Po:..u-s I , ; - j u L i - r
Ralph M. Parsons L a b o r a t o r y
For Water Resources and Hydrodynamics
Department o f C i v i l E n g i n e e r i n g
Massachusetts I n s t i t u t e o f Technology
WAVE TRANSMISSION BY OVERTOPPING
by
Ralph H. Cross
Charles K. S o l l i t t
T e c h n i c a l Note No. 15
December 1970
Prepared Under
C o n t r a c t No. DACW-72-68-C-0032
C o a s t a l E n g i n e e r i n g Research Center
U.S. Army Corps o f Engineers
Washington, D.C.
DSR P r o j e c t 71135
ABSTRACT
This report presents a theory for ocean wave transmission
past breakwaters by overtopping, based on an evaluation of the
energy content of the overtopping water.
While several co-
efficients are subject to further investigation, the data shows
that the general form of the equations developed is correct.
Comparison with large-scale model tests reinforces this belief,
and comparison of an intermediate theoretical result predicting
the volume of overtopping water with published data again shows
reasonable agreement.
An envelope curve for the transmission
coefficient, based on all available data, gives a simple tool for
preliminary design estimates of the transmission coefficient.
ACKNOWLEDGEMENTS
This study o f wave t r a n s m i s s i o n past breakwaters was
sponsored by t h e C o a s t a l E n g i n e e r i n g Research Center o f t h e U. S.
Army Corps o f Engineers
under C o n t r a c t No. DACW-72-68-C-0032.
Contract a d m i n i s t r a t i o n was p r o v i d e d by t h e M.I.T. D i v i s i o n o f
Sponsored Research under DSR 71135.
The work was c a r r i e d o u t i n t h e Ralph M. Parsons L a b o r a t o r y
f o r Water Resources and Hydrodynamics o f t h e Department o f C i v i l
Engineering a t M.I.T.
Mr. Ken W i l s o n , a Research A s s i s t a n t i n
the L a b o r a t o r y , a s s i s t e d w i t h t h e data r e d u c t i o n and numerous o t h e r
essentials.
INTRODUCTION
The d e s i g n f o r a h a r b o r i n an exposed l o c a t i o n g e n e r a l l y
i n c l u d e s a breakwater t o p r o v i d e an area s h e l t e r e d from t h e
waves.
As breakwaters a r e u s u a l l y designed t o p e r m i t some over-
topping b y . t h e waves d u r i n g severe storms, i t becomes necessary
to p r e d i c t t h e c h a r a c t e r i s t i c s o f t h e waves so t r a n s m i t t e d
into
the h a r b o r , t o assure t h a t t h e wave a c t i o n i n t h e s h e l t e r e d
i s w i t h i n acceptable l i m i t s .
area
This r e p o r t presents a theory f o r
wave t r a n s m i s s i o n by o v e r t o p p i n g , based on an e v a l u a t i o n o f t h e
energy c o n t e n t o f t h e o v e r t o p p i n g w a t e r .
While t h e c o e f f i c i e n t s
f o r r e f l e c t i o n and r e g e n e r a t i o n a r e open t o q u e s t i o n , t h e
l a b o r a t o r y d a t a shows t h a t t h e g e n e r a l form o f t h e e q u a t i o n s
developed i s c o r r e c t .
Comparison w i t h l a r g e - s c a l e model t e s t s
r e i n f o r c e s t h i s b e l i e f , and comparison o f a t h e o r e t i c a l p r e d i c t i o n o f t h e volume o f o v e r t o p p i n g w a t e r w i t h a p o r t i o n o f t h e
p u b l i s h e d data a g a i n shows reasonable agreement.
An envelope curve
f o r t h e t r a n s m i s s i o n c o e f f i c i e n t , based on data f r o m s e v e r a l sources,
gives a simple t o o l f o r p r e l i m i n a r y e s t i m a t e s o f t h e t r a n s m i s s i o n
coefficient.
THEORY
When a wave o v e r t o p s a b r e a k w a t e r , some o f t h e i n c i d e n t wave
energy i s r e f l e c t e d , some i s d i s s i p a t e d , and some i s t r a n s m i t t e d
past t h e breakwater i n t o t h e s h e l t e r e d area.
Overtopping
occurs
when t h e wave runup on t h e seaward f a c e o f t h e breakwater exceeds
the
l e v e l o f t h e breakwater c r e s t .
A s l u g o f w a t e r from each
i n c i d e n t wave o v e r t o p p i n g t h e s t r u c t u r e f l o w s across t h e t o p
and down t h e l e e f a c e o f t h e b r e a k w a t e r ;
are
t h e t r a n s m i t t e d waves
generated i m p u l s i v e l y by t h i s w a t e r mass, and thus t h e energy
c o n t e n t o f these t r a n s m i t t e d waves must be d e r i v e d f r o m t h e overtopping water.
