BED SHEAR STRESS COEFFICIENT
WITHIN THE SURF ZONE

Carlos Severino Veitia Garcia

NAVAL POSTGRADUATE SCHOOL

Monterey, California

THESIS

BED SHEAR STRESS COEFFICIENT
WITHIN THE SURF ZONE

by

Carlos Sever! no Veitia Garcia

Sep tembe r 1977

Thesis Advisor

E . B . Thornton

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Bed Shear Stress Coefficient
within the Surf Zone

5. TYPE OF REPORT & PERIOO COVERED

Master's Thesis;
Septembe r I 977

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7. AUTHORfa;

Carlos Severino Ve i t i a Garcia

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Naval Postgraduate School
Monterey, California 93940

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20. ABSTRACT (Contlnua on ravaraa alda II naeaaaary and Idantlty by block numbar)

An analytical formulation of the bed shear stress coef-
ficient inside the surf zone is derived using the concept
of radiation stress. A truncated Rayleigh p.d.f. is used
to describe the wave field inside the surf zone and provides
the input to calculate the variation of wave energy and long-
shore current as a function of wave height, water depth and
distance to shore. The wave set-up is approximated using a

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sinusoidal wave solution. Field measurements of longshore
current and waves within the surf zone are used to calculate
the bed shear stress coefficient. The data consist of 647
data points selected from LEO program and 62 data points
from Ingle (1966) observations, all taken along the Southern
California coast. Frequency distributions and statistics
are calculated for the bed shear stress coefficient. A mean
bed shear stress coefficient to two significant decimal
places is found to be 0.01.

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Bed Shear Stress Coefficient
within the Surf Zone

by

Carlos Severino Veitia Garcia
Commander, Venezuelan Navy
B.S., United States Naval Postgraduate School, 1976

Submitted in partial fulfillment of the
requirements for the degree of

MASTER OF SCIENCE IN OCEANOGRAPHY

f rom the

NAVAL POSTGRADUATE SCHOOL
September I 977

o-/

ABSTRACT

An analytical formulation of the bed shear stress coef-
ficient inside the surf zone is derived using the concept
of radiation stress. A truncated Rayleigh p.d.f. is used
to describe the wave field inside the surf zone and provides
the input to calculate the variation of wave energy and long-
shore current as a function of wave height, water depth and
distance to shore. The wave set-up is approximated using a
sinusoidal wave solution. Field measurements of longshore
current and waves within the surf zone are used to calculate
the bed shear stress coefficient. The data consist of 647
data points selected from LEO program and 62 data points
from ingle (1966) observations, all taken along the Southern
California coast. Frequency distributions and statistics
are calculated for the bed shear stress coefficient. A mean
bed shear stress coefficient to two significant decimal
places is found to be 0.01.

TABLE OF CONTENTS

INTRODUCTION ------------------ 12

A. STATEMENT OF THE PROBLEM ---------- 12

B. HISTORICAL REVIEW ------------- 12

C. OBJECTIVES OF THE STUDY ---------- 16

THEORY --------------------- 17

A. INTRODUCTION ---------------- 17

B. WAVE SET-UP INSIDE THE SURF ZONE ------ 19

C. WAVE FIELD INSIDE THE SURF ZONE ------ 23

D. LONGSHORE CURRENT VELOCITY ---------29

DATA ----------------------33

A. LEO DATA ------------------33

1. Rip Currents --------------35

2. Angle of Wave Approach - - - - - - - - - 35

3. wind------------------35

4. Foreshore Slope ------------35

5. Wave Period --------------35

6. Doubtful Data -------------36

B. SOURCES OF ERROR --------------36

1. Breaker Angle -------------36

2. Beach Slope and Surf Zone Width - - - - 37

3. Wave Period --------------38

4. Breaker Height -------------38

5. Longshore Current ___________ 39

C. INGLE DATA -----------------39

IV. RESULTS --------------------41

A. LEO DATA ------------------ 42

B. INGLE DATA ----------------- 42

C. COMPARISON OF RESULTS ----------- 45

D. CORRELATION WITH INDEPENDENT VARIABLES - - - 47

V. CONCLUSIONS ------------------49

APPENDIX A: Littoral Environment Observations - - - - 51

BIBLIOGRAPHY ---------------------53

INITIAL DISTRIBUTION LIST --------------55

LI ST OF TABLES

Bottom Friction Coefficients Proposed by

Various Investigators ------------- |5

Selected Statistics for Distribution of

Coefficients According to Breaker Type

(LEO Data) -------------'------ 47

Correlation of Coefficient with Wave

Period Statistics ---------------48

LI ST OF FIGURES

1. Definition of Longshore Current Variables - - - - 13

2. Comparison of Longshore Current Models ------ |8

3 . Comparison of Wave Set-up Solutions - - - - - - - 2 2

4. Truncated Rayleigh p.d.f. - - - - - - - - - - - - 24

5. Location Maps ------------------34

6. Frequency Distribution of Coefficient Values

(LEO Data) --------------------43

7. Frequency Distribution of Coefficient Values

(Ingle Data) -------------------44

LIST OF SYMBOLS

D
E

g

h

hi

H

H,

S

S. .
i J

Uw

Wave celerity

Wave celerity at breaking

Bed shear stress coefficient

Speed of wave energy propagation

n, + h , total depth of water

Energy density

Acceleration due to gravity

Local depth below sti I I water level

Depth below sti I I water level at breaking

Local mean wave height

Significant wave height at breaking

Local significant breaker wave height
within surf zone

Index corresponding to horizontal coordinate in
X-d i reef i on

Index corresponding to horizontal coordinate in
Y-d i rect i on

Bottom slope

Excess of momentum flux tensor (radiation stress)

Water particle velocity due to wave motion

Mean velocity component paral lei to the beach

Horizontal coordinate perpendicular to the beach

Width of the surf zone

Horizontal coordinate paral lei to the beach

a

a\

n

Y

P
T,

Incident wave angle

Incident wave angle at breaking

Mean water surface elevation

Mean water surface elevation at breaking

Ratio breaking wave height to depth of
water at breaking

Water dens i ty

Bottom shear stress

10

ACKNOWLEDGEMENTS

I would like to express my appreciation to Dr. Edward
B. Thornton for his encouragement, patience and assistance
during this study.

