R 64·13
0 \ 1 I :_, 2
EXPERIMENTAL STUDY OF
LONGSHORE· CURRENTS ON A PLANE BEACH
by
C. J. Galvin Jr.
P. S. Eagleson
HYDRODYNAMICS LABORATORY
Report No. 63
Prepared Under
Contract No. DA-49-055-eng-62-9
COASTAL ENGINEERING RESEARCH CENTER
U.S. Department of the Army
Corps of Engineers
Washington, D.C.
and
NR 083-157
Contract Nonr 1841 (74)-4
OFFICE OF NAVAL RESEARCII
Earth Sciences Division
Geophysics Branch
Oceanography Section
Washington, D.C.
May 1964
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R 64-13
HYDRODYNAMI CS LABaRA'IORY
Department of ,Civil Engineering
Massachusetts Institute of Technology
EXPERIMENTAL STUDY OF
LONGSHORE CURRENTS ON A PLANE BEACH
by
C.J. Galvin, Jr. and P.S. Eagleson
April 1964
Report No. 63
Prepared Under
Contract No. DA-49-055-eng-62-9
COASTAL ENGINEERING RESEARCH CEN'lER
NR 083-157
Contract Nonr 1841 (74)-4
D.S. Department of the Army
Corps of Engineers
OFFICE OF NAVAL RESEARCH
Earth Sciences Division
Washington, D.C.
Geophysics Branch
Oceanography Section
Washington, D.C.
,
,
,
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i
ACKNOWLEDGEMENTS
This study of the mechanics of longshore current generation on
straight beaches was sponsored jointly by the Coastal Engineering
Research Center (formerly the Beach Erosion Board) of the Department
of the Army, Corps of Engineers under Contract Number DA-49-055-eng62-9 and by the Office of Naval Research, Earth Sciences Division,
Geophysics Branch, Oceanography Section under Contract Nor~ 1841 (74)-4.
Administration of these contracts was provided by the Division of
Sponsored Research of MIT under DSR 9266 and DSR 8840 respectively.
The work was performed in the Hydrodynamics Laboratory of the
Department of Civil Engineering at the Massachusetts Institute of
Technology by Cyril J. Galvin, Jr., Research Assistant in Geology and
Geophysics under the supervision of Dr. Peter S. Eagleson, Associate
Professor of Civil Engineering.
R.L. Bernstein and R.M. Males, undergraduate part-time assistants,
aided in taking the data and did much of the data reduction. G.A. Tlapa,
Research Assistant in Civil Engineering, assisted with the machine
computation which was performed on the IBM 1620 in the Civil Engineering
Computer Laboratory.
ii
ABSTRACT
Measurements are made of the characteristics of breaking wa'les
and the resulting longshore currents for 34 combinations of wave height
(up to 0.22 feet), period (0.90 to 1.50 seconds), and breaker angle
(up to 32 0 ) , along a 20 foot test section of a 30 foot plane, smooth
concrete beach with a 0.104 slope.
Techniques are developed to measure the distribution of longshore
current velocity and of mean water level in the surf zone, and the
measurement of breaker point and angle for plunging waves on a laboratory
beach is standardized.
The nature of the growth of the longshore current in the lee of an
impermeable obstacle is studied. In addition to the expected increase
of the current velocity downstream of such an obstacle the mean water
level is found to increase and the breaker position and runup limit to
move shoreward. It is not clear to what extent these latter effects
are influenced by the scale of the apparatus used.
V
fl
iii
TABLE OF CONTENTJ
Page No.
ACK NOWI,ED GEMENTS
l\J3~-> 'LRll.C
'r
'k{ E ut,
ry
l:m\j']Ir';N'I'~)
LIS~e OF JPIGURES
LIST OF TABLES
LIS T OF SYMBOLS
INTRODUCTION
I.
AoProblel'! Jtatemerrt
BoScope of this Investigation
PREVIOUS INVESTIGATIONS
110
Ao Field Studies
B. Laboratory Studies
Co Analytical Studies
EXPERIMENTAL PROGR.M1
III.
EXPERU'1ENTAL EQUIPMENT AND PROCEDURES
IV.
A. Run Conditions
B. Wave Profile Gage
C. Measurement of Wave Height, Speed and Shape
Do Measurement of Breaker Point, Breaker Angle
and Runup Limit
E. Change in Mean Water Level
F .. Velocity Measurement
G. General Basin Conditions
H. Coordinate Systems and Definition
EXPERIMENTAL RESULTS
V.
A,. Wave Height, Shape and Speed
B. Breah:er and Runup
C. Mean lrvater Level
D. Longshore Current Velocity
E. Breaker Height and Depth
F. General Longitudinal Nonuniformity
VI.
ANALYTICAL CONSIDERATIONS
A. Qualitative Description of Longshore Current
Formation
B.. Approximate Energy Budget of Typical Test
Co Effect of External Circuit
D. Momentum and Enerl:~Y Analysis of Putnam et ale (12)
E. Revised Momentum Analysis
F. Empirical Correlation
VII ..
S1JllllJ~AI{Y AND CONCLUSION~3
A. Summary
B.. Conclusions
RFJI'miENCES
VIII ..
APPENDIX - :3U]v[MAllY OF DATA
i
ii
iii
iv
vi
vii
1
1
2
2
2
3
3
5
7
7
7
12
14
18
19
23
24
26
26
31
32
33
36
39
42
42
44
48
50
52
56
61
61
61
62
1\-1
"
iv
I,IST OF Fl(}llfiK;
Figure No.
1.
2.
,3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
1,3.
14.
I
r; ~
!.CJ •
17.
18.
19.
20.
21.
22.
2,3 •
24.
25.
26.
27.
28.
,30.
,31.
32.
33.
Title
Wave Basin
Plan View of Basin
Cross-Section of Beach and Basin
Platinum Wire Resistance Wa're Gage
Wave Gage Calibratioll Curve
Test Section with IrmtrLUnent Frame
Phase Lag Measurement for Celerity
Determination
Breaker Point Definition
Breaker Locator
Definition Sketch
Location of Taps for Damped Piezometers
Piezometer Well
Velocity Probes
Calibration of Veloci t;y Probes with
Surface Floats
Definition Sketch - Coordinate System
Definition Sketch
Definition Sketch
Wave Height Erwelope Through Surf Zone
Definition Sketch - Wave Shape
Wave Speed in Surf Zone
Wave Speed at Breaking
Mean Water Level in ,'jurf Zone
Frequency Distribution of Set-Dp in MWL
Velocity Distribution in Longshore Current
Longshore Variations for Conditions of
Test 4
Breaker Height and Breaker Depth
Location of Maximum and Minimum Longshore
Current Velocity
Location of Maximum and Minimum Mean
Water Level
Location of Maximum and Minimum Breaker
Distance
Location of Maximum and Minimum Hunup
Distance
Location of Maxim1Jll1 and Minimum Breaker
Angle
Shape 0 f Shaa] :iYlF~ Wav f? Near B:l'paJd 11[;
Mean Water Level in ~;ur I' !:al1e for Hun 111-2
Page No.
8
9
9
11
11
12
1,3
13
15
18
20
20
22
22
25
25
25
28
28
30
3(1
34
.34
35
.35
38
40
40
41
4l
~3
h3
117
v
Figure No.
34.
35.
36.
37.
38.
Title
Control Volume for Momentum Analysis
Variation in Ratio of Longshore Velocity
at Breaker to Mean Longshore Velocity
Variation of Mean Longshore Current
Velocity with Distance
Idealized Distribution of Longshore
Current Velocity
Empirical Correlation of Field and
Laboratory Data
Page No.
53
57
57
58
&J
vi
LIST OF TABLES
Table No.
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
Title
Page No.
Types of Experimental Data
Run Conditions
Repeatability and Operator Variation for G
b
Measurement (Test IV-2)
Operator Variation in Average Breaker Angle
Dye and Float Velocity in Longshore Currents
Shape Factor vs. Position on Beach
Breaker Angle from Small Amplitude Theory
and From Observation
Net Particle Motion in Shoaling Zone
(Test rv-4)
Estimates of Breaker Depth-Breaker Height
Ratios
Energy Budget for Uniform Longshore Current
Friction Factor, Reynolds Number and Relative
Roughness for Data from Putnam, et ale (12)
Comparison of C sin G with V
b
b
6
7
17
17
21
27
32
51
52
vii
LIST OF SYMffiLS
a
Wave amplitude, 1/2 H
Vertical distance between mean water
level and trough elevation
1\
A
w
Area of longshore current channel
Cross-sectional area of breaker carried
into surf zone
ft.
ft.
ft.2
Cross-sectional area of triangular
breaker ~ high and Lb cos Gb long
Breaker distance, horizontal distance
between SWLine and breaker position
ft.
C
Wave speed
ft/sec.
d
Water depth, mean water level to beach
surface
ft.
Mean water depth at breaking, defined as
mb
ft.
Mean water depth between toe of beach
and plunger
ft.
Constants in revised momentum analysis
Elevation of mean water level above
still water level
ft •
b
e
.
E
Energy flux
Rate of energy dissipation in wave
breaking
Rate of energy dissipation by longshore current per foot length of beach
E.1
.
E
.
P
E
r
ft-lbs/sec.
ft-lbs/sec.ft.
ft-lbs/sec.ft.
Rate of energy brought to beach per
foot of beach length
ft-lbs/sec.ft •
Rate of energy dissipation in runup
and backwash
ft-lbs/sec.ft •
energy flux off test beach in reflected
wave per foot of test beach
Darcy-Weisbach friction factor
ft-lbs/sec.ft.
Acceleration due to gravity
ft/sec 2
Head loss due to friction
ft.lbs/lb.
viii
LIST OF SYMPOLS (cont Id)
Mean water level at breaking from solitary
wave theory
h
H
k
k
x
ft.
Trough elevation above beach
Wave height, crest elevation minus trough
elevation
ft.
absolute roughness
ft.
Local horizontal distance between top of
beach and SWLine
ft.
ft.
Proportionality constant in empirical
correlation
L
Wave length
ft.
.,.
m
Beach slope
ft/ft.
;;
MWL
Mean water level, average water surface
elevation
p
proportionality constant
P
Energy flux from plunger per foot of crest
length
Mass flux in longshore current
c·
LJ
ft.
b
ft-lbs/sec .ft.
cJ
ft 3 /sec.
f
~/~
Hypothetical mass flux through wave
r
R
o
Runup distance, horizontal distance between
SWLine and runup limit
ft •
"
r
w
Hydraulic radius of longshore current channel:
~/2
ft.
JR
Reynolds number of longshore current, 4 VR/v
S
Surface force 'in breaking
Ibs.
SWL
Still water level. Free surface of fluid
at rest
ft.
SWLine
L
ft 3 /sec.
w
{3
y
p
Intersection of SWL with beach, the x-coordinate
axis
a
t
Time
sec.
T
Wave period
sec.
u
Particle velocity in head of bore
ft/sec.
v
Average longshore current velocity
ft/sec.
L I S T OF SYMBOLS ( c o n t ' d )
Vj_^p
V
Longshore c w r e n t v e l o c i t y i n d i c a t e d by p r o b e
breaker p o s i t i o n
,V
at
f t /s e c .
Longshore component o f v e l o c i t y i n r e t i i r n f l o w
o u t o f s u r f zone
ft/sec.
Average v e l o c i t y o f f l o a t s a t x = 10 and l 6 f t ,
ft/sec.
^10 ^16
V'
v^A
x
H o r i z o n t a l c o o r d i n a t e d i s t a n c e measured
positively
a l o n g beach f r o m upstream t r a i n i n g w a l l
f t .
y
D i s t a n c e measured p o s i t i v e l y o f f s h o r e f r o m SWLine
f t ,
y'
H o r i z o n t a l c o o r d i n a t e , measured f r o m t o p o f beach
f t .
z
V e r t i c a l c o o r d i n a t e measured p o s i t i v e l y up
f r o m SWL
f t .
subscripts
b
Location at
breaker
d
L o c a t i o n between beach t o e and wave g e n e r a t o r
f
R e f e r s t o f l o a t , or f r i c t i o n
L
Refers t o longshore
o
Deep w a t e r c o n d i t i o n s
r
R e f e r s t o runup l i m i t
W
Wave a t b r e a k i n g i n e m p i r i c a l c o r r e l a t i o n
w
Wave a t b r e a k i n g i n momentum a n a l y s i s
Y
S p e c i f i c weight o f f l u i d
lbs/ft^
p
Mass d e n s i t y o f f l u i d
slugs/ft^
CT
Wave shape f a c t o r
6
Angle between wave c r e s t and x - c o o r d i n a t e
current
^.^A^^
degrees
I
A.
INTRODUCTION
Problem Statement
Longshore c u r r e n t s f l o w p a r a l l e l t o t h e s h o r e l i n e and r e a c h t h e i r
h i g h e s t v e l o c i t i e s between t h e p o i n t o f wave b r e a k i n g and t h e s h o r e line.
They a r e d r i v e n b y t h e l o n g s h o r e component o f m o t i o n a s s o c i a t e d
w i t h waves w h i c h approach t h e shore l i n e o b l i q u e l y .
Longshore c u r r e n t s a r e o f i n t e r e s t t o e n g i n e e r s and g e o l o g i s t s
because t h e y e r o d e , t r a n s p o r t and d e p o s i t s e d i m e n t . Along beaches
b o r d e r i n g oceans and l a r g e l a k e s t h e s e c u r r e n t s a r e capable o f t r a n s p o r t i n g hundreds o f thousands o f c u b i c yards o f sand p a s t a g i v e n p o i n t
d u r i n g an average year (see Johnson (1)", Brebner and Kennedy ( 2 ) ) .
This
i s an a n n u a l l a y e r one square m i l e i n area and s e v e r a l i n c h e s t h i c k .
On n a t x j r a l beaches a q u a s i - e q u i l i b r i u m o f t e n e x i s t s between t h e
r a t e a t w h i c h sediment i s s u p p l i e d f r o m r i v e r s and c l i f f s and t h e r a t e
a t w h i c h i t i s t r a n s p o r t e d away b y l o n g s h o r e c u r r e n t s , so t h a t n e t
e r o s i o n o f t h e beach f a c e i s s l i g h t . An example o f t h i s e q i o i l i b r i u m
i s the w e l l s t u d i e d coast o f Southern C a l i f o r n i a , Here, a l l o f t h e
beaches s t u d i e d b y Handin (3, p.5U) w i t h t h e e x c e p t i o n o f one l 5 m i l e
s t r e t c h , "had been i n e q u i l i b r i a d u r i n g h i s t o r i c t i m e p r i o r t o
c o n s t r u c t i o n o f a r t i f i c i a l b a r r i e r s " . Sand r e a c h e s t h e s e beaches
" c h i e f l y f r o m streams and t o a much l e s s e r e x t e n t f r o m e r o s i o n o f sea
c l i f f s and t h e sea f l o o r " (U, p . 2 ^ ) . P o s s i b l y t h i s i s t y p i c a l o f
most c o a s t s f o r i t has been e s t i m a t e d t h a t c l i f f e d s h o r e l i n e s o f t h e
w o r l d erode back a t an average r a t e o f o n l y 1 cm per year and c o n t r i b u t e on a w o r l d w i d e b a s i s about 1/8 km^ per y e a r t o t h e sea ( 5 ) .
Rivers,
on t h e o t h e r hand, b r i n g t o t h e sea 5 t o 10 km^ o f sediment each y e a r .
C o a s t a l e n g i n e e r i n g works ( b r e a k w a t e r , g r o i n s , j e l t i e s , p i e r s ,
dredged c h a n n e l s , e t c . ) d i s t u r b t h e e q u i l i b r i u m , a c c e l e r a t i n g l o c a l
e r o s i o n and d e p o s i t i o n a l o n g beaches and i n h a r b o r e n t r a n c e s .
Engineers
a r e t h e r e f o r e i n t e r e s t e d i n t h e mechanics o f l o n g s h o r e c u r r e n t s i n o r d e r
t o p r e d i c t a n d , i f p o s s i b l e , e l i m i n a t e t h e unwanted e f f e c t s o f t h e i r
s t r u c t u r e s on t h e e n v i r o n m e n t . Beach E r o s i o n Board T e c h n i c a l R e p o r t
No. I4 (6) summarizes and e v a l u a t e s t h e l i t e r a t u r e on t h e e n g i n e e r i n g
importance o f longshore c u r r e n t s .
C o a s t a l processes i m p a r t d i s t i n c t i v e c h a r a c t e r i s t i c s t o t h e s i z e
d i s t r i b u t i o n o f beach sand (7) as w e l l as t o t h e morphology o f t h e
b o d i e s i n w h i c h t h i s sand o c c u r s .
Because t h e sand i n some sandstones
has l a i n , o f t e n more t h a n once, on beaches d u r i n g p a s t g e o l o g i c t i m e ,
knowledge o f l o n g s h o r e c u r r e n t s i s v a l u a b l e i n i n t e r p r e t i n g t h e r o c k
record o f the p a s t .
The l a n d f o r m s produced b y l o n g s h o r e cTJi-rents s p i t s , h o o k s , b a r s , and s t r a i g h t beaches •- are sometimes, as f o s s i l
forms, important petroleum reservoirs ( 8 ) .
"Numbers i n p a r e n t h e s i s
report.
r e f e r to l i s t of references
1
a t t h e end o f t h e
B,
Scope o f t h i s I n v e s t i g a t i o n
The u l t i m a t e aim o f s t u d i e s o f l o n g s h o r e c u r r e n t s by e n g i n e e r s
o r g e o l o g i s t s i s t h e q u a n t i t a t i v e p r e d i c t i o n o f sediment t r a n s p o r t b y
longshore c u r r e n t s .
Because sediment t r a n s p o r t i s p r i m a r i l y r e l a t e d
t o t h e v e l o c i t y o f the t r a n s p o r t i n g c u r r e n t , i t i s f i r s t necessary t o
p r e d i c t t h i s f l u i d v e l o c i t y b e f o r e a p p r o a c h i n g t h e more d i f f i c i a l t t a s k
o f p r e d i c t i n g sediment t r a n s p o r t .
This p r e d i c t i o n o f v e l o c i t y should
be c o n s i s t e n t w i t h t h e b a s i c e q u a t i o n s o f m o t i o n , energy and c o n s e r v a t i o n
o f mass. S o l u t i o n o f t h e s e e q u a t i o n s f o r such a complex p r o b l e m r e q u i r e s
t h e i r considerable but judicious s i m p l i f i c a t i o n .
Therefore, the l o g i c a l
approach t o u n d e r s t a n d i n g sediment t r a n s p o r t by l o n g s h o r e c u r r e n t s i n ¬
v o l v e s f i r s t t h e a c c u r a t e d e s c r i p t i o n o f t h e phenomena g e n e r a t i n g t h e s e
c u r r e n t s , t h e n t h e p r o p e r f o r m u l a t i o n o f s i m p l i f i e d eqxiations o f m o t i o n
t o p r e d i c t t h e v e l o c i t y o f t h e c u r r e n t s and an e x p e r i m e n t a l v e r i f i c a t i o n
o f t h i s p r e d i c t i o n , and f i n a l l y t h e i n v e s t i g a t i o n o f r e l a t i o n between
l o n g s h o r e c u r r e n t v e l o c i t y and sediment t r a n s p o r t .
This i n v e s t i g a t i o n deals w i t h the i n i t i a l steps o f experimental
d e s c r i p t i o n o f l o n g s h o r e c u r r e n t s and t h e a n a l y t i c a l p r e d i c t i o n o f l o n g shore c u r r e n t v e l o c i t y .
Sediment m o t i o n i s n o t t r e a t e d .
The e x p e r i m e n t a l phase i n c l u d e s measurements under c o n t r o l l e d l a b o r a t o r y c o n d i t i o n s
o f phenomena a s s o c i a t e d w i t h l o n g s h o r e c u r r e n t s f l o w i n g on a p a r t i c \ 3 l a r
p l a n e , smooth c o n c r e t e b e a c h .
The a n a l y b i c a l phase i n c l u d e s an e m p i r i c a l
r e l a t i o n between l o n g s h o r e c u r r e n t v e l o c i t y and wave c o n d i t i o n s a t
b r e a k i n g on l a b o r a t o r y and n a t u r a l beaches, and a p r e l i m i n a r y e x a m i n a t i o n
o f the equations o f motion f o r longshore c u r r e n t s .
II
A.
Field
PREVIOUS INVESTEGATIONS
Studies
Johnson ( 9 ) stmimarizes t h e e a r l y work p r i o r t o I919 i n w h i c h some
i n v e s t i g a t o r s recognized the fundamental importance o f breaker angle
and wave h e i g h t . C o r r e l a t i o n o f l o n g s h o r e c u r r e n t v e l o c i t y w i t h t h e s e
v a r i a b l e s f o r o v e r a thousand q u a l i t a t i v e o b s e r v a t i o n s a l o n g t h e c o a s t
o f S o u t h e r n C a l i f o r n i a was l a t e r d e m o n s t r a t e d b y Shepard ( 1 0 ) .
Of p a r t i c u l a r I n t e r e s t and i m p o r t a n c e i s t h e r e p e a t e d o b s e r v a t i o n
t h a t mean l o n g s h o r e c u r r e n t s are unsteady and n o n - u n i f o r m on n a t u r a l
beaches, ( 1 1 ) , ( 1 2 ) .
The u n s t e a d i n e s s i s u s u a l l y a t t r i b u t e d t o t h e
s t o c h a s t i c natvire o f t h e i n c i d e n t wave t r a i n .
Non-uniformity w i l l o f
course resu3.t f r o m c o a s t a l s t r u c t u r e s or f r o m v a r i a t i o n s i n t h e n e a r shore h y d r o g r a p h y . Shepard (10) a t t r i b u t e s n o n - u n i f o r m i t y t o t h e
p r e s e n c e o f r i p c u r r e n t s , s t a t i n g " . . . . most l o n g s h o r e c u r r e n t s can be
shown t o be r e l a t e d t o r i p s because i n v a r i a b l y t h e y can be t r a c e d t o
a l o c a l i t y where t h e c u r r e n t t u r n s seawai-d i n t o a r i p " . D u r i n g h i s
2
l o n g s h o r e c t t r r e n t measiirements on S o u t h e r n C a l i f o r n i a beaches, r i p
s p a c i n g v a r i e d f r o m s e v e r a l hvmdred t o a few t h o u s a n d f e e t .
Q u a n t i t a t i v e o b s e r v a t i o n s were made b y Putnam, Kxxnk and T r a y l o r
(12) and by Inman and Quinn ( I 3 ) , B o t h groups o f i n v e s t i g a t o r s g i v e
meastirements o f b r e a k e r a n g l e , wave h e i g h t and p e r i o d , beach s l o p e
and l o n g s h o r e c u r r e n t v e l o c i t y f o r c e r t a i n C a l i f o r n i a b e a c h e s .
Their
e x p e r i m e n t a l t e c h n i q u e s w i l l be d i s c u s s e d l a t e r .
Inman and Quinn (I3) f o u n d t h e v a r i a b i l i t y i n l o n g s h o r e c u r r e n t
v e l o c i t y t o be such t h a t t h e s t a n d a r d d e v i a t i o n o f t h i s v a r i a b l e
(measured a t 15 s t a t i o n s r o t i g h l y 3OO f e e t a p a r t ) u s u a l l y equaled
o r exceeded t h e mean o f t h e measurements. I n o t h e r words i t was
n o t uncommon on some r e l a t i v e l y p l a n e n a t i a r a l beaches ( T o r r e y P i n e s
and P a c i f i c Beach, C a l i f o r n i a ) f o r l o n g s h o r e c u r r e n t v e l o c i t y t o
oppose t h e l o n g s h o r e component o f m o t i o n i n t h e b r e a k i n g w a v e s .
B.
Laboratory Studies
E a r l y l a b o r a t o r y work was concerned w i t h l i t t o r a l d r i f t , t h e
m a t e r i a l moved by l o n g s h o r e c u r r e n t s .
The l i t e r a t u r e abounds w i t h
d e s c r i p t i o n s o f model s t u d i e s t o d e v e l o p s o l u t i o n s f o r l i t t o r a l d r i f t
problems on p a r t i c u l a r shore l i n e s however t h e s e o f f e r l i t t l e t o w a r d
a f u n d a m e n t a l u n d e r s t a n d i n g o f t h e p r o b l e m . Savage (ih)
summarizes
p r e s e n t f i e l d and l a b o r a t o r y knowledge o f t h e r a t e o f t h e l i t t o r a l
d r i f t t r a n s p o r t r a t e and d i s c u s s e s t e c h n i q u e s u s e f u l i n l a b o r a t o r y
studies o f longshore c u r r e n t s .
One o f t h e f i r s t f i x e d - b e d s t u d i e s o f t h e l o n g s h o r e c u r r e n t s
themselyes i s t h a t o f Putnam e t a l (12) i n w h i c h beach s l o p e and
roughness were v a r i e d i n a d d i t i o n t o a l l t h e wave parameters a t
breaking.
More r e c e n t l y Brebner and Kamphuis (l5) have made a s e r i e s o f
f i x e d - b e d s t u d i e s on smooth p l a n e beaches b u t • u n f o r t u n a t e l y t h e wave
c h a r a c t e r i s t i c s were n o t measured a t b r e a k i n g , o n l y i n deep w a t e r ,
C.
Analytical
Approaches
P r o b a b l y t h e f i r s t a n a l y t i c a l a t t a c k on any a s p e c t o f t h e l o n g s h o r e
c u r r e n t p r o b l e m was an a t t e m p t t o r e l a t e t h e l i t t o r a l d r i f t t r a n s p o r t
r a t e t o t h e l o n g s h o r e component o f power s u p p l i e d by t h e w a v e s .
The i d e a
was i n i t i a l l y advanced by Munch-P e t e r son ( i n I91J4 a c c o r d i n g t o Svendson
(16)). S i n c e t h a t t i m e , E a t o n (1?), C a l d w e l l ( 1 8 ) , and o t h e r s have
c o n t r i b u t e d t o i t s e v a l u a t i o n . Savage (I9) g i v e s a c o m p i l a t i o n o f f i e l d
and l a b o r a t o r y d a t a w h i c h shows an o r d e r o f magnitude s c a t t e r i n t h e
l a b o r a t o r y d a t a b u t good agreement among t h e two a v a i l a b l e s e t s o f f i e l d
measurements (18) , F i e l d meastirements o f Johnson (20) i n c l u d e a l l b u t
t h e angles needed b y t h i s method, b u t even assuming t h e most f a v o r a b l e
v a l u e f o r t h e m i s s i n g a n g l e s , t h e d a t a do n o t come w x t h x n an o r d e r o f
magnitude o f t h e r e l a t i o n d e f i n e d p r e v i o u s l y ( 1 8 ) .
