has not been treated consistently through-
out the analysis. The Pp, 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 Pog 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 Pg, and Q is heavily dependent on the 14 plotted points
of Komar, the error due to using equation (4) for breaker speed in calculating
Pog is concluded to be less than 25 percent and may be negligible.

Moreover, the Komar (1969) data for energy flux probably include an over-
estimate of the breaker angle. This is because breaker angle, Gp, was
computed from the angle at the sensor, a, by

G

: Dagens
Sn Coy ere

where Ch 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 gen-
eral 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, Pg.,
is proportional to any one of the following groups of wave variables (App. A):

(@) B27" sin Aor
(b) Eve (cos opi) ca sin 209

(ec) H2 T sin Op COS A,

H3
(d) (2| sin a,

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, Ppg,. The indicated proportionality constant is 7,500 for tradi-
tional units (Q in cubic yards per year and Pg, 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 varia-
tions 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.

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

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. II, 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.

KOMAR, P.D., ''The Longshore Transport of Sand on Beaches,'' unpublished Ph.D.
Thesis, University of California, San Diego, Galhtea lO 69F

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.

i)

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 Geophystcal 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
o£ (Cavady Engineers), Vols 1 1969 pp mols 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 Unton, Vol. 31,
1950, pp. 555-565.

SAVILLE, T., Jr., "Discussion of Laboratory Determination of Littoral Trans-
port 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 Trans-
ported 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
Protectton Manual, 3d ed., Vols. I, II, and III, Stock No. 008-022-00113-1,
U.S. Government Printing Office, Washington, D.C., 1977, 1,262 pp.

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 eMIgS3%

WIEGEL, R.L., Oceanographic Engineering, 1st ed., Prentice-Hall, Englewood
Gila INedloi |S Koule

14

APPENDIX A
DERIVATION OF LONGSHORE ENERGY FLUX FACTOR

This appendix derives four formulas for longshore energy flux, Peg, 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 devel-
opment in Wiegel (1964) where more detailed derivation is provided.

leEquations for (Pio.

The derivation for Pp 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 a, with the shoreline when in deep water
will refract to make an angle a; at some shallower depth (Fig. A-2), where a7
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 untt length of wave crest at point i is given by

Ee Cafe
= eT GLE (A-1)

where C is the wave group velocity, C is the wave phase velocity,
n=C,/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)

22
Ee a (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-1), the energy flux in the direction of wave travel, at

POIs, fom crest lengthy, by, is (see eq. 7 in Galvin and Vitale, 1976)

pt? b; = E} Cg bz (A-3)

15

Table 4-7. Longshore Energy Flux,Pg, 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, (A E sin 2a)
C (% EB sin 2a,)
KRC, (Ys iE sin 2a)
(2C) (KR C.J" C, (AE sin 20,)

no subscript indicates a variable at the specified depth
where simall-amplitude theory is valid
Cy = group velocity (sce assumption 1b, Table 4-9)
= decpwater

water depth

= wave period

C,

d

H = significant wave height
T

a

= angle between wave crest and shoreline

’ : COS Qo
Kp = refraction coefficient , | eS
cos @

Table 4-8. Approximate Formulas for Computing Longshore

Energy Flux Factor, Pp,, Entering the Surf Zone

P
(ft.-Ibs./sec./ft.

Data Required

g
Stibeach front) | (ft.-sec. units)

4-35 32.1 H, 5? sin 2a, H, a,
18.3 H,* (cos a,)'!* sin 2a,| H,,a,
20:5) Mukie sin Gp COS &, Te Ho» G+ &p,

100.6 (Hj /T) sin a, T, Hy. 4,

H, = deepwater
= breaker position

Hy,
H = siynificant wave height
T = wave period

a

= angle between wave crest and shoreline

Sce Table 4-7 for equivalent small amplitude equations and
Table 4-9 for assumptions uscd in deriving Pe, from Po.

4-97

Figure A-l1. Page 4-97 of the SPM, showing four
formulas for longshore energy flux,
Po, and four formulas for longshore
energy flux factors, P»..

Orthogonal 1
: Orthogonal 2

Wave Crest

-”
a
Pa
s
=a

Tangent to Crest
at ij

Bottom
Contours

Breaker
Line

Surf
Zone

Figure A-2. Definitions for conservation of energy flux for
shoaling wave (after Walton, 1972).