I am deeply indebted to the Coastal Engineering Research
Center of the Department of the Army, which made possible
this study by providing the necessary data.

I am also very deeply thankful to my wife, Frine, for
her understanding and for typing the original (at no cost!)
despite her little knowledge of the English language, and
my three beautiful daughters Dubhe, Ka r I a and Deneb.

11

I . I NTRODUCTI ON

A. STATEMENT OF THE PROBLEM

It is known that when sea waves or swe I I approach a
straight coastline at an oblique angle a mean current is
generated parallel to the shoreline, see Figure I. Such
longshore currents are of prime importance for both
coastal engineering and for aiding in the strategic planning
of Naval inshore warfare operations.

An accepted theory of longshore currents on plane
beaches is developed in terms of the momentum flux due to
the waves directed down coast being balanced by the shear
stress associated with the mean flow. The formulation of
the bed shear stress requires the specification of a bed
shear stress coefficient. The purpose of this thesis is
the determination of the bed shear stress coefficient to De
used in the longshore current formulas.

The study will also help in the analysis of sediment
transport. The shear stress does work on the bottom in
moving sediments. Several authors have formulated sediment
transport in terms of the bed shear stress which in turn
requires an appropriate bed shear stress coefficient.

B. H I STORICAL REV I EW

I nman and Quinn (1952), using the momentum approach for
the prediction of longshore current by Putnam, Munk and

12

■ * I I I

•/.•.•.•r/.V-;.' ST ILL WATER LINE-'.'v

BiL

> CURRE NT

BREAKER LINE

Ot,

WAVE CREST

\

SWL

Figure I. Definition of Longshore Current Variables

13

Taylor (1949), showed that in order to fit theory with obser-
vations, the bed shear stress coefficient must be permitted
to vary with the longshore velocity over a wide range of
3 1/2 orders of magnitude.

Bretschne i der (1954) found that the spectral limitations
of wave growth, under the action of steady wind in shallow
water with a typical sandy bottom, suggested a value for the
friction coefficient of between 0.01 and 0.02. Also, he
found that the observed damping of swell propagating over a
smooth, leveled, impermeable sea bed was consistent with a
value of the coefficient of between 0.034 and 0.097.

Longuet-H i gg i n s (1970), using the concept of radiation
stress, developed a relationship for prediction of the
theoretical maximum longshore current just inside the break-
ing and proposed a friction coefficient of the order of 0.01
He concluded, on the basis of the finding of Bret sc hne i de r
(1954), Prandtl (1952) and Mi kuradse's experiment with
roughened pipes, that is was not "...unreasonable to expect
a friction coefficient of the order of 0.01."

Table I was taken from Sonu (1975); it summarizes some
values of the friction coefficient proposed by various inves'
tigators. The values reported were obtained from measure-
ments outside the surf zone or from laboratory experiments.
It can be seen from this table that the range of values is
relatively wide and the test conditions varied.

14

TABLE I. Bottom Friction Coefficients

Proposed by Various Investigators

Friction
Coe f f i c i ent

Wave Wave
Height Period
( meters ) ( sec )

Test
Cond i t i on s

Authors

0.01

0.030-0.089

0.030-0.040

0.01

0.03 -0.15

0.09 -0.50

Arbitrary Arbitrary

0.23-0.51 2.88-3.96

5.4^

0.01 -0.40

0.002-

0. 100

0.03 -0.18

1.77-2

10

8.4*

0.88-2.58

-15.5

.05-1.60 7.4-12.5

(UT/2ttti)-4~ 20

U: near bottom velocity

r\: ripple height

Sha I low water
steady state
wave generation

Gu I f of Mexico;
depths 3.4-5.2 m
slope 0.00035-
0.00-41

Ni igata, Japan;
depths 2.25-
2.75 m
slope 0.018

Osci I lati ng water
channel , turbu-
lent boundary
I ayer

Wave f I ume,
laminar boundary
layer

Hiyshizu, Japan;
depths 13-10 m
slope 0.0060

Takahama, Japan;
depths 10-7 m
slope 0.0057

Wave flume study;
derived from
energy dissipa-
tion in sand
ri pple vortices

Bretschnei der
( 1954, a)

Bretschneider
( 1954, b)

Kishi (1954)

Jonsson ( I 966)

Iwagaki and
TsuchiyaC 1966)

Iwagaki and
Kaki numa( I 966)

Iwagaki and
Kakinuma( I 966)

Tunsta I I and
Inman (1975)

Note: ^Equivalent values at 1 0 m depth

15

C. OBJECTIVES OF THE STUDY

The bed shear stress coefficients previously determined
are based on a very I imited set of field data or on labora-
tory studies which used as a mode I simple sinusoidal waves
which are not typical of the randomness found in nature.
The objective of this study is to analyze existing sets of
field observations obtained in the surf zone and by using
the best available theory attempt to determine a reasonable
value of the bed shear stress coefficient. For this purpose
a fairly large data set obtained for the Channel Island
Littoral Environment Observations (LEO) Program was used as
well as a set of observations by Ingle (1966) taken at various
locations along the Southern California coast. It is
expected that the data obtained and the theory applied will
ultimately contribute to the establishment of a reasonable
value of the bed shear stress coefficient and to a more
accurate prediction of the longshore current velocity across
the surf zone .

I I . THEORY

A. INTRODUCTION

Several models have been proposed for the distribution
of the longshore current velocity across the surf zone on a
plane sloping beach. The so I ut i on , wh i c h uses pure sinusoids
to describe the waves and no lateral shear stress, gives a
velocity distribution which is triangular shaped with both
a peak velocity and a discontinuity at the breaker point
dC3owen, I 969) , Thornton (1969) and Lon guet-H i gg i ns (1970)].
This is unreasonable since there are no discontinuities in
nature. A second model including lateral shear stress tends
to smooth out the discontinuity at the breaker I ine and pro-
duces a smoother velocity distribution with the maximum
velocity occurring closer to shore. However, no criterion
to predict an optimum lateral shear stress coefficient is
as yet available. This introduces an added complexity to
the problem.