T h i s energy c o n t e n t can be c o n v e n i e n t l y e v a l u a t e d i n s t e p s :
first,
t h e energy c o n t e n t o f t h e water as i t crosses t h e b r e a k w a t e r
c r e s t must be e s t i m a t e d ;
must be d e t e r m i n e d ;
next, the f r i c t i o n
and f i n a l l y ,
(and p e r c o l a t i o n ) l o s s e s
t h e r e g e n e r a t i o n process must be
studied.
The energy c o n t e n t o f t h e o v e r t o p p i n g w a t e r mass can be e s t i m a t e d
by assuming t h a t t h e t o t a l energy c o n t e n t o f t h e o v e r t o p p i n g w a t e r i s
the
same as t h e energy o f t h a t p o r t i o n o f t h e wave runup t h a t would l i e
above t h e b r e a k w a t e r c r e s t were t h e b r e a k w a t e r f a c e extended t o - a h i g h e r
e l e v a t i o n as shown i n F i g . 1.
At maximum runup, f l o w v e l o c i t i e s a r e
e s s e n t i a l l y z e r o , and a l l t h e energy o f t h i s w a t e r i s i n t h e f o r m o f
p o t e n t i a l energy.
By knowing t h e shape and p o s i t i o n o f t h i s h y p o t h e t -
i c a l runup wedge, t h i s p o t e n t i a l energy can be c a l c u l a t e d .
W i t h t h e above assumptions s t a t e d , t h e s o l u t i o n t o t h e problem can
be o u t l i n e d as f o l l o w s :
I.
The shape o f t h e runup wedge i s s p e c i f i e d as an n - t h
degree p a r a b o l a f r o m t h e f i r s t seaward wave t r o u g h t o t h e
p o i n t o f maximum runup on t h e extended b r e a k w a t e r f a c e .
- 2 -
II,
Mass c o n s e r v a t i o n r e q u i r e s t h a t t h e volume o f runup
above t h e s t i l l water l e v e l (SWL) equals
in
t h e t r o u g h , below t h e s t i l l water
t h a t "removed"
l e v e l , over t h e
r e g i o n from t h e f i r s t seaward wave t r o u g h ( a t maximum
runup) t o t h e breakwater.
III.
The energy c o n t a i n e d i n t h e runup wedge i s e v a l u a t e d
from t h e n e t energy f l u x i n t o a c o n t r o l volume e n c l o s i n g
t h e runup wedge and t h e p a r t i a l s t a n d i n g wave system j u s t
seaward o f t h e breakwater.
IV.
The o v e r t o p p i n g energy i s e v a l u a t e d as t h e p o t e n t i a l
energy o f t h a t p o r t i o n o f t h e runup wedge e x t e n d i n g above
the a c t u a l breakwater
c r e s t e l e v a t i o n a t maximum runup
on t h e extended f a c e .
V,
The o v e r t o p p i n g water volume i s s i m i l a r l y e v a l u a t e d as
t h a t p o r t i o n o f t h e runup wedge l y i n g above t h e breakwater
c r e s t a t maximum runup.
VI.
The t r a n s m i t t e d wave energy i s e v a l u a t e d by a c c o u n t i n g
for
t h e n e t energy f l u x i n t o a c o n t r o l volume which en-
closes t h e o v e r t o p p i n g f l o w and t h e t r a n s m i t t e d wave t r a i n .
The above g r o u p i n g p r o v i d e s a c o n v e n i e n t means o f summarizing
the a n a l y t i c a l d e t a i l s o f t h e t h e o r y w h i c h f o l l o w s .
While t h e a n a l y s i s i s f o r a two-dimensional
section of the
s t r u c t u r e , w i t h waves a r r i v i n g a t normal i n c i d e n c e t o t h e s t r u c t u r e ,
i t should apply a l s o t o waves a r r i v i n g n e a r l y p e r p e n d i c u l a r t o t h e
breakwater.
- 3 -
I.
P a r a b o l i c Runup Wedge
At maximum runup, t h e shape o f t h e runup wedge i s assumed t o
be a p a r a b o l a w i t h i t s v e r t e x a t t h e b o t t o m o f t h e f i r s t wave t r o u g h .
The c o r r e s p o n d i n g e q u a t i o n i s
Y = MX'^ - A
where Y i s t h e water s u r f a c e e l e v a t i o n above t h e SWL, X i s d i s t a n c e
from t h e t r o u g h shoreward,
(Fig.
and A i s t h e a m p l i t u d e a t t h e t r o u g h
1 ) . This equation s a t i s f i e s the c o n d i t i o n s of surface c o n t i n u i t y
and s l o p e a t i t s v e r t e x and approximates
observed
i n the laboratory.
t h e shape o f runup wedges
I t w i l l be assumed t h a t maximum runup
occurs i n phase w i t h t h e extremum i n t h e p a r t i a l s t a n d i n g wave p r o f i l e .