Putnam e t a l (12) c o n s i d e r t h e f l u x o f mass, momentum and e n e r g y
i n t o a c o n t r o l volume o f d i f f e r e n t i a l l e n g t h a l o n g t h e s h o r e bounded
by t h e beach and t h e b r e a k e r l i n e .
They d e v e l o p two e x p r e s s x o n s , b o t h
f o r c o n d i t i o n s w h i c h are s t e a d y and u n i f o r m i n t h e mean:
1.
From energy
flux:
i n which
V
mean l o n g s h o r e c u r r e n t v e l o c i t y
m " beach s l o p e
C^=» wave group v e l o c i t y a t b r e a k i n g
wave energy per u n i t o f s u r f a c e a r e a a t b r e a k i n g
0^= a n g l e o f wave i n c i d e n c e a t b r e a k i n g
p = f l u i d mass d e n s i t y
d^= mean w a t e r l e v e l a t b r e a k i n g
f
= Darcy-Wiesbach r e s i s t a n c e c o e f f i c i e n t
s - f r a c t i o n o f b r e a k i n g wave energy a v a i l a b l e f o r
longshore current generation.
2.
Prom momentum f l u x :
V - I
[(1 .
h
V.
s i n e J ^ / 2 „ 13
[2]
i n which
V
b
•= wave p a r t i c l e v e l o c i t y a t b r e a k i n g
8 m
cos 9^
volume r a t e o f i n f l o w a c r o s s b r e a k e r
T
wave p e r i o d
h
line
The m a j o r assumptions i n v o l v e d i n t h e d e r i v a t i o n o f E q . [2]
i
are:
•- momentxam e q u a t i o n can be w r i t t e n i n terms o f t i m e average
quantities
ii
- f l o w i s uniform i n the longshore d i r e c t i o n
iii
- no shear e x i s t s on t h e c o n t r o l volume f a c e a t t h e
breaker l i n e
i v - 0 e x i t s t h e c o n t r o l voltmie across t h e b r e a k e r l i n e w i t h
velocity v.
Putnam e t a l t h e n a p p l y s o l i t a r y wave r e l a t i o n s h i p s
f o r Q^,
v^,
and use t h e f i e l d and l a b o r a t o r y measurements mentioned p r e v i o u s l y t o
e v a l u a t e s and f .
Inman and Quinn ( 1 3 ) c a l c i a a t e d t h e Darcy f f r o m E q . [ 2 ] u s i n g t h e i r
f i e l d measurements as w e l l as t h e f i e l d and l a b o r a t o r y d a t a o f Putnam
e t a l ( 1 2 ] ' , Assuming an e x p o n e n t i a l r e l a t i o n between f and v t h e y
o b t a i n e d t h e l e a s t squares f i t :
f
= 0.38k
V
•^•^-^
Brebner and Kamphuis (l5) measured o n l y deep w a t e r wave
and p e r f o r m e d m t a t i p l e r e g r e s s i o n analyses w h i c h y i e l d e d :
[3]
characteristics
H
V - m
sin^/^(m)
[ s i n (1.65 e^) + a i s i n (3.30 6^) J
[k]
Most o f Ikh s e p a r a t e o b s e r v a t i o n s l i e w i t h i n + 10 ° / o o f t h i s r e l a t i o n .
No a t t e m p t was made t o check i t a g a i n s t t h e d a t a o f o t h e r i n v e s t i g a t o r s . .
I n a r e v i e w o f t h e o r i e s o f l o n g s h o r e c u r r e n t m o t i o n Bruxin ( 2 1 ) c o n c l u d e s t h a t t h e momentum approach i s t h e b e s t a v a i l a b l e f o r p l a n e beaches
b u t suggests t h a t f o r beaches b o r d e r e d b y b a r s , a c o n t i n u i t y r e l a t i o n
between t h e f l u i d b r o u g h t i n b y t h e b r e a k i n g waves and t h a t c a r r i e d o u t
by t h e r i p s m i g h t be u s e f u l .
Ill
EXPERIMENTAL PROGRAM
A...
As was s t a t e d e a r l i e r , t h e u l t i m a t e o b j e c t i v e o f t h e r e s e a r c h
program o f w h i c h t h i s i s a p r e l i m i n a r y r e p o r t , i s t o f o r m u l a t e a
5
m a t h m a t i c a l .nodel g o v f c r r i n g t h e g e n e r a t i o n o f l o n g s h o r e c u r r e n t s w h i c h
i s c o n s i s t e n t w i t h t h f f x f i d a m e n t a l laws o f f l u x d mechanics and xs hence
independent o f s c a l e .
I n o r d e r t o dete>-mine t h e p h y s i c a l l y i m p o r t a n t ïeat^es
this
complex phenomena and t o add t o t h e body o f d a t a ^ ^ ^ f ^ ^ ^ ^
J'^^J?^!^
a s e t o f e x p e r i m e n t s were p e r f o r m e d on a l a b o r a t o r ; / beac I n t h e Hydro
dynamics L a b o r a t o r y o f t h e Department o f C : r / i l l ^ ^ ' g - ^ ^ ' ^ ^ ^ ^ y ' \ , , / ; 3 \ , , e
a r T e x t e n s i v e p r e l i m i n a r y i n v e s t i g a t i o n , .nethods were
' f ^^^^ '"^^^f
'
f o u r t v o e s o f d a t a , c l a s s i f i e d f o r convenience as t y p e s A , B, b a a u
{^B S i e 1 ) , E n ^ g y d i s s i p a t i o n i n t h e b r e a k i n g wave, p e r h a p s t h e
l o ^ t i m p o r t a n ; d e p e x S n t v a r i a b l e i n t h i s s t u d y , cotiLd r,ot be d x r e c t l y
measured, f o r a p p r o p r i a t e i n s t r u m e n t s do n o t e x x s t .
One o r more o f t h e f o u r t y p e s o f d a t a a r e i n c l u d e d i n UU r u n s ,
i d e n S f i e d f o r c o n v e n i e n c e by a unique combination o f t h r e e independent
i ï ï a b l e s : 9 , , H . , and T . Four v a l u e s o f 9 ^ , f «
-"g^%X'?ïao°)
g e n e r a t o r anS s h o r e l i n e , d i v i d e t h e r u n s i n t o S e r i e s I ( 0 ° ) * / ^ ^ ! ° ^>
111(27°), and I V ( 5 1 ° ) . Each s e r i e s , e x c e p t I , i n c l u d e s about a dozen
t e s t s i d e n t i f i e d b y a u n i q u e c o m b i n a t i o n o f wave p e r i o d ( T ) and wave
height i n f r o n t o f generator ( H ^ ) ,
TABLE 1
D
TYPES OF EXPERIMENTAL DATA
Principal
measuring d e v i c e
Q u a n t i t y measm^ed
Symbol
parallel wire
r e s i s t a n c e gage
wave h e i g h t
wave speed
wave f o r m
H
G
o
breaker
breaker p o s i t i o n
breaker angle
b
locator
tape
runup l i m i t
damped
piezometers
change i n MWI.
due t o waves
miniature current
meter, f l o a t s
l o n g s h o r p --'iirrejit
velocity
Tables A l and A 2 o f t h e Appendix summai^ize t h e d a t a a v a x l a b l e i r o m
t h e s e e x p e r i m e n t s a c c o r d i n g t o d a t a t y p e , s e r i e s , and t e s t number.
^or.-i«<. IT (ro9 «« 0
0 °°!) was
was done
done oonnllyy t o check a few cases when no longiihorc
Jierxes
c u r r e n t s would be expectn<1.
IV
A.
EXPERIMENTAL EQUIPMENT AND PROCEDURES
Rvn C o n d i t i o n s
The experiments were conducted i n a model b a s i n k'? f t . by 22 f t .
by l,k f t , c o n t a i n i n g a movable, p l u n g e r - t y p e wave g e n e r a t o r 20 f t .
l o n g and a smooth p l a n e c o n c r e t e beach .30 f t , l o n g and 13 f t . w i d e w i t h
a 1 on 10 s l o p e (see F i g u r e s 1 , 2, and 3 ) .
O f f s h o r e w a t e r d e p t h was
k e p t a t I . l 5 f t . Waves, g e n e r a t e d a t known angles t o t h e b e a c h ,
advanced i n t o t h e beach between t r a i n i n g w a l l s c u r v e d t o match t h e
r e f r a c t i o n p a t t e r n o f a wave o f i n t e r m e d i a t e p e r i o d i n each s e r i e s .
The upstream t r a i n i n g w a l l extended up t h e beach beyond t h e swash l i n e
c o m p l e t e l y b l o c k i n g l o n g s h o r e motions i n t h e s u r f z o n e .
The downstream
t r a i n i n g w a l l t e r m i n a t e d s h o r t o f t h e shore l i n e t o p r o v i d e an e x i t
from the t e s t s e c t i o n f o r the longshore c u r r e n t .
The f l o w e v e n t u a l l y
r e t u r n e d t o t h e beach b y p a s s i n g under t h e p l u n g e r .
Table 2 l i s t s t h e r a n g e o f t h e independent v a r i a b l e s i n t h e s e
experiments.
The v a l u e s were l a r g e l y d e t e r m i n e d b y a v a i l a b l e space and
equipment.
B.
Wave P r o f i l e Gage ( l y p e A Equipment)
"Wave h e i g h t , speed, and f o r m were o b t a i n e d u s i n g a p l a t i n u m w i r e
r e s i s t a n c e wave gage and Sanborn r e c o r d i n g o s c i l l o g r a p h . Model 1^0.
The wave gages (see F i g u r e 3) d i f f e r f r o m p r e v i o u s models (Dean, ( 2 2 ) j
W i e g e l , (23)) p r i n c i p a l l y i n t h e a d d i t i o n o f c o n n e c t o r s between t h e
s e n s i n g w i r e s and t h e c a b l e l e a d i n g t o t h e r e c o r d e r .
For use i n and
near t h e s u r f zone, t h i s c o n n e c t i o n i s made by s o l d e r i n g t h e 0,036
i n c h diameter platinvmi w i r e
TABLE
2
RUN CONDITIONS
0 ° , 1 0 ° , 27°, 51°
0.90 t o 1.50 s e c ,
0,05 t o 0.21 f t .
Wave g e n e r a t o r a n g l e
Wave p e r i o d
Wave h e i g h t a t
(H,)
generator
Beach s l o p e
im)
Beach s u r f a c e
Water d e p t h a t
generator
Water t e m p e r a t u r e
L e n g t h o f t e s t beach
(between t r a i n i n g w a l l s )
Opening between downstream t r a i n i n g
w a l l and SW L i n e
Mean e l e v a t i o n o f p l u n g e r
base above f l o o r
Training w a l l curvature
Series
I
II
III
IV
7
O.lOU f t . / f t . o v e r a l l average
0.109 f t . / f t . n e a r s h o r e average
smooth c o n c r e t e
1.15 f t .
1U°
to
23°
^22 f t .
2.2 f t .
o.ho
f t .
Cwvature
n o t needed
f o r r e f r a c t i o n o f 1.25
none
f o r r e f r a c t i o n o f 1,50
sec.wave
sec.wave
Fig. 1
Wave B a s i n
Basin
Fig,
2
Wolll
P l a n View o f B a s i n
PLUNGER AT
MEAN ELEVATION
X 2
Fig,
3
VERTICAL
EXAGGERATION
Cros;:.-oection o f Beach and B a s i n
9
i n an Amphinol f e m a l e j a c k (80 M C 2 F) w h i c h f i t s a male p l t i g (80 M G 2M)
on t h e c a b l e . The base o f t h e f e m a l e j a c k was covered w i t h a smooth
convex c o a t o f beeswax.
These m o d i f i c a t i o n s p r e v e n t t h e c a l i b r a t i o n
f r o m d r i f t i n g due t o t h e c o l l e c t i o n o f s p r a y f r o m b r e a k i n g waves a t t h e
c o n n e c t i o n , and t h e y a l s o make t h e w i r e s e a s i l y d e t a c h a b l e f r o m t h e
l o n g l e n g t h s o f c a b l e . The p a r a l l e l , O.O36 i n c h d i a m e t e r w i r e s o f
t h e s e gages c a n t i l e v e r 5-1/2 i n c h e s beyond t h e connector and are h e l d
p a r a l l e l and 3/16 i n c h a p a r t b y two l u c i t e spacers (see F i g u r e h) >
C a l i b r a t i o n . Gages were c a l i b r a t e d b y l o w e r i n g them i n t o s t i l l
w a t e r i n one c e n t i m e t e r i n c r e m e n t s over a range o f 6 or 7 c m . , and r e c o r d i n g on Sanborn paper t h e change i n r e s i s t a n c e between t h e w i r e s .
A p l o t o f gage e l e v a t i o n ( i n cm. above an a r b i t r a r y datum) a g a i n s t
r e c o r d e r d e f l e c t i o n ( i n mm.) i s r e a s o n a b l y l i n e a r and c o n s t a n t f o r t i m e
i n t e r v a l s on t h e o r d e r o f 10 hours (see Figvire 5) • The s l o p e o f t h i s
c u r v e , i n f e e t o f w a t e r per mm. o f Sanborn p a p e r , i s t h e c a l i b r a t i o n
c o n s t a n t used t o o b t a i n wave h e i g h t .
This s t a t i c c a l i b r a t i o n was checked b y h a r m o n i c a l l y o s c i l l a t i n g
t h e gage a known v e r t i c a l d i s t a n c e i n s t i l l w a t e r , f o l l o w i n g t h e p r o cedure o f Dean (22).
I n s i x t e s t s , d i f f e r e n c e s between s t a t i c and
dynamic c a l i b r a t i o n s were 2 per c e n t o r l e s s , w h i c h approaches t h e o r d e r
t o w h i c h t h e a m p l i t u d e o f t h e v e r t i c a l o s c i l l a t i o n can be r e a d .
—Gage p e r f o r m a n c e . Because t h e gages were used i n s h a l l o w w a t e r
on a c o n c r e t e beach c o n t a i n i n g aliominum channels i n t h e beach s u r f a c e ,
t h e e f f e c t s o f t h e s e s u r f a c e s on t h e c a l i b r a t i o n were d e t e r m i n e d .
L i n e a r i t y o f c a l i b r a t i o n was n o t a f f e c t e d b y t h e f r e e s \ i r f a c e i n s h a l l o w
w a t e r as l o n g as t h e p r o b e was 3 mm. below t h e f r e e s t i r f a c e . The c a l i b r a t i o n c u r v e o b t a i n e d w i t h t h e ends o f t h e two w i r e s i n c o n t a c t w i t h
t h e same c o n c r e t e s u r f a c e was i n d i s t i n g u i s h a b l e f r o m a n o r m a l c a l i b r a t i o n
c u r v e . C a l i b r a t i o n made i n s h a l l o w w a t e r was a f f e c t e d by t h e aluminum
channels a t d i s t a n c e s o f l e s s t h a n 2 i n c h e s , and t h e r e f o r e , gages were
n o t used over t h e c h a n n e l s . The l u c i t e s p a c e r s , as l o n g as t h e y were
e n t i r e l y below o r above t h e f r e e s u r f a c e , d i d n o t a f f e c t t h e s l o p e o f
t h e c a l i b r a t i o n c u r v e , b u t p o s i t i o n o f t h i s ctirve was t r a n s l a t e d s l i g h t l y i f s p r a y wet t h e upper s p a c e r .
D u r i n g a few r u n s , t h e p l a t i n v m i w i r e s were a c c i d e n t l y b e n t , b u t
a f t e r s t r a i g h t e n i n g , a r e c a l i b r a t i o n showed no more t h a n n o r m a l v a r i a t i o n
f r o m t h e o r i g i n a l c a l i b r a t i o n . U n p u b l i s h e d experiments b y J . A . Hoopes
showed t h e r e t o be l i t t l e v a r i a t i o n i n s e n s i t i v i t y f o r l a r g e changes i n
w i r e s p a c i n g . A 100 per cent i n c r e a s e i n s p a c i n g ( f r o m l / U t o l / 2 i n c h )
decreased t h e s e n s i t i v i t y o n l y 15 per c e n t . I n normal u s e , s p a c i n g does
n o t change.
The w i r e s o f t h e gages are r e l a t i v e l y s h o r t and s t i f f j no
v i b r a t i o n or severe b e n d i n g under t h e a c t i o n o f t h e waves o c c u r s .
Gages were never w i p e d w h i l e i n use s i n c e c a l i b r a t i o n s made b e f o r e
and a f t e r a t e s t w i p i n g were i n d i s t i n g u i s h a b l e , one f r o m t h e o t h e r . Water
s u r f a c e d u r i n g t h e s e r i i n s was c l e a n , and t h e gages were i n w a t e r o n l y
w h i l e i n use.
10
M)
5 1/2 inch
3 / l 6 inch
i_i
' 1
Connection
0.036 inch diameter platinum
Lucite spacers
Fig. k
P l a t i n u r n W i r e R e s i s t a n c e Wave Gage
37 5
s
37,0
>-
<
t
360
cr
<
UJ
>
O
CD 35,0
<
xon
z
o
34,0
O
10 0 5 A M ,
•
12 18 P.M.
A
5 30RM.
>
Ui
330
32 0
10
Fig. 5
20
DISPLACEMENT
I
30
40
50
OhJ SANBORN PAPER (mm)
Wave Gage C a l i b r a t i o n Curve
11
I n f ' e n e r a l , these t e s t s o f the performance o f p a r a l l e l w i r e
r e s i s t a n c e gages r e p e a t and extend t e s t s r e p o r t e d by W i e g e l and Dean,
and s u b s t a n t i a t e Dean's c o n c l u s i o n t h a t a c t u a l wave h e i g h t w i l l n o t
d i f f e r by more t h a n 3 t o 5 per cent f r o m measured wave h e i g h t .
O s c i l l o g r a p h . D u r i n g t h e t e s t s o f t h e wave gage, an i m p o r t a n t
p o s s i b l e s o u r c e o f e x p e r i m e n t a l e r r o r was n o t e d .
The s t y l u s o f t h e r e c o r d e r i s h e l d b y a s p r i n g a g a i n s t t h e r e c o r d i n g p a p e r , and i f t h e s p r i n g
t e n s i o n i s t o o h i g h , f r i c t i o n between s t y l u s and paper can cause l a r g e
e r r o r s , p a r t i c u l a r l y when l o w paper speeds and s m a l l d e f l e c t i o n s o f t h e
stylus permit s t i c k i n g to occur.
For t h i s r e a s o n , most wave gage d a t a
were t a k e n a t h i g h e s t paper speed (100 mm./sec.) and about two t h i r d s o f
f u l l s c a l e d e f l e c t i o n , under w h i c h c o n d i t i o n s t h e r a p i d r e l a t i v e m o t i o n
between s t y l u s and paper minimizes t h e f r i c t i o n e f f e c t . Whenever r e c o r d i n g s were made a t low speeds, t e n s i o n on t h e s t y l u s was c h e c k e d .
Recording d a t a near t h e edges o f t h e Sanborn paper was a v o i d e d when
p o s s i b l e because t h e response o f t h e s t y l u s i n t h i s r e g i o n i s s l i g h t l y
nonlinear,
C.
Measurement
o f Wave H e i g h t , Speed and Shape ( I > p e A Procedure)
I n use, wave gages were suspended f r o m a p o i n t gage mounted on a
r o l l i n g cross b a r w h i c h t r a v e l e d on an aluminum f r a m e e x t e n d i n g over a
6 f t . b y 20 f t , area o f t h e t e s t beach (see F i g u r e 6 ) . Wave h e i g h t w&s
o b t a i n e d b y a v e r a g i n g t h e r e c o r d e d wave h e i g h t o f t w e n t y s u c c e s s i v e waves
p a s s i n g one c a l i b r a t e d wave gage. Wave shape ( t e m p o r a l v a r i a t i o n o f w a t e r
s u r f a c e e l e v a t i o n a t t h e gage p o s i . t i o n ) c o u l d be o b t a i n e d f r o m t h e s e same
Figur-e
6. Test S e c t i o n w i t h I n s t r u m e n t
12
Frame
r e c o r d i n g s . Envelopes o f wave h e i g h t were o b t a i n e d by moving ^he c a l i b r a S d S g e s l o w l y f r o m o f f s h o r e , throiogh t h e b r e a k e r z o n e , and i n t o t h e
T u S u r r e g L n . Wave speed was o b t a i n e d b y a v e r a g i n g t h e r e c o r d e d phase
l a g (As i n F i g u r e 7) o f 20 waves p a s s i n g two gages f i x e d 0.25 f t . a p a r t
on a l i n e p a r a l l e l t o t h e d i r e c t i o n o f wave t r a v e l .
These d a t a were always t a k e n a t a s t a t i o n i n t h e m i d d l e o f t h e
t e s t beach and u s u a l l y a l s o a t t h e upstream and downstream ends ^ ^ J ^
beach (see A p p e n d i x , Table A U ) . A t t h e s e s t a t i o n s , wave speed and h e i g h t
were measured a t between s i x and t e n l o c a t i o n s a l o n g a l i n e e x t e n d i n g
ïrom t h e t o p o f t h e r u n u p r e g i o n t o o f f s h o r e o f t h e b r e a k e r .
AS
F i g u r e 7.
Phase Lag M e a s u r é d b y 2 Wave Gages, Gages Spaced 0.25 f e e t
A p a r t i n P l a n e Normal t o D i r e c t i o n o f Wave P r o p a g a t i o n .
Shape Shown i s t h a t o f a Bore Forming f r o m t h e Hroken Wave.
L
Figure 8 .
^1'
OF
Breaker P o i n t D e f i n i t i o n , P o s i t i o n o f t h e F i r s t
Appearance o f Dark L i n e i n t h e S h o a l i n g Wave i s
D e f i n e d as t h e Breaker P o s i t i o n .
13
D.
Meastrement o f Breaker P o i n t . B r e a k e r _ A n g l e _ j i n ^ ^
jTyye
B Eqviipment and P r o c e d ü r e J
D e f i n i t i o n s . As t h e b r e a k e r b e g i n s t o p l u n g e , t h e c r e s t o v e r r e a c h e s
t h e concave f r o n t o f t h e w a v e . When v i e w e d f r o m a b o v e , t h i s c o n c a v i t y
f o r m s a g r a y i s h , t r a n s l u c e n t band a l o n g t h e wave f r o n t and i s s e p a r a t e d
f r o m t h e c l e a r w a t e r t o t h e back o f t h e wave by a d a r k l i n e m a r k i n g t h e
v e r t i c a l segment o f t h e wave f a c e (see F i g u r e 8 ) .
The p o s i t i o n where t h i s
d a r k l i n e f i r s t appears i n t h e shoreward moving wave i s d e f i n e d as t h e
b r e a k e r p o i n t ( y ' } . The b r e a k e r a n g l e (9^^) i s d e f i n e d as t h e a n g l e
between t h e w a v e ^ c r e s t a t t h e b r e a k e r p o i n t and t h e mean shore l i n e .
The average maximum p o s i t i o n a t t a i n e d b y t h e b o r e w h i c h forms a f t e r t h e
wave b r e a k s i s d e f i n e d as t h e runup l i m i t ( y ' ) .
The c o o r d i n a t e , y ' , i s
h o r i z o n t a l , p e r p e n d i c u l a r t o t h e s t i l l w a t e r l i n e , has i t s o r i g i n a t
t h e t o p o f t h e beach and i s p o s i t i v e g o i n g o f f s h o r e (see F i g u r e 3) .
Breaker l o c a t o r . Breaker p o i n t and b r e a k e r a n g l e were measured
v i s u a l l y w i t h a b r e a k e r l o c a t o r c o n s i s t i n g o f two h o r i z o n t a l p l a t e s
whose f r o n t edges had been f i x e d i n t h e same v e r t i c a l p l a n e . The l o w e r
o f t h o s e two p l a t e s i s opaque l u c i t e and p i v o t s f r o m a p r o t r a c t o r w h i c h
can-move a l o n g t h e t r a v e l i n g cross bar (see F i g u r e 9 ) . A second l u c i t e
p l a t e i s f i x e d on dowels 0.25 f e e t above t h e f i r s t .
W i t h t h e p l a t e s p a r a l l e l t o t h e shore l i n e ( 9 = 0 ) , breaker p o i n t
was measured b y s l i d i n g t h e b r e a k e r l o c a t o r o f f s h o r e u n t i l t h e d a r k l i n e
d e f i n i n g t h e b r e a k e r p o i n t j u s t d i s a p p e a r e d b e n e a t h t h e opaque p l a t e .
The p o s i t i o n o f t h e b r e a k e r was t h e n r e a d f r o m a t a p e on t h e c r o s s b a r .
The measurement was r e p e a t e d b y s l i d i n g t h e l o c a t o r i n f r o m o f f s h o r e
and n o t i n g t h e f i r s t appearance o f t h e d a r k l i n e .
The average o f t h e s e
two r e a d i n g s was r e c o r d e d as t h e b r e a k e r p o s i t i o n ( y ^ ' ) «
W i t h the breaker l o c a t o r a t the breaker p o i n t , breaker angle (9^)
was measured by p i v o t i n g t h e p l a t e s u n t i l t h e i r f r o n t edges f e l l i n t h e
same v e r t i c a l p l a n e as t h e wave c r e s t a t b r e a k i n g . The a n g l e was t h e n
r e a d d i r e c t l y f r o m t h e p r o t r a c t o r on t h e b r e a k e r l o c a t o r .
I n measurements o f b o t h y ' and 9 , t h e o b s e r v e r ' s l i n e of s i g h t
k e p t v e r t i c a l b y l o o k i n g Sown p a s t t h e l o c a t o r i n such a way t h a t t h e
f r o n t edges o f t h e two p l a t e s c o i n c i d e d . 9 and, y ' were m e a s u r é d b y two
o b s e r v e r s at each o f 5 s t a t i o n s a l o n g t h e t e s t s e c t i o n o f t h e b e a c h .
was
Runup l i m i t .
The d i s t a n c e between t h e beach t o p and r u n u p l i m i t
( v ' ) was measured w i t h a t a p e b y two o b s e r v e r s a t t h e 5 s t a t i o n s where
yi'^and 9^ were measured.