A random-sea model developed by Collins (1972) circum-
vents the difficulty of the lateral shear stress coefficients
and allows the statistical input of the sea state as de-
scribed by a Rayleigh distribution. Figure 2 compares the
velocity distribution resulting from the various models: the
non lateral stress model, the lateral shear stress model for
a coefficient equal to 0.4 (Longuet-Higgins, 1970), and the

17

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LU

LU

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18

random-sea model. The velocities are referenced to the peak
velocity, Vm, of the non lateral shear stress model, which
according to Long uet-H i gg i n s (1970) is given by

c s i na.

w 57T9 u c b

Vm = T6c7 Hb s ~c—

and V is the longshore component of the mean current velocity

Bowen (1969), Thornton (1969) and Long uet-H i gg i n s (1970)
attributed the generation of longshore currents, due to an
oblique wave approach, to the longshore component of the
momentum flux (radiation stress) of the water waves. The
calculation of the wave-induced longshore current velocities
and changes in mean water level requires the specification
of the radiation stresses as a function of the location and
wave properties in the nearshore region.

B. WAVE SET-UP INSIDE THE SURF ZONE

As waves approach the coast and shoal there is a change
in the momentum flux of the waves which is balanced by a
change in the mean water level. Outside the surf zone there
is a set-down while inside the surf zone, after breaking,
there is a set-up or superelevation of the water level. The
wave set-up is important because both the local wave height
and speed are functions of the total local water depth which
is unknown. The change in mean water level required to
balance the excess momentum flux of the waves must be
determined first. A convenient form of The x-component
(shoreward) of The momentum flux equation integrated over

19

depth and averaged in time, derived by Lon gu et-H i gg i n s and
Stewart (1962), for describing the wave set-up is given by

3S

__ + pg(h + T1) -1 = 0

( I )

which says that the change of excess momentum flux due to
wave action ("radiation stress") is balanced by a change in
the mean water level. It is assumed in the derivation that
the net local mass flux perpendicular to shore is zero so
that there is no contribution from the mean motion to the
momentum flux perpendicular to the shore, and that the mean
stresses are negligible.

Inside the surf zone it is assumed that the radiation
stress tensor can be expressed in terms of the energy and
wave speed in the same form as in shallow water. This implies
that the breaking waves are of the spilling type and that
even under breaking waves the waier particle motion retains
much of its organized character as described by linear wave
theory. Using the shallow water approximation. that the group
velocity equals the phase velocity and that the angle of
incidence equals zero, the radiation stress term reduces to

-,2

Sxx = T6 P9Y2(h+n)

(2)

Substituting (2) into (I) and integrating gives the mean
water elevation of the form

n = K(hb-h) + nb ,

(3)

20

whe re

K =

(4)

+ 378 Y2

and the mean water elevation at breaking is given by

YHK

6 *

(5)

H is a single breaker height which corresponds to the

significant wave height observed at the breaking position.

For this study, it was assumed that the waves fol lowed

the breaking index,

Hh
Y = ° = 0.78 ,

b

derived from a modified solitary wave theory by Munk (1949)
The total local depth of water, D, is obtained by combining
the local mean water super elevation and the local depth,

D = n + h .

(6)

A sinusoidal description of the waves was used to solve
for the wave set-up in order to get a closed form analytical
solution and to circumvent the difficult numerical solution
of equation (I) required dv the random-sea model. Collins
CI972) compared the sinusoidal solution to the random sea
solution as shown in Figure 3. The effects of the random-sea
model is to smooth the waves set-up curve. The magnitude of
the difference between the two solutions is very small but

21

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the percent difference can be large as the depth of water
approaches zero. The total depth of water is important in
prescribing the breaker point or the limits of integration
on the Rayleigh distribution for the random-sea model.
In the calculation of longshore currents using the random
sea model it is the area under the Rayleigh distribution
that is used so that small errors on the limits generally
cause only even sma I ler errors in the area. Hence, it is
felt using sinusoidal wave descriptions to calculate wave
set-up is a reasonable approximation.

C. WAVE FIELD INSIDE THE SURF ZONE

A description of the wave field is required in the long-
shore current calculation because knowledge of it is needed
for specifying the horizontal wa+er particle velocities and
for determining the longshore component of the radiation
stresses in an irregular wave field. Inside the surf zone
the waves are unstable and the fluid motion loses seme of
its ordered character; but Thornton (1976) points out that
most of the water particle motion in the body of the fluid
is coherent with the surface and can be considered wave-in-
duced and not turbulent, particularly for spilling type
breakers.

In this study, a truncated Rayleigh distribution as shown
in Figure 4 is used to give a statistical description of the
wave field as described by Collins (1972) and Battjes (1974).
The basic assumption is that at each depth a limiting breaker

23

Q.

x:

CD
CD

>■

CC

T2
0)

+-

(0

u

c

3

CD

1_
Z3

cn

I

a

24

height can be defined which cannot be exceeded by the indi-
vidual waves of the random field, and that those wave
heights which in the absence of breaking would exceed the
breaker height are reduced by breaking to the value of the
local breaker height. That is, the energy corresponding to
the height in excess of the local breaker height is assumed
to be dissipated. The limiting breaker height decreases as
depth dec reases .

In describing the Rayleigh distribution, a fictitious,
or reference, local energy per unit area, denoted as E , is
defined. The reference energy density refers to that energy
density that would exist if breaking had not occurred nad
accounts for shoaling and refraction transformation. Battjes

(1974) also defines reference wave heights H and their mean
3 r

2

square value H which is related to E according to

E = q- pg H '
r 8 a r

(7)

The reference wave heights are assumed to be Rayleigh distri-
buted .