Using l i n e a r wave t h e o r y t o d e s c r i b e t h e wave m o t i o n seaward o f t h e
f i r s t t r o u g h , A becomes
A = A. + A
1
r
= A^ ( 1 + k^)
where
= i n c i d e n t wave a m p l i t u d e
A^ = r e f l e c t e d wave a m p l i t u d e
and
k
r
= reflection
coefficient*
The runup e q u a t i o n may be w r i t t e n i n d i m e n s i o n l e s s
form by d i v i d i n g
t h r o u g h by A, i . e . .
where L
i s t h e v a l u e o f X where Y = R, t h e runup h e i g h t ;
thus,
* The symbol c o n v e n t i o n used t h r o u g h o u t t h i s paper i s lower case l e t t e r s
r e p r e s e n t d i m e n s i o n l e s s q u a n t i t i e s and c a p i t a l l e t t e r s r e p r e s e n t
dimensional equivalents,
- 4 -
ML„
R
= R + A
and
Y ^ R + A /iL \
_
The f o l l o w i n g d i m e n s i o n l e s s q u a n t i t i e s a r e d e f i n e d t o s i m p l i f y
subsequent a l g e b r a :
(H^^, H, X^, X^, and S a r e d e f i n e d i n F i g . 2 ) .
Y R
y , r , uh ^ , hu , l1 = -,
= 2L
1
X,X ,X , S , l
^
,
^
^R
—^ ,
H
^1
^2
,
^R
A
-
AS
,
^R
^
,
^R^ ^R
As a r e s u l t , t h e d i m e n s i o n l e s s runup s u r f a c e becomes
y = (r + l)x - 1
II.
(1)
Conservation o f Mass
R e f e r r i n g t o F i g . 2, mass c o n s e r v a t i o n r e q u i r e s t h a t t h e volume
of water c o n t a i n e d i n t h e runup wedge between x = x^ and x = 1 equal
volume m i s s i n g from t h e v o i d between x = 0 and x = x.^. That
1
is.
2
ydx - ^
= 0
0
Evaluating the i n t e g r a l y i e l d s ,
r + 1
or
- 1
sr
+ 1
2_
2
r - n
n + 1
(2)
The p h y s i c a l l i m i t s on n a r e n - 1,0 f o r t h e runup s u r f a c e t o be concave upward and n < r f o r p o s i t i v e b r e a k w a t e r s l o p e s .
Equation
(2) a p p l i e s s t r i c t l y t o impermeable s l o p e s ,
and
w i l l y i e l d c o n s e r v a t i v e l y h i g h e s t i m a t e s f o r o v e r t o p p i n g volumes
on permeable
III.
breakwaters.
Energy A n a l y s i s - Seaward Face
As t h e t r a n s m i t t e d wave energy comes from t h e o v e r t o p p i n g
w a t e r , t h i s o v e r t o p p i n g energy, E^ i s u n a v a i l a b l e t o form
r e f l e c t e d wave.
the
For t h e c o n t r o l volume of F i g . 3, t h e energy f l u x
over a wave p e r i o d T can be w r i t t e n ,
Power i n - Power out
Power l o s s
The power i n i s t h a t o f t h e i n c i d e n t wave.
The
power out i n c l u d e s
t h a t o f t h e r e f l e c t e d wave and t h e o v e r t o p p i n g w a t e r .
The
losses
i n c l u d e f r i c t i o n losses on t h e s l o p e and l o s s e s i n t h e r e g e n e r a t i o n
of
the r e f l e c t e d wave.
I f PE i s d e f i n e d as t h e p o t e n t i a l energy o f t h e e n t i r e runup
wedge above SWL
a t maximum runup, then PE - E^ i s t h a t p o r t i o n o f t h e
runup energy w h i c h r e t u r n s seaward v i a rundown.
P a r t o f t h i s r e t u r n i n g energy i s l o s t t o s u r f a c e f r i c t i o n
entrance losses i n t h e rundown process.
and
Independent s t u d i e s a t
M.I.T. have enumerated t h e l o s s e s f o r a s l u g of f l u i d r e l e a s e d down
a smooth s l o p e , (Sy, 1969,
K i n g , 1970).
These s t u d i e s i n d i c a t e d t h a t
r e c o n v e r s i o n o f rundown energy t o r e f l e c t e d ( o r t r a n s m i t t e d ) wave
energy i s v e r y i n e f f i c i e n t :
at
i f E i s t h e p o t e n t i a l energy o f t h e s l u g
t h e onset of rundown, then t h e r e c o n v e r s i o n t o wave energy r e s u l t s
i n energy l o s s e s equal t o k. E where k
0.65
for
< k
<
i s a loss c o e f f i c i e n t
and
0.85
a smooth s l o p e of 1:1.0.
R e c a l l i n g t h e energy f l u x r e l a t i o n s h i p s from l i n e a r wave t h e o r y .
- 6 -
Equation ( 3 ) becomes
E^
h%
"^r^g "
where
C
(PE - E^)k^
= -
(4)
= energy p r o p a g a t i o n r a t e
E. = i n c i d e n t wave energy
\
1
2
(group
velocity)
density
A 2
^i
Y = s p e c i f i c weight o f f l u i d
E^ = r e f l e c t e d wave energy
2
L e t T = ^ = Hgve
C
Have
^i
density
"^r
length
celerity
PE
pe =
E
e
o
where pe and e^ a r e d i m e n s i o n l e s s p o t e n t i a l e n e r g i e s , and A =
A ^ ( l + k^) as p r e v i o u s l y d e f i n e d .