The c o n v e r s i o n o f y ' and y ^ , d i s t a n c e s measured
f r o m t h e % e a c h t o p , t o r and b , d i s t a n c e s measured f r o m s t i l l w a t e r l e v e l ,
is outlined l a t e r .
R e p e a t a b i l i t y and o p e r a t o r v a r i a t i o n .
The p o s i t i o n o f t h e b r e a k e r
p o i n t and runup l i m i t and t h e magnitude Of t h e b r e a k e r a n g l e a r e v i s i b l e
over o n l y a s m a l l f r a c t i o n o f t h e wave c y c l e , and t h e q u a n t i t i e s t h e m s e l v e s
Ih
f l u c t u a t e f r o m wave t o wave. The average p o s i t i o n o f t h e s e q u a n t i t i e s
was meastired v i s u a l l y i n a s t a n d a r d manner by two o b s e r v e r s , b u t i n s p i t e
o f t h e s t a n d a r d i z a t i o n , t h e o b s e r v a t i o n s are s t i l l somewhat s u b j e c t i v e .
I n e v a l u a t i . n g t h e s e measurements i t i s necessary t o know what i s
t h e agreement c"^ong r e p e a t e d measurements o f t h e same q u a n t i t y by one o b s e r v e r ( r e p e a t a b l l j l y ) and what i s t h e d i f f e r e n c e between measurements o f
t h e same q u a n t i t y t y d i f f e r e n t o b s e r v e r s ( o p e r a t o r v a r i a t i o n ) . I f i t can
be assumed t h a t t h e average v a l u e s o f t h e q u a n t i t i e s can be measured i n
t h e i n t e r v a l o f o b s e r v a t i o n (about 1 m i n u t e ) , t h e n t h e r e p e a t a b i l i t y and
o p e r a t o r v a r i a t i o n g i v e measur-es o f t h e accuracy o f t h e method.
Over a s i x week i n t e r v a l , y ' , y ' , and 9 were measured 3 t i m e s b y
one observer f o r t h e c o n d i t i o n s o f lest
I V 2.
I n Table B, t h e r e p e a t a b i l i t y o f y ' and y ' seems about t h e same, and t h e maximum range i n any o f
t h e 9 s e t s o f r e p e a t e d y ' and y ' r e a d i n g s i s 0 , l 6 f e e t .
The i m p o r t a n t
q u a n t i t y here i s y ' - y ' , t h e d i s t a n c e between b r e a k e r p o i n t and r u n u p
l i m i t w h i c h i s on t h e o r d e r o f 2 f e e t i n these t e s t s . I f i t i s assumed
t h a t any measurement i s w i t h i n 0.08 f e e t o f t h e mean, t h e n t h e 'most
p r o b a b l e ' e r r o r i n y^ - y ^ ( T o p p i n g , (2I4)) i s (0.08^ + 0.082)1/2 o r
0.11 f e e t o r about 5 per cent o f y ^ I n a l l b u t k o f t h e 38 t e s t s i n v o l v i n g t y p e B d a t a t h e measurements were r e p e a t e d by a second o b s e r v e r .
The o p e r a t o r v a r i a t i o n , i n d i c a t e d by Table 3, haa-maximum ranges o f 0.23 f e e t f o r measurements o f
y ' and 0,09 f e e t f o r y ' . I n t e s t s I I 2, I I I 2 , I V 2, t h e a b s o l u t e v a l u e
o r o p e r a t o r v a r i a t i o n averages 0,12 f e e t f o r y ^ and O.O3 f e e t f o r y ^ ,
b u t t h e d i f f e r e n c e s betweem t h e average y ' and y ' o f each o b s e r v e r m
t h e s e 3 t e s t s i s o n l y O.Oi; f e e t and 0.02 f e e t . T h i s i n d i c a t e s t h a t y ^
i s subject t o greater i m c e r t a i n t y than y ^ , but t h a t systematic d i f f e rences between o b s e r v e r s a r e s m a l l .
The e x p e r i m e n t a l d a t a p r e s e n t e d
i n t h i s r e p o r t (see Appendix) i s based on t h e average j.^ and y ^ o f t h e
two o b s e r v e r s .
I n Table 3, t h e r e p e a t a b i l i t y o f measurements o f 9^ i s shown t o be
w i t h i n 8.5 d e g r e e s , o r 3.5 degrees i f one measurement i s e l i m i n a t e d ,
f o r 5 s e t s o f 3 r e a d i n g s . For t h e same d a t a , t h e o p e r a t o r v a r i a t i o n
ranges up t o 6 d e g r e e s . U n l i k e measurements o f y ^ and y ' , t h e r e i s a
s y s t e m a t i c d i f f e r e n c e between t h e measurements o f t h e two o b s e r v e r s
(see Table k ) , b u t s i m i l a r t o t h e t r e a t m e n t o f y ^ and y ^ d a t a , t h e
v a l u e s o f t h e a n g l e p r e s e n t e d i n t h i s r e p o r t a r e t h e average 9^ o f t h e
two o b s e r v e r s .
This c h o i c e o f 9 ^ , a c r i t i c a l v a r i a b l e i n t h i s s t u d y ,
i s explained l a t e r .
16
TABLE
3
R E P E A T A B I L I T T AND OPERATOR V A R I A T I O N POR 9^ MEASUREMENT
Measurement
y ' (in f t , )
Observer
CJG
•
y^ ( i n f t . )
RLB"'*'
CJG
(TEST I V
2)
9^ ( i n d e g . )
RLB
CJG
RLB
D i s t a n c e along
Beach i n f e e t
k
0.5U
0.52
2.82
2.76
3.0U,2.79
2,91
7
O.Iió
2.73
2.73
0.li2
0.37
27
0.39
2.68
2.70
2.7h
2.66
15
2.U2
2.ii9
2.i;9
2.31
2.32
2.33
18
0.32
0.31
0.17
0.35
0.3^,0.19
0.31
2.71,2.62
2.29
26
28.5
O.iió
2.86
2.86
11
23.5
32
0.50
30
29
30.5
26
28
26
31
26.5
29
28
29
0.25,0.2130.5
29
0.29
33.5
32.5
'"Measurements t a k e n on 8/3I/63, 9/6/62, I O / I I / 6 2
Average d i f f e r e n c e y ^ -- y ^ - ' 2 , 2 5 f t .
Measixrements t a k e n on 9/6/62
TABLE
h
OPERATOR VARIATION I N AVERAGE BREAJ<ER ANai.,E
Series
Date
_ _ _ _ _
III
IV
b
"
b
A 9^
xn degrees
-^^^y-j^2
March I 9 6 3
S e p t . 1962
AV ( R L B ) '
b
AV
2 . 6 U ~ ~
2.39
3.87
(CJG)
17
m
9, AVE
degrees
"T^
8
17
E.
Change i n Mean Water Level^^jJT^j^^e J J ^ J k ^ ^
D e f i n i t i o n s . The eleA^ation o f t h e s u r f a c e of a motionl^ess body
o f w a t ë F T s ~ d ë f I n e t i as t h e s t i l l w a t e r l e v e l (oWI,) . The i n t e r s e c t i o n
of t h e s t i l l w a t e r l e v e l with a }ilane boaoh i s c a l l e d t h e s t i l l water
l i n e (wSWIAne) , a terin approxlin-itely e q u i v a l e n t t o s h o r e l i n e .
The time
average e l e v a t i o n o f a movjng water jna^face i s d e f i n e d as t h e mean water
l e v e l (MWL) . The e l e v a t i o n oJ' t h o mean water l e v e l measured p o s i t i v e l y
upward f r o m t h e r r b i l l w a t e r l e v e l i d e f i n e d as t h e setup e l e v a t i o n , e,
(see F i g i i r e 10) .
e = m\.
~ GWl.
[51
L i t t l e i s known o f e, except t l i a t i t depends on wave energy b r o u g h t
t o t h e beach and i s d i r ; t i n g u i : ï h e d from w i n d or t i d a l s e t u p .
Variation
i n t h e energy s u p p l i e r ! by wave" t o n a t u r s l beaches causes t h e
Figure 10.
D e f i n i t i o n Sketch
MWL t o f l u c t u a t e . Because wave setup i s a meas\ire o f one mechanism o f
energy d i s s i p a t i o n i n t h e s u r f zone, njeasurements were made t o e v a l u a t e
i t s importance.
Piezometer c o n d u i t s and w e l l s . E i g h t 10 f e e t l o n g p i e z o m e t e r c o n d u i t s
were imbedded f l u s h w i t h t h e s u r f a c e o f the smooth c o n c r e t e b e a c h , and
extend i n a d i r e c t i o n perpendicular t o the s h o r e l i n e at h f o o t i n t e r v a l s
a l o n g t h e beach (see F i g u r e 2 ) . The c o n d u i t s , c o n s i s t i n g o f n e s t e d
aluminum c h a n n e l s , c o n t a i n 6 smootlxLy p o l i s h e d l / l 6 - i n c h p i e z o m e t e r t a p s
so t h a t h o f t h e s e t a p s l i e i n or near t h e s u r f zone (see F i g u r e 11) ,
Three e i g h t h i n c h I D l y g o n t u b i n g l e a d s f r o m each p r e s s u r e t a p t i i r o u g h
t h e piezometer c o n d u i t and a m a n i f o l d t o a p i e z o m e t e r w e l l , w h i c h i s a
U~inch ID t u b e w i t h a hook gage t o measure w a t e r l e v e l (see F i g u r e 12) .
The l a r g e r a t i o o f w e l l - t o - t u b i n g d i a m e t e r and t h e
f l o w i n t h e t u b i n g (]F( 10) so damps t h e f l u c t u a t i o n o f
i n t h e piezometer w e l l t h a t c a r e f u l o b s e r v a t i o n w i t h a
f l a s h l i g h t i s r e q u i r e d t o see any p e r i o d i c r e s p o n s e t o
the basin.
The ^/iscous damping and d i a m e t e r r a t i o are
Ifi
highl^y v i s c o u s
the water l e v e l
hook gage and
wave m o t i o n i n
such t h a t 5
minutes are r e q u i r e d f o r t h e w a t e r l e v e l i n t h e t u b e t o a d j u s t t o a
change o f one mm. i n t h e SWL o f t h e b a s i n .
Measurement o f e. From t h e d e f i n i t i o n o f e q u a t i o n 5, i t i s n e c e s s a r y
t o measure MWL t o o b t a i n e. SWL was measured i n t h e piezometer w e l l b y
r a i s i n g t h e hook gage t o t h e r e f l e c t i o n o f a f l a s h - l i g h t s h i n i n g on t h e
water s u r f a c e .
The wave g e n e r a t o r was t h e n s t a r t e d and a l l o w e d t o r u n
f o r 2,5 m i n u t e s . Then t h e l o c a l mean p r e s s u r e a t t h e t a p s was o b t a i n e d
b y c o n n e c t i n g t h e d i f f e r e n t t a p s , one a t a t i m e , t o t h e w e l l t h r o u g h
t h e m a n i f o l d , and measuring t h e s t e a d y s t a t e l e v e l i n t h e w e l l w i t h
hook gage and f l a s h l i g h t . Steady s t a t e l e v e l was a t t a i n e d i n t h e w e l l
when s u c c e s s i v e r e a d i n g s o f t h e w a t e r s u r f a c e , t e n minutes a p a r t , had
a v a r i a t i o n o f no more t h a n 0.1 mm.
A f t e r some p r a c t i c e , r e a d i n g s c o u l d be made b y d i f f e r e n t o b s e r v e r s w i t h an o p e r a t o r v a r i a t i o n o f no more t h a n 0.2 mm. P r e c i s i o n was
0.1 mm. The t i m e o f each measurement was always r e c o r d e d .
The f i n a l SWL measured a t t h e end o f t h e t e s t was almost always
l o w e r t h a n t h e i n i t i a l SWL. This drop i n SWL was r e c o r d e d t h r o u g h t h e
t e s t a t a damped piezometer w e l l i n a q u i e t s e c t i p n o f t h e b a s i n ( w a t e r
l e v e l p o i n t gage o f F i g u r e 2 ) .
From t h e r e c o r d o f t h i s p o i n t gage, t h e
' e v a p o r a t i o n c o r r e c t i o n ' , as much as 1.0 mm. i n a d a y ' s r u n c o u l d be
computed f o r t h e d i f f e r e n t t i m e s o f p r e s s u r e t a p measurements.
Indic a t i o n s f r o m t h e o r y and experiment are g i v e n l a t e r t h a t t h e p i e z o m e t e r
t a p b e n e a t h t h e waves measures t h e mean h y d r o s t a t i c head, and t h u s
i n d i c a t e s MWL.
F.
V e l o c i t y Measurement
( l y p e D Equipment and Procedure)
V e l o c i t y i n t h e l o n g s h o r e c u r r e n t was measured by j o i n t use o f
f l o a t s and a v e l o c i t y probe ( p r o p e l l o r - t y p e m i n i a t u r e c u r r e n t meter) .
The f l o a t s were r e c t a n g u l a r s o l i d s o f s o f t w o o d 5 / 8 - i n c h by 5 / 8 - i n c h b y
5 / l 6 - i n c h w h i c h f l o a t e d about I/I4 above w a t e r when w e t . The v e l o c i t y
p r o b e was an Armstrong M i n i f l o w m e t e r AWE I83A ISSUE A w h i c h counts
e l e c t r o n i c a l l y t h e r e v o l u t i o n s o f a jewel-mounted p r o p e l l e r i n a 5 / 8 - i n c h
d i a m e t e r housing (see F i g u r e I 3 ) . When a c t u a t e d b y u n i f o r m f l o w i n t h e
d i r e c t i o n o f t h e p r o p e l l e r a x i s , these c o i m t s are c o n v e r t a b l e t o v e l o c i t y
u s i n g t h e manufacturer's c a l i b r a t i o n c u r v e . This c u r v e was r e p e a t e d l y
checked i n a t o w i n g t a n k and f o u n d t o be a c c u r a t e t o w i t h i n 2 per c e n t .
However, t h e response o f t h e probe depends upon i t s o r i e n t a t i o n
w i t h r e s p e c t t o t h e v e l o c i t y v e c t o r o f t h e f l u i d , and s i n c e f l u i d
v e l o c i t i e s i n t h e s u r f zone v a r y i n an unknown manner, b o t h i n magnitude
and d i r e c t i o n , t h i s i n s t r u m e n t cannot be used d i r e c t l y t o measure l o n g s h o r e c t i r r e n t v e l o c i t y . L i n e a r r e l a t i o n s were f o u n d t o e x i s t between
t h e a p p a r e n t v e l o c i t y g i v e n by t h e probe when a l i g n e d w i t h t h e l o n g s h o r e
c \ i r r e n t and t h e mean s u r f a c e v e l o c i t y o b t a i n e d b y t i m i n g t h e t r a v e l o f
t h e f l o a t s . C a l i b r a t i o n s were developed f o r S e r i e s I I , I I I , and I V by
comparing t h e maximum i n d i c a t e d v e l o c i t y a t a s t a t i o n w i t h l o c a l f l o a t
19
7
X2
VERTICAL
EXAGGERATION
FEET
F i g . 11
L o c a t i o n o f Taps f o r Damped Piezometers
0
MOUNT FOR HOOK GAGE
FEET
C
TO BASIN AND
PIEZOMETER TAPS
PIEZOMETER
WELL
MANIFOLD
F i g . 12
Piezometer W e l l
20
v e l o c i t y (see F i g u r e l i ; ) . D u r i n g t h e s e t e s t s t h e probe was a t m i d - d e p t h ,
midway between t h e b r e a k e r l i n e and t h e SWLine.
C e r t a i n c a u t i o n s must be observed when u s i n g t h e s e c a l i b r a t i o n curves
1.
The s c a t t e r
i s n o t random b u t r e p r e s e n t s i n p a r t , a v a r i a t i o n i n
%•
2.
They may be i n e r r o r when a p p l i e d t o probe p o s i t i o n s ( i n t h e
y d i r e c t i o n ) d i f f e r e n t f r o m t h a t used i n c a l i b r a t i o n . I n o t h e r w o r d s ,
i n d i c a t e d v e l o c i t y v a r i a t i o n s i n t h e y d i r e c t i o n may be a c t u a l l y due t o
a changing c a l i b r a t i o n . This i s p a r t i c u l a r l y l i k e l y i n s h o r e o f t h e
SWLine where t h e p r o b e i s o f t e n exposed d u r i n g p a r t o f t h e wave c y c l e .
I n 31 t e s t s w h i c h i n c l u d e d t h i s t y p e o f d a t a , v e l o c i t y was measured
w i t h t h e probe a t l e a s t
and u s u a l l y 7, s t a t i o n s a l o n g t h e beach and
a t each s t a t i o n 3, ,U, o r 5 measurements were made on a l i n e p e r p e n d i c u l a r
t o t h e s h o r e l i n e i n t h e s u r f zone (see A p p e n d i x , Table A 7 ) . Each i n d i c a t e d p r o b e v e l o c i t y i s t h e average o f 50 seconds o f c o i m t i n g . Whenever
p o s s i b l e , probe e l e v a t i o n was a t t h e e s t i m a t e d mean d e p t h .
The d i s t a n c e t r a v e l e d by t h e f l o a t s i n an i n t e g r a l number o f wave
p e r i o d s (U or 5) was measured w i t h a t a p e f o r two s e c t i o n s o f t h e
beach i n each o f t h e 31 t e s t s .
By assuming t h e v e l o c i t y v a r i e d l i n e a r l y
between t h e s e two s e c t i o n s , t h e mean f l o a t v e l o c i t i e s w e r e a d j u s t e d m
a l l t h e t e s t s t o comnion p o s i t i o n s 10 f e e t and l 6 f e e t f r o m t h e upstream
training wall.
The necessary a d j u s t m e n t was made as s m a l l as p o s s i b l e
by t h e i n i t i a l c h o i c e o f t h e s e c t i o n over w h i c h t o measure t h e f l o a t
travel.
Each f l o a t v e l o c i t y i s t h e average o f 10 measurements d u r i n g
w h i c h measurements t h e f l o a t s d i d n o t ground on t h e b e a c h .
TAHLl!; 5
DYE Al'JD FLOAT VEI.0C1TY IN LONCSHOtdjl GUIdÜilNTS
Test
Dye Velocity'""
ft/sec.
Ill
2
III
I;
III
6
lo31
1.33
1.'13
1.15
0.97
\ a c h number i s t h e average o f 5 measui'ements
over t h e downstream s e c t i o n o f b e a c h .
Midp o i n t o f dye p a t c h was t i m e d .
21
Float Velocity
ft/sec.
1.33
1.32
1.2)4
1.23
1.00
F i g . 13
V e l o c i t y Probes
a
FLOAT VELOCITY, Vf, IN FEET PER SECOND
F i g . Ih
C a l i b r a t i o n o f V e l o c i t y Probes w i t h
Surface Floats
22
I t has been n o t e d (10) t h a t f l o a t s vised i n t h e s u r f a r e s u b j e c t t o
s u r f b o a r d i n g e f f e c t s w h i c h make them t r a v e l f a s t e r t h a n t h e w a t e r
particle velocity.
To t e s t t h i s e f f e c t , races were h e l d between a
s m a l l q u a n t i t y o f m a l a c h i t e green-dyed w a t e r and t h e f l o a t s .
The
c o u r s e was a s i x f o o t s t r e t c h over t h e downstream end o f t h e b e a c h ,
and t h e wave c o n d i t i o n s were t h o s e o f t e s t I I I I4 and I I I 6.
The f r o n t
edge o f t h e dye won 6, l o s t 2 , and t i e d 2 f o r t e s t I I I U, and won 7,
l o s t 2, and t i e d 1 f o r t e s t I I I 6. I t appears f r o m t h e s e r e s u l t s t h a t
t h e e f f e c t s o f d i f f u s i o n on dye balances t h e e f f e c t o f h i g h e r s u r f a c e
v e l o c i t y and s u r f b o a r d i n g on t h e f l o a t s i n t h e s e t e s t s . Measured dye
and f l o a t v e l o c i t i e s show o c c a s i o n a l d i f f e r e n c e s ( T a b l e 5)«
G.
General Basin Conditions
Care was t a k e n t o p e r f o r m t h e d i f f e r e n t s e r i e s and t y p e s o f exp e r i m e n t s under as n e a r l y i d e n t i c a l c o n d i t i o n s as p o s s i b l e (see Table 2 ) .
The p l u n g e r t o t h e wave g e n e r a t o r was l e v e l e d a f t e r each change i n wave
g e n e r a t o r p o s i t i o n and t h e mean e l e v a t i o n o f t h e p l u n g e r m a i n t a i n e d
w i t h i n 0.01 f e e t .
The a n g l e between p l u n g e r and s h o r e l i n e (9 ) was
measured w i t h t a p e and t r a n s i t .
T r a i n i n g w a l l s were c u r v e d f o r r e f r a c t i o n a t t h e 10 and ^2 p o s i t i o n b u t n o t f o r t h e 27° p o s i t i o n . For a l l
e x p e r i m e n t s t h e upstream t r a i n i n g w a l l s extended t o t h e t o p o f t h e beach
and t h e e x i t w i d t h f o r t h e l o n g s h o r e c u r r e n t s between t h e downstream
t r a i n i n g w a l l and t h e s h o r e was m a i n t a i n e d t o w i t h i n a f e w t e n t h s o f a
f o o t . SWL was k e p t t o w i t h i n O.OO3 f e e t o f a c o n s t a n t v a l u e , a v a r i a t i o n w h i c h i s an o r d e r o f magnitude l e s s t h a n t h e v a r i a t i o n i n b a s i n
f l o o r e l e v a t i o n . Water t e m p e r a t u r e i n t h e s e e x p e r i m e n t s ranged f r o m
Ih
t o 23 G,
No a t t e m p t was made t o s t u d y t h e e f f e c t o f changing t h e b a s i n
g e o m e t r y . Wave h e i g h t j u s t o f f s h o r e o f t h e b r e a k e r showed i n some cases
a pronqunced s e c u l a r i n c r e a s e i n t h e f i r s t f e w minutes a f t e r t u r n i n g on
t h e wave g e n e r a t o r and a g r a d u a l approach t o s t e a d y c o n d i t i o n s l a s t i n g
20 m i n u t e s o r s o .
The d a t a p r e s e n t e d i n t h i s r e p o r t were o b t a i n e d a t
l e a s t 25 minutes a f t e r t h e s t a r t o f t h e wave g e n e r a t o r .
Wave h e i g h t , H ^ , was measured i n f r o n t o f t h e p l u n g e r t o d e f i n e
t h e wave h e i g h t s a s s o c i a t e d w i t h each o f t h e t e s t s , and t h e r e s u l t s a r e
l i s t e d i n Table A 1 .
i s t h e average h e i g h t o f t h e t h i r d t h r o u g h t e n t h
waves t o pass t h e wave gage l o c a t e d i n f r o n t o f t h e p l u n g e r a f t e r t h e
s t a r t o f the plunger.
The f i r s t c o u p l e o f waves were s m a l l e r due m a i r l y
t o random s t a r t i n g p o s i t i o n o f t h e p l u n g e r and t h e d i f f e r e n c e between
group and phase v e l o c i t i e s . A c c u r a t e measixrements o f p l u n g e r f r e q u e n c y
showed t h a t even a t t h e f i r s t complete r e v o l u t i o n , a f t e r s t a r t i n g , t h e '
wave g e n e r a t o r had a t t a i n e d i t s uteady s t a t e f r e q u e n c y . R e f l e c t i o n s
f r o m t h e beach and t r a i n i n g w a l l s became measurable a f t e r about 10 or
12 w a v e s .
These
measui-ements v a r i e d about 7 per c e n t f o r d i f f e r e n t
p o s i t i o n s r e l a t i v e t o t h e p l u n g e r , and v a r i e d l e s s t h a n 2 per cent f o r
r e p e t i t i o n s o f t h e measureinerit a t t h e same p o s i t i o n .
23
li.
C o o r d i n a t e Systems and D e f i n i t i o n
The p r i n c i p a l c o o r d i n a t e system i s a r i g h t - h a n d e d , r e c t a n g t i l a r
x - y - z system (see F i g u r e l 5 ) i n w h i c h x f o l l o w s t h e s t i l l w a t e r l i n e
(SWLine) , y i s p e r p e n d i c x i l a r t o x and l i e s i n t h e p l a n e o f t h e s t i l l
w a t e r l e v e l (SWL), and z i s normal t o t h e x y p l a n e .
The o r i g i n o f
t h i s system i s t h e i n t e r s e c t i o n o f t h e SWLine w i t h t h e f i r s t p i e z o m e t e r
c o n d u i t on t h e beach ( a p p r o x i m a t e l y e q i i i v a l e n t t o t h e upstream t r a i n i n g
w a l l ) . X i s measured p o s i t i v e l y i.n t h e d i r e c t i o n o f l o n g s h o r e c u r r e n t
f l o w ( t h e downstream d i r e c t i o n ) , y i s measured p o s i t i v e l y t o w a r d t h e
wave g e n e r a t o r ( t h e o f f s h o r e d i r e c t i o n ) and z i s measured p o s i t i v e l y
upwards.
As d e f i n e d e a l i e r SWL i s t h e e l e v a t i o n o f t h e w a t e r s u r f a c e i n t h e
b a s i n when t h e f l u i d i s n o t i n m o t i o n , and SWLine i n t h e i n t e r s e c t i o n
o f t h e SWL w i t h t h e b e a c h .
A second c o o r d i n a t e p a r a l l e l t o t h e y a x i s b u t o r i g i n a t i n g a t t h e
t o p o f t h e beach i s d e f i n e d as t h e y ' c o o r d i n a t e .
The d i f f e r e n c e
between y and y ' i s d e f i n e d as k (see F i g u r e l 6 ) ,
The v a l u e o f k
was c a l i b r a t e d a g a i n s t SWL. as measured by t h e water l e v e l p o i n t g a g e ,
and as a f u n c t i o n o f x . I n t h i s way meastirements o f t h e v a r i o u s t y p e s
o f d a t a c o u l d be r e f e r r e d t o a l o c a l and t e m p o r a l v a l u e o f SWLine, s i n c e
t h i s q u a n t i t y f l u c t u a t u e s s l i g h t l y dui-ing a r u n due t o decrease i n SWL
and a l s o v a r i e s a l o n g t h e beach due t o s l i g h t uneveness o f t h e beach
surface.