The Rayleigh wave height distribution is

clipped at H = H in accordance with the assumption that the
height of a breaking wave equals the local breaker height,
H , in order to obtain an approximation to the actual wave
height distribution. Then, the mean energy per unit area at
a fixed point, taking account of breaking, is calculated from

E = 3 pg h

2

(8)

25

The variance is calculated from the pdf of H,

/

H2 p (H) dH ,

(9)

where

H
s

d[l-exp(-H2/Hr2)]

/

+ H 2 exp(-H 2/H 2) ,
s K s r '

(10)

/

H2d[l -exp (-H2/H )]

+ H 2 exD(-H 2/H 2) ,
s s r '

(II)

H2 = [I -exp (-H 2/H 2)] H 2 .
r s r r

(12)

The clipped Rayleigh distribution implies that all waves
from H to infinity that were previously larger than H now
are reduced to the same height, H . Therefore, the total
probability (percent) of waves having the height H is given
by

/

p (H) dH .

26

The contribution to the variance is given by

oo y

p(H) dH = H exp(- -At) ,

Hr2

which is the term on the right of equation (II). H is the
local breaker height, which inside the surf zone is assumed
to be given by

Hs = YD .

(13)

The local mean wave height H can be expressed in a similar
manner in terms of H and H by means of the clipped Rayleigh
distribution,

/

H

Hd[l -exp(-H2/H )] + H exp(-H 2/H )
K r sKsr

(14)

where again the term on the right represents the percent of
waves greater than H in the original distribution which now
have the height H . Integrating (14) gives

^r- I erf (H /H )
2 r s r

rms rms

(15)

n which

H = (H 2)i/2

r r
rms

and the error function being defined as

erf ( p ) =

/F

/

exp ( -t ) d t .

The error function was calculated using the rational approx
mation of Abramowitz and Stegun (1965),

erf(p) = l-[(a.t + a9t2 + at3) exp(-p2)] + e(p)

where
t

z =

l/( l+zp),

.47047,

H /(H 2) l/2
s r

a. = .3480242,
a2 = .0958798,
a = .7478556.

The largest error using this approximation is
e(p) <_ 2.5 x I 0~5 .

In the observations used for comparison with the theory,
the breaker height is measured visually. It is assumed
that an observer visually measures the significant breaking
wave height defined as the average of the highest one-third
fraction of the wave heights. The difficulty in applying
this definition to the present problem is that the defini-
tion applies to a point measurement or a statistically homo-
geneous (spatial) wave field and in this problem the waves
are defined as varying spatially as they shoal shoreward.

28

In order to define the significant wave height for a
spatially varying ( nonhomogeneou s ) wave field, it is
assumed that the observer measures waves when spatial ly
one-third of the waves have broken; hence, the reference
wave height can be specified from the clipped Rayleigh d i s
tr i but i on

exp (-H 2/H 2) = 0.333
r b r

and H 2 = - H 2/Ln (0.333) .

r b

(16)

D. LONGSHORE CURRENT VELOCITY

The derivation of the longshore current starts with the
y-momentum flux equation

dS

xy

8x

+ Tb"T£ = ° '

(17)

Where the lateral shear stress, Tf, is neglected (17) reduces
to

8S

xy

dx

+ Tb = 0 ,

(18)

which says that the change of y-component (longshore)
momentum flux due to waves in the x-d i reef ion is balanced by
the bottom stress, t, , in the y-direction. Assuming that
the amplitude of the wave motion | U w | is much greater than
the mean current velocity, V, then (Thornton, 1969)

^b = P Cf I 0w I V .

(19)

29

The excess of momentum flux of the waves, or "radiation
stress" component is given by

C

S = E s i na cosa -J*- ,
xy C '

which inside the surf zone reduces to

S = E sina cosa
xy

(20)

under the assumption that C = C for sha I low water. Combin

9

ing equation (8) and (12) gives the mean energy per unit
a rea as

E = i pg H 2[l-exp(-H 2/H 2)]
8 a r s r

(21 )

and

2 2 2

Sv = r pq H Ll-exp(-H /H )J sina cosa . (22)
xy 8r s r

The variables H and a can be expressed as d i f f e ren t i a b I e

s r

functions of x (distance from shore). Recalling that it was
assumed inside the surf zone

H = yD ,
s ' '

where the total depth is The sum of the sti I I water depth
plus the set-up

D = n + h ,

30

from equations (3), (5), (6), and (13) H can be expressed

as

H = H. ( K - ^=0 + h(x) (Y-YK) .
s b 16 ' '

(23)

Application of Snell's law allows the local breaker
angle, a, to be expressed in terms of the known breaking
angle a, and +he breaking celerity C,,

C
sma = 7=r- s i n a u .

Cb b

(24)

Using the sha I low water approximation for wave speed

C = (gh)

1/2

and

then

r (n HbJ/2

Cb - (9 — )

s i na = ( h r—) s i na, .

H , b

b

(25)

Hence, from equations (23) and (25) it is seen that both
H_ and a are now expressed as functions of h, which in turn
is a function of x.

The bed shear stress coefficient is determined by com-
bining equation (18) and (19),

C

as

xy I
f 9x p I Uw I V *

(26)

31

The mean horizontal water particle velocity amplitude is
expressed using linear theory (Battjes, 1974),

Uw

H C
7T D '

(27)

where H is given by equation (15),

The change in the radiation stress is given by

2 2 2

xy E^J/2 • rn Yh Sin V-l/2 .„ 2^hsin ab,
—L = 7^) s.nab(h - ) S(l — - ) +

b b b

where

sina cosa r , ..... . . , 2 ,, . 2,

+ _ pgS(y-YK)Hs exp(-Hs /Hp ) ,

(28)

E = i pg H 2[l-exp(-H 2/H 2)]
83r s r

a nd

si na co sa =

(rr-) '/2 s inaK(h
H . b

b

yh sin a

b 1/2

32

III. DATA

A. LEO DATA

Data from the Littoral Environment Observation (LEO) pro-
gram established by the Coastal Engineering Research Center
was used in this study. In the LEO program, nearly simul-
taneous observations of breaker conditions (height, period,
angle of approach and type), local winds, longshore currents,
foreshore slope, width of the surf zone and rip currents were
made daily during the period under consideration. The long-
shore current was determined by observing the direction and
measuring the distance parallel to shore that a dye packet
injected into the surf zone traveled in one minute. Appendix
A provides the set of instructions followed during the obser-
vations.