Substitution
o f t h e above i n t o
Equation ( 4 ) y i e l d s
I-
G
T
^
\
( 1 - k ^)
1
(1+ k )^
r
where k^ i s 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 overtopped s t r u c t u r e .
This
w i l l be taken as a l i n e a r i n t e r p o l a t i o n between Miches r e f l e c t i o n coe f f i c i e n t , k^, f o r a s t r u c t u r e whose h e i g h t exceeds maximum runup, and
zero r e f l e c t i o n f o r no s t r u c t u r e .
That i s ,
(h + h, )
k
= k
—
(h + r )
- 7 -
(6)
E q u a t i o n ( 6 ) may be I n t e r p r e t e d as a r e d u c t i o n i n t h e r e f l e c t i o n coe f f i c i e n t by a f a c t o r which i s p r o p o r t i o n a l t o t h e f i c t i t i o u s b r e a k water e x t e n s i o n .
P r e s e n t l y a v a i l a b l e i n f o r m a t i o n about p a r t i a l
r e f l e c t i o n from o v e r t o p p e d s t r u c t u r e s does n o t j u s t i f y
a more
elaborate r e l a t i o n s h i p .
The p o t e n t i a l energy o f t h e runup wedge must be e v a l u a t e d
relative to i t s ability
t o r e t u r n work t o t h e r e f l e c t e d wave system.
The r e c o n v e r s i o n o f runup p o t e n t i a l energy t o wave energy occurs
i n two s t e p s : (1) The p o t e n t i a l energy o f t h e s l u g i s c o n v e r t e d t o
k i n e t i c energy as t h e s l u g f a l l s t o t h e w a t e r s u r f a c e , and ( 2 ) t h i s
k i n e t i c energy i s c o n v e r t e d t o f l u i d m o t i o n i n t h e r e - e n t r a n c e
vicinity.
Much o f t h e induced f l u i d m o t i o n i s t u r b u l e n t and i s
e v e n t u a l l y l o s t t o v i s c o u s d i s s i p a t i o n (as accounted f o r by t h e l o s s
coefficient, k^).
of
the
The r e m a i n i n g m o t i o n c o n t r i b u t e s t o a component
t h e r e f l e c t e d wave.
I t i s i m p o r t a n t t o note,however, t h a t once
c e n t e r o f g r a v i t y o f t h e s l u g f l o w has e n t e r e d t h e w a t e r , t h e
s l u g i s n e u t r a l l y bouyant and t h e r e f o r e has no f u r t h e r p o t e n t i a l t o
accelerate.
The r e c o n v e r s i o n energy i s l i m i t e d t o t h e k i n e t i c energy
gained by t h e s l u g i n f a l l i n g
t o the water surface.
Consequently, a t
maximum runup t h e p o t e n t i a l energy s h o u l d be measured r e l a t i v e t o t h e
water surface.
Due t o t h e u n c e r t a i n t y o f p o s i t i o n o f t h e w a t e r s u r -
f a c e d u r i n g r e - e n t r a n c e , t h e p o t e n t i a l energy w i l l be e v a l u a t e d w i t h
r e s p e c t t o t h e SWL.
I t f o l l o w s d i r e c t l y t h a t t h e d i m e n s i o n l e s s p o t e n t i a l energy o f t h e
runup wedge a t maximum runup i s s i m p l y (see F i g . 2 ) ,
- 8 -
2
Z_
2
pe
dx - s r
At X = x^, y = 0
1/n
= (^TT)
thus
and c o m p l e t i n g t h e above i n t e g r a t i o n y i e l d s
1
( r + 1)2n + 1
pe =
1 (r
2 ( r + 1)
n + 1.
+
1)
2 + 1/n
1 ( r + 1)
1 + 1/n
sr
1 -
(7)
(r + D^^M
IV.
Overtopping Energy
As d e s c r i b e d e a r l i e r , t h e o v e r t o p p i n g energy i s e q u a l t o
the p o t e n t i a l energy o f t h a t p o r t i o n o f t h e runup wedge e x t e n d i n g
above t h e breakwater
respect t o t h e SWL,
crest.
The p o t e n t i a l energy i s e v a l u a t e d w i t h
thus
1
e
o
=
At X = X
Completing
2 •
(y - h^)
y = h
(h^ +
dx
h, + 1
b
thus x =
2 [r + 1
t h e above i n t e g r a t i o n y i e l d s
- 9 -
1/n
^
h^ +
1
2
'o
j
2 +
1/n
iJV^)
( r + 1)2
2n + 1
^ r + 1 /
+ 1 \
2 ( r + 1)
n + 1
1/n
+
r + 1
^ 1/n
h, + 1
1 - b
!
, r + 1 /'
^
1
( 1 - h^^)
I
1 +
(8)
( r - h ^ ) ^ ( r + 2h^)
)
S u b s t i t u t i o n o f Equations (7) and ( 8 ) i n t o E q u a t i o n ( 5 ) completes
the energy c o n s e r v a t i o n a n a l y s i s on t h e seaward s i d e o f t h e b r e a k water.