The r u n u p l i m i t y ' was d e f i n e d e a i - l i e r as t h e average d i s t a n c e f r o m
t h e t o p o f t h e beach t o t h e uppermost p o s i t i o n o f t h e r u n u p .
The r u n u p
d i s t a n c e r i s d e f i n e d here as t h e h o r i z o n t a l d i s t a n c e between SWLine
and y ' (see F i g u r e l 6 )
r
= k^ - y ;
[6]
The b r e a k e r p o s i t i o n y ' was d e f i n e d e a r l i e r as t h e d i s t a n c e between
t h e t o p o f t h e beach and t h e b r e a k e r p o s i t i o n .
The b r e a k e r d i s t a n c e b
i s d e f i n e d here as t h e d i s t a n c e between t h e b r e a k e r p o s i t i o n and t h e
SWLine (see F i g u r e l 6 )
b = y ^ " k ^
[7]
There are t h r e e q u a n t i t i e s measured i n t h e z c o o r d i n a t e : d , t h e
v e r t i c a l d i s t a n c e between I"iWI, and t h e beach s u r f a c e (see F i g u r e 1 0 ) ,
2h
*
F .fïST r
tOnF/Tt
C C' Kl't VIT*
j
y
F i g u r e 1$.
D e f i n i t i o n S k e t c h - C o o r d i n a t e System
Figure l 6 .
SORW
R u n OP
D e f i n i t i o n Sketch
^ C N E .
S l . A^,NiC3
^.'i';-N <.!. \<
Figure 1?.
D e f i n i t i o n Sketch
25
LCMFL
e, t h e l o c a l v e r t i c a l d i u t a n c o behween oWL and MWI., and I I , t h e v e r t i c a l
d i s t a n c e between wave c r e s t and t r o u g h ,
e i s measured p o s i t i v e l y up
f r o m SWI., and d ir. measured p o : ; i t : n ; e l y down f r o m MWI..
S e v e r a l pai-bs o f t h e beach are f r e q u e n t l y r e f e r r e d t o (see F i g u r e 1?)
The s h o a l i n g ?,one l i e s l)etween t h e p o s i t i o n a t w h i c h t h e wave ceases t o
be a deepwater wave and t h e bi-eaker p o i n t , Tlie sui-f zone l i e s between the
brealter p o i n t and the runup l i m i t .
The breaker r e g i o n i s i n t h e o f f shore segment o f t h e s u r f zone between the b r e a k e r p o i n t and t h e p o i n t
where t h e b r e a k i n g wave re-Torms an t h e runup b o r e .
The r u n u p r e g i o n
I S t h e i n s h o r e segment o f t h e surJ' zone between t h e p o i n t a t w h i c h
t h e runup b o r e formji and t h R r'unuj) i i m i t .
The s u b s c r i p t
r e f e r ; ; t o q u a n t i t i e f ; measm-ed a t t h e b r e a k e r p o s i t i o n ,
t h e s u b s c r i p t ^, reïerr.
t o q u a n t i t i e s meanured on t h e f l a t , o f f - b e a c h
p o r t i o n o f t h e b a s i n , an<l t h e s u b s c r i p t
r e f e r s t o h y p o t h e t i c a l deep
w a t e r q u a n t i t i e s o b t a i n e d from small a m p i i t u d e t h e o r y .
V
KXPERIMFMT/IT, RESIJI/K
_ Ï M s s e c t i o n p r e s e n t s and d i s c u s s e s the t r e n d s observed i n t h e b a s i c
variables.
A t a b u l a t i o n of the o r i g i n a l d a t a appears i n t h e Appendage.
^'
^^SXeJjgight, Shape, and Speed
( I ^ ^ e A Data)
^ . M|ieheight.
l l i e r e s i s t a n c e wave gage was used t o measure wave
h e i g h t a t b r e a k i n g (R ) and envelopes o f wave h e i g h t t h r o u g h t h e s u r f
zone; t o t e s t t h e uni^^ormity o f wave h e i g h t c o n d i t i o n s over t h e b a s i n ;
t o measure r e f l e c t i o n f r o m t h e b e a c h ; and t o e s t a b l i s h t h e wave h e i g h t s
w h i c h d e f i n e t h e t e s t numbers o f the e x p e r i m e n t s .
A c o n t o u r map o f t h e wave h e i g h t s i n t h e b a s i n s i m i l a r t o t h a t
p r e s e n t e d b y Savage (Ih) showed t h a t f o r s t e a d y s t a t e c o n d i t i o n s o f
t e s t 111 2, wave h e i g h t on t h e f l a t p o r t i o n o f t h e b a s i n v a r i e d w i t h
l o c a t i o n about 10 o r 15 p e r c e n t f r o m a mean v a l u e , and t h a t i n t h e
^ ^ n " ! u ^ ° ' ' ^ " ^ ' ^ b r e a k i n g , t h e h e i g h t s were g e n e r a l l y h i g h e r a t
X = 10 t h a n a t e i t h e r end o f t h e b e a c h .
w . ? ' ' ' ' ^ ^ ^ ^ f " ^ * envelopes were made by t r a v e r s i n g f r o m o f f s h o r e o f t h e
b r e a k e r p o s i t i o n t o t h e runup l i m i t .
'Pyplosü. p r o f i l e s (see F i g u r e 18)
r e v e a l a s l i g h t i n c r e a s e i n wave h e i g h t t h r o u g h t h e s h o a l i n g zone t o
a s ^ ' ï p T '^r^
^ decrease i n h e i g h t
tLt^
r
Pl""ff ^
i n c r e a s e due t o t h e s p l a s h - u p , and t h e n a
s t e a d y decrease as t h e wave moves toward the r u n u p l i m i t .
The l o c a t i o n
o f t h e b r e a k e r p o s i t i o n j u s t i n s l i o r e o f the maximmn h e i g h t a t t a i n e d
^^'^^
expected f r o m the way b r e a k e r p o i n t i s d e f i n e d
m t h i s i n v e s t i g a t i o n , (see S e c t i o n I V )
°^
^"''^^"P^
H e i g h t s o r s u c c e s s i v e waves a t t h e b r e a k e r p o s i t i o n f l u c t u a t e d
r a n d o m l y about a mean and t h e s e f l u c t u a t i o n s may be due t o random m o t i o n
o f t h e b r e a k e r p o s i t i o n . Because t h e wave gage i s s t a t i o n a r y , random
changes i n t h e breal-rer p o s i t i o n may s h i f t t h e p o s i t i o n o f t h e gage r e l a t i v e t o t h e envelope (see F i g u r e 18).
A t t e m p t s t o c o r r e l a t e t h e h e i g h t o f a wave a t b r e a k i n g w i t h t h e
h e i g h t o f t h e runup b o r e f o r m e d f r o m t h i s wave a f t e r b r e a k i n g l e d t o
t h e c o n c l u s i o n t h a t no e a s i l y demonstrated c o r r e l a t i o n e x i s t s .
From meastirements o f t h e s p a t i a l v a r i a t i o n i n t h e envelope o f wave
h e i g h t s d u r i n g t e s t I I I - - 2 , t h e r e f l e c t i o n c o e f f i c i e n t o f t h e beach
( r e f l e c t e d wave a m p l i t u d e d i v i d e d b y i n c i d e n t wave a m p l i t u d e ) was f o u n d
t o be 0.05,
Wave shape. Mean w a t e r l e v e l a t a p o i n t i s t h e t i m e averaged
w a t e r s u r f a c e e l e v a t i o n a t t h a t p o i n t and wave h e i g h t H i s t h e d i f f e r e n c e
between t h e maximum and minimum e l e v a t i o n o f t h a t w a t e r s u r f a c e .
The
shape o f t h e wave, as used h e r e , i s t h e t e m p o r a l v a r i a t i o n o f t h e w a t e r
surface at a p o i n t .
For waves w i t h s i n u s o i d a l shape, t h e mean w a t e r l e v e l occurs a t
an e l e v a t i o n above t h e wave t r o u g h o f a^ = 1/2 H , b u t f o r f i n i t e a m p l i t u d e waves,
< l / 2 H . A shape f a c t o r , o , (see F i g u r e 19) h a v i n g
a maximum v a l u e o f 0,5 i s thus d e f i n e d :
[8]
a
T h i s r a t i o i s i m p o r t a n t i n computing t h e t o t a l d e p t h f r o m wave
c r e s t t o beach s u r f a c e , and a l s o as an i n d i c a t i o n o f t h e shape o f t h e
wave. By g r a p h i c a l l y a v e r a g i n g t h e wave shape r e c o r d e d w i t h wave gage
and o s c i l l o g r a p h , shape f a c t o r s were o b t a i n e d f o r d i f f e r e n t p o s i t i o n s on
t h e beach (see Table 6 ) .
TAHL,E
6
SHAPE FACTOR VS. POSITION ON BEACH
Test
Position
III
2
several f e e t offshore of
brealcer p o i n t
O.37
III
2
near t h e b r e a k e r p o i n t
O.3U,
near t h e b r e a k e r p o i n t
0.32
II
III
3
2
0.35
' i n runup r e g i o n
27
to
0,U5
0,23
to
0,l45
F i g . 18
F i g . 19
Wave Height Envelope Through S u r f
D e f i n i t i o n S k e t c h - Wave Shape
28
Zone
There i s some evidence t h a t a, l i k e H, v a r i e s e r r a t i c a l l y w i t h
p o s i t i o n on t h e b e a c h , and even w i t h t i m e , b u t t h e d a t a i n d i c a t e t h a t
b e f o r e a p p r o a c h i n g t h e b r e a k e r p o i n t , shape f a c t o r i s h i g h , a t t h e
b r e a k e r p o i n t i t reaches a minimum, and i n t h e runup r e g i o n t h e shape
f a c t o r i s again h i g h .
However, t h e two l o c a t i o n s o f h i g h a c o r r e s p o n d
t o d i f f e r e n t wave shapes. The o f f s h o r e wave shape i s r o u g h l y s i n v i s o i d a l
w h i l e t h a t i n t h e r u n u p r e g i o n , as w e l l as t h e b r e a k e r r e g i o n , i s
a p p r o x i m a t e l y t r i a n g u l a r . Wave shape o f a wave near b r e a k i n g d i d n o t
r e s e m b l e t h a t o f a s o l i t a r y wave i n t h e s e e x p e r i m e n t s , p o s s i b l y because
o f t h e s m a l l d i s t a n c e between t o e o f t h e beach and t h e b r e a k e r p o s i t i o n .
I n t h e a n a l y s i s o f t h e i r f i e l d d a t a Inman and Q\jinn (I3)
assume a =
•> 0.25,
implicitly
Wave s p e e d . The speed o f t h e wave a t b r e a k i n g , as measured w i t h
t h e p a x r o f wave gages and t h e o s c i l l o g r a p h , changes r e l a t i v e l y l i t t l e
as t h e wave approaches t h e b r e a k e r p o i n t .
I t t h e n decreases t o a l o w e r
v a l u e i n t h e runup r e g i o n and c o n t i n u e s t o decrease t o t h e runup l i m i t ,
(see F i g u r e 20),
I n a l l cases where p r o f i l e s o f t h e wave speeds were
made t h r o t i g h t h e s u r f zone, o n l y s l i g h t e x t r a p o l a t i o n o f t h e c u r v e o f
C v e r s u s y i s n e c e s s a r y t o r e a c h t h e r-unup l i m i t , and always t h e wave
speed i n d i c a t e d t h e r e by t h i s e x t r a p o l a t i o n i s w e l l above zero ( a b o u t
1 f o o t p e r second o r 1/2 t o I/I4 o f t h e b r e a k e r speed) . Because measurements a r e made n e a r l y t o t h e r u n u p l i m i t , t h i s i n d i c a t e s t h a t t h e d i s t a n c e
over w h i c h t h e runup b o r e d e c e l e r a t e s t o z e r o a t t h e runup l i m i t , i s
on t h e o r d e r o f t h e gage s p a c i n g (0.25 f e e t ) on t h e l a b o r a t o r y b e a c h .
The wave i s d i f f i c u l t t o d e f i n e a t , and j u s t a f t e r , t h e p o i n t a t w h i c h
t h e p l u n g i n g c r e s t s t r i k e s t h e w a t e r . Wave speed t h e r e has l i t t l e meaning
due t o t h e r a p i d changes i n shape w i t h y , a l t h o u g h one may be measured
f r o m t h e r e c o r d i n g s . When i n s i m u l t a n e o u s u s e , t h e p a i r o f wave gages
v i s u a l l y i n t e r f e r e d e l e c t r o n i c a l l y w i t h each o t h e r , e s p e c i a l l y a t g r e a t e r
depths o f submergence, causing s m a l l h i g h f r e q u e n c y o s c i l l a t i o n s t o be
superimposed on t h e r e c o r d e d wave t r a c e . This e f f e c t was almost u n n o t i c e a b l e
f o r t h e s h a l l o w bores i n t h e r u n u p , b u t was s i g n i f i c a n t when measuring
b r e a k e r speed. However, most o f t h e u n c e r t a i n t y i n wave speed a t t h e
b r e a k e r p o i n t i s caused by p e c u l i a r i t i e s i n t r o d u c e d when one o f t h e gages
was l o c a t e d t o o near t h e p o i n t where t h e p l u n g i n g c r e s t s t r i k e s t h e w a t e r ,
o r b y t h e s u p e r p o s i t i o n o f s m a l l r e f l e c t e d waves. S i n c e each measurement
o f l o c a l wave speed i s t h e average o f 20 i n d i v i d u a l measurements, t h e
e f f e c t o f s h i f t i n g breaker p o s i t i o n i s lessened.
The wave speed a t b r e a k i n g i s an i m p o r t a n t q u a n t i t y i n t h e a n a l y s i s
o f l o n g s h o r e c u r r e n t s (see E q . [2] f o r example) t h u s i t i s u s e f i f L t o be
a b l e t o p r e d i c t i t . The wave l e n g t h s and depths o b t a i n e d i n t h e s e
e x p e r i m e n t s p l a c e t h e b r e a k i n g waves i n t h e c n o i d a l range b u t near t h e
commonly quoted t r a n s i t i o n ( ^ b / L ^ ^ 0.02) f r o m c n o i d a l t o s h a l l o w w a t e r
waves. For t h i s r e a s o n and s i n c e t h e c n o i d a l r e l a t i o n s a r e so cumbersome,
t h e measured wave speed a t b r e a k i n g , G , , w i l l be compared w i t h t h a t o f t h e
s o l i t a r y wave (2h):
29
§
a:
OJ
QUJ
UJ
BREAKER
POINT-
Ü
ij
z
O
ac
o
UJ
UJ
Q.
1.125 SEC
(/)
UJ
X=2
A X = 2 FEET
X = 10.3 O X = 10.3 FEET
X = 17 O X= 17 FEET
^ = 085
>
0.17 FEET
cr = 0.5 IN RUNUP REGION
= 0.3 IN BREAKER REGION
-1.0
-0.5
0,0
0.5
DISTANCE
Fig.
20
1.0
FROM
1.5
2.0
SW L I N E , y, IN F E E T
Wave Speed i n Svrf
Zone
o SERIES n
O SERIES m
Ö SERIES m
2.0
3,0
[gH(,(l-o-t0)]'''2jN FEET PER SECOND
Fig.
21
Wave Speed i n B r e a k i n g
30
2.5
•1
3.0
y (FT)
1/2
C = [gdd-f
|)]
•
[cp]
i n which
a = a m p l i t u d e o f s o l i t a r y wave (maximum d i s p l a c e m e n t above
s t i l l water)
Using t h e n o t a t i o n o f F i g . I 9 , E q . [9] becomes:
1/2
C -
i n which
[ g H ( l » o + p)]
[10]
,
[11]
P =^ H
S^catter°ïsT^i^^' ' " f ' f
"i^^h E q . [10] evaluated at b r e a k i n g ,
as i n t i e L b L a w f ° '
comparisons i n the f i e l d (2^) as w e n
v
i n - the
' t ï r runup r""^^^
v aa xl ii üS iitvy ^ m
e g i o n"^^^"^
as w^e° l l .
^^SSi£E™a£Ë„£Hi2E
^ ° ^° i n d i c a t e i t s approximate
(lype B data)
c r i b S r S e wid't'h"? Ï h ' ' """l^P ' ' ^ ' " ^ ' ^ ^
geometrical f a c t o r s desSows
1 . . ^ ^ ^
which the longshore c u r r e n t
llows
I n a d d i t i o n the runup e l e v a t i o n , mr, i s a measure o f the
p o t e n t i a l energy stored t e m p o r a r i l y i n the runup r e g i o ^ and tóe breaker
depth d^ = mb IS an important v a r i a b l e d e s c r i b i ^ b r e a k e r p o s i t i o n !
^ ' c
As described e a r l i e r there were consistent d i f f e r e n c e s i n Q (nr.
o ^ n ' ' ' - ' '
^-tween the measurements o f S o o b s e ^ v l s
the t h e o r y , p a r > t i c u l a r l v i n L r L .
error i s most s i g n S S a ^ t !
?t
31
k " ^ ^ f ^ ^ d values agreed best w i t h
^"^^^
^"'^^^
Table 7 compares t h e average o b s e r v a t i o n w i t h t h e t h e o r e t i c a l
o b t a i n e d f r o m F i g u r e h o f Johnson, 0'Bj:'ien and I s a a c s (26) .
TAHLE
value
7
BREAKER MGLE FROM SMALL A^ffLI'fUDE IHBORY
AND FROn 0BSER.VA'I10M
Test
No.
Series
0- "
.
m
I I
theory
observation
m
theory
jv
observation
theory
degrees
2
3
5
h
h
h
Iu5
5.5
6
12.5
Iii.5
11.5
10.5
11
11.5
10
8
h
9 '
10
11
12
13
observi'
o
a.5
3.5
9
li.5
11
U.5
5.^
11.5
a
11
12. c;
8
23
18
17
10.5
13.5
3.5
10.5
15.5
13
17
17.5
17
18;:
The o b s e r v a t i o n s , w h i c h are t h e averages o f 2 r e a d i n g s o f each o f
t h e two o b s e r v e r s , agree w e l l w i t h t h e o r y .
The agreement i s p o s s i b l y
b e t t e r t h a n s h o u l d be e x p e c t e d , f o r f i n i t e a m p l i t u d e e f f e c t s p r o b a b l y
ought t o decrease r e f r a c t i o n , t h u s r e s u l t i n g i n observed v a l u e s o f 0
h i g h e r t h a n computed. However, o f t h e 26 o b s e r v a t i o n s i n Table 7, 1^
are g r e a t e r and 12 are l e s s t h a n t h e t h e o r e t i c a l v a l u e s .
Mean Water L e v e l
(Type C d a t a )
On a l a b o r a t o r y beach on w h i c h waves a r e b r e a k i n g , i t i s o b s e r v e d
t h a t t h e SWLine i s p e r m a n e n t l y under w a t e r and t h a t t h e s h o r e l i n e o s c i l l a t e s between t h e runup l i m i t and a l o w e r e l e v a t i o n w h i c h i s above t h e
SWLine. The d i s t a n c e over w h i c h t h e i n t e r f a c e o s c i l l a t e s i s on t h e o r d e r
o f 1/h r f o r t h e s h o r t e s t p e r i o d waves o f t h i s s t u d y (T = 1.0 seconds)
and perhaps r f o r t h e l o n g e s t p e r i o d waves ( T = 1.5 s e c o n d s ) .
The w a t e r
f r o m t h e bore l e a v e s t h e r u n u p r e g i o n by a smooth d r a i n i n g so t h a t
d e f i n i t i o n of the receding shoreline is d i f f i c u l t .
C a l i b r a t e d wave gages were p l a c e d i n t h e s u r f zone b e f o r e t h e wave
g e n e r a t o r war. s t a r t e d and the [ W , was r e c o r d e d .
Then, a f t e r r u n n i n g t h e
3'''
29
1?
19
9
6
7
8
" a l l measurements a t x = 10 f e e t , and t h e average o f two o b s e r v e r s
t o the nearest h a l f degree.
0.
.
21
23
wave g e n e r a t o r f o r 2^ m i n u t e s , t h e bore p a s s i n g t h e gage was r e c o r d e d .
By g r a p h i c a l i n t e g r a t i o n o f t h e s e r e c o r d s t h e MML was e s t a b l i s h e d and b y
s u b t r a c t i n g t h e SWL, t h e l o c a l v a l u e o f e was o b t a i n e d . A r e p r e s e n t a t i v e
comparison o f t h e MWL d e t e r m i n a t i o n by t h i s " e x a c t " t e c l m i q u e and f r o m
t h e damped p i e z o m e t e r e i s g i v e n i n F i g . 22.
I t i s concluded t h a t t h e
p i e z o m e t e r s g i v e a s a t i s f a c t o r y v a l u e o f MWL even i n t h e s u r f z o n e »
A l s o shown on F i g . - 22 i s t h e envelope o f wave h e i g h t w h i c h f o r t h i s
r u n i n d i c a t e s a a i n t h e s u r f zone o f between 1/3 and l / 2 .
A u s e f u l r u l e f o r computing d e p t h o f b r e a k i n g on t h i s l a b o r a t o r y beach
was o b t a i n e d f r o m a n a l y s i s o f t h e curves o f e v e r s u s y . Such curves
t y p i c a l l y show t h a t e decreases i n t h e o f f s h o r e d i r e c t i o n , becoming z e r o
b e f o r e t h e b r e a k e r p o s i t i o n and v e r y s l i g h t l y n e g a t i v e a t t h e b r e a k e r
p o s i t i o n . F i g u r e 23 i s a h i s t o g r a m o f t h e f r e q u e n c y o f e^ m a g n i t u d e s ,
where e^ i s t h e d i f f e r e n c e between MWL and SWL a t t h e b r e a k e r p o i n t .
A mean e i s about -1mm (0.003 f e e t ) and an average b r e a k i n g d e p t h i s
about 0.12 f e e t ; t h e r e f o r e t o a good a p p r o x i m a t i o n , t h e b r e a k e r d e p t h
d^ i s t h e s t i l l w a t e r d e p t h a t t h e b r e a k e r p o i n t .
For t h i s r e a s o n ,
b r e a k e r d e p t h i s d e f i n e d i n t h i s r e p o r t as
d^ « m b .
D,
Longshore C u r r e n t V e l o c i t y
[12]
( l y p e D data)
'Transverse D i s t r i b u t i o n :
I n t h e y ( o f f s h o r e ) d i r e c t i o n t h e maximum
measured v e l o c i t y a t a g i v e n x ( l o n g i t u d i n a l p o s i t i o n ) was u s u a l l y f o u n d
a f e w t e n t h s o f a f o o t o f f s h o r e o f t h e SWLine and t h e minimum v a l u e
occurred at the breaker p o i n t .
This i s i l l u s t r a t e d i n F i g . 2I4. I t must
be remembered however t h a t t h e s e d i s t r i b u t i o n s are t h e r e s u l t o f v e l o c i t y
p r o b e r e a d i n g s made a t m i d - d e p t h and t h a t t h e probes were c a l i b r a t e d
a g a i n s t s u r f a c e f l o a t measurements o n l y when l o c a t e d i n t h e c e n t e r o f t h e
s u r f z o n e . Comparisons o f s u r f a c e f l o a t and dye v e l o c i t i e s p r e s e n t e d
e a r l i e r i n d i c a t e t h e s u r f a c e v e l o c i t y t o be c l o s e t o t h e mean v e l o c i t y
t h u s t h e probe v e l o c i t i e s a t t h e c e n t e r o f t h e s u r f zone are p r o b a b l y
good r e p r e s e n t a t i o n s o f t h e mean. Near t h e shore l i n e and t h e b r e a k e r
l i n e t h e accuracy o f the probe readings
i s unknown.
V e r t i c a l and L o n g i t u d i n a l D i s t r i b u t i o n
O b s e r v a t i o n s o f t h e n e t p a r t i c l e paths i n t h e s h o a l i n g zone were
made u s i n g dye and paper f l o a t s .
The observed p a t t e r n o f t h e c u r r e n t s
i s complex and changes w i t h t h e t e s t c o n d i t i o n s . R e s u l t s o f a t y p i c a l
s e r i e s o f o b s e r v a t i o n s f o r b o t t o m , m i d - d e p t h , and s u r f a c e about 1.5 f e e t
o f f s h o r e o f t h e b r e a k e r p o i n t are g i v e n i n Table 8, For t e s t I V 6 t h e r e
was a n o t i c e a b l e upstream d r i f t a t x = 15 and I 7 a t t h e same d i s t a n c e
offshore.
33
s
\
.30
T"l.00
sec
"0.19
Beach
slope".109
ft
O Piezometer
well r e a d i n g s of
A
determination
Wove
-o W a v e
gage
MWL
of MWL
envelope
.20
e n v e l o p e of wove c r e s t s
.10
-.20
,30
•40 I
-10
1.0
Y
F i g . 22
2.0
D i s t a n c e from S W L i n e , In
feet
Mean Water L e v e l i n Siarf Zone
10
O
UJ
O
UJ
tr
5 h
Ll
0
3
-2
-I
e|~,, IN
F i g . 23
0
mm
Frequency D i s t r i b u t i o n o f Set-Up i n MWL
3h
3.0
I
I
V, LONGSHORE CURRENT VELOCITY, IN FEET PER ^ C O I ^ '
1.0
0
10
0
LO
0
1,0
o»
1,© 0 !
LO
breaker
line
o Velocity measurement
TEST E - 3
e Breaker position
6
9
11
13
15
17
X, DISTANCE FROM UPSTREAM TRAINING WALL, IN FEET
F i g . 21;
Velocity Distribution
i n Longshore C u r r e n t
H- 1,0
UJ 0.5
0
? ÜJ 1.0
. H 0.5
0
20
-.1^
CD
10
0'
°0.2
UJ
- y
0,1
0
1.0
0.5
0
Y =0.5 FT
• Y = l,0 FT
Y = 0.5FT
i Y = l,0 FT
,SWL
SWL
SW
1.5
UJ K
UJUJ 1.0
U- CA)
§ tE 0,5
. ÜJ
0
12
IN
FEET
16
TEST 114
12
IN FEET
= 10°
F i g . 25
16
TEST i n 4
4
IN
FEET
, = 27°
Longshore V a r i a t i o n s f o r C o n d i t i o n s o f
Test 1|
35
12
16
TEST
EE 4
TABLE
8
NET PARTICLE MOTION I N SHOALING ZONE
(TEST I V h)
bottom
mid-depth
surface
1
onshore ( f a s t )
onshore ( f a s t )
onshore ( v e r y slow)
onshore ( f a s t )
5
downstream ( s l o w )
downstream and o f f shore ( s l o w )
onshore
X (ft)
8
11
15
18
o f f s h o r e and downstream
s l i g h t l y offshore
downstream (no
onshore)
downstream o n l y
downstream o n l y
onshore ( f a s t )
onshore and down^
stream ( f a s t )
downstream and
onshore
downstream and
s l i g h t l y onshore
"1.5 f t . o f f s h o r e o f b r e a k e r .