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.

Pao Ped, = total wave power between orthogonals (A-4)

From the geometry of Figure A-2, it is obvious that b; 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 1 in all respects except being displaced along
the coast by a distance s. Therefore, at any point, i, in the wave path

b; = Ss cos af (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

PIS 1 (COS Cy (A-6)

In the Appendix, i indicates any point on the path where equations (A-1)
to (A-5) are valid. With this understanding, the subscript 7% is dropped in
the derivation below. Since P has both a magnitude and a direction at any
point, a longshore component, Pp, can be defined. Using equations (A-1) and
(A-6), this longshore component of energy flux can be written

Pp = (ECg cos a) sina (A-7)
By use of the identity

Sin’2)va)-=) 24cos a) Sina (A-8)

equation (A-7) can be written as the first of the four alternate forms for
Py given in Table 4-7 of the SPM, i.e., equation (4-31) in the SPM:

1 aa
ey Cg (45 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,
E, is proportional to the local height squared, H7?, (eq: “A-2).° “The local
height can be related to the deepwater height, Ho» by equations (7.8) and
(7.9) in Wiegel (1964)

RKsH, (A-10)

where) Kp Yas) the refraction coefficient

cos a, 1/2
Cael (eee (Ney)
cos a
and K, is the shoaling coefficient
(e 1/2
Ka | (A-12)
&g

K, for later use may also be approximated by the breaker height index
Ko ae (A-13)

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

jen Ss 2 2a
E Ke Ke EO
cos a (¢ =
mf, (EO (A-14)
cos a C O
g
so Po becomes
Pp, = K2 k2 C_E_ sin a cos a (A-15)
& 1 SS ep GO

By substituting equations (A-11) and (A-12) into equation (A-15) and canceling
nkemee rms:

Po = Coo iB, COS a, sina (A-16)

At €

€
Sch PN | ap ee (A-17)
C 2 . d
sinh (4t ia

where d is the water depth. To a good approximation, n is equal to 0.5 in
deepwater, ) ie. , d/l, > 0.5), and” ny ais equally to (mOhintshallowswatemrcdenccn,
d/L < 0.04. (Exact values of n at these limits are 0.5117 at d/L equal to
0.5, and 0.9795 at d/L equal to 0.04.) Equation (A-16) can be further modi-
fied using Snell's law (where C is the local wave speed given by equation
(2-3) in the SPM),

C sin %) (A-18)

and equations (A-8) and (A-17) to get the second of the equivalent forms for
Po given in Table 4-7 of the SPM, i.e., equation (4-32) in the SPM:

eae (¢ ER sin 2) (A-19)

The third equivalent form for Pg is equation (4-33) in Table 4-7 of the SPM.
It is obtained from equation (A-15) by using equations (A-12) and (A-8),

= (2 Le | gs A-20
Pare Ke (2 E sin 2a) (A-20)
The fourth and last of the equivalent forms of Py is equation (4-34) in Table
4-7 of the SPM. It is obtained by substituting equations (A-8), (A-11), and
(A-18) into equation (A-9),

2 G a
Pe = (2) ‘) Cg (G E sin 2a) (Aq21))
R

2) sEquations for. (Poin.

Up to this point, all results are for small-amplitude linear theory. How-
ever, the assumed relation between longshore transport and energy flux in the
surf zone requires that Py be evaluated at the breaker line where small-
amplitude theory is less valid. To indicate approximations for waves entering
the surf zone, the symbol Pp, will be used in place of Py. This approxima-
tion is called the energy flux factor, Pp,, in the SPM, and like Py alte vals
measured in units of energy per second per unit length of shoreline; e.g., foot-
pounds per second per foot. One expression for Pp, will be derived from each
of the four equivalent expressions for Pg (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 Pp. In foot-pound-
second units, and for saltwater (w = 64 pounds per cubic foot), the energy
density is, from equation (A-2),

wH2

ie ee

8

R

8 H2 (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) 2.2/— 1m thesSPM));.

The equation for wave speed near breaking, (C,, is an approximation.
Several approximations are possible, but the solitary wave approximation
used in obtaining the SPM equations for Pgs is as follows (symbols defined
in Fig. A-3). The first approximation for the speed of the solitary wave is

20

Figure A-3. Definition of terms for breaker
speed equation.