V.
O v e r t o p p i n g Volume
The o v e r t o p p i n g volume i s e v a l u a t e d f r o m F i g . 2 as
volimie i n t h e runup wedge which extends above t h e b r e a k w a t e r
that
crest.
That i s ,
^
V =
s ( r - h, )2
(y - h^) dx
Evaluating the i n t e g r a l y i e l d s
V =
r + 1
+ 1
h, + 1
b
1 . r + 1
+ 1
( 1 + h^)
1 -
1 + 1/n •)
1/n-l
r + 1-
s ( r - h^)(9)
- 10 -
VI.
T r a n s m i t t e d Wave Energy
Energy c o n s e r v a t i o n
requires
t h a t d u r i n g each wave p e r i o d
the n e t energy accumulated i n t h e c o n t r o l volume sketched i n F i g . 4
sums t o z e r o , i . e . ,
power i n - power o u t = power l o s s
(10)
The energy f l u x i n t o t h e c o n t r o l volume i s s i m p l y t h e o v e r t o p p i n g
energy d i v i d e d by t h e wave p e r i o d .
The power l o s s equals t h e o v e r t o p p i n g
energy f l u x m u l t i p l i e d times t h e rundown l o s s c o e f f i c i e n t , k^.
The
energy f l u x p a s s i n g o u t o f t h e c o n t r o l volume i s t h a t a s s o c i a t e d w i t h
the t r a n s m i t t e d wave.
Although the regeneration
process i s q u i t e com-
p l e x , i t r e p e a t s p e r i o d i c a l l y a t t h e frequency o f t h e i n c i d e n t wave,
and
thus t h e fundamental mode o f t h e t r a n s m i t t e d wave has t h e same f r e -
quency as t h e i n c i d e n t wave.
mode c o n t a i n s
Experiments a t M.I.T. i n d i c a t e t h a t t h i s
most o f t h e t r a n s m i t t e d wave energy.
As a f i r s t
m a t i o n , t h e n , h i g h e r harmonics may be n e g l e c t e d and t h e power
approxileaving
the c o n t r o l volume d u r i n g one wave p e r i o d i s s i m p l y E^C^ where
YA.
^t = ^
2
\
'
%
A.
= -7- = t r a n s m i s s i o n
t
A,
1
Equation (10) y i e l d s
and k
E
T
But E = A.
0 1
coefficient.
S u b s t i t u t i n g t h e above i n t o
YA.2
,
E
1
k
C = — k
- 2 • t • ^g
T
2
(1 + k )
r
2
^t
L„. e and T = — , where
R o
C
length.
- 11 -
t
= t r a n s m i t t e d wave
I f t h e w a t e r depth on t h e l e e s i d e o f t h e b r e a k w a t e r i s t h e same on
the seaward s i d e , then
= L and
\' = '%hr
+
(1-V
(11)
The problem as d e f i n e d i n c l u d e s f i v e d i m e n s i o n l e s s unknowns:
n, r , k^, k^ and s ( o r L ^ ) .
However, o n l y f o u r e q u a t i o n s have been
p r e s e n t e d t o f a c i l i t a t e t h e s o l u t i o n t o t h e above.
The p e r t i n e n t
e q u a t i o n s a r e ( 2 ) , ( 5 ) , ( 6 ) , and ( 1 1 ) . A l l o t h e r q u a n t i t i e s o f i n t e r est
a r e f u n c t i o n s o f t h e f i v e fundamental dependent v a r i a b l e s l i s t e d .
A f i f t h r e l a t i o n s h i p i s needed t o c o m p l e t e l y s p e c i f y t h e problem.
a l t e r n a t i v e r e q u i r e m e n t s have been e x p l o r e d t o s a t i s f y
Two
t h e need f o r a
f i f t h equation.
The f i r s t
the
a t t e m p t r e q u i r e d t h a t t h e runup wedge be t a n g e n t t o
extended breakwater s l o p e a t t h e h e i g h t o f maximum runup.
a c o n d i t i o n which has been observed f o r g e n t l e s l o p e s .
This
This i s
restraint,
however, y i e l d s runup h e i g h t s exceeding those observed by a f a c t o r o f
1.5 and g r e a t e r .
The second a t t e m p t s p e c i f i e d t h a t t h e d i s t a n c e f r o m t h e b r e a k water t o the f i r s t
t r o u g h be e q u a l t o h a l f a m o d i f i e d wave l e n g t h .
The
m o d i f i e d wave l e n g t h i s computed f r o m l i n e a r wave t h e o r y as a f u n c t i o n
of w a t e r depth on t h e b r e a k w a t e r s l o p e , and i s d e f i n e d as t h e i n t e g r a t e d
average wave l e n g t h i n t h e i n t e r v a l between t h e f i r s t
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 r e a k w a t e r s l o p e .
t r o u g h and t h e
This requirement
reduces t o known r e s u l t s f o r t h e two extreme c o n d i t i o n s o f v e r t i c a l w a l l s
and h o r i z o n t a l s l o p e s .