The q u a l i t a t i v e o b s e r v a t i o n s o f T a b l e 8 show t h a t a t t h e u p s t r e a m
end o f t h e beach f l u i d i s drawn r a p i d l y i n t o t h e l o n g s h o r e c u r r e n t and
that-%he l o n g s h o r e qqmponent o f m o t i o n appears t o _ i n c r e a s e w i t h d i s t a n c e
from the t r a i n i n g w a l l .
The component o f n e t onshore f l u i d m o t i o n d e creases w i t h ^ d i s t a n c e ' T r o m ^ . t h e
indicating a reduction i n
mas^^transf'er ' a c r o s s , t h e b r e a k e r as t h e l o n g s h o r e c u r r e n t becomes
e s t a b l i s h e d . There were no s t r o n g n e t o f f s h o r e motions o b s e r v e d . I n
o t h e r w o r d s , t h e f l u i d i n t h e s u r f zone appears t o r e m a i n t h e r e .
The
s u r f a c e v e l o c i t y i n t h e s h o a l i n g zone shows t h e most c o n s i s t e n t s h o r e ward t r e n d .
A t t h e downstream end o f ' t h e b e a c h , some o f t h e f l u i d , i n s t e a d o f
f l o w i n g o f f t h e t e s t beach i n t h e l o n g s h o r e c u r r e n t , f l o w s o f f s h o r e along
t h e downstream t r a i n i n g w a l l .
This i s p a r t o f a g e n e r a l c o u n t e r - c l o c k w i s e
p a t t e r n i n t h e t e s t area bounded b y t h e t r a i n i n g w a l l s , t h e p l u n g e r and
t h e beach.
Some examples o f t h e l o n g i t u d i n a l g r o w t h o f t h e mean l o n g s h o r e v e l o c i t y
i n t h e s u r f zone a r e shown i n F i g u r e 25.
E.
Breaker H e i g h t and D e p t h
Because o f t h e i m p o r t a n c e o f b r e a k e r c o n d i t i o n s i n t h e g e n e r a t i o n
o f l o n g s h o r e c u r r e n t s a b r i e f r e v i e w i s g i v e n here o f t h e f i n d i n g s o f
o t h e r i n v e s t i g a t o r s as w e l l as t h o s e o f t h i s s t u d y .
T h e o r e t i c a l l y and
e x p e r i m e n t a l l y determined values of the breaker height t o depth r a t i o ,
YL^/d^, are g i v e n i n Table 9'
36
/ /
TABLE
ESTimTES
Reference
Mc Cowan (2?)
Davies (28)
L a l t o n e (29)
I p p e n and
K u l i n (30)
I p p e n and
K u l i n (30)
L a l t o n e (29)
I v e r s o n (31)
II
II
II
L a r r a s (32)
II
II
Eagleson (33)
9
O F BREAI<:ER D E P T H -
Wave Type
Theory or
BREAKER H E I G H T R A T I O S
observation
Slope
solitary
solitary
solitary
theory
theory
theory
0
0
0
0.78
0.827
< 0.71J4
solitary
laboratory
0
< 0.72
solitary
cnoidal
oscillatory
II
II
II
oscillatory
II
II
oscillatory
laboratory
theory
laboratory
II
II
II
laboratory
It
II
II
laboratory
II
II
II
II
II
II
II
laboratory
field
0.0227
0
0.10
0.05
0.033
0.020
0.091
0.020
0.010
0.067
0.066
0.098
0.100
0.139
1.20
< 1.33
Putnam e t al(12)
"
tl
II
II
II
II
II
II
II
II
II
II
II
II
II
Series I I , I I I ,IV o s c i l l a t o r y
oscillatory
Munk (25)
0.11i3
O.llil;
O.2I4I
0.260
0.100
7
l.Olt
0.8a
0.76
0.82
0,86
0.75
0.68
1.12
0.739
0,733
0.772
0.7^8
0.7^2
0.6^8
0.71a
0.727
1.18
0.52 t o 1.28
I p p e n and K u l i n (30) p o i n t o u t t h a t a l t h o u g h c o n s t a n t d e p t h t h e o r i e s
g i v e a r e a s o n a b l e d e s c r i p t i o n o f t h e s h o a l i n g o f o s c i l l a t o r y wavesto a
p o i n t c l o s e t o b r e a k i n g , a s m a l l s l o p e causes a g r e a t d e p a r t u r e f r o m
t h e o r y f o r t h e s o l i t a r y wave.
The v a l u e s g i v e n i n Table 9 p o i n t t h i s
o u t . L a l t o n e ' s h i g h o r d e r a p p r o x i m a t i o n s (29) g i v e a l i m i t i n g v a l u e
o f H|^/d^ < 0.71a f o r t h e s o l i t a r y wave on a h o r i z o n t a l b o t t o n and f r o m
over a hundred such measurements I p p e n and K i i l i n (30) f o u n d a maximum
v a l u e o f 0.72.
For t h e o s c i l l a t o r y wave L a l t o n e (29) g i v e s a l i m i t o f
1.33 f o r Hj^/d^ on a h o r i z o n t a l b o t t o m , t h e a c t u a l v a l u e depending upon
t h e wave f o r m . Hundreds o f o b s e r v a t i o n s on s l o p e s o f a w i d e r a n g e do
n o t exceed 1.28.
I n any p a i - t i c u l a r case, t h e b r e a k i n g wave shape (and
hence H , / d ^ ) w i l l depend upon t h e i n i t i a l shape b e f o r e s h o a l i n g b e g i n s
and upon t h e beach s l o p e ,
Miche (3a) i n c o r p o r a t e s
breaking to obtain
t h e wave p e r i o d , T, i n t o t h e t h e o r y o f wave
37
SERIES IT
0
0.01
002
SERIES I E
0.03
0
0.01
F i g . 26
002
SERIES E I
0
0.03
0.01
0O2
PUTNAM, et al, ( I Z )
0.03
0
Breaker H e i g h t and Breaker Depth
001
0O2
0.03
004
0.05
L
O.ia tanh
O
L
[13]
O
l n which
L
O
^
« 5.12
Uk]
The l a b o r a t o r y d a t a o f t h i s s t u d y and o f Putnam e t a l (12) a r e compared
w i t h ' E q . [13] I n F i g . 26.
For s m a l l v a l u e s o f C I ^ A Q E q . [13] i s seen
t o be g i v e n c l o s e l y b y t h e r e l a t i o n s h i p l / P ^ " 0.88.
S i n c e t h e range
o f v a r i a b l e s was about t h e same i n t h e two s e t s o f e x p e r i m e n t s i t w o t i l d
appear as t h o u g h d i f f e r e n t b r e a k i n g c r i t e r i a had been employed.
F.
General L o n g i t u d i n a l N o n u n i f o r m i t y
One i m p o r t a n t r e s u l t o f t h e l a b o r a t o r y s t u d y i s t h e o b s e r v a t i o n
t h a t s u r f c o n d i t i o n s are non-uniform i n the x d i r e c t i o n .
The d i f f e r e n t
types- o f d a t a , c o n s i d e r e d as a w h o l e , r e i n f o r c e t h i s i n t e r p r e t a t i o n .
Histograms were c o n s t r u c t e d f o r each o f 5 c l a s s e s o f d a t a showing
where t h e maximum and minimum v a l u e s o f t h e g i v e n v a r i a b l e o c c t i r r e d on
t h e t e s t s e c t i o n o f t h e beach.
F i g u r e s 2?, 28, 29, 30, and 31 each show
6 h i s t o g r a m s : t h e maximum and minimum p o s i t i o n o f t h e g i v e n v a r i a b l e f o r
a l l t e s t s i n each o f t h e 3 s e r i e s .
I n cases where t h e maximum or m i n i mum o c c u r r e d a t two l o c a t i o n s , b o t h were p l o t t e d .
F i g u r e 27 shows t h a t v e l o c i t y maxima occur p r e d o m i n a t e l y downstream
and t h e minima upstream f o r a l l s e r i e s .
F i g u r e 28 shows t h a t e i n c r e a s e s
(MWL r i s e s ) i n t h e downstream d i r e c t i o n a l s o .
S e r i e s I I I w h i c h does n o t
d e m o n s t r a t e t h i s i s based on 3 r u n s o n l y , and t h e y a l l f o r t h e same t e s t
(III 2).
F i g u r e s 29 and 30 show an i n v e r s e c o r r e l a t i o n between b and r .
In
g e n e r a l , b i s a minimum and r a maximum a t t h e downstream end o f t h e
t e s t b e a c h , and t h e r e v e r s e h o l d s a t t h e upstream e n d .
I t appears f r o m t h e s e k f i g u r e s t h a t l o n g s h o r e c u r r e n t s c a r r y more
f l u i d w i t h a h i g h e r v e l o c i t y on t h e downstream end o f t h e b e a c h , and
t h a t because w a t e r l e v e l i s r a i s e d downstream, t h e b r e a k e r p o i n t moves
c l o s e r t o s h o r e and t h e runup l i m i t moves h i g h e r up t h e b e a c h .
I f water
l e v e l i s r a i s e d a t t h e downstream end o f t h e b e a c h , t h e r e s h o u l d be a
l o c a l decrease i n 0^ because t h e waves have a s l i g h t l y g r e a t e r d i s t a n c e
t o r e f r a c t b e f o r e b r e a k i n g . The h i s t o g r a m o f F i g u r e 31 p r e s e n t no c l e a r
39
LOCATION OF V ^ j n
LOCATION OF V ^ a x
y = 0 5 FEET
SERIES
H
SERIES
y = 1.0 FEET
SERIES I I
e,=io=
9^=10=
I5r
•
HI
SERIES HI
10
10
3
e, = 2 7 '
O
o
,= 2 7 °
LÜ
CC
SERIES
nr
SERIES m
UJ 10
3
O
O
,= 51°
ë 51
0
n
JIL
5
10
15
X, IN FEET
F i g . 27
20
5
10
15
5
20
X, IN FEET
L o c a t i o n o f Maximum and Minimum Longshore
Current V e l o c i t y
X, IN
F i g . 28
10
15
FEET
20
Lai
5
10
X, IN
15
20
FEET
L o c a t i o n o f Maximuir, and Minimum Mean
Water L e v e l
LOCATION OF b .
LOCATION OF b„
LOCATION OF r ^ o x
LOCATION OF rrnin
15
SERIES H
SERIES I I
10
e, = io=
JIL
XIJ
X I
SERIES m
>o
10
3
SERIES
X I
= 27°
X I
>Ü I5r
15
SERIES 12
10
e, = 5i =
O
n
m
5
10
X, IN FEET
F i g . 29
m
10
3
O
o
2
LU
=1
15 r
15
20
iLU
10
UJ
tr
^
r
SERIES
nr
r>
5
10
15
r-n
20
5
X, IN FEET
L o c a t i o n o f Maximum and Minimum Breaker
Distance
10
15
X, IN FEET
F i g . 30
20
5
10
15
20
X, IN FEET
L o c a t i o n o f Maximum and Minimum Runup
Distance
evidence o f t h i i s , Perhaps d i f f r a c t i o n o f waves arotind t h e end o f t h e
downstream t r a i n i n g w a l l d i m i n i s h t h i s e f f e c t , o r perhaps t h e e f f e c t
i s t o o s m a l l t o measure.
Samples o f t h e d a t a c o n t a i n e d i n t h e appendix are p l o t t e d i n F i g u r e
25 t o show t h e r e l a t i o n o f t h e measured v a r i a b l e s i n t h e l o n g s h o r e
direction.
The t h r e e t e s t s are i d e n t i c a l except f o r t h e p l u n g e r a n g l e
©d. The curves i l l u s t r a t e f o r s p e c i f i c cases t h e dependence o f V and
dV/dx on e^, t h e i n t e r r e l a t i o n between
and b ( t h u s d, ) and between
b and r , and t h e i n c r e a s e i n e w i t h x .
The v a r i a t i o n o f V w i t h x and 9^ i s t y p i c a l l y i l l u s t r a t e d i n t h e
l o w e r s e t o f c u r v e s . V v e r s u s x was t y p i c a l l y convex upward i n S e r i e s I I ,
s t r a i g h t i n S e r i e s I I I and convex downward i n S e r i e s I V , V i n c r e a s e s
markedly w i t h 0^.
The t y p i c a l i n c r e a s e o f e a l o n g t h e beach amounts
t o a f e w mm. i n 20 f e e t . As u s u a l , b r e a k e r h e i g h t i n t h e c e n t e r o f t h e
beach i s h i g h e r t h a n a t e i t h e r end, and t h e i n c r e a s e d h e i g h t a t x = 10
f e e t i s r e f l e c t e d i n i n c r e a s e d b i n S e r i e s I I and I I I , b u t n o t I V . I f
t h e r e i s a c o n s t a n t p , t h e n b s h o u l d i n c r e a s e i f H, d o e s .
The t y p i c a l l y
minimum v a l u e s o f r a t t h e upstream end o f t h e beach and maximum v a l u e s
a t t h e downstream end are d e m o n s t r a t e d i n a l l t h r e e t e s t s .
The i n c r e a s e
e f f e c t on b i s a l s o p r e s e n t b u t l e s s c l e a r , perhaps due t o t h e v a r i a t i o n
in K .
VI
A.
ANALYTECAL CONSIDERATIONS
Q u a l i t a t i v e D e s c r i p t i o n o f Longshore C u r r e n t F o r m a t i o n
From t h e e x p e r i m e n t a l r e s u l t s o f t h e p r e c e d i n g s e c t i o n t h e r e
emerges a q u a l i t a t i v e vmderstanding o f t h e f o r m a t i o n o f l o n g s h o r e
c t i r r e n t s under t h e s e l a b o r a t o r y c o n d i t i o n s a n d , b y e x t e n s i o n , t o o t h e r
beaches as w e l l :
The b r e a k e r , f o r m e d l a r g e l y o f f l u i d w i t h d r a w n f r o m t h e s u r f z o n e ,
c o n t r i b u t e s mass, momentum, and energy t o t h e svirf z o n e .
The mass c o n t r i b u t i o n i s a t e m p o r a r y one, b e i n g l a r g e l y w i t h d r a w n t o f o r m t h e s u c ceeding b r e a k e r .
The energy c o n t r i b u t i o n i s p a r t l y s p e n t i n i m m e d i a t e
d i s s i p a t i o n , b u t a s i g n i f i c a n t amount remains t o f o r m t h e b o r e w h i c h
moves up t h e beach d i s s i p a t i n g some o f i t s k i n e t i c energy b y t u r b u l e n c e
and b o t t o m f r i c t i o n , and c o n v e r t i n g most o f t h e remainder i n t o p o t e n t i a l
energy a t t h e r u n u p l i m i t .
Most o f t h i s p o t e n t i a l energy i s d i s s i p a t e d
i n t h e runback when t h e w i t h d r a w i n g f l u i d meets t h e n e x t b o r e , .
The
l o n g s h o r e c u r r e n t i s a s i d e e f f e c t o f t h i s p r o c e s s . Energy n o t d i s s i p a t e d i n t h e b r e a k e r and rixnup i s a v a i l a b l e t o m a i n t a i n t h e l o n g s h o r e
current.
The s h o a l i n g wave a t b r e a k i n g c o n t a i n s f l u i d p a r t i c l e s drawn f r o m
t h e l o n g s h o r e c u r r e n t and moving i n t h e l o n g s h o r e d i r e c t i o n .
The wave
f o r m i m p a r t s an a d d i t i o n a l l o n g s h o r e component o f momenttmi t o t h i s mass
and r e t u r n s i t t o t h e s u r f zone d u r i n g b r e a k i n g . The f l \ a i d i n t h e s u r f
U2
LOCATION OF SIN 6^
LOCATION OF SIN Gb
15
>Ü
z 10
Ixl
SERIES
H
O
ÜJ
J U
CC O'
u.
I5|
n
SERIES HE
>-
o
z 10
UJ
Z)
0d = 2 7 "
O
ÜJ
(r
U-
>o
0
15,
SERIES BZ:
u '01
O
UJ
cc
Ü.
0
5
F i g . 31
F i g . 32
10
15
X, IN FEET
EO
5
10
15
X, IN FEET
L o c a t i o n o f Maximum and Minimum Breaker
Angle
Shape o f S h o a l i n g Wave Near B r e a k i n g
^3
20
zone t h u s a c c e l e r a t e s I n t h e l o n g s h o r e d i r e c t i o n u n t i l l o n g s h o r e
r e s i s t a n c e f o r c e s b a l a n c e t h e r a t e o f momentum a d d i t i o n .
B.
Approximate Energy Budget o f l y p i c a l
Test
This s e c t i o n a t t e m p t s an o r d e r o f magnitude a n a l y s i s o f t h e energy
budget f o r t e s t I I I 2 on t h e l a b o r a t o r y b e a c h .
The c o n d i t i o n s o f t h i s
t e s t are o u t l i n e d i n t h e Appendix and r e p e a t e d here f o r r e f e r e n c e :
"
27°
=
0.19
= 1.57
f t .
=
i n which
and
b y s u r f a c e "^'^floats
r ^ ^ = 0.95 f t .
V ^ l ° = 1.76 f t . / s e c .
b _ = 1.52 f t .
av
m
= 0.109
16
1.00 s e c .
d ^ = 1.15 f t .
T
ft./sec.
r e f e r t o l o n g s h o r e cviri'ent v e l o c i t i e s as measured
a t x « 10
and a t x = 16* r e s p e c t i v e l y .
Energy I n p u t
_A.ccording t o s m a l l a m p l i t u d e t h e o r y , t h e power s u p p l i e d (P)
Tonit w i d t h o f wave c r e s t o f h e i g h t 2 a i s
P
i _
„
(1 + ^ _ ^ _ _ )
per
[15]
For t h e c o n d i t i o n s o f t e s t I I I 2, waves t r a n s m i t shoreward
0.85
^
sec.
p e r f o o t o f p l u n g e r , and t h i s power i s d e l i v e r e d t o a beach segment t h a t
i s l o n g e r t h a n t h e p l u n g e r by t h e r a t i o l / c o s 9 ^ . C o n s e r v i n g e n e r g y ,
t h i s means t h a t O.77
' j ! ; ^ ^ ' i s t h e power s u p p l i e d p e r f o o t o f s h o r e l i n e .
This i s t h e power i n p u t f r o m waves
( E f ) p e r f o o t o f beach a t
breaking.
According t o the analysis o f the preceding s e c t i o n , t h i s incoming
power d i s s i p a t e s by t h e f o l l o w i n g mechanisms:
1.
2.
3o
h'
5.
Breaker d i s s i p a t i o n
Rvmup and backwash d i s s i p a t i o n
Longshore c u r r e n t b o t t o m f r i c t i o n .
K i n e t i c energy f l u x o f f t h e t e s t ' beach and o f f s h o r e o f t h e
breaker i n the retvirn f l o w ,
R e f l e c t i o n and e v e n t u a l d i s s i p a t i o n o f f s h o r e .
kh
Reflection
The r e f l e c t i o n c o e f f i c i e n t ( r e f l e c t e d wave h e i g h t d i v i d e d b y i n c i d e n t
wave h e i g h t ) was measured t o be 0.0^ f o r n o r m a l i n c i d e n c e on t h i s
1:10
b e a c h . S i n c e wave energy i s p r o p o r t i o n a l t o t h e s q u a r e o f t h e h e i g h t
t h e r a t e o f energy r e f l e c t e d ,
, w i l l be g i v e n b y
È
Breaker
- ,0025 E. - 0
Dissipation
The r a t e o f energy d i s s i p a t i o n a t b r e a k i n g i s unknown b u t a f a i r
e s t i m a t e can be made f r o m t h e d a t a a l r e a d y o b t a i n e d . From e x a m i n a t i o n
o f wave measurements i n t h e s u r f zone f o r t h i s t e s t i t appears t h a t
i n i t i a l b o r e h e i g h t i s on t h e o r d e r o f one h a l f t h e b r e a k e r h e i g h t and
t h a t t h e i n i t i a l b o r e speed i s t h e same as t h e b r e a k e r c e l e r i t y . Assuming
t h e wave and b o r e energy per u n i t o f s u r f a c e area t o be p r o p o r t i o n a l t o
a m p l i t u d e s q u a r e d t h i s i n d i c a t e s a r a t e o f energy d i s s i p a t i o n a t b r e a k i n g ,
\ » of:
- 0.75 [ \ - \ ]
Ê,
= 0.58
L3
S ec.
= 0.75 [ 1 - .0025] \
- 0.75
\
per f o o t o f s h o r e l i n e
L o n g s h o r e C u r r e n t Bottom F r i c t i o n
Assuming t h e r e s i s t a n c e t o m o t i o n o f t h e l o n g s h o r e c u r r e n t t o f o l l o w
t h e w e l l known laws f o r s t e a d y u n i f o r m f r e e s u r f a c e f l o w s we can e s t i m a t e
t h i s rate of dissipation from:
1
i n which
R = hydraulic radius
f
o f s u r f zone = d^/2
= D a r c y - Weisbach r e s i s t a n c e c o e f f i c i e n t
Q = volume r a t e o f f l o w i n l o n g s h o r e
current.
For t h e d a t a o f t h i s t e s t , IR = ^j^crk
x 10^^ w h i c h , w i t h k = l O ' ^ f t .
f o r smooth c o n c r e t e , g i v e s 1=0.03. Then
ii5
w = (; {)>; - i - l . ! . — ' • •
't'
i;ec.
per Toot o f
ohoreline.
K i n e t i c Energy F l u x
C o n s i d e r i n g a s e c t i o n o f bea-h a t w l i L - l i t h e long;-;hore c u r r e n t i.^;
l u d f o r n t h e r e w i l l be n o n e t f l u x o f e ü e r g y o u t o f a u n i t c o n t r o l
- c l u n i e . Energy f l u x seaward aorons t h e b r e a k e r l i n e w i l l be n e g l e c t e d
i n corre^^pcndence w i t h t h e e a r l i e r o i : , ; o v v a t i o n t h a t n^a:^:: f l \ i x seaward
across t h e b r e a k e r i s r d n i n a l .
Runup and Bacicwash D i s s i p a t i o n
F i g u r e 33 shows a s k e t c h o f t h e s u r f z o ï i e above SWL., The an.ouTit
o f p o t e n t i a l e n e r g y , E , above SW, c o n t a i n e d i n t h e s u r f z o n e a t t h e
i n s t a n t o f naximujn run^ap must b e a p p r o x i m a t e l y t h a t o f t h e t r i a n g l e
i n d i c a t e d i n F i g . 3 3 . A;;suming t h a t t h i ; ; energy must be r e s t o r e d
once e v e r y wave p e r i o d , t l i e mean r a t e o f energy s u p p l y t o t h e r t i n u p ,
Ep, i s given by:
*
\ /
\ r density x
Ep = (mean e l e v a t i o n o f volume) (volume) { - ^ ^ ^ p e r i o d '
Using t h e dimensions o f F i g . 33
E
p
= C.07'^
'
"^^^ per f t . o f s h o r e l i . n e
sec
We w i l l now assiime t h a t E r e p r e s e n t s a l l t h e wa'. e energy r e m a i n i n g
a f t e r t h e b r e a k i n g and ruB.up has o c c u r r e d . We f u r t h e r ' assume t h a t t h e
f r i c t i o n a l d i s s i p a t i o n , E , i s d i v i d e d e q u a l l y between runup and b a c k w a s h , and t h a t s i n c e backwash w i l l occ\ir a p p r o x i m a t e l y normal t o t h e
s h o r e l i n e , energy i s s u p p l i e d t o t h e l o n g s h o r e c x i r r e n t o n l y d u i ' i n g rurr
u p . We can t h e n w r i t e f o r r u n u p :
e
•
o
E,
o
•
E. - F. - E,. = E^
1
!•)
^
i
p
and f o r baclwash
= 0
C^l_COUATlON
F i g u r e 33.
üF
t,
Mean Water L e v e l i n S u r f Zone f o r Run I I 1 - 2
Combining the^e two r e l a t i o n s we may e l i m i n a t e t h e unJknown r a t e o f
d i s s i p a t i o n , E ^ , i n t h e runup and backwash t o o b t a i n :
E. - E^ - E^ = 2 E
E n e r g y Budget
Table 10 g i v e s a summary o f t h e e s t i m a t e d d i s p o s i t i o n o f t h e i n coming wave e n e r g y .
hi
TAHLE
10
ENERGY BUDGET FOR UNIFORM LO-NGSHORE CURRENT
^
Credit
p e r " f t ! o f shoreline)
I n c i d e n t Wave Energy
O.77
Reflection
Breaker D i s s i p a t i o n
Runup and Backwash D i s s i p a t i o n
Longshore C u r r e n t D i s s i p a t i o n
Residual (net f l u x )
ÏDTALS
Debit
^
Q [-0
0 iJi
ó'n^
^0 ^
0,77
0.77
I t xs demonstrated b y t h i s a n a l y s i s t h a t t h e l o n g s h o r e c u r r e n t
g e n e r a t i o n xs a r e s i d u a l phenomenon u t i l i z i n g o n l y 6-1/2 ° / o o f t h e
i n c i d e n t wave energy and o n l y 26 %
o f t h a t energy w h i c h s u r v i v e s t h e •
i n i t i a l breaking process(m t h i s t e s t ) .
C*
E f f e c t of External Circuit
The t e s t a r e a o f t h e b a s i n i s bounded by t h e p l u n g e r , t h e t r a i n i n g
w a l l s , and t h e beach (see F i g u r e 2 ) . The e x t e r n a l a r e a o f t h e b a s i n i s
t h a t p a r t o f the basin which i s not the t e s t area.
The e x t e r n a l c i x - c m t
xs t h e p a t h f o l l o w e d by t h e f l u i d i n f l o w i n g f r o m t h e t e s t beach t o t h e
p l u n g e r and back t o t h e t e s t beach t h r o u g h t h e e x t e r n a l a r e a .