The data used for this study cover a period from May 1972
to September 1975 and refer to stations: 5703, 5706, 5707,
5713, 5714 and 5715, located within the confines of Point
Mugu Naval Air Station, 60 miles northwest of Los Angeles,
California (location 6 in Figure 5).

From these stations. 4,632 observations were considered
of which only 647 data points were used in the analysis. The
following criteria were discussed to eliminate observations
which were not consistent with the application of the theory
or were simply erroneous:

33

o

3 <"' <c

"O 0

CD (D

E <D

1- CD

O

*+- c

i- 0

CD 4-

Q_ CD

c

CD —

s- +-

CD c

2 =J

X

01

C II

0

— ^r

4-

ra -~

> (T3

1- U

C/3

0) —

Q_

01 c

<

-Q O

«>

O 2

2!

XT (0 — »

O

(J 4- VO

—

— C vD

h-

-C I'D CD

<C

5 00 —

CJ

0

+- II «.

_l

.XI <D

K^i —

4- CD

•

m ■* c

IP,

(0 -C —

0 U —

CD

O ru

s_

0)

3

(TJ CD •

cn

• - 01

—

C 01 C

u_

L- <V 0

O O —

•+- C +-

— 1X3 i"D

— L. -1-

^ 1— u~>

CJ

II O

C UJ

1_ CN —J

x: ■ ■%. 11

4- 4-

3 c o

J —

n c • -

CL T3

4- —

c v —

4- C

C <D — ?

. —

— 0 "3

3 4

I . Rip Cu rrents

It was beyond the scope of this study to account for
any modifying effect of the longshore current system by rip
currents. Hence, all observations noting the presence of
a rip current were systematically deleted.

2 . Angle of Wave Approach

The theory used in the derivation of the formulas
employed in this study assumes that the angle between the
direction of wave approach and the depth contour must be
different from 90 degrees in order for a longshore current
to be generated. Hence, all observations in which the wave
direction was reported as being perpendicular to the shore-
I ine were neglected.

3. Wi nd

Shepard and Inman (1950) suggested the importance
of the wind in generating longshore current; they also indi-
cate that it is difficult to separate the wind generated
current contribution from the current generated by the waves.
Thus, observations where the wind speed was reported as
being greater than ten mi les per hours were not considered.

4 . Fo res ho re Slope

Observations where the foreshore slope was reported
as being greater than ten degrees were neglected since such
large values are not consistent with what is usual ly observed
on the beaches under consideration.
5 . Wave Period

Arbitrarily, to keep the study restricted to sea and
swell of relatively short period, all observations where the

35

wave period was reported as greater than 20 seconds were
neg I ected .

6. Doubtf u 1 Data

Al I observations in which the reported data were
considered to be incorrect due to either mistakes of the
observer or the typist, such as longshore currents in ex-
cess of six feet per second, direction of approach greater
than 180 degrees, distance of dye injection greater than
600 feet, etc., were systematically rejected.

B. SOURCES OF ERROR

Considering the interest in longshore currents, it is
somewhat surprising that there are relatively few sets of
adequate field measurements of longshore currents and the
simultaneous wave parameters in the surf zone. After a
search of the literature it was concluded that little has
been achieved for devising electronic equipment designed
for gathering longshore current and associated wave informa-
tion on a routine basis. Hence, as in the case of the LEO
data, most of the observations must rely on the good
judgment and personal abilities of the observers. This
introduces a subjectivity factor which ultimately affects
the final results.

I . Brea ker Angle

Galvin and Nelson (1967) suggested that tne variable
most difficult to measure with necessary accuracy is the
angle of wave approach or wave direction. Galvin and Savage
(1966) suggesled that when using a visual compass referenced

36

to a baseline to measure the breaker angle, the errors may
easily be + two degrees, leading to a relative error which
is very large for sma I I breaker angles but which decreases
as the breaker angle increases.

In the LEO observations a protractor was used for
determining the breaker angle as shown in Appendix A. This
system is completely visual using the unaided eye to estab-
I ish the perpendicular to the shore and introduces a greater
human factor. Hence, it is a good assumption to attach an
accuracy less than that suggested by Ga I v i n and Savage to
such measurements.

2 . Beach Slope and Surf Zone Width

In describing longshore currents which flow within
the surf zone, accurate knowledge is required of the beach
profile including both the beach slope and the width of the
surf zone. The approach used for the LEO data was to assume
a plane beach inside the breaker line. The nearshore sub-
aqueous slope was computed using the observed surf zone
width and the observed breaker height. Thus an uncertainty
factor for the beach was introduced. Referring to LEO obser-
vation instructions in Appendix A, the observation of the
surf zone width "... is based upon the judgment of the obser-
ver; man-made or natural features in the surf (e.g., a pier)
may aid in this observation." Again, a subjectivity factor
is involved.

The observed foreshore slope could not be used because
it proved to be unrepresentative of the beach prof i le within

37

the surf zone. The calculated water depths inside the break-
er line using this slope were systematically greater than
the calculated breaker depths.

3 . Wave Period

Galvin and Nelson (1967) point out that under favor-
able conditions the wave period can be measured with reason-
able consistency in the field by visual observation. Although
this parameter was not used directly in the computation it
is interesting to notice that their suggestion agrees quite
well with the LEO observations since the range of periods
found fa I I into the expected values for the shorel ine under
con s i de rat i on .

4 . Brea ke r Height

In the LEO program, the breaker height observation
is based solely on the judgment of the observer. Known
dimensions of natural or man-made features on the shoreline
or in the surf zone are used as references for estimating
the wave breaking height. Galvin and Savage (1966) sug-
gested a relative error in breaker height measurement of
+_ 25 percent. They arrived at this figure by comparison of
breaker height measurements made with pressure gages,
oscillographs and visual observations, although the measure-
ments were not made simultaneously. Hence, in the light
of their finding it can be concluded that at least the same
error should be expected in the LEO data in which the obser-
vations are solely visual.