Imposing t h i s r e s t r a i n t , however, y i e l d s
runup
h e i g h t s which f a l l s h o r t o f those observed e x p e r i m e n t a l l y by a f a c t o r
- 12 -
of 0.7
and l e s s .
similarly
Corresponding
transmission c o e f f i c i e n t s
are
low.
To s a t i s f y t h e immediate need f o r a f i f t h e q u a t i o n , t h e
authors have r e l i e d on S a v i l l e ' s e x p e r i m e n t a l r e s u l t s (B.E.B. T.M.
as an i n p u t f o r runup h e i g h t s .
#64)
As a s p e c i f i c example, f o r a smooth
impermeable s l o p e o f 1:1.5, wave h e i g h t t o water depth r a t i o s i n
excess of 3.0,
and wave cambers near 0.05,
h e i g h t i s t w i c e t h e i n c i d e n t wave h e i g h t .
r a t i o s f o r h i g h breakwater
t h e a p p r o p r i a t e runup
Use
c r e s t s (^^/^ ^ 0.5)
of S a v i l l e ' s runup
y i e l d s good c o r r e l a -
t i o n between e x p e r i m e n t a l and t h e o r e t i c a l t r a n s m i s s i o n c o e f f i c i e n t s .
However, f o r r e l a t i v e l y low breakwater
c r e s t s (h, / r < 0.05)
b
Saville's
data u n d e r e s t i m a t e s t h e e q u i v a l e n t runup h e i g h t , and p r e d i c t e d t r a n s m i s s i o n c o e f f i c i e n t s are somewhat low.
I t s h o u l d be p o i n t e d out t h a t
t h i s l a t t e r category i s of l i t t l e i n t e r e s t f o r p r a c t i c a l
breakwater
design.
Summarizing, t h e f i v e d i m e n s i o n l e s s
and s may
unknowns, n, r , k^, k^,
be s o l v e d u s i n g S a v i l l e ' s e x p e r i m e n t a l runup h e i g h t s and
Equations ( 2 ) , ( 5 ) , ( 6 ) , and ( 1 1 ) . Equations (7) and (8) must be
u t i l i z e d t o f i n d pe and e i n terms o f t h e f i v e unknowns. An i t e r a ^
o
t i v e procedure
i s employed i n seeking a s o l u t i o n which s a t i s f i e s a l l
f i v e conditions simultaneously.
most q u i c k l y accomplished
u n i t y u n t i l Equations
The a u t h o r s have found t h a t t h i s i s
by i n c r e m e n t i n g n up from a minimum v a l u e of
( 2 ) , (5) and
(6) are s a t i s f i e d and t h e n s o l v i n g
d i r e c t l y f o r t h e t r a n s m i s s i o n c o e f f i c i e n t and t h e o v e r t o p p i n g volume.
A s i m p l e d i g i t a l computer program has been w r i t t e n i n FORTRAN IV G t o
expedite t h i s
solution.
- 13 -
EXPERIMENTAL EQUIPMENT
The experiments were performed a t M.I.T, by Lamarre
( 1 9 6 7 ) , u s i n g a g l a s s - w a l l e d wave flume 2,5 f e e t w i d e and 105 f e e t
l o n g , u s i n g a c o n s t a n t w a t e r depth o f 1.5 f e e t .
A t t h e f a r end o f
t h e flume was an i m p e r v i o u s beach a t a 5% s l o p e .
The wave g e n e r a t o r was o f t h e f l a p t y p e , 2.5 f e e t h i g h , and
hinged a t t h e b o t t o m ;
t h e t o p was moved back and f o r t h by a r o d
a t t a c h e d t o a crank arm h a v i n g v a r i a b l e speed and e c c e n t r i c i t y .
The smooth, impermeable breakwater was l o c a t e d 51.5 f e e t
from t h e wave g e n e r a t o r and was 1.3 f e e t h i g h .
I t c o u l d be r a i s e d by
s m a l l increments t o a maximum h e i g h t o f 1.77 f e e t by adding b l o c k i n g
underneath.
These b l o c k i n g s were made o f wood, shaped and i n s t a l l e d
i n such a way as t o m a i n t a i n c o n s t a n t f r o n t and r e a r s l o p e s o f 1 v e r t i c a l
t o 1.5 h o r i z o n t a l .
wide.
The h o r i z o n t a l c r e s t o f t h e breakwater was 0.33 f e e t
Roughness was o b t a i n e d when d e s i r e d by adding f l a t t e n e d
expanded
m e t a l l a t h sheets on t o p o f t h e smooth s u r f a c e s .
The i n s t r u m e n t s used i n t h e experiment were p a r a l l e l - w i r e
r e s i s t a n c e t y p e wave gages connected by a w h e a t s t o n e b r i d g e t o a twochannel Sanborn r e c o r d e r .
The wave gages c o n s i s t e d o f two v e r t i c a l
stain-
l e s s s t e e l w i r e s 1/8" i n diameter and 1 f o o t l o n g , mounted f r o m above
3/4" a p a r t , and p a r t i a l l y immersed i n t h e w a t e r .