Co even (A-23)
where
h = Hy, + d,, - a, (A-24)
By dividing with Hh the wave speed becomes
C, = KV, (A-25)

where K is a dimensional term depending on two ratios and g (see eq. 11 in
Galvin, 1969)

Ki = ve Be ap) (A-26)

The depth-to-height ratio, 8, was assumed equivalent to the usual coastal
engineering rule-of-thumb (f = 1.28). The relative depression of the trough,
o, was known to vary from 0.15 to 0.40, and a standard value of 0.28 was used
for o giving a wave speed of (eq. 4)

C. = 8.02VH, (feet per second)

It should be noted that there is recent evidence that both 8 and o
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

= Op (A-27)

where the subscript b refers to breaker zone conditions.

Z|

Substituting the approximations represented by equations (4), (A-22), and
(A-27) into equation (A-9) yields Py, approximations for each of the four
equations for P, in Table 4-7.

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

Me 1/2 ele Din (a
P 2(8.02 He ) mn (8 He) sin 2

Qs °

SY oll Hp’? sin 2a, (foot-pounds per second per foot) (A-28)

b

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 Po (eq. A-19) can be reduced to

Pave 16.0 ae He sin 20, (foot-pounds per second per foot) (A-29)

Hp is put in terms of H, using equations (A-10) and (A-11) to get
We COS a. 1/4
| = ——— WD =
He (KH) (A-30)

In this equation, cos a, equals 1.0 to a good approximation. For example,
even if oa, has a high value of 20° (exceeded less than 5 percent of the time
on straight beaches), (cos 20°) 1/2 =. 0n08).

So, using this approximation and substituting equation (A-30) into equation
(A-29),

Pos = 16.0 H°/2 (cos a jivh Ki/2 sin 20 (foot-pounds per second
O O s O
per foot) (A-31)

The shoaling coefficient, Kgs, is approximated by the breaker height index.
Hp/HZ » obtained from the experimental work of Iversen (1952). Related data are
in Figure 2-65 of the SPM. H} is the unrefracted deepwater water height. If
the slope and period are known, a steepness can be computed, and H,/H! 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 H,/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 Iversen'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:

Boe = Ore Bie (cos on) sin 20. (foot-pounds per second
per foot) (A-32)

22

The deepwater wave velocity, (C,, can be approximated by (see eq. 2.37
in Wiegel, 1964)

at

Cy Sor 5.1/2) 1 (feet per second) (A-33)

Using this along with equations (A-8), (A-11), and (A-22), the third expression
for Py, can be obtained from equation (A-20):

cos oF l 2
Die ~ | Ga2 aI) q (8 Hd (2 COS oa sin a)

= LOLS 1 Hé cos a, sin a, (foot-pounds per second per foot) (A-34)

This is equation (4-37) in the SPM.
The final equation for Po, equation (A-21), meGdueesm to

coS a

=
2 92 Ms (( te
ae Syl r) (8.02 He ) (8 He) (3 cos a Sin °0)

eo 1/2
P 2(8.02 H}/?) ee

H3
= 10055 (=) COS Oo, sin O (foot-pounds per second per foot) (A-35)

by use of equations (4), (A-8), (A-11), (A-22), and (A-33). The cos Oo, term
in many cases, is close enough to 1.0 to be ignored. Therefore, equation (A-35)
can be further reduced to:

H3
poe = 100.5 | sin a (foot-pounds per second per foot) (A-36)

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

The four approximate equations for Py, (eqs. A-28, A-32, A-34, and A-36)
are given in Table 4-8 of the SPM. Unlike the four exact equations (Pp, 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 Py, that agree within a factor of 2. Figures
4-36 and 4-37 in the SPM show that variation of Pp, by a factor of 2 is within
the scatter of the data defining the dependence of Q on Pg,.

All results in this appendix, including the Py, 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.

So Summary .