There must be a f a v o r a b l e energy g r a d i e n t i n t h e e x t e r n a l c i r c u i t
under s t e a d y s t a t e c o n d i t i o n s w h i c h t h e energy l e v e l a t t h e end o f t h e
t e s t s e c t x o n t o be h i g h e r t h a n a t t h e b e g i n n i n g . S i n c e t h i s e x t e r n a l
fhS^^fv.""^ ^
° ^ ^he apparatus used i t must be e s t a b l i s h e d
t h a t x t has a n e g l i g x b l e e f f e c t on t h e experiments b e f o r e t h e i r
g e n e r a l i t y i s assured.
For t e s t I I I - 2 t h e l o n g s h o r e c u r r e n t l e f t t h e t e s t s e c t i o n o f t h e
beach w x t h c o n s x d e r a b l e k i n e t i c e n e r g y .
The k i n e t i c energy f l u x a t t h e
begxnnxng o f t h e e x t e r n a l c i r c u i t i s t h u s
Ê,
= I V3A « 0.88
^
£ttl^
sec
From p r e v i o u s ^ a j g u l a t i o n s t h e t o t a l r a t e o f energy s u p p l i e d b y t h e
wave maker xs I 7
( i . e . 0.85 t i m e s 20) t h u s s l i g h t l y more t h L
5 / o o f t h i s energy l e a v e s t h e t e s t beach as k i n e t i c energy o f t h e
longshore c u r r e n t .
This i s augmented b y t h e energy i n waves w h i c h
d i f f r a c t around t h e downstream t r a i n i n g w a l l and t h e p o t e n t i a l energyc a r r i e d o u t b y t h e l o n g s h o r e c u r r e n t i n t h e runup r e g i o n . Most o f t h e
d i f f r a c t e d energy i s d i s s i p a t e d on t h e b e a c h , and p r o b a b l y most o f t h e
p o t e n t i a l energy i n t h e r u n u p i s l o s t i n immediate d i s s i p a t i o n , as i t i s
on t h e t e s t s e c t i o n o f t h e b e a c h . As a f i r s t a p p r o x i m a t i o n assume t h a t
t h e k i n e t i c energy f l u x i n t h e l o n g s h o r e c u r r e n t i s t h e t o t a l energy
supplied to the external c i r c u i t .
The c o n d i t i o n s i n t h e t e s t a r e a may be a f f e c t e d b y t h e n e c e s s i t y
t o s u p p l y energy t o overcome l o s s e s i n t h e e x t e r n a l c i r c u i t , b u t t h e
f a c t t h a t t h e energy i s needed a t a r a t e w h i c h i s o n l y 5 p e r c e n t o f t h e
r a t e a t w h i c h i t i s produced suggests t h a t t h e e f f e c t i s q u i t e s m a l l .
To d e m o n s t r a t e t h a t t h i s 5 per c e n t i s s u f f i c i e n t t o r e t u r n t h e f l u i d
t h r o u g h t h e e x t e r n a l c i r c u i t t o t h e p l u n g e r , an a n a l y s i s o f p o s s i b l e
e n e r g y d i s s i p a t i o n i n t h e e x t e r n a l c i r c u i t was made f o r t h e c o n d i t i o n s
o f Test I I I 2.
The c u r r e n t w h i c h l e a v e s t h e t e s t beach c o n t i n u e s t o f l o w a l o n g
t h e 8 f o o t s t r e t c h o f beach downstream o f t h e t r a i n i n g w a l l , s p r e a d i n g
o u t as i t f l o w s and d i m i n i s h i n g i n v e l o c i t y .
A t t h e end o f t h e beach
i t meets t h e b a s i n w a l l , makes a r i g h t a n g l e t u r n , and f l o w s o f f t h e
beach w i t h a s t i l l l o w e r v e l o c i t y throvigh a l a r g e r a r e a .
The f l u i d
t h e n moves s l o w l y over a r e l a t i v e l y u n o b s t r u c t e d p a t h t o t h e p l u n g e r
and passes under t h e p l u n g e r .
There appear t o be f o u r p o s s i b l e mechanisms o f e n e r g y d i s s i p a t i o n :
F r i c t i o n a l d i s s i p a t i o n i n the c o n t i n u a t i o n of the longshore c u r r e n t ,
head l o s s due t o t h e r i g h t a n g l e t u r n a t t h e b a s i n w a l l , f r i c t i o n a l
l o s s t h r o u g h t h e r e m a i n d e r o f t h e e x t e r n a l c i r c u i t , and head l o s s a t
the plunger.
For t h e g i v e n c o n d i t i o n s t h e volume f l u x o f f t h e beach i s Q = O.3O
c u b i c f e e t ' p e r s e c o n d , and t h e v e l o c i t y head a t x = 10 f e e t i s O.Oi; f e e t .
The k i n e t i c energy f l u x i n t h e l o n g s h o r e c u r r e n t i s 0.88 f t . l b s / s e c .
Longshore c u r r e n t • l o s s . F r i c t i o n a l d i s s i p a t i o n p e r f o o t o f beach
due t o t h e f l o w o f t h e l o n g s h o r e c u r r e n t on t h e t e s t beach i s 0.0^ f t . l b s / s
I t i s a c o n s e r v a t i v e a s s u m p t i o n t h a t energy i s l o s t a t t h e same r a t e
o v e r t h e 8 f o o t s e c t i o n o f beach i n t h e e x t e r n a l c i r c u i t .
This means
a l o s s o f O.liO f t . l b s / s e c .
Loss due t o t u i - n . For r i g h t a n g l e t u r n s i n smooth p i p e s , head
l o s s i s a b o u t 0.2 o r 0.3 t i m e s v e l o c i t y h e a d . Assume t h a t t h e head l o s s
x n t h x s case i s 1/2 a v e l o c i t y head and t h a t t h e v e l o c i t y head i n
q u e s t i o n i s t h a t o f x - 10 on t h e t e s t b e a c h .
^turn - ^ n - l
k9
0.37
^ ^ r ^ ^
sec,
j ' r i c t i o i i lo::.:' a f t e r tur}-.. Assume t h e Qioid i s c o r j c e n t r a t e d i n a
cliannel which i ; : 1 f o o t deep,
f e e t w i d e , and hO f e e t l o n g i n r e t ' o r n i n g
t o p l u n g e r . An a p p r o p r i a t e ]te7^,old:; n - m l e r f r o m t l i e g i v e n Q, a r e a , a n d '
d e p t h i s 6 x 1 0 ' ' , ai.d a;:;;ui;.ing :;!!,ooth f l o w , f i s about 0 . 0 3 6 ,
;ec,
Flow under p l u n g e r .
The mean ele-. a t i o n o f t h e base o f t h e 20 f o o t
p l u n g e r iv, O.J4Ü f e ë t T " A head l o s e o f s e v e r a l v e l o c i t y heads tl'iro\agh
t h i s v a r y i n g o r i f . i ce w i l l r e s u l t i ; . energy lo;:;: l e s s t h a n 1 0 " f t . l b s / s e c
Suiimiary. Tl-d.:. a n a l y s t : : , w l u l e u n c e r t a i n i n p a r t i c u l a r s , demons t r a t e s t h a t t h e k i n e t i c energy ;iup] l i e d t o t h e e x t e r n a l c i r c u i t i s o f
t h e o r d e r needed t o overcome t h e l o o s e s t h e r e .
According to the calcul a t i o n s o u t l i n e d above, energy i s :-,upplied a t a r a t e o f 6.88 f t . l b s / s e c ,
and d i s s i p a t e d a t about t h a t r a t e , c h i e f l y on t h e c o n t i n u a t i o n o f t h e
beach.
The head l o s s due t o b o t t o m f r i c t i o n over a UO f o o t r e t u r n p a t h
i s l e s s t h a n IQ-^t f t . l b s / l b whereas t h e p r e c i s i o n o f t h e p i e z o m e t e r s
used t o measure w a t e r l e v e l i s 3 x l ( r 3 f e e t .
I t i s concluded t h a t e x t e r n a l c i r c u i t l o s s e s must produce an energy
g r a d i e n t on t h e t e s t beach w h i c h i s a f u n c t i o n o f t h e p a r t i c u l a r l a b o r a t o r y apparatus u s e d j however, t h e magnitude o f t h i s g r a d i e n t i n t h e
p r e s e n t case i s so s m a l l as t o be n e g l i g i b l e . There remains t h e p o s s i b i l i t y t h a t t h e t r a n s i t i o n between t e s t area and t h e e x t e r n a l a r e a a t t h f
downstream t r a i n i n g w a l l may a f f e c t c o n d i t i o n s on t h e downstream end o f
t h e t e s t beach,
°*
Momentum and Energy Analyses o f Putnam e t a l (12)
I n v i e w o f t h e o b s e r v a t i o n s and c a l c u l a t i o n s r e p o r t e d above, t h e
a n a l y s e s o f Putnam e t a l (12) summarized i n Chapter I I , mav be examined
more c r i t i c a l l y .
Friction
Factor
Since t h e Moody p i p e f r i c t i o n :iiagram i s a p p l i c a b l e as a f i r s t appr 0'/. I
m a t i o n t o f u l l y developed f l o w i n open c h a n n e l s , and s i n c e t h e a n a l y s i s
t r e a t s l o n g s h o r e c u r r e n t s as a u n i f o r m open channel f l o w , e x i s t i n g
diagraiTBought t o pi o v i d e an o r d e r o f magnitude check on t h e f r i c t i o n
f a c t o r compute<l froin t h o a n a l y : ; i s .
The co|,ij,ari:-;on i s r.ade i n Table 1 1 .
TABLE i J
raiCTION
1'^ACTOR, KEWOLDS MUMBKi,' AMD I f M . A T I V E
ROUGHNESS TOR DATA 1'ROM i'UTNAM, ET Al.
(12)
iBeach
]R
U RV
Relative
Roughness
k
Field
> 10
L a b , smooth cone.
10
L a b , smooth cone.
10
Lab, I/I4
lO''
g r a v e l on c o n e .
Darcy f
Eq[2]
Moody
Diagram
k
> i;000
1000 - 5000
5OÜ
25
0.06
O.Oli;
0.06
0.32
0.02
0.025
3.08
0.07
I t i s t o be expected t h a t t h e c o e f f i c i e n t s c a l c \ i L a t e d f r o m E q . [2]
w i l l be h i g h e r t h a n t h o s e f o r s t e a d y u n i f o r m f l o w as g i v e n by t h e Moody
d i a g r a m . The magnitude o f t h e d i f f e r e n c e r a i s e s q u e s t i o n s c o n c e r n i n g
e i t h e r t h e v a l i d i t y or t h e u s e f u l n e s s o f E q . [ 2 j .
Energy D i s t r i b u t i o n
Using t h e v a l u e s o f f computed f r o m E q . [ 2 ] Putnam o b t a i n e d t h e
f a c t o r , s, f r o m E q . [ 1 ] . T h i s c a l c u l a t i o n showed t h e l o n g s h o r e c u r r e n t
t o d i s s i p a t e f r o m l ^ o / o t o 3 3 ° / o o f t h e energy i n t h e b r e a k i n g wave i n
comparison w i t h t h e 6 1/2 ° / o c a l c u l a t e d e a r l i e r f o r a t y p i c a l t e s t
o f t h i s i n v e s t i g a t i o n . The c o n s i s t e n c y o f t h e energy b a l a n c e c o n t a i n i n g
t h i s l a t t e r p e r c e n t a g e suggests t h e p r o b a b l e a p p l i c a b i l i t y o f t h e Moody
f and t h e l i k e l y hood o f an e r r o r i n Putnam's f o r m u l a t i o n o f E q . [ 2 ] .
Momentum F l u x
I n t h e d e r i v a t i o n o f t h e e q u a t i o n s f o r o o n s e r v a t i o n o f momentum,
Putnam (12) assumed t h a t t h e mass o f f l \ r L d b e i n g i n j e c t e d i n t o t h e s u r f
zone e n t e r s w i t h an x component o f v e l o c i t y g i v e n b y C j ^ . s i n
and
t h a t t h e mass f l u x o u t o f t h e s u r f zone across t h e b r e a k e r l i n e occurs
w i t h v e l o c i t y V . I n t h i s formulatocnC^ s i n Qv, must be g r e a t e r t h a n V
f o r maintenance o f t h e l o n g s h o r e c u r r e n t .
A l l b u t one o f 3h t e s t s i n
S e r i e s I I , I I I and I V f o r w h i c h t h e d a t a are a v a i l a b l e show t h a t C b . s i n ö ^ ^
< V (see Table 12).
For almost a l l t e s t s , C^ s i n
< l / 2 V and
i n some cases C^ s i n 0^ < 1/5 V . C^ was computed \ i s i n g E q . [10] e v a l u a t e d
at the breaker.
51
Since t h e o b s e r v a t i o n s o f t h i s s t u d y i n d i c a t e t h e b r e a k i n g wave t o
be made up p r i m a r i l y o f f l u i d t a k e n f r o m t h e s u r f z o n e , i t appears
necessary t o assiame t h a t t h e w a t e r p a r t i c l e i n t h e b r e a k i n g wave has an
x-component o f v e l o c i t y w h i c h i s b e t t e r a p p r o x i m a t e d b y V + Cb s i n 9.|^.
I n t h i s case C^ s i n 0^ need n o t be l a r g e r t h a n V .
E.
R e v i s e d Momentum A n a l y s i s
I n t h i s s e c t i o n a r e v i s e d momentum e q u a t i o n f o r t h e n o n - t i n l f o r m
f l o w o f longshore currents i s presented.
The c o n t r o l volume i s t h e same
as t h a t used b y Putnam, Munk, and T r a y l o r (see F i g u r e 3h)f b u t l o n g s h o r e
c u r r e n t v e l o c i t y i s assumed t o average V over t h e s u r f zone and t o be V^^
at the breaker l i n e .
Shear i s assumed t o be n e g l i g i b l e a t t h e f l u i d
i n t e r f a c e a l o n g t h e breaker l i n e .
TABLE 12
COMPARISON OF C^ s i n 9^ WITH V
b
b
Series
Test
2
3
h
II
ft/sec
0.20
0.22
0.15
0.82
0.68
0.83
0.53
0.52
0.18
O.lU
7
0.07
0.06
10
11
12
13
V""
C, s i n 9,
b
b
ft/sec
6
8
9
III
o.3h
C,n s i n 0,b
ft/sec
0.76
0.65
0,53
0.i;2
0.30
0.22
0.27
O.lU
O.lii
0.50
0.67
0.71
O.liO
0.1;9
0.22
0.8ii
0.60
O.OU
0.21
0.U5
o.ih
O.lil
0.06
- 0.3,
p =
0.85, R^ =
"V i s f l o a t a t X = 16 f e e t .
52
V
ft/sec
1,76
1,60
1,61
l,ii2
1,23
0,56
0.96
1,23
1.U9
1.77
-
"C^ computed f r o m [ g H^^ (1 - a + p ) ] " ^ ^ ^
a
IV
mb
C, s i n 9,
b
b
ft/sec
1,1;0
1,01
0,89
0,79
0.36
0,1|8
0.58
0,93
0,96
1,10
0,20
0.82
V
ft/sec
2.15
1.89
1.91
1.81
0.91
0.65
1.20
1.57
1,88
1,96
0,18
l.ii6
. F i g u r e 3k.-
C o n t r o l Volume f o r Momentum A n a l y s i s
I n t h e n o n - u n i f o r m c a s e , t h e r e a r e 3 momentum f l u x e s i n t h e
x - d i r e c t i o n t o be b a l a n c e d b y shear on t h e c o n t r o l s u r f a c e s . They a r e
t h e x-component o f momentum f l u x o f t h e f l u i d e n t e r i n g t h e c o n t r o l volume
i n t h e b r e a k i n g wave, t h e x-component o f momentum f l u x o f t h e f l u i d
l e a v i n g a c r o s s t h e b r e a k e r l i n e , and t h e change i n momenttmi f l u x i n t h e
l o n g s h o r e c u r r e n t . The mass f l u x o u t across t h e b r e a k e r l i n e i s d i f f e r e n t
f r o m t h e mass f l u x i n across t h e b r e a k e r l i n e by t h e amount t h a t t h e
mass f l u x i n t h e l o n g s h o r e c u r r e n t changes i n t h e x - d i r e c t i o n . The
x-component o f t h e f l u i d v e l o c i t y i n t h e b r e a k i n g wave i s p C^ s i n Q.^+ V^
where p i s a c o n s t a n t < 1 d e f i n e d as t h e average shoreward p a r t i c l e
v e l o c i t y i n t h e b r e a k e r d i v i d e d by C^. A^^ i s a c r o s s s e c t i o n a l area o f
t h e f l u i d i n t h e b r e a k i n g wave w h i c h e n t e r s t h e s u r f z o n e .
Momenttmi f l t i x i n
across b r e a k e r l i n e
Momentum f l u x o u t
across breaker l i n e
(V^ + pC^ s i n 9^) cos 9^ dx
A p
, w ^ cos 9^ dx - Aj^ pdV)V^
Net Momentum f l v i x o u t
= A^ p V dV
i n longshore current
Shear on b o t t o m
^ p V2 (b + r ) dx
Surface forces
associated w i t h
wave b r e a k i n g
[16]
[17]
[18]
[19]
[20]
53
is
By e q u a t i n g t h e momentum f l u x e s t o t h e f o r c e s , a momentum e q u a t i o n
obtained f o r the non~uniform f l o w o f longshore currents
pC^ s i n 2 9 , = f ^
T
. 2 ^
. ^
T V ( 1 - V-) ^
[21]
i n w h i c h V ' equals V ^ / V , and i s assumed t o be l e s s t h a n or e q u a l t o 1 .
The l e f t hand s i d e o f E q u a t i o n 2 1 i s t h e wave t e r m , t h e f i r s t t e r m on
t h e r i g h t hand s i d e i s t h e f r i c t i o n t e r m , t h e second t e r m on t h e r i g h t
hand s i d e i s t h e b r e a k i n g t e r m , and t h e t h i r d t e r m on t h e r i g h t hand
s i d e i s a c o n v e c t i v e a c c e l e r a t i o n t e r m . S, i n t h e b r e a k i n g t e r m ,
i n c l u d e s unknown f o r c e s a s s o c i a t e d w i t h t h e b r e a k i n g , i m p a c t , and
s p l a s h o f t h e wave, and i t cannot now be expressed a n a l y t i c a l l y .
For
l a c k o f a b e t t e r s o l u t i o n , i t w i l l be assumed p r o p o r t i o n a l t o t h e
momentum f l u x i n t h e e n t e r i n g wave and w i l l be absorbed i n t h e q u a n t i t y
p . Assumed n e g l i g i b l e a r e t h e shear f o r c e on t h e f l u i d i n t e r f a c e along
t h e b r e a k e r l i n e , and p o s s i b l e p r e s s u r e f o r c e s due t o a g r a d i e n t o f
MWL i n t h e x - d i r e c t i o n .
The s h o r t e n e d e q u a t i o n i s t h u s
p
s i n 2 9^ = ^ ^
T
D i f f e r e n t i a t i n g w i t h respect
[23] becomes
+ 2 ^
T V (1 - V ) g
t o x and s o l v i n g f o r d ^ V / d x ^ ,
[22]
equation
where
D2=2-^T
w
[25]
Suppose l o n g s h o r e c u r r e n t v e l o c i t y i s n e g l i g i b l e .
The w a t e r i n t h e s u r f
zone s t i l l has an average p a r t i c l e v e l o c i t y due t o t h e b o r e .
The p a r t i c l e
v e l o c i t y a t t h e head o f t h e b o r e i s g i v e n (35) i n n o t a t i o n o f t h i s r e p o r t ,
by
5i|
[26]
w h e r e C i s t h e l o c a l c e l e r i t y o f t h e b o r e i n t h e r u n u p r e g i o n , and f o r
m e a s u r e m e n t s o f t h e s e t e s t s , has a n a v e r a g e v a l u e > 1/2
C (see F i g u r e
20), S i n c e u m u s t go t o z e r o o n c e e a c h h a l f p e r i o d , a n a v e r a g e u i s
^ v ^ 5 ' - 3 ^ b
t27]
I n t h e s e t e s t s u ^ ^ has a r a n g e f r o m a b o u t 0.3
t o 0.6 f e e t p e r s e c o n d ( s e e
F i g v i r e 21) a n d V u p t o 2 f e e t p e r s e c o n d .
T h e r e f o r e , t h e p o s s i b l e change
i n Ji a l o n g t h e b e a c h w i l l b e l e s s t h a n a n o r d e r o f m a g n i t u d e , a n d f o r t h e
given r e l a t i v e roughnesses, f i s a small q u a n t i t y which should not v a r y
b y m o r e t h a n 20 p e r c e n t a l o n g t h e 20 f o o t t e s t b e a c h .
For t h e s e r e a s o n s ,
t h e v a r i a t i o n o f f w i t h x w i l l b e n e g l e c t e d a n d e q u a t i o n 23 r e w r i t t e n as
" ^ ( V - V , ) g
= [ D , ^ - ^ ) - D , 2 V l g
[26]
where
B e c a u s e V, i s d e f i n e d as t h e a v e r a g e x - c o m p o n e n t o f t h e v e l o c i t y
a t t h e b r e a k e r w i t h w h i c h t h e r e t u r n f l o w escapes o f f s h o r e , a t t h e u p s t r e a m t r a i n i n g w a l l ( x - 0),
= 0.
But s i n c e t h e b r e a k i n g wave
e n t e r s t h e s u r f zone w i t h a l o n g s h o r e component o f m o t i o n p r o p o r t i o n a l
t o t h e l o n g s h o r e component o f t h e b r e a k e r v e l o c i t y , V = p C
s i n 9, a t
the upstream t r a i n i n g w a l l .
°
The s h a p e o f V ( x ) w i l l b e c o n v e x u p w a r d s as l o n g as t h e r i g h t h a n d
s i d e o f E q . [28] i s p o s i t i v e .
2 D Y/b
i s on t h e o r d e r o f ( f / d ^ ) p
s i n ©b n e a r t h e u p s t r e a m t r a i n i n g w a l l , w h i c h c a n become a v e r y s m a l l
number f o r t e s t s i n S e r i e s I I h a v i n g s m a l l 9 ^ .
Thus, f o r t h e case
of small
t h e s i g n o f d^V/dx^ w i l l depend m a i n l y on t h e r e l a t i o n
b e t w e e n dV-^^/dx a n d d V / d x .
Because V goes f r o m p
s i n 0, t o V
a l o n g t h e b e a c h , and
goes f r o m 0 t o n e a r V j ^ ^ ^ , i t i s p o s s i b l e ^
t h a t d V ^ / d x i s l a r g e r t h a n d V / d x o v e r t h e u p s t r e a m end o f t h e b e a c h
a n d t h u s d ^ V / d x ^ may b e i n i t i a l l y p o s i t i v e .
F i g u r e 35 shows t h e r a t i o o f t h e v e l o c i t y a t t h e b r e a k e r p o s i t i o n
i n d i c a t e d b y t h e p r o b e (V-^p) t o t h e l o c a l mean v e l o c i t y i n t h e s u r f z o n e
55
i n d i c a t e d b y t h e probe ( V ) , a l l f o r a g i v e n x . V^p i s t h e sum . o f t h e
c o n s t a n t p C-5 s i n 0, and t h e v a r i a b l e v e l o c i t y V ^ , and i n c o r p o r a t e s
p o s s i b l e e r r o r due t o t h e probe r e s p o n d i n g d i f f e r e n t l y a t t h e o u t e r
edge o f t h e s u r f zone .from t h e c a l i b r a t e d response i n t h e m i d d l e o f
t h e s u r f zone.
Because p
s i n 9^ and t h e p o s s i b l e e r r o r i s p r o b a b l y c o n s t a n t
a l o n g t h e beach, t h e v a r i a t i o n shown on F i g u r e 35 can be a t t r i b u t e d
to v a r i a t i o n i n V^.
This v a r i a t i o n i n d i c a t e s t h a t
i n c r e a s e s more
s w i f t l y t h a n V i n t h e x - d i r e c t i o n , b u t does n o t i n d i c a t e whether t h e
r a t e o f i n c r e a s e i s l a r g e enough t o malte d^V/dx^ p o s i t i v e .
Neglecting the v a r i a t i o n i n f , the f r i c t i o n term of equation[22]
grows as t h e square o f t h e v e l o c i t y , b u t s i n c e t h e r i g h t hand s i d e o f
[ 2 2 ] equals a c o n s t a n t , t h e v e l o c i t y must approach an upper l i m i t .
T h e r e f o r e , d^V/dx^ w i l l always become n e g a t i v e i f t h e beach i s l o n g
enough, and V w i l l t e n d t o w a r d a maximum.
E q u a t i o n [ 2 2 ] p r e d i c t s t h a t V ( x ) may be concave upward i n i t i a l l y ,
and must approach a c o n s t a n t e v e n t u a l l y . F i g u r e 36 shows t y p i c a l
v e l o c i t y v a r i a t i o n w i t h x f o r 2 t e s t s o f each s e r i e s .
Series I I t e s t s
show i n d i c a t i o n s o f b e i n g concave upward, w h i c h as p o i n t e d o u t above,
i s t o be expected f o r t h e s m a l l 9^ o f t h i s s e r i e s .
The o t h e r two
s e r i e s demonstrate curves w h i c h r e a c h a maximum.
V ^ p / V , and t h u s "^i^/^, was n o t observed t o be g r e a t e r t h a n 1 i n
t h e s e t e s t s w h i c h , a c c o r d i n g t o e q u a t i o n [ 2 2 ] , s h o u l d be t h e case f o r
curves w h i c h pass t h r o u g h a maximum. Pronounced o f f s h o r e f l o w was
f r e q u e n t l y observed a t t h e downstream end o f t h e beach however.
F i g t i r e 37 demonstrates an i d e a l i z e d v e l o c i t y d i s t r i b u t i o n o f
l o n g s h o r e c u r r e n t s on t h e l a b o r a t o r y b e a c h .
The curves r e p r e s e n t a
composite o f t y p i c a l c o n d i t i o n s f r o m t h e p l o t s o f v e l o c i t y d i s t r i b u t i o n
f o r 30 t e s t s , b u t no s i n g l e t e s t demonstrated a l l t h e f e a t u r e s d i s p l a y e d on t h e f i g u r e .
F.