EXPERIMENTAL PROCEDURE
S i x d i f f e r e n t wave p e r i o d s were i n v e s t i g a t e d .
Once t h e
f r e q u e n c y o f t h e wave g e n e r a t o r was a d j u s t e d t o t h e p r o p e r v a l u e , and
b e f o r e i n s e r t i n g t h e breakwater i n t o t h e f l u m e , t h e e c c e n t r i c i t y o f
t h e d r i v e r crank was s e t a t f o u r d i f f e r e n t p o s i t i o n s and t h e c o r r e s p o n d i n g
- 14 -
" i n c i d e n t " waves generated a t each s e t t i n g were recorded.
A f t e r t h e r e c o r d i n g o f these i n c i d e n t waves was
completed,
the breakwater was i n s t a l l e d , and two wave gages were mounted a t
d i s t a n c e s o f one and two wavelengths
l i n e o f the s t r u c t u r e .
r e s p e c t i v e l y beyond t h e c e n t e r -
The f o u r d i f f e r e n t i n c i d e n t waves a l r e a d y
measured were then reproduced and t h e t r a n s m i t t e d waves r e c o r d e d .
A l l t h e experiments were r e p e a t e d a f t e r adding t h e expanded m e t a l
sheet f o r roughness.
The complete s e r i e s o f t e s t s was made f o r each i n c r e a s e i n
the h e i g h t o f t h e breakwater.
A f t e r t h e breakwater was a t i t s
maximum h e i g h t , i . e . , when t h e r e was no more o v e r t o p p i n g , t h e s t r u c t u r e was p u l l e d o u t o f t h e w a t e r , t h e f r e q u e n c y o f t h e wave-maker changed,
and t h e process r e p e a t e d f o r f i v e more p e r i o d s .
EXPERIMENTAL RESULTS AND DISCUSSION
The t r a n s m i t t e d wave h e i g h t s measured by gages 1 and 2 were
averaged
t o o b t a i n an e s t i m a t e o f " t h e " t r a n s m i t t e d wave h e i g h t .
to t h e presence o f t h e h i g h e r harmonics
Due
i n t h e t r a n s m i t t e d wave system,
t h e r e was g e n e r a l l y some v a r i a t i o n between t h e wave h e i g h t s measured
at t h e two l o c a t i o n s ;
cent.
t h i s v a r i a t i o n t y p i c a l l y amounted t o 5 t o 15 p e r -
The i n c i d e n t wave h e i g h t i s s i m i l a r l y taken as t h e average o f
the two wave gages, b u t t h e d i f f e r e n c e o n l y amounts t o a p e r c e n t o r two.
S i m i l a r e f f e c t s were n o t e d by t h e U. S. Army Corps o f Engineers
(1965)
i n t h e Dana P o i n t model t e s t s .
The d a t a a r e shown i n F i g s . 5 t h r o u g h 10 as k
- 15 -
vs H, /R f o r a l l
e x p e r i m e n t a l runs.
The
curves p l o t t e d are s o l u t i o n s t o Eq. 11
and d e p i c t t h e o r e t i c a l bounds f o r smooth and rough s u r f a c e s .
The
smooth s u r f a c e curve corresponds
a
l o s s c o e f f i c i e n t k„
t o a runup r a t i o R/H.
= 1.8,
^ ,
£ - 0,6,
and an
' i n t r i n s i c " c o e f f i c i e n t of r e f l e c -
t i o n (as d e s c r i b e d by Miche) p = 0.8.
The rough s u r f a c e curve c p r r e s -
ponds t o R/H^ = 1.6,
0.7.
= 0.8
and p =
The runup r a t i o s used are those i n d i c a t e d by our own s t u d i e s .
They are somex^hat lower than those suggested
by S a v i l l e (R/H.
-
2.0)
and p r o b a b l y i n c l u d e some s c a l e e f f e c t s as w e l l as p e c u l i a r i t i e s o f
the e x p e r i m e n t a l
apparatus.
The l o s s c o e f f i c i e n t v a l u e s f o l l o w d i r e c t l y f r o m Sy's e x p e r i ments.
felt
He found f o r a smooth 1:1 s l o p e an average k
t h a t t h e r e g e n e r a t i o n process
==0.7.
I t is
i s more e f f i c i e n t f o r h o r i z o n t a l
momentum t r a n s f e r (as i n a wave g e n e r a t o r f l a p ) than f o r v e r t i c a l
momentum t r a n s f e r .
Since t h e h o r i z o n t a l component o f rundown momentum
i n c r e a s e s f o r d e c r e a s i n g slopes one might expect s m a l l e r l o s s coe f f i c i e n t s f o r more g r a d u a l s l o p e s .
There i s , o f course, a t r a d e o f f
t o s u r f a c e f r i c t i o n on v e r y g r a d u a l slopes b u t f o r smooth 1:1.5
a l o s s c o e f f i c i e n t equal t o 0.6
t o 0.8
seems a p p r o p r i a t e .
slopes
This i s increased
f o r t h e e q u i v a l e n t roughened s l o p e .