This appendix provides the derivation for eight equations given in the
SPM. These eight equations are the four equivalent equations for the long-
shore energy flux, Pp (Table 4-7), and the four nonequivalent equations for
the longshore energy flux factor, Pog (Table 4-8) given in the SPM. The
correspondence between equation numbers in this appendix and the SPM is as
follows:

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

L s
SPM This Appendix nt 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, Py, given
in Table 4-7 of the SPM are developed from small-amplitude linear theory (see
App. A), and each expression for P, contains the wave energy density, E,
as a linear factor. The energy density per wave is

AW,
E = aoe (B-1)

where w is the weight density of seawater, and H is the wave height measured
from crest to trough (see Fig. B-1). For a group of waves, N in number, each
with wave height H;, for i= 1 toN, the average energy density is given by

(E) = Sa
v ; ;
GA Ne eth et
ep a 2 :
= 3 Crns) (B-2)
where ee is the root-mean-square (rms) wave height given by
2 1 S 2
(Hoe ag = als (B-3)
t=1
Stil l- n(t)7
Water Time, t
Level

Figure B-l. 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
—E and hence P,. In conditions where a uniform train of periodic waves exists,
approximated in some laboratory experiments, all of the H; are equal. Thus,
by equation (B-3), H- is identical to Hyg. In such cases H; is inter-
changeable with Hyg 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 Hy »g 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, H,, assumed equivalent to the mean of the highest

(55)

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-1), E is
proportional to H2, most of the energy is carried by the higher waves. So
H, is a better representative wave height than the mean wave height in most

s
coastal engineering problems.

However, the significant wave height, H,, does not equal the rms wave
heighitoy Hy as7 SOU Hoy) Cannot be used stomevaluates vr) (or, | Bo directly. s™uhuse
the relation between H, and Hymg must be determined. This can be done for
restrictive conditions following Longuet-Higgens!' (1952) work (see also Kinsman,
Ios (SACs SoA!) ehovel eases.) WO\Y/S)) ¢

To develop a relation between H, and Hyg, it is assumed that the total
wave energy density, E, as given in equation (B-1), depends on the potential
and kinetic energy densities, Ep and Ey, respectively, both given by

Ss jae LY 2 =

where H = Hyg. 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 condi-
tions 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

Ce) i ING) ene nas (B-5)

where A(w) is the spectrum function (possibly complex though only the real
part of the integral in eq. B-5 is taken) of the amplitude of n(t) with
respect to the frequency w. At any time, twice the amplitude equals the wave
height, H;, measured from trough to crest (see Fare: Bal) he

26

If the spectrum function, A(w), is appreciable only within a single,
narrow frequency band of wavelength 2n/w, around frequency Wo, then

MICE), = ein? wie uA) oOo) Me ia

elot Bt) (B-6)

Here, the factor aov represents a carrier wave of wavelength 21/w, 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 H
is between H and H + dH is given by

P(H) dH = -d [o> H/ Meme )* |

(2) e7 H/Hyme)* ay (B-7)

Hie

Envelope Function, B(t)

Stil I-
Water

Time,t
Level

Water Surface Level Function »/(t)

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, H, is equal to the probability that a wave height will
exceed H and is given by

Sy P(t) dH

AS)
I

= 97 Cl/Hyms )* (B-8)

Cal

Thus , “fe
oc) “

The mean value, Hp», of the wave heights larger than H is, using
equations (B-7) and (B-8),

ey
Be eaip

=e Mms)? pale Hams) 7] (B-10)

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

Hymg give

Hp H 1 (H/Hye) 2 2
= ao ome rms 0 - (H/Hyms)
Hams Hyms “(eek Tiss © dH
Il VWire2 il foe) yf 7
S yee & dh x
n (3) oes fie dx
2.
= Ee) a vn 1- —~ s2 Loe don (B-11)
p 2p He ©

where a = [2n(1/p)]!/2 and x = H/Hymg+ The ratio of the significant wave
height, H,, to the rms height is found by letting p = 1/3 (from the defini-
tion of H,), which gives

H
SE ANG
Hyams
~ /2 (B-12)
since /2 = 1.414. Thus,
2 D
er 2 ar (B-13)

From equation (B-2), E « H* and from equation (A-9) of Appendix A,
Po « E. This means that if the significant wave height is used to compute
or P,, the result will be approximately twice what it should be using
the rms wave height.

Since coastal engineers are more familiar with H, than Hyg, all long-
shore transport predictions in the SPM are designed so that H, is used when
the equation requires a wave height. (To emphasize that the resulting Pp, is
approximately twice the theoretical longshore energy flux, Py, 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 Py,. The resulting line in Figure
4-37 yields equation 4-40 of the SPM:

QS (eS 02) Pry. (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 Pyg
had opposite signs; see App. C) were reported in terms of significant wave
height. Thus, the values for Pgg 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 Py, 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 5 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 Hymns, 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 Hg to Hye
are all valid at the same time, the basic relation between Q and Poy, (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 P,, predict Q when used in equation (B-14).