Empirical Correlation
An e m p i r i c a l c o r r e l a t i o n i s p o s s i b l e between two groups o f meastired
v a r i a b l e s f o r some o f t h e a v a i l a b l e f i e l d and l a b o r a t o r y d a t a . I n one
f o r m , t h i s c o r r e l a t i o n i s between t h e mass f l u x i n t h e l o n g s h o r e c u r r e n t ,
Q L , and a h y p o t h e t i c a l mass f l u x , i d e n t i f i e d as Qj^, e q u a l t o t h e p r o d u c t
o f a v e l o c i t y , C^ s i n 9|^, and a t r i a n g u l a r area o f h e i g h t , H-j^, and l e n g t h
L k cos 9, .
Q^ = A^V ° 1/2 m b^ V
56
[30]
O
5
10
15
20
X, in feet
F i g . 35
in
V a r i a t i o n i n R a t i o o f Longshore V e l o c i t y
a t Breaker t o Mean Longshore V e l o c i t y
2
1,5 -
0
5
10
15
Distance along b e a c t i , in feet
F i g . 36
20
O
5
10
Distance along beocti, In feet
V a r i a t i o n o f Mean Longshore C n i ' r e n t
V e l o c i t y w i t h Distance
15
Distance along beach
F i g . 37
I d e a l i z e d D i s t r i b u t i o n o f Longshore
Ctirrent V e l o c i t y
58
= (1/2
\ %
Using Eqs. [11]
\
Using E q . [10]
- l A
and [12]
'
cos 9^)C^ s i n 9^
[31]
Oj^ becomes:
Pb'
[32]
^
Q may be w r i t t e n :
w
g T H^Ml
"
+ P^) s i n
2
[33]
9^
- ^ a t a f r o m Putnatf^-et a l . (12), and Inman and Quinn (13t-'together
w i t h d a t a f r o m t h i s i n v e s t i g a t i o n are p l o t t e d as Q^ and % on F i g u r e 38.
Some s e l e c t i o n o f t h e d a t a has been made. Only t h o s e p o i n t s have been
t a k e n f r o m Inman and Quinn f o r w h i c h t h e mean v e l o c i t y exceeds t h e
standard d e v i a t i o n o f the v e l o c i t y , thus e l i m i n a t i n g the t e s t s f o r
w h i c h an upstream l o n g s h o r e c u r r e n t was most s i g n i f i c a n t . I t i s p o s s i b l e
t h a t ^ r t o f t h e r e a s o n why t h e d a t a o f Inman and Qiiinn f a l l below t h e
l i n e q - = 1 i s t h a t even i n t h e s e l e c t e d t e s t s ,
the v e l o c i t i e s l i s t e s by
Inman and Quinn a r e t h e average over 1$ s t a t i o n s , some o f w h i c h a r e
d i r e c t e d upstream.
Tests 13 and 1? o f t h e Putnam, e t a l . f i e l d d a t a were e l i m i n a t e d
because o f h i g h f o l l o w i n g w i n d s . A l l l a b o r a t o r y d a t a o f Putnam e t a l .
and o f S e r i e s I I , I I I and I V are p l o t t e d .
For a l l d a t a , t h e v a l u e s o = 0.3,
= 0.85 were used, and i n
e v a l u a t i n g d a t a i n S e r i e s I I , I I I , and I V , e q u a t i o n [30] was used
i n s t e a d o f [33], i n order t o e l i m i n a t e s c a t t e r i n t r o d u c e d by e r r a t i c
values of H^.
E q s . [32] and [ 33] can be combined t o o b t a i n
59
^
Q,,,, IN CUBIC FEET PER SECOND
2
[i^
[3W
or
V - K-j^ g m T s i n
[35]
2 0^
l^i-om t h e p l o t o f t h e d a t a on F i g u r e 38, QR, f o r some s e r i e s o f t e s t s , can
be a p p r o x i m a t e d b y a c o n s t a n t on t h e o r d e r o f 1, and f r o m t h e v a l u e s o f
1 13, and a
+
found i n t h i s i n v e s t i g a t i o n ,
o r d e r o f 1, so t h a t
has a v a l u e near
i s a l s o a c o n s t a n t on
1.
- H i e average K f o r t h e I 6 u s a b l e f i e l d t e s t s o f Putnam, e t a l . i s
1.02.
I n d i v i d u a l K ' s are w i t h i n 25 per c e n t o f t h e average K i n 13 o f
t h e 16 t e s t s , and t h e use o f K^-y- tends t o be c o n s e r v a t i v e a t h i g h e r
v e l o c i t i e s , t h a t i s , p r e d i c t e d v e l o c i t i e s a r e h i g h e r t h a n measured
velocities.
V I I - SUMMARY AND CONCLUSIONS
A.
Summary
The p u r p o s é o f t h i s i n v e s t i g a t i o n has been t o d e s c r i b e , b y e x p e r i m e n t ,
the c h a r a c t e r i s t i c s o f longshore currents i n order t o f a c i l i t a t e the
development o f a r a t i o n a l m a t h e m a t i c a l model f o r t h e i r p r e d i c t i o n .
The e x p e r i m e n t s d e s c r i b e d were p e r f o r m e d on t h e 20 f o o t t e s t s e c t i o n
o f a p l a n e , smooth c o n c r e t e beach w i t h a 1 on 10 s l o p e .
The e x p e r i m e n t s
p r o v i d e d d a t a c o n c e r n i n g b r e a k i n g waves and l o n g s h o r e c u r r e n t s , and t h e s e
o b s e r v a t i o n s l e d t o an a n a l y s i s o f energy d i s s i p a t i o n i n t h e s u r f z o n e ,
an a n a l y t i c a l d e s c r i p t i o n o f t h e non-xaniform f l o w o f l o n g s h o r e c u r r e n t s ,
and an e m p i r i c a l c o r r e l a t i o n between t h e v e l o c i t y o f l o n g s h o r e c u r r e n t s ,
t h e wave c o n d i t i o n s and beach g e o m e t r y .
B.
Conclusions
1,
The energy r e q u i r e d f o r maintenance o f a u n i f o r m l o n g s h o r e
61
c u r r e n t i s l e s s t h a n 10 / o o f t h e energy o f t h e b r e a k i n g wave.
2. Most o f t h e f l u i d i n j e c t e d i n t o t h e s u r f zone when a wave
breaks has been drawn f r o m t h e s u r f zone and hence a l r e a d y has a
l o n g s h o r e v e l o c i t y , Y-^, when i t becomes p a r t o f t h e brealcer.
3.
Downstream o f an o b s t a c l e t h e mean l o n g s h o r e c u r r e n t i n c r e a s e s
f i r s t a t an i n c r e a s i n g r a t e w i t h d i s t a n c e and t h e n a t a d e c r e a s i n g
r a t e , approaching a c o n s t a n t v e l o c i t y a t l a r g e d i s t a n c e s f r o m t h e
obstacle.
U. The w i d t h ( i n t h e o f f s h o r e d i r e c t i o n ) o f t h e l o n g s h o r e
c u r r e n t expands i n t h e d i r e c t i o n o f f l o w downstream o f t h e o b s t a c l e .
5. Q u a n t i t a t i v e p r e d i c t i o n o f t h e g r o w t h i n mean l o n g s h o r e
c u r r e n t v e l o c i t y downstream o f an o b s t a c l e appears p o s s i b l e f r o m
momentum f l u x c o n s i d e r a t i o n s .
6, From an e m p i r i c a l c o r r e l a t i o n o f b o t h f i e l d and l a b o r a t o r y
d a t a , t h e mean v e l o c i t y o f u n i f o r m l o n g s h o r e c u r r e n t s i s g i v e n
approximately by
V = g m T e i n 2 6,
VIII
REFERENCES
1.
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2.
Brebner, A., and Kennedy, R . J , , L i t t o r a l d r i f t i n Lake O n t a r i o Harbors
E n g . J o u r n . E n g . I n s t , of Canada, [|.2, No. 9
3«
Handin, J . W . , The s o u r c e , t r a n s p o r t a t i o n , and d e p o s i t i o n of beach
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U.
Emery, K . O . , The s e a o f f Southern C a l i f o r n i a , John Wiley and Sons,
New Y o r k , I 9 6 O .
5.
Kuenen, P . H . , Marine geology, John Wiley and S o n s , New York, 19,50.
6,
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Report No. k j Beach E r o s i o n Board, I 9 6 1 .
62
Technical
7.
F r i e d m a n , G.Mo, D i s t i n c t i o n between dtrne, beach, and r i v e r sands
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8.
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9.
Johnson, D . W . , Shore processes and shore l i n e d e v e l o p m e n t ,
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John
10.
Shepard, F . P . , Longshore c u r r e n t o b s e r v a t i o n s i n S o u t h e r n C a l i f o r n i a ,
Beach E r o s i o n Board Tech Memo No. 1 3 , 1 9 ^ 0 .
11.
Shepard, F . P . , and Inman, D . L . , Nearshore w a t e r c i r c u l a t i o n r e l a t e d
t o b o t t o m t o p o g r a p h y and wave r e f r a c t i o n . T r a n s . Am. Geophys.
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12.
Putnam, J . A . , Munk, W . H . , and T r a y l o r , M . A . , The p r e d i c t i o n o f
l o n g s h o r e c u r r e n t s , T r a n s . Am. Geophys. U n i o n , 3 0 , p p .
33J-3k$,
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13.
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lU.
Savage, R . P . , L a b o r a t o r y d e t e r m i n a t i o n o f l i t t o r a l - t r a n s p o r t
r a t e s , J . Waterways and Harbors D i v . , Am. Soc. C i v i l
Engineers, 88, p p . 69-92. 1962.
15.
B r e b n e r , A . and Kamphuis, J . W . , Model Tests on t h e R e l a t i o n s h i p
between Deep-Water Wave C h a r a c t e r i s t i c s and Longshore
C u r r e n t s , C . E . Research R e p o r t N o . 3 I ,
Queen's U n i v e r s i t y
a t K i n g s t o n , O n t a r i o , Augvist I 9 6 3 .
16.
Svendson, S . , M u n c h - P e t e r s o n ' s l i t t o r a l d r i f t f o r m u l a . B u l l .
E r o s i o n B o a r d , h, p p . I - 3 I , 1 9 ^ 0 .
17.
E a t o n , R . O . , L i t t o r a l p r o c e s s on sandy c o a s t s , i n P r o c . F i r s t C o n f .
on C o a s t a l E n ^ . , e d i t e d b y J.W. Johnson, p p . U i Ó - l ^ i ; , C o u n c i l
on Wave Research, Richmond, C a l i f , 1 9 ^ 1 .
18.
C a l d w e l l , J . M . , Wave a c t i o n and sand movement near Anaheim Bay,
C a l i f o r n i a , Beach E r o s i o n Board T e c h n i c a l Memo. No. 6 8 , 1 9 ^ 6 .
19.
Savage, R . P . , L a b o r a t o r y S t u d y o f t h e E f f e c t o f Groins on t h e
Rate o f L i t t o r a l T r a n s p o r t : Eqtiipment Development and
I n i t i a l T e s t s , Beach E r o s i o n Board T e c h n i c a l Memo. N o . l l i i
1959
~ ~
—
63
—
Beach
20.
Johnson, J.W., Sand t r a n s p o r t by l i t t o r a l c u r r e n t s , P r o c e e d i n g s ,
F i f t h I t y d r a u l i c C o n f . , S t a t e U n i v . o f I o w a , S t u d i e s o f Eng.
B u l l . 3h, 1952.
21.
Rruun, P . , Longshore C u r r e n t s and Longshore Troughs,
G e o p h y s i c a l Res, 60, p p . IO65-IO78, I963.
22.
Dean, R.G. and U r s e l l , F , I n t e r a c t i o n o f a F i x e d , Semi-Immersed
C i r c u l a r C y l i n d e r w i t h a T r a i n o f S u r f a c e Waves, T e c h n i c a l
Report No. 37, Hydrodynamics L a b o r a t o r y , M . I . T . , Cambridge,
Mass. 1959.
~
~
23.
W i e g e l , R . L , , P a r a l l e l w i r e r e s i s t a n c e wave m e t e r , i n C o a s t a l
E n g i n e e r i n g I n s t r u m e n t s , e d i t e d by R . L . W i e g e l , p p , 39-h3>
C o u n c i l on Wave R e s e a r c h , Richmond, C a l i f . , 1956.
2U.
D a i l y , J,W. and S t e p h a n , S.C. J r . , The S o l i t a r y Wave, i t s C e l e r i t y ,
I n t e r n a l V e l o c i t i e s and A m p l i t u d e A t t e n u a t i o n T e c h n i c a l
Report No. 8, Hydrodynamics L a b o r a t o r y , M I T , Cambridge,
" M a s s . , 1952.
25.
Munk, W . H . , The s o l i t a r y wave t h e o r y and i t s a p p l i c a t i o n t o s u r f
p r o b l e m s , N . Y . A c a d . S c i ^ 51^ PP. 376-!|2i|, 1 9 i i 9 .
26.
Johnson, J . W . , O ' B r i e n , M . P . , and I s a a c s , J . D . , G r a p h i c a l cons t r u c t i o n o f wave r e f r a c t i o n d i a g r a m s , Hydrodynamic O f f i c e
Pub. No. 605, 19Ü8.
27.
McCowan, J . ,
Journ.
On t h e H i g h e s t Wave o f Permanent Type, P h i l , Mag.
38,,
p , 351, I89U
28.
D a v i e s , T , V . , S y m m e t r i c a l , F i n i t e A m p l i t u d e G r a v i t y Waves, U . S .
N a t i o n a l Bureau o f S t a n d a r d s , G r a v i t y Waves, NBS C i r c u l a r
521,
1952 p p . 55-60.
29.
L a l t o n e , E . V . , Higher A p p r o x i m a t i o n t o N o n l i n e a r Water Waves and
t h e L i m i t i n g H e i g h t s o f C n o i d a l , S o l i t a r y and S t o k e s ' Waves,
Beach E r o s i o n Board T . M . No. 133, W a s h i n g t o n , I963.
30.
I p p e n , A . T , , and K u l i n , G . , S h o a l i n g and b r e a k i n g c h a r a c t e r i s t i c s o f
t h e s o l i t a r y wave, MIT Hydrodyn. L a b . TR No. l 5 ,
1955,
31.
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o f S t a n d a r d s , G r a v i t y Waves, NBS C i r c u l a r 521, 1952, p p .
9-32,
32.
L a r r a s , J , , E x p e r i m e n t a l Research on t h e B r e a k i n g o f Waves,
Annales des Fonts e t Chavissees, V o l . 122, p g . 525> 1952.
33.
E a g l e s o n , P . S . , P r o p e r t i e s o f S h o a l i n g Waves by Theory and
E x p e r i m e n t , K-ans. A . G . U . , 375 PP. 565-572, 1956
6k
3h.
Miche, R , , Movements ondtQatoires de l a mer
Ann
Pnr.+ .
a
35.
ping beach, J . K l m d M e c h a m . „ ,
65
y
302-316,
i960.
APPENDIX
SUMMARY OF DATA
A"I
APPENDIX
TAHLE
A1
DEFINITION OF TEST NUMBERS
Test Number
Period (T)
'
Waveheight a t
Plunger
(seconds)
1
0.90
0.211
2
1.00
0.191
3
1.125
0.167
k
1.25
0.1ii3
$
1.375
0,121
6
1.50
0.105
7
1.25
o.o5o
8
1.25
0.098
9
1.25
0,12U
10
1.25
0.130
11
1,25
0.156
12
i.5o
0.062
13
1.0
0.110
lU
1.0
0.073
15
1,0
O.liió
16
1.0
0.213
Series
9,
I
0°
II
10°
III
27°
IV
51°
A-2
(H^)
(feet)
TAHLE
A2
DATA AVAILABLE
Test
Series
Data
Type
I
A
B
II
C
D
1
X
2
X
A
B
X
3
X
X
5
6
C
X
X
X
X
X
X
X
X
X
S
7
8
X
^ ^
X
9
X
X
10
X
X
11
X
X
12
X
X
13
^
lU
^
15
*
16
X
X I n d i c a t e s data couplet©
X
X
X
TAHLE A 3
AVERAGE BREAKER AND RUNUP DATA,
AND FLOAT VELOCITIES
Test
b
'^b
av
II
ft
av
av
ft
degrees
^f
^10
ft/sec
^f
^16
ft/sec
2
1.62
0.177
3
0.1^-7
O.l'i-'
1.07+
1.15+
l.OU
h.h
7
0.62
0.13'>
0,128
0.066
6.3
a.1
3.6,
6
1.^3
1.33
I.2U
1.17
1.01
1.06+
0.62
3.2
8
9
10
11
12
0.87
1.21
1.07
0.77
h.8
1.03
1.03
1.08+
0,78
U.7
o.u5
0.25
0.27
0.35
I4.6
O.UO
0.71
h.7
0,75
1.1
5.9'
0.38
0.21
0.52
0.8U
0,21
o.ii5
_
i.Ul
1,81
1.76
1.60
h
:?
0.095
0.132
0.117
0.76
0.98
0.157
0.083
0.107
_
_
-.
0.166
3
1.52
1.51
h
l.hh
0.165
0.157
1.13
0.123
6
I.OU
7
8
0.68
9
10
11
12
1.11
1.33
1.55
0.77
0.9U
0.113
O.07I4
0.093
0.121
0.1U5
0.169
0.08);
0.102
0.95
1.05
1.05
1.05
1.06
0,61
13
III
ft
av
r
1
2
13
l.hh
0.85
A-4
0.75
0.89
5.7
IU.5
12.1
10.9
9.3
8.1
7.1
7.0
1.10
0,76
10.5
11.1
12.7
5.0
0.67
10.3
l.OL
0.73
0.65
0.U2
0.60
1.57
1.U3
. 1.29
1.32
1.06
0.60
0.86
1.16
0.83
0.82
0.68
0.53
0.52
0.3U
0.50
0.67
1.61
1.U2
1.23
0.56
0.96
1.23
1.27
l.h?
1.35
1.77
-
-
TAHLE A 3 ( c o n t ' d )
AVERAGE BREAKEi^ AND RUNUP DA'IA,
AND I'XOAT VELOCITIES
Series
Test
b
av
2
3
k
6
7
8
9
10
11
12
13
15
r
^b
ft
l.ijO
1.15
1.22
1.32
0.91
0.69
0.83
1.19
1.27
1.29
0.57
0.88
1.11
%
av
av
ft
ft
0.8k
0.153
0.125
0.133
0.88
1.02
1.17
0.90
0.59
0.79
0.86
O.lhk
0.099
0.075
0.090
0.130
0.138
O.lUl
0.97
1.12
0.062
0.096
0.121
0.5U
0.72
0.79
A-5
^f
av
^f
10
degrees
ft/sec
30.7
22.1
1.83
1.77
1.72
19.3
16.0
7.3
13.a
16.8
20.a
22.2
25.0
8,8
22.5
19.0
1.71
l.OU
0.87
1.20
l.hh
• 1.73
1.93
O.iiO
1.31
^16
ft/sec
2.15
1.89
1.91
1.81
0.91
0.65
1.20
1.57
1.88
1.96
0,18
l.ii6
TABLE
Ak
WAVE HEIGHT AND CELERITY AT BREAKING
Series
Test
3
X
II
S e r i e s: I I I
^b
IV
"b
ft
«b
ft
3
0.192
_
—
0.130
3.9k
-
-
0.192
-
_
-
-
-
0.162
2.50
2.30
•-
0.155
3.20
—
—
-
-
0.176
3.55
3.22
2.66
h
ft/sec
ft
6
-
7
0.023
-
8
0.075
-
9
o.iui
10
11
0.15U
0.171
-
-
-
-
2
S e r i e si
10
ft/sec
-
_
ft
ft/sec
0.179
2.60
3
0.230
2.91
-
-
0.20U
k
0.170
3.28
-
-
0.185
0.218
3.12
0.192
2.21
3.16
6
-
2.88
1.78
7
0.0^5
1.88
1.15
0.166
2.90
8
0.157
2.70
0.157
0.091
0.126
0.167
0.101
-
0.125
9
0.211
3.26
2.01
0.152
10
0.198
2.37
0.167
3.08
11
0.221+
2.79
2.6o
O.II49
0.170
2.7a
2.89
0.167
0.237
3.38
12
0.065
0.088
—
—
13
2.76
0.109
2.39
1.58
2.79
0.107
2.2a
A-6
—
2.22
TABLE
I k
(cont'd)
WAVE HEIGHT AND CELERITY AT BREAKING
ft
1
2
3
13
ft
ft/sec
ft
ft/sec
ft
ft/sec
„
-,
-
_
-
0.177
0.168
0«186
0.187
-
2.50
0.198
-
-
_
k
0.130
_
3.6U
„
-
_
-
_
0.160
3.92
2.86
5
O.lSk
-
-
-
-
-
0.032
0.102
0.186
0.177
0.177
-
„
_
-
_
_
„
_
0.1ii7
_
0.130
_
I.I4O
k
$
3
2
7
8
9
10
11
17
A-7
-
-
. _
_
_
3.25
_
mSLE
A 5
LOCAL BREAKER POINT, BREAKER ANGLE
AND RUNUP L I M I T
X
(II
b
r
ft
ft
Idegrees
2)
(II
h
1.38
0.88
7
1.67
1.08
b
r
ft
ft
3)
(II
h)
1;
1.37
0.75
3.5
7
6
1.58
1.06
7.5
1.1;9
1.13
3
1
1.09+
6
1.59
1.12+
5
5
1.11
7.5
1.25 • 1.22+
li.5
.8
8.5
0.96
ii.5
5
15
1.91;
1.02
7
1.70
1.21
8
(II
5)
1.03
6)
(II
I;
7)
3.5
7
2
0.66
0.56
1.5
1
1.09+
5.5
0.77
0.75
3.5
5
1.16
1.19+
3.5
0.58
0.76
3
.8
1.13
1.09
6
0.53
0.55
1;.5
1.22
-1.5
1.22
0.87
7
1.38
1.28+
2
1.11
0.96
11
1.35
1.09+
6.5
1.22
15
1.16
1.19
3
18
0.92
0.96
8
(II
(II
0.1;7
l.lil
8)
1.13
0.57
h
(II
^b
degrees
2.5
1.05
(II
ft
2
1.82
l.Oli
ft
1.01
11
1.28
(II
r
1.I;6
1.68
18
(l e g r e e s
b
(II
9)
(IJ
ll
10)
k
0.82
0.59
3.5
1.19
0.7U
k
1.29
0.73
3
7
7
0.91
0.70
3
1.35
0.99
3
1.39
1.02
3.5
i
11
0.96
0.91
5.5
i.iiU
1.12
5
1.51
1.09
6
6
15
0.85
0.91
U.5
1.07
1.21
Ul5
1.12
1.21
3.5
iB
18
0.82
0.75
7.5
1.01
1.07
7
1.05
1.11
7
(II
(II
11)
12)*
(II
\
13)
k
l.lt5
0.80
2
0.80
0.60
k
1.00
0.61;
2
1
1
1.62
1.12
3
0.67
0.72
2
1.06
0.82
3.5
1
11
1.78
1.12+
6
0.79
0.83
0.5
0.96
0.69
5.5
5
15
1.32
1.22+
1;
0.73
0.80
0
l.Ol;
0.83
6
,B
18
1.03
1.11;+
8.5
0.81
0.97
-1
0.83
0.77
6.5
b
by CJG, A l l o t h e r 0^ by RLB
b
A-8
TAHLE
A 5 (cont'd)
LOCAL BREAKER POINT, BREAKER ANGLE
AND RUNUP L I M I T
X
(III
h
b
r
ft
ft
degrees
2)
1.58
1.68
(III
11
15
18
(III
k
7
5)
0.y5
(III
0.91;
5
1.03 11
1.26 1.11+ 12.5
1.05 1.19 10
1.12 1.00
8
1.27
11
15
18
( I I I 8)
h
7
0.93
11
15
18
k
7
11
15
18
ft
3)*
1.58
degrees
(III
6)
1.10 0.96
11
16
15
9
7
1.23
1.12+
1.22+
1.13
11.5
7.5
5.5
9)
(III
0.85 0.72
a.5
l.ll;
0.89
8
0.89 ' 0.75
7
6
1.16 0.96
0.98 1.05
1.05 0.90
10
11
12
1.5U
0.96
12.5
1.5)4 1.06 10
1.75 1.09 11;
i . J i l 1.22+ l l ; . 5
1.52 1.11;+ 12.5
12)
0.83 0.1;7
h)
1.38
0.67
0.85
11.5
(III
ft
0.73
0.61;
11)
ft
9
1.21
8.5
r
0.90
o.hh
0.63
0.71; 0.66
0.65 0.76
0.62 0.58
1.35
0.80
1.U3 0.99
1.39 0.99
1.17 1.18
1.29
(III
1.09
9.5
11.5
12
9.5
10.5
1;.5
7
7
6.5
11
9
12
11.5
12
13)
9
0.89
0.1t6
0.73
0.61
8
1.07
0.90
0,96
0.73
0.92
2
3
0.76
0.82
3
0.95 0.70
0.97 0.75
0.82 0.71
A-9
12
10)
0.61;
'0^ hy CJG. A l l o t h e r 0^ hy RLB.
degrees
( I I I 7)
0.98
0.93
b
1.51 0.97
1.51; 1.10
1.35 1.16
i.ko
1.12
9.5
9
0.75 0.95
0.85 0.73
(III
ft
0.98
(III
0.62
r
0.86
1.69 1.01
1.68 1.11+
1.37 1.19
1.23 1.07
0.81
9.5
0.83 13.5
1.62 1.00 16.5
l.ho
1.07 16
1.31 1.05 17
7
b
9.5
10
11
11
10
TABLE
A 5 (cont'd)
LOCAL BREAKER POINT, BREAKER ANGLE
AND RUNUP L I M I T
X
b
r
ft
ft
degrees
2b)
(IV
(IV
h
1.56
0.71
26
7
1.U9
0.91
30.5
11
15
18
1.60
1.39
1.16
1.12+
0.92
0.96
31
h
1.67
7
11
15
18
i.m
1.31
0.96
1.25
(IV
k
7
11
15
18
(IV
k
7
11
15
18
8)
1.02
1.05
0.89
0.59
0.62
11)
1.63
1.62
1.08
1.06
1.06
ft
ft
degrees
3)*
32.5
21
19
16
18
l.li;
1.1U+ 6
19
17
15.5
19
13.5
(IV
1.01
1.20
1.12+
1.22+
6)^
1.16 1.00
0.8U 0.91
1.00 0.89
0.82
0.9U
0.7U
0.81
i.o5
by CJG. A l l
l.kk
0.81
1.39
1.06
1.08
1.00
0.91
1.12+
1.00
0.71
11
degrees
0.90
1.09
1.12+
1.21
0.87
21
20
20.5
Ik
21
3.5
0.88
0.86
6.5
0.72
0.58
0.56
0.56
8.5
7
0.1i9
0,6h
l5.5
0.51
0.58
7
lk.$
16
Ik
23
23
21
1.52
0.8U
2k
1.53
1.08
21
1.18
1.12+
25
17 .