38

5 . Longshore Current

In the LEO program, the current speed was determined
by using a dye as a tracking agent. This also adds an uncer-
tainty factor due to the diffusivity characteristic of the
dye.

C. INGLE DATA

A set of 62 field observations made along the Southern
California coast (Fig. 5), taken from Ingle (1966) and per-
sonal communication, were selected using the same criteria
used for selecting the LEO data. Despite the size of Ingle
sample, about ten percent the size of the LEO data, its
analysis is important since the Ingle observations are more
accurate than the LEO observations. Thus, the Ingle results
serve as a reference comparison to the results obtained using
the LEO data.

The parameters H , period, a and V were taken directly
from a summary appendix in Ingle (1966). The beach profiles
and the distance shoreward from breaker zone in which the
longshore current velocities were recorded also were avail-
able. However, the positions of the breaker were not avail-
able. Thus, the parameter h, was obtained from the relation-
ship h = 1.28 H , ; and the local depth below still water
level, h, and the beach slope were scaled out from the beach
profiles presented in the publication.

It should be pointed out that in the Ingle data the break'
er heights were measured by sighting on either a graduated
pole held at an approximate still-water line and the horizon,

39

a graduated pole held in the zone of breaking waves, or a
piece of cardboard with a slit and graph paper along one
edge. Breakers less than 2 feet in height were estimated
while standing in the breaker zone. In the LEO program,
breaker height observations were based solely on the judg-
ment of the observer on the shore. For measuring the
breaker angle, Ingle observers used a Brunton compass while
standing in the surf zone, supplemented by sights taken
from positions elevated above the beach; LEO observers used
a protractor, as shown in Appendix A, with the observer on
the beach. For measuring other parameters, both LEO and
Ingle observers used essentially the same techniques. It
is important to mention, that in the case of the Ingle
observations, the beach slope and the position of the obser
vations in the surf zone were better than those of the LEO
observations since in the former an ordinate and abscissa
arrangement of wooden stakes allowed workers to position
themselves in the surf zone; also most people involved in
Ingle observations were wel I trained personnel.

40

I V. RESULTS

Equations (3), (5), (6), (12), (15), (16), (21), (22),
(23), (25), (26), (27) and (28) presented in the theory sec-
tion were used with the parameters H, , a, , x, , x and V from

b b b

LEO and Ingle field observations to solve for the bed shear
stress coefficient. The coefficient calculated is based on
data at specific locations within the surf zone and not for
mean conditions. It should be mentioned that since the co-
efficient is not determined by direct measurement, it there-
fore not only reflects bed shear stress, but also any errors
and uncertainties in measurement.

The significance of the assumption that the observer
measured waves correspond to a clipped Rayleigh distribution
when spatial ly one-third of the waves have broken, used for
computing the reference wave height H , was tested using
other assumptions. It might just as logically be argued
that the significant wave height might correspond to the
point where half the waves have broken, in which case

exp(-H 2/H 2 = 0.5 .
s r

This assumption was used for computing new values for the
coefficient. The relative difference between the coeffi-
cients thus calculated and the original ones was determined;
the variability was found to be of the order of ten percent,

41

which is relatively low. Hence, it can be said that the
bed shear coefficient calculations are not very sensitive
to the assumed definition of the significant wave height.

A. LEO DATA

Figure 6 depicts the frequency distribution of values
obtained for the coefficient and selected statistics of the
distribution. The variability in coefficient values, as
represented by the standard deviation of the distribution,
is a measure of the consistency of the calculation of the
coefficient from the field observations. The mean of the
distribution is 0.008, while the standard deviation is 0.010
This suggests there is a large spreading in the results.
However, it should be noticed that more than 90 percent of
the calculated values fall between 0.001 and 0.020 and that
the distribution has less spread than a Gaussian distribu-
tion for the same standard deviation.

B. INGLE DATA

Figure 7 depicts the frequency distribution of values
obtained for the coefficient and selected statistics of the
distribution. The mean of the distribution is 0.014 and
the standard deviation 0.01 I , which suggests again a large
spreading in the results. However, 95 percent of values
lie between 0.001 and 0.030 and again the distribution has
less spread than a Gaussian distribution for the same
standard deviation.

42

2 1 0-,
2 0 0.
1 8 0.

1 60.

1 4 0.

>

o

2 120.
UJ

^10 0.

o

UJ

a. 80

LL

6 0.

40.
20.

.0 01 .0 0 6 .0 11 .0 16 .0 2 1 .0 2 6 .0 3 1 .0 3 6 .0 4 0 .0 4 5 .0 5 0 .0 5 5 .0 6 0

COEFFICIENT

Mean 0.008 Skewness 3.694

Variance 0.00009 Kurtosis 17.380

Std. Deviation 0.00972 Minimum 0.00!

Coef, Variation 1.19004 Maximum 0.080

Figure 6. Frequency Distribution of Coefficient Values
( LEO Data) .

43

1 J -
1 2 -

1 1 -

10 -

9 -

> 8 1

o

z

LU 7 "

O

LU 6 '

K

"- 5 ■

4

3 -

2 ■

1 ■

1

1

i 1

_____

.001

.008 .016 .023 .031

COEFF ICI ENT

.0 3 8

.04 5

Mean 0.0I4 Skewness

Variance 0.000I Kurtosis

5 1 d . Deviation 0.0M2 Minimum

Coef. Variation 0.3I09 Maximum

I .584
2. 649
0. 00 I
0.056

F i

gure 7. (-requency Distribution of Coefficient Values
(Ingle Data).

44

C. COMPARISON OF RESULTS

Despite the difference in sample size between the LEO
and Ingle data used for the calculations, some comparisons
can be made. A simple way of comparing both sample results
is by looking at their mean and standard deviation. The
mean and standard deviation corresponding to the values of
the coefficient for the Ingle data are both larger than the
values obtained for the LEO data. There is a relative dif-
ference of 75 percent between the mean of the coefficient
values of the two samples; but, the relative difference
between the two standard deviations is only ten percent.
This says that the distribution values for the coefficient
in both samples is nearly the same, although for the Ingle
data the values for the coefficient were somewhat larger
and with more spread than for LEO data which might be ex-
pected for the sma I ler sample size.