(os)

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 Py), (Fig. 4-37 of the SPM) presents only the field
data. The empirical relation,

QS (5 10?) Pa. (C-1)

where Q is measured in cubic yards per year and Py, is measured in foot-
pounds 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-1) is summarized
Tn eS (UDA).

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 long-
shore 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 long-
shore energy flux factor, Pp, (denoted as Ep in Watts' paper), was computed
from linear theory

H\* tou
Pay 2 SU Ie Me = =
Qs 8 LH E M @ | n 7 Sina cos a (C-2)
where
w = weight density of seawater
L = wavelength
H = wave height
M = a function of d/L where d is the water depth
= wave period
a = 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),
fgg &
1 L
iM sey 1 + ————_—_ (C-3)

Substituting the value for w, taking the shallow-water approximation
that the factor [1 - M(H/L)*] ~ 1, and evaluating L from linear theory by

We 3 2 (4)
L = ae T+ tanh Tia

Ip =eoelelogtanh (225) (C-4)

equation (C-2) becomes

21d

Il

P= 41 TH? n tanh (
Rs

) t sin a cos a (foot-pounds per day per foot) (C-5)
Note that Watts (1953) computes Pp, in units of foot-pounds per day per foot
of beach front, which is converted in the SPM to units of foot-pounds per sec-
ond 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_ con-
dition where the measured longshore transport and the computed longshore energy
flux are in opposite directions. The longshore transport rate, Q, was meas-
ured 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, Pp,, was com-
puted 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 direc-
tion 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 long-
shore energy flux, Pps, was computed by Saville (1962) using the following
equation (T. Saville, CERC, personal communication, 1974)

wLH2
Po =

1e Z
a 3 n T sin a cos a (C-6)

3 |

No mention is made in Moore and Cole (196C) 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 Pgs is larger by nearly an
order of mggnitude 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-1).

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 trans-
port 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. Inte-
grating 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

B= om See (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 _ Ec
unit beach length “vg °°S °

Assuming no energy dissipation until breaking, the energy flux per unit beach
length at the breaker zone is

(EC, cos a) 4, = ECy cos a (C-8)

Equation (C-8) is then multiplied by the sin ap, where op, is measured in
the surf zone or calculated from a using Snell's law

sin a (C-9)

to give the energy flux factor
Pog = Be, cos ot) 5 sin o, (C-10)

In (C-9), Ch is evaluated from phase speed near breaking assumed to be (Komar,
1969)

Cre (2e28 yet) ae (C=)

Note that Komar uses rms wave height to compute values for Pp,. 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, Komar's values for Py,
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 Pes 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 Pgs, 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-1) 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 grav-
ity 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), sedi-
ments with specific gravities of 1.1 and 1.4 were used. Thus, these 16 light-
weight data points were not included in Figure 4-36.

The. longshore energy flux, Py,, for the laboratory data was computed,
using equation (32) of Das (1972)

ably? buoy NK2 si C212
Von” Be Su Oui) SPMMEON eeenD (con)

Hp is the deepwater wave height, Lo is the deepwater wavelength, and N is
the number of waves per day, given by

ae (86,400 seconds per day) (C-13)

where the period, T, is measured in seconds, and Kp 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 H, comes from

H (C-14)
(a) *
oO

where H! is the deepwater wave height unaffected by refraction, the ratio
H/H; is obtained from Wiegel's tables (SPM, App. C, Table C-1), and the

33

refraction coefficient, Kp, is given by equation (7-12) of Wiegel (1964)

COSMO WI A2:
PegeS (fee (C-15)

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

J can be calculated from
Lo,
sin 9 = T sin (C-16)
and ap can be calculated from
Lp
sin op = ino Oto (Gai)
O

so that equation (C-15) can be evaluated to give the refraction coefficient,
Kp, at the breaker position.

To summarize, 23 field data points were used to derive equation (C-1), the
relation between Q and Pgs. The values for Pp, 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 Pgg 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-11) and (C-9) to
get the ap. It is again emphasized that only field data were used to compute
equation (C-1), and the equations for P,, require the use of the observed
Significant height.

34

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