1.06
1.13
19
18
1.08
0.82
22
(IV
0.69
23.5
0.57
0.65
0.60
0.52
0.58
O.UO
o.5i
0.5U 0.52
A-10
ft
( I V 7)
12)*
o t h e r 6^ by RLB.
ft
( I V 10)
2U.5
26
26
25
r
1.U5
1.U2
1.08
0.98
1.16
1.03
( I V 9)
0.8U
0.79
0.81
0.71
0.82
b
(IV
2i|.5
1.02 o.ih
21.5
1.08 1.12+ 15
1.00 0.76
23
1.11 0.97
26.5
(IV
1.11
1.29
1.12+
r
1.56
33.5
( I V 5)"''
b
13)
10
1.10
0.68
15
0.79
0.69
22
22.5
5
0.82
0.80
19.5
15 t o 9
0.82
0.69
25.5
0.86
0.82
23
k
to-l
TiBLE
A 5 (cont'd)
LOCAL BREAKER POINT, BREAKER ANGLE
AND RUNUP LIMIT
X
(IV
b
r
ft
ft
b
degrees
15)
1.29
1.07
1.09
1.18
0.9U
0.7li
0.85
1.12
0.8U
0.73
18.5
23
Ih
20.5
19
^0^ by CJG. A l l other 0^ by RLB.
A-11
Tkm£
16.1
CHANGE I N MEAN SEA LEVEL ON BEACH (SERIES I )
rap
y
e
y
e
y
e
y
e
ft
cm
ft
cm
ft
cm
ft
cm
( I U)
1)
(I
12 f t
0 ft
12 f t
0 ft
1
0.22
1.13
o.UU
1.08
0.27
1.03
0.U9
0.85
2
0.72
0.87
0.9U
0.83
0.77
0.U2
0.99
0.7U
3
1.22
0.37
0.08
1.27
-0,27
1.U9
0.62
h
2.72
-0.10
i.UU
2.9U
-0.22
2.77
-0.06
2.99
-0.12
(I
lU)
(I
6)
1
0.25
0.U5
0.U7
0.39
0.28
0,60
o.5o
0,10
2
0.75
-o.o5
0.97
0.02
0.78
-o.Uo
1.00
0
3
1.25
-0.15
1.U7
-O.IO
1.28
-O.UO
1.50
-0,UO
h
1.75
-o.oU
2.97
-0.08
2.78
-0.10
3.00
-0.10
(I
(I
15)
16)
1
0.28
0.63
o.5o
0.50
0.35
1.15
0.57
1.30
2
0.78
0.16
1.00
-0.02
0.85
0.6U
1.07
0.90
3
1.28
-0.10
1.50
-o.iU
1.35
0.25
1.57
-0.33
h
2.78
-0.05
3.00
-0.05
2.85
0.09
3.07
-0.2U
X
(I
0 f t
=
U ft
8
16 f t
12 f t
2)
1
0.27
l.OU
0.27
1.00
0.35
2
0.77
0.52
0.77
0.52
-
3
1.27
2.77
0.08
1.27
-0.18
1.35
-0.08
2.77 -0.22
2.85
h
ft
A- 12
0.U9
0.92
0.37
0.7;
0.99
0.73
-
-
-0.20
1.U9
0.70
1.37
-0.12
2.99
-0.12
0.98
-0.1'
2. 87 -0.01
TAHLE
A 6.2
CHANGE I N MEAN SEA LEVEL ON BEACH (SERIES I I AND I V )
On a l l p i e z o m e t e r c h a n n e l s ,
For
p i e z o m e t e r t a p s have t h e f o l l o w i n g s p a c i n g :
t a p number
1
distance from tap
n o . 1, i n f e e t
^
2
3
U
5
6
-^^0
2.5
5.0
8.0
s e r i e s I I and I V , t h e y c o o r d i n a t e o f t a p n o . 1 a t v a r i o u s x v a l u e s
X o f p i e z o m e t e r channel, i n f e e t :
y o f t a p n o . 1,
in feet:
h
8
O.ll
0.18
12
l6
20
0,30
0.25
0.27
is
0.30
e cm
Test
II
II
II
2
3
k
^"^P
no.
X = U ft
8 f t
12 f t
16 f t
20 f t
-
0.58
1.06
-
-
-
1
-
2
0,i;7
3
-0.32
-0.10
k
-O.lU
-0.13
-
0.02
-0.16
5
6
-O.Oi;
1
1.12
2
0.1;6
3
-0.23
-0.10
0.U6
-0,08
k
-O.li;
-0.15
-O.lU
-O.OU
-0.05
-0.02
-
-O.OU
-0.12
-
5
6
-0.05
1
o.Uo
2
0.30
-
3
-O.lU
0
il
-0.08
0
5
6
0
-O.OU
0.10
-0.20
-0.09
-0.01
_
o.Ui
-0,20
-O.OU
-o.i5
-0.o6
-0.03
0.73
-0.22
-0.12
-0.09
-O.OU
0.82
-0.25
-0.08
-0,13
-0.08
-O.OU
-0
-0.01
A-13
0.17
-0.30
-0.12
-0.07
-0.03
_
0.10
-0.32
-0.09
-0.05
-0.02
-0.16
-0.25
-0.08
-0.03
0.03
21; f t
28 f t
-0.30
0.22
-0.3U
-0.11
-0.05
-
-0.10
-0.30
-O.lU
0.76
-0.08
-O.OU
0,01
-
28
2k
-
-0.05
0.10
-0.03
0.01
0
(
TAHLE
A 6.2
(cont'd)
CHANGE I N MEAN SEA LEVEL ON BEACH (SERIES I I AND I V )
e
Test
II
tap
no.
5
8
-
0.72
—
O.Uo
-
-0.25
-0.16
-
-0.2U
-0.12
-0.02
6
-0.06
-0.15
1
2
0.02
0
3
-0.28
6
II
20 f t
2
-
7
16 f t
-
-0.19
II
12 f t
1.18
h
6
8 ft
1
3
II
a f t
X
-O.Ik
cm
-0.17
-0.08
-0.03
-0.20
-
-O.lU
-0.19
-0.12
-0.09
0
-0.07
-0.03
-0.01
-
0.62
-0.01
-0.16
-0.05
-0.01
-0.10
-0.20
-
-0.09
-0.17
-0.10
-0.05
-O.OU
-O.OU
0.03
-
-0.U3
-0,15
-0.09
-0.05
0.10
1
-
-
2
-0.09
-
-0.13
3
-O.OU
u
0.01
-0.05
0
-0.03
0.00
-
0.03
0
6
0.03
O.OU
0.01
0.01
0,02
0
O.OU
0.02
1
-
-
0.U8
2
0.06
-0.10
3
-0.07
U
-0.05
6
-O.OU
-
-
O.OU
-O.OU
0
-0.18
-
—
-0,03
-0.01
-O.lU
-0.06
-0.12
-0.10
-0.2U
-0.12
-O.OU
-0.03
-0.05
-0.03
-0.12
-O.OU
-0.03
-0.05
-O.OU
O.OU
0.03
A-14
o.o5
-0.22
0,02
-0.03
0.01
_
-0.05
-0.02
_
28 f t
2U f t
-
-
-
-
O.OU
1.19
-0.20
-0.13
-0.08
-
-0.06
_
-
-0.03
-0.08
0.02
0.0k
0.03
0.02
-
-
0.05
-0.10
-0.01
—
-0,05
-O.OU
-0.02
-
_
-
-0,03
TABLE
A 6.2
(cont'd)
CHANGE I N MEAN SEA LEVEL ON BEi\CH (SERIES I I AND I V )
e
Test
II
9
tap
no.
1
2
3
0.78
0.22
-o.o5
k
-0.09
16
ft
-
-
0.70
-
0.37
-
-0.20
-0.10
-O.OU
-0,20
-0,10
-0.05
-0.05
-0.03
0.63
1.07
0.86
2
1.1)4
0.32
-
0.27
-
3
-O.OU
0.18
-0,13
-o.lil
0.10
0.03
-0,11
-0,07
I
20
f t
-0.12
-0,19
-0.08
-0.03
0
0.30
0.30
-0.10
-0.10
-0.07
-0,03
6
-
O.OU
-O.OU
-0.07
-0.09
-0.05
-0,00
1
2
1.02
0.62
0.87
0,73
0.31
-
-
-
-o,ou
-0.09
-0.1);
0
0
-
-
-0.19
-0,12
-0.09
-0.03
4
6
f t
-0.05
3
IV
12
6
-0.08
IV h
ft
-0.20
-0.08
-0,03
-0.02
-
IV 2
8
=U f t
X
cm
-0.09
-
-0.05
O.lU
0.30
-
-
0.35
-0,10
-0.08
1
2
3
0.62
-O.OU
-0.17
k
-0.08
-0.15
6
-0.02
-o,iU
-
0
6
-0.02
-0.15
-O.Oy
-0,07
-
-
-
-
-
-0.05
-0.02
-0.03
A~15
_
2U
f t
0,01
-0.01
-0.03
-0.02
-.
0.06
-O.lU
-0.03
-O.OU
0
0.56
-0.50
-0,07
28
f t
0.01
-
-0.01
0.7U
0.5U
0.20
-0.05
-0.05
-0.05
0.20
0.18
0.06
O.OU
-
-0.05
0.08
-O.oU
_
-0.12
0.02
-
-O.lU
-
-0.05
0
-
-0.05
-0.02
-
-
0
miLE
A 6.2
(cont'd)
CHANGE I N MEAN SEA LEVEL ON BEACH (SERIES I I AND I V )
e cm
Test
IV
7
"^^P
no.
9
8 f t
12 f t
16 f t
20 f t
0.35
_
0.12
0.18
-
-
-
-
-0.12
0.03
-O.OU
-O.OU
_
-0.03
-0.02
0
-O.OI
1
2
3
-0.10
-0.05
U
-O.OU
5
_
-0.01
-0.10
_
_
0.78
0.18
-0.10
-0.07
0.51
-0.13
-0.07
6
IV
X = U ft
1
2
3
U
oM
5
6
0.65
-0.09
-
-
-0.01
-
-
A-16
-
0.60
-0.07
-O.OU
-O.OU
-0.02
-
-0.02
-
-0.01
-
2h f t
-0.05
-0.05
-0.05
-0.02
-
0.12
-0.22
-0.07
-0.02
-
28 f t
0.03
o.o5
0.17
-o.o5
TiMjE
A
6.3
CHA.N(S: I N MEAN SEA LEVEL ON BEACH (SERIES I I I ) '
tap
no.
Test
III
20
ft
e
y
e
y
8
7
6
7
e
7
e
ft
cm
ft
cm
ft
cm
ft
cm
ft
cm
ft
cm
ft
cm
ft
cm
0.28
-0.2k
ft
2.00
-0.2ii
1.81
-0.16
1.75
1.26
0.32
1.3ii
-0.23
-
-
6.81i
-0.02
5
3.76
6
6.76
-0.05
-0.05
d, = l . l i i
d
-
0
1.62
0.01
1.39
1.5
-0.17
1.51
h.O
0.ii3
ii.Ol
-0.13
-0.05
-
-
-
0.10 0.82
1.60 -0.08
ii.10 -0.02
7.00 -0.02
0.37
-
0.10 1.U6
1.60 -0.15
U.10 -O.Oii
7.00 -0.02
-
0.31
0.81
1.31
2.81
5.31
0.10
-0.56
-0.U9
-0.08
-0.09
0.72
0.h2
1.22
0.U2
0.11
0.U2
0.12
1.12
0.19
1.05
0.31
1.10
0.26
0.96
2
0.61
0.79
0.62
0.h2
-
-
-
-
-
-
3
1.11 0.30
2.61 -0.13
5.11 -0.13
1.12
0.08
1.19
2.62 -0.17
2.69
-
-
d^ =
2
3
k
1.18
0.22
1.87
ii.37
7.37
ft
1
1
ft
7
0.ii3
2c
28
ft
e
1.26
5
6
2li
ft
7
h
k
III
16
ft
e
- 0.99
2b
12
8 ft
h ft
7
2a
3
III
0 ft
-
-
-0.12
-0.20
1.31
-0.27
2.81
-0.19
1.26 -0.37
2.76 -0.17
-
5.31
-0.12
-
-
-
-
-
-
-
-
-
-
-
-
-
-
—
—
1.72 -0.27
3.22 -0.05
5.72 -0.05
8.72 -0.08
ft
0.78
0.72
0.30
1.22 -0.15
2.72 -0.07
0.23
0.73
1.23
2.73
0.85
0.30
-0.15
-0.12
0.30
0.U5
0.80 1.30 -0.15
2.80 -0.10
0.U2 0.65
0.92
0
1.U2 -0.10
2.92 -0.10
0.37
1.37
2.87
0.75
-0.20
-0.10
S e r i e s I I I data f o r e a r e 3 runs of t e s t I I I 2, a t d i f f e r e n t s t i l l water l e v e l s ,
± O.OU f e e t .
0.92 -0.30
1.U2 -0.20
2.92 -0.10
0.U2 -0.06
0.92 -0.27
1.U2 -0.12
2.92 -0.05
j
good to
coordinates
only
0.83 O.I45
1.33 0.37
I.83 0.78
3.33 0.17
TAHLE
A 7
TWO-DIMENSIONAL VELOCITY DISTRIBUTION
Series
II
Test
2
3
y
X =
ft
ft
9
11
13
15
17
ft
ft
ft
ft
ft
ft
0.5);
0.60
o.5o
0.37
0.)49
0.3U
0.52
0.57
0.68
0.k7
0.)49
0.36
0.25
0.33
0.U3
0.83
O.hh
1.33
0.17
0,33
0.35
0.25
1.83
0.19
0,2U
-0.17
0.17
0.32
0.31
0.19
0.83
0.58
0..'i7
0.)|2
0.77
0.53
0.28
0.25
0.16
0.35
0.h2
o.li5
0.2h
0.23
0.35
0.21
0.32
0.U6
0.51
0.59
0.38
0.57
0.38
0.61;
0,3h
0.U9
0.28
0.16
0.h9
0.51
0.60
0.71
0.68
0.38
0.19
0.U3
0.29
0.51
0.29
0.52
0.29
0.53
0.50
1.33
1.83
0.15
0.2U
0.18
•-0.17
0.18
0.2h
0.32
0.31
0.37
0.1(2
0.5)4
0.33
0.83
0.32
0.32
0.30
0.37
O.hB
0.)40
1.33
1.83
0.19
0.15
0.19
0.18
0.18
0.18
0.2ii
0.31
0.16
0.3)1
0.17
0.51
0.28
0.16
0.53
0.50
0.67
0.31
0.27
0.27
-0.17
0.27
0.30
0,28
0.21
0,21
0,51
0.52
0,h7
0.)4l
0.3)1
0.37
0.50
0.62
0.50
0.6)1
0.26
0.18
0.31
0.h3
O.I49
0.3)1
0.16
0.27
0,17
0.15
0.19
0.28
0.16
o.)i5
0.33
0.17
0.15
0.19
0.2h
0.15
0.2)4
0,27
0.19
0.33
0.U5
0.I;)4
o.)_a
O.ho
0.)42
0.27
0.38
0.37
0.50
0.83
0.27
0.16
0.U7
0.26
0.37
0.27
0.38
O.lil
O.Jil
0.27
0.28
0.28
0.35
0.25
0.33
0.83
1.33
1.83
6
6
-0.17
0.33
h
2
-0.17
1.33
0.2)4
A-18
o.)45
0.17
0. 32
TAHLE
A 7
(cont'd)
TWO-DIMENSIONAL VELOCITY DISTRIBUTION
Test
6
9
11
13
15
17
ft
ft
ft
ft
ft
ft
ft
0.0
0.13
0.16
0.20
0.2i|
0.23
0.23
0.83
o.lil
O.lil
o.lil
0.13
O.lil
0.17
0.13
0.16
0.12
0.10
0.21
0.15
1.33
0.13
0.13
0.13
0.23
0.16
0,12
0.33
0.0
0.19
-0.17
0.0
O.lil
0.15
o.lil
0.23
0.13
0.25
0.13
0.33
0.28
0.37
0.29
0.83
O.lil
0.20
0.28
0.15
0.15
0.13
0.15
0.21
0.17
0.15
0.23
1.33
0.0
0.25
0.16
0.15
0.15
O.lil
0.13
0.15
O.lil
0.32
0.21
0.16
o.i5
y
ft
II
7
8
-0.17
1.83
9
10
11
X =
2
o.lil
0.16
0.25
0.31
0.30
0.28
0.i|2
0.33
0.12
0.31
0.26
0.30
0.36
O.iil
0.1i3
0.60
0.83
0.27
0.29
0.35
0.39
0.ii9
0.i|2
1.33
0.17
0.17
0.25
0.26
0.27
1.83
0.17
0.19
0.16
0.31
o.i5
0.16
0.17
O.lil
O.lil
0.15
-0.17
0.13
0.19
0.17
0.3i|
0.3ii
0.30
o.ii5
0.33
0.32
0.30
0.28
0.38
0.i|6
0.62
0.83
0.27
0.27
O.iiO
0.ii3
O.ilO
1.33
0.18
0.20
0.26
0.27
0.27
1.83
0.17
0.22
0.16
0.31
0.16
0.17
0.51
0.50
0.17
O.lil
o.lil
o.lil
-0.17
O.lil
0.27
0,30
0.35
0.149
0.51
0.33
0.15
0.25
0.27
0.31
0.39
o.51i
0.63
0,72
0.83
0.29
0.28
0.28
0.29
0.57
0.58
0,57
1.33
0.27
0.27
0.30
0.39
O.iil
o.ii5
0.17
0.17
0.20
0.20
0.19
-0.17
1.83
' 0.23
0.19
0.17
A-19
TABLE
A7
(cont'd)
TWO-DIMENSIONAL VELOCITY DISTRIBUTION
Series
III
Test
2
X =
ft
ft
•
7
9
11
13
15
17
ft
ft
ft
ft
ft
ft
-
1.68
i.a9
1.76
1.55
2
_
1.08
1,23
1.20
-
1,68
1.62
0.77
1.20
-
i.a9
-
1.6a
1.22
-
1.57
0.98
-
1.15
1.20
0.63
-
-
-
I.U7
1.76
-
1.67
1.50
-
-
0.93
0.92
1.97
-
0.35
-
-
-
-
-
_
1.20
1.32
1.17
1.65
1.69
1.10
1.03
1.57
1.61
1.57
1.69
1.56
0.30
1.10
0.53
0.80
1.06
-0.06
3
h
y
0.2U
0.5U
0.8U
1.2U
-
-
1.30
1.51
-
1.23
i.ao
-
-
1.2a
i.a2
-
0.98
1.6U
-
-
0.26
0.U6
_
0.76
1.06
1.1;6
1.73
-
1.10
1.57
1.30
1.23
1.59
1.32
0.55
0.59
0.79
0.85
0.93
1.15
1,32
-
1.33
1.38
i.a5
1.71
1.75
1.61
i.a5
1.25
1.27
1.53
i.a9
-
1.0a
1,07
1.11
1.32
-
0.62
o,6a
0.81
-
-
0,35
o.a3
0.87
0.52
i.ao
0.98
0.82
1.20
0,63
i.ai
1.35
1.16
1.51
1.66
1,36
1.53
1.33
0.69
0.31
0.88
1.33
1.38
1.30
0.96
0.27
0.3a
0.39
o.a5
-
-0.06
0.2U
-
-
1.00
0.5U
0.814
-
-
1.2U
_
1.6a
_
-
0.97
1.06
A-20
0.67
0.32
0,83
1.38
1.70
1.63
0.76
TkSLE
A7
(cont'd)
TWO-DIMENSIONAL VELOCITT DISTRIBUTION
Series
Test
y
ft
III
X =
2
ft
7
9
11
13
15
17
f t
f t
ft
ft
ft
ft
0.78
1.21
1.27
1.19
0.77
0.1;3
0.82
0.59
0.32
0.97
1.22
1.27
1.13
0.70
0.36
0.61;
0.36
0.95
0.73
0.90
o.kk
1.03
0.i;6
0.90
0.88
0.70
O.kk
0.95
0.70
0.1;6
0.15
0.17
0.15
0.80
1.25
1.30
0.71;
1.30
0.57
-0.06
0.71+
0.99
0.2k
1.07
1.12
0.97
0.60
1.30
1.29
1.02
0.%
o.Bk
1.1k
0.32
-0.06
0.57
0.96
0.2k
0.5U
o.dk
0.59
i.ik
0.23
1.53
0.15
-0.07
0.95
_
0.22
-
1.2k
1.12
0.95
0.51
1.6i|
0.23
0.51;
o.dk
11
0.88
-0.06
0.2k
o.Sk
O.Qk
1.2k
1.6k
1.07
1.25
1.27
1.28
1.06
0.59
A-2I
0.85
0.66
0.27
0.20
1.00
1.13
1.17
1.02
0.56
0.25
0.85
1.07
1.15
1.05
0.75
0.i;5
0.1;9
1.23
0.75
0.27
1.1;2
1.10
0.72
0.30
0.36
1.21
1.27
1.30
0.98
1.3U
1.38
1.58
1.75
1.58
1.53
1.72
l.iil;
l.kk
1.75
1.63
1.15
0.67
1.15
1.01;
0.81;
1.23
0.92
1.35
1.19
0.80
1.71;
1.36
0.90
miLE
A7
(cont'd)
TWO-DIMENSIONAL V E L O C I T Y D I S T R I B U T I O N
Series
Test
y
ft
IV
2
X
=
2
ft
15
17
ft
ft
ft
ft
ft
ft
2.07
1.85
i.ia
0.70
2.18
2.oil
1.U6
0.80
2.17
2.05
1.U2
0.80
2.00
0.53
0.57
1.07
0.89
1.67
1.66
0.^9
1.09
0.53
l.lh
i.5i
0.78
0.50
1.73
1.71
1.07
0.56
1.88
1.61
1.00
0.^1
1.73
1.63
0.81
0.U8
0.65
1.75
0.99
0.69
1.23
0.95
0.^0
1.5l
1.39
0.97
Ü.Ü8
1.63
1.73
0.95
0.u8
1.91
1.53
0.81
0.51
2.oil
1.79
0.98
1.96
1.80
1.08
0.60
1.73
1.62
1.03
0.6U
1.10
0.82
0.^0
0.52
l.[i2
1.33
0.90
1.62
1.56
0.82
0.50
1.93
I.I45
0.78
0.55
1.82
1.71;
0.99
0.51
1.7ii
1.56
1.05
0.60
1.57
1.U9
1.03
0.57
1.05
0.92
0.97
0.77
0,U8
0.98
1.02
0.86
0.77
0.)42
0.85
0.146
0.82
0.78
1.81
0.97
0.5U
0.U7
0.50
o.hh
o.Uo
0.31
1.00
0.92
0.81
0.)i?
0.63
1.31
O.uli
0.)ii|
o.9h
0. 5o
0 J|2
0.93
0,52
o.Uo
0.28
0.78
1.28
1.78
0,31
0.81
0.31
0.81
1.31
1.81
0.31
0.81
1.31
7
13
1.75
1,53
1.19
0,58
1.31
1.81
6
11
1.7U
1.75
1.18
1.23
0.92
0.^6
1.78
h ,
9
1.U9
1.U9
1.26
0.28
0.78
1.28
3
6
O.bO
o.kQ
0.U3
0.h3
A-2 2
0.53
o.ki
0.57
O.hh
0.77
0.52
0.h2
0.8U
0,61
0,.)|1
0.78
0.68
0.U7
,0.71
0.61
0.);7
TAHLE
A 7
(cont'd)
TWO-DIMENSIONAL VELOCITY
Test
y
ft
IV
8
9
10
11
12
13
0.28
0.78
1.28
1.78
0.31
0.81
1.31
1.81
0.31
0.81
1.31
1.81
X
=•
ft
1.12
o.kö
O.kS
0,h6
1.15
0.77
0.36
0.ii9
1.26
0.9U
0.U8
0.51
2
DISTRIBUTION
6
9
11
13
15
17
ft
ft
ft
ft
ft
ft
1.2ij.
1.11
0.97
0.51
O.iil
1.12
0.U3
0.1i2
1.27
0.81
0.U2
o.ko
1.148
l.lil
1.57
1.08
0.55
o.kQ
0.55
0.U6
0.81
1.59
1.J49
0.78
1.85
1.12
0.72
0.h9
0.i|8
0.U9
1.59
0.86
O.J48
1.53
i.ho
1.05
0.51
1.77
1.66
1.12
0.52
1.92
2.16
1.56
1.00
1.7ii
1.11
0.57
0.59
0.53
0.51
0.53
0.51
o.ko
0.ii7
0.50
1.07
0.9U
0.k3
0,kk
1.25
1.27
o.5u
0.k9
l.kh
1.32
1.19
0.78
0.31
0.81
1.31
1.81
0.51
0.31
0.67
0.81
0.U3
0.63
0.53
1.31
0.i;3
O.iiO
0.i|2
0.31
0.81
1.09
1.15
0.50
1.31
0.88
1.31
1.09
0.73
O.I46
O.Uó
1.81
O.hh
o.hh
0.k2
1.22
1.00
0.56
A-23
iM
0.91
0.55
0.U6
0.97
0.U3
O.ko
1.60
1.33
0.6k
o.ki
1.90
0.ii9
1.50
0.99
0.53
0.U3
0.92
0.U8
0.39
l.k9
I.2U
0.70
0.ii9
l.Uit
1.2k
0.72
0.U7
1.87
1*59
0.92
0.51
1.71
1.50
2.08
1.92
1.25
0.61
0.U6
0.U7
0.1;8
1.57
1.09
0.57
0.i;2
0.93
0.56
1.80
1.59
1.19
0.65
0.50
o.kh
O.Ul
1.51
1.11
0.58
0.U6