It was stated, when comparing both sets of data the
Ingle observations were more accurately taken and more
reliable than the LEO observation. Hence, the results ob-
tained with the Ingle data would be expected to be better
than the results obtained with the LEO data. To test if
there is any statistical difference between the two sets
of data, a hypothesis test about the two means obtained was
made. The central I imit theorem states that, if x is the
mean of a random sample of size n taken from a population
having the mean u and the finite variance a2, then

7 = iizE

a/n~

45

is the value of a random variable whose distribution function
approaches that of the standard normal distribution a s n -»■ °° .
The variances of the population are unknown, but since both
samples are fairly large, it is justifiable to approximate
the population variances with the samples variance. Thus, a
test statistic can be stated as

z =

x, - xL

V?

The hypothesis to be tested is the null hypothesis, p.-y, = 0,
against the alternative hypothesis ]i . -y , > 0, where u repre-
sents the mean of the population. The evaluation, for the
data available, of the z statistic was found to be equal to

z =

0.014 - 0.00.

0.0001 + 0.00009 I 12

= 4.53 .

62

647

For a level of significance of 0.001 the z statistic for the
normal distribution is 3.49. Since the value obtained for
the test statistic is larger than the critical value of 3.49,
the null hypothesis is rejected with great confidence; and it
can be concluded that the difference between both means is
statistically significant and cannot be attributed to chance.
Therefore the results obtained with Ingle data are better
than the results obtained with the LEO data.

46

D. CORRELATION WITH INDEPENDENT VARIABLES

Attempts were made to correlate the calculated coeffi-
cients with the independent variables, breaker type and wave
period, which were recorded in the field but which were not
used directly in the computations. Analysis showed nothing
conclusive regarding the correlation of the coefficient to
the breaker type since the distribution of breaker types
among the data was very uneven; the spi I l/plunge type repre-
sented 72 percent of the data and the spilling type 20 per-
cent. Table II shows some selected statistics of the a i s-
tribution of coefficient values for various breaker types.

TABLE II. Selected Statistics for Distribution of
Coefficient According to Breaker Type
(LEO Data)

No.
Observat ion

Mean

Variance

Std. Dev.

Coef . Var.

Ra nge

Minimum

Max i mum

S kewness

Ku rtos i s

Spilling
130

0.0090

0.0001

0.01 07

I . I 880

0.0790

0.001 0

0.0800

3.720

I 7.60

Plunging Surging Spill/Plunge

454

27
0.0075
0.00004
0.00609
0.0150
0.0230
0.0010
0.0240
I .397 I
I .0553

25
0.0098
0.000 I 5
0.01 204
0.23321
0.05300
0.001 00
0.05400
2.90456
7.38989

0.00788
0.00009
0.00946
I .2031 8
0.07500
0.0CI 00
0.07600
3.74730
I 8.8567

47

A simple linear regression between the calculated coef-
ficients and the observed period gave the selected statistics
of Tab I e III.

TABLE III. Correlation of Coefficient with
Wave Period Statistics

Correlation (R)

Std. error of estimate

R squa red

Significance

I n tercep t

Slope

LEO Data

- .05370
.00970
.00288
.08574
.01 088

- .00023

INGLE Data

- .08479
.01 120
.007 I 9
.2561 7
.01 892

- .00042

The negative sign of the correlation coefficient indi-
cates that there is an inverse relationship; that is, the
value of the coefficient tends to become sma I I er as the
period increases. However, this relationship is very weak
as indicated by the absolute value of R which in both cases
is much smaller than one. This result is not surprising
since waves in shal low water become non-dispersive or in-
variant of period.

48

V. CONCLUS IONS

An analytical solution for the bed shear stress coeffi-
cient was derived using the concept of radiation stress.

The best theory for calculating the variation of wave
energy and longshore current, and the resulting bed shear
stress coefficient, was to use the truncated Rayleigh p.d.f.
for the statistical description of the wave field inside the
surf zone. A sinusoidal approximation of the waves was used
to calculate the wave set-up. Calculations of the coeffi-
cient were made by using suitable sets of data obtained
during the LEO observation program and Ingle (1966) observa-
tions along the Southern California coast.

Variability in the results obtained for the coefficient
values were expected due to subjectivity and uncertainties
in the techniques used in the data collection. This is the
first test of the bed shear stress coefficient using fairly
large setsof field measurements within the surf zone. Even
with the uncertainties involved, the analysis resulted in a
fairly good agreement between the mean of the calculated
coefficients in this work and the values obtained by various
investigators for the bed shear stress coefficient for dif-
ferent test conditions and outside the surf zone.

It was shown that the dependence of the coefficient on
the wave period is negligible, in agreement with the assump-
tion that waves inside the surf zone are non-dispersive or

49

period invariant. Since one of the biggest differences
between Pacific and Atlantic coast waves is the period, it
may be concluded that the calculated coefficient is not
ocean dependent .

Since it was initially concluded that Ingle's data was
of higher quality than the LEO data, it is assumed the
coefficient values using Ingle's data is therefore more
reliable. In any event, the mean value of the two data
sets are the same to two significant decimal places. There'
fore, it is concluded that a reasonable value for the bed
shear stress coefficient within the surf zone is 0.01.

50

APPENDIX A
LITTORAL ENVIRONMENT OBSERVATIONS

CERC Form No. 113-72-8 Mar 72 has been designed for keypunching onto computer cards directly (small numbers above each box represent card column numbers) .
It is recommended a pencil be used. All data should be recorded carefully and legibly. Errors should be corrected by first erasing erroneous data as
write overs usually produce illegible data. Make remarks as necessary on the form but record only data in the boxes provided. All observations mist be
made at the same point on the beach every time (in front of the reference pole) .

STATION IDENTIFICATION:

Each site in the "Littoral Environment Observation" study has been assigned a numerical code consisting of 5 digits. The first two digits define the state

or territory in which the site is located and the remaining 3 digits define the particular beach or park, within the state or territory. A space has also

been provided to write in the name of the particular beach or park at which the observation is taking place.

DATE:

Indicate in the spaces provided the year, month and day on which an observation is made.

TIME:

Indicate the time at which the observations are being made. The 24-hour system of reoording time has been selected in order to eliminate any confusion be-
tween AM and PM. The hour "00" refers to midnight, "07" to 7:00 AM, "13" to 1:00 PM, etc.

SURF OBSERVATIONS :

a. Wave Period - Record the time in seconds for eleven (11) wave "crests" to pass seme stationary point. Eleven "crests" will include ten complete
waves (crests and troughs). Ine first (1) "Crest" selected for observation is recorded as time zero and the eleventh (11) "crest" will be the stop or cut
time. Record this time in seconds in the spaces provided.

b. Breaking Wave Height - This observation is based solely on the judgment of the observer. Natural or manmade features on the shoreline or in the
surf zone whose dimensions are known may aid in judging the height of a wave. Otherwise the observer's best estimate will be sufficient. Pecord the breaker
height to the nearest tenth of a foot.

c. Breaker Angle - To determine the direction from which the waves are approaching the beach use the protractor on this reverse side of the data form.
The 0-180° line should be oriented along the shoreline; use the protractor to site the direction from which the waves are approaching when they are first
breaking.

d. Type of Breaking Wave:

Spilling - Spilling occurs when the wave crests becomes unstable at the top and the crest flows down the front face of the wave producing an irregu-
lar, foamy water surface, (see figure 1)

Plunging - Plunging occurs when the wave crest curls over the front face of the wave and falls into the base of the wave producing a high splash and
much foam 'figure 2)

Surging - Surging occurs when the wave crests remains unbroken -while the base of the front face of the wave adcances up the beach itsejerfigurs: 2Y

Spill/Plunge - A combination of both spilling and plunging occurring simultaneously.

WIND OBSERVATIONS :

a. Wind Speed - A wind meter is provided to each observer and it is recommended that the instructions provided with the meter be followed to obtain
wind speed measurements.

b. Wind Direction - After the approximate orientation of the beach with respect to north has been defined the observer can determine the direction
"frcm wnich" the wind is coming.

FORESHORE SLOPE:

For measurement of the foreshore one must 'use either the clipboard/ inclinometer or the Abney hand level. Observations should be made as close to mid-swash
as possible. Using the clipboard/inclinometer place it en the appropriate edge and record the angle -where the ball comes to rest. Using the Abney hand
level place it on a straight edge and level the bubble; record the indicated angle.

WIDTH OF SURF ZONE:

This observation is based solely on the judgment of the observer. Estimate in feet the distance from the shoreline to the line of the most sea/ard breakers

(not to be confused 'with white caps) .

LONGSHORE CURRENT:

a. Dye Distance - Dye packet should be injected just shoreward of the breakers, if possible. Driftwood or any other floating object should be used if
dye is not available. Estimate the distance from the shoreline to point of injection and record this distance in feet.

b. Current Speed - Mark the beach in line with the injected dye and make a second mark to indicate the dye movement after one minute has lapsed. Pace
the distance between these marKS and record this distance in feet.

c. Current Direction - when looking seaward, if the dye has moved to the left record -1, to the right record +1, and no longshore movement record 0.

RIP CURRENTS:

Rip currents are defined as seaward moving channels of water which return the water that has been piled up along the shore by meaning waves. Rip currents
are fed by feeder currents, water moving along the shore (see figure 4) . Two currents join and extend out in what is known as the "neck", where the water
rushes through the breaker zone in a narrow lane. Beyond the breaker zone the current spreads out and dissipates in what is called the "head". If such
rip currents are present estimate their spacing in feet. If no rips are present record 0.

BEACH CUSPS:

Cusps are semicircular or crescent shaped cutouts in the beach face (see figure 5) . If such shapes are observed record the distance between the "horns" of

the cusds which indicate the spacing. Where the spacing is irregular estimate the average spacing. If no cusps are present record 0.

FIGURE 2.
PLUNGING WAVE

. , > ,

t

a»Ea«f« ;:■•{

Sr4; f g

FIGURE 4.

*'"*>*;

FIGURE 5 BEACH CUSPS

51

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52

BIBLIOGRAPHY

Abramowitz, M. and Stegun, I. A., Handbook of Mathematical
Functions, National Bureau of Standards, 1964.

Ba I s i I I i e, J. H . , Surf Observations and Longshore Current
Prediction, Technical Memorandum No. 58, U . S. Army,
Coastal Eng. Research Center, November 1975.

Battjes, J. A., Computation of Set-up, Longshore Currents,

Run-Up and Overtopping Due to Wind-Generated Waves, 1975

Bowen, A. J., "The Generation of Longshore Currents on a

Plane Beach", Journal of Marine Research, v. 37, p. 206-
15, 1969.

Bretsch ne i d er , C. L., Field Investigations of Wave Energy
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54

INITIAL Dl STRIBUTION LIST

Defense Documentation Center

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Li brary (Code 0 I 42)

Naval Postgraduate School

Monterey, California 93940

Department Chairman, Code 68
Department of Oceanography
Naval Postgraduate School
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Dr. Edward B. Thornton, Code 68Tm
Department of Oceanography
Naval Postgraduate School
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CDR. Carlos Severino Veitia Garcia

Embajada de Venezuela

Ag regadu r i a Naval

2409 Ca I i torn i a St. , N.W.

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Scripps Institution of Oceanography

University of California

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No . Cop i es
2

55

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Naval Oceanography and Meteorology
National Space Technology Laboratories
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Dean, College of Marine Studies
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University of Delaware
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21. Coastal Studies Institute
Louisiana State University
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22. Mr. Thorndike Saville, Director
Coastal Engineering Research Center
Department of the Army

Fort Belvoir, Virginia 22060

56

Thesis
V36?5

c.l

172325
Veitfa Garcia

Bed shear stress
coefficient within
the surf zone.

Th

V3

c

Thesis

V3625

c.l

Veitia Garcia

Bed shear stress
coefficient within
the surf zone.

172325

ZTZL stress coefficient within the

3 2768 002 05416 5

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