The i n t r i n s i c c o e f f i c i e n t o f r e f l e c t i o n i s a f u n c t i o n o f s u r -
f a c e roughness and p e r m e a b i l i t y . Miche suggests
f o r smooth impermeable s l o p e s , and p ^ 0.33
s e q u e n t l y a v a l u e of 0.8 was
t h e rough s l o p e .
a v a l u e of p =
f o r rubble slopes.
chosen f o r the smooth s l o p e and 0.7
For deep w a t e r wave cambers H^/L^ < 0.06
- 16 -
and
0.8
Confor
1:1.5
s l o p e s , Miche's t h e o r y y i e l d s
= p,
For low b r e a k w a t e r s H^/R < 0.3± t h e t h e o r y u n d e r e s t i m a t e s
the t r a n s m i s s i o n c o e f f i c i e n t ;
i t appears r e a s o n a b l e t h a t t h e
assumptions u n d e r l y i n g t h e t h e o r y a r e l e a s t v a l i d i n t h i s r e g i o n , and
moreover, t h e values o f K
and k
used may n o t be a p p l i c a b l e .
How¬
ever, t h i s range o f H^^/R i s o f l i t t l e p r a c t i c a l i n t e r e s t , as t h e
t r a n s m i s s i o n c o e f f i c i e n t s a r e t y p i c a l l y 0.3 t o 0.6.
For h i g h e r b r e a k w a t e r s (H^/R >0.5±), t h e t h e o r y g e n e r a l l y
overestimates
the transmission c o e f f i c i e n t , p a r t i c u l a r l y f o r the
s m a l l e r v a l u e s o f r e l a t i v e d e p t h , H/L.
For the most o f t h e range o f
H/L however, t h e t h e o r y p r o v i d e s an "upper envelope" f o r k^, and thus
is
u s e f u l f o r p r e l i m i n a r y design
estimates.
I t i s i n t e r e s t i n g t o n o t e on F i g s . 5 t h r o u g h 10 t h a t t h e
i n c i d e n t wave steepness, H^/L has l i t t l e e f f e c t on t h e t r a n s m i s s i o n
c o e f f i c i e n t except a t t h e lowest v a l u e s o f E^/L.
As one m i g h t e x p e c t , adding
roughness t o the slopes reduces t h e
t r a n s m i s s i o n c o e f f i c i e n t f o r t h e l a b o r a t o r y d a t a , p r o b a b l y by r e d u c i n g
t h e runup and i n c r e a s i n g k .
There i s a s i g n i f i c a n t amount o f s c a t t e r i n the d a t a ;
can be a t t r i b u t e d t o s e v e r a l
1.
this
sources.
Wave gage i n a c c u r a c i e s ;
these gages t y p i c a l l y a r e o n l y
good t o 5 t o 10 p e r c e n t .
2.
L a t e r a l resonance e f f e c t s can b i a s r e a d i n g s
t h e channel c e n t e r l i n e .
taken
along
For s e v e r a l r u n s , i t was n o t i c e d
t h a t t h e wave c r e s t s were not u n i f o r m across t h e channel.
For t h e 1.5 second waves, the channel w i d t h i s c l o s e t o a
- 17 -
q u a r t e r - w a v e - l e n g t h and f o r the 1.0
half-wave-length.
second waves, a
For the o t h e r p e r i o d s , however,
e s p e c i a l l y w i t h s t e e p e r waves, the p a r t i a l b r e a k i n g
o c c a s i o n a l l y observed on t h e s t r u c t u r e can
h i g h e r harmonics
3,-
capable o f l a t e r a l
generate
resonance.
B r e a k i n g o f t h e waves on t h e s t r u c t u r e , observed f o r
the s t e e p e r waves, causes an a d d i t i o n a l energy l o s s n o t
accounted
f o r by t h e t h e o r y .
Because o f t h e s t r o n g
dependence o f b r e a k e r c h a r a c t e r i s t i c s on the backwash
from t h e p r e c e d i n g wave, no v e r y u n i f o r m e f f e c t o f breaki n g can be
Besides
expected.
t h e s c a t t e r , another shortcoming o f t h e d a t a i s t h a t t h e
ranges o f r e l a t i v e d e p t h , H/L,
and wave steepness, EjL,
cover those found i n t h e p r o t o t y p e .
Breakwaters
do n o t
are t y p i c a l l y
fully
built
i n 15 t o 40 f e e t o f w a t e r , and t h e i n c i d e n t waves t y p i c a l l y have wave
l e n g t h s from 200 t o 400 f t ; thus H/L
0.2
i n the p r o t o t y p e , w h i l e H/L
0.45.
ranges from a p p r o x i m a t e l y 0.04
i n t h e experiments ranges from 0.16
to
to
S i m i l a r l y t h e wave steepness H^/L, i n the p r o t o t y p e under s t o r m
c o n d i t i o n s w i l l be a p p r o x i m a t e l y 0.04
t o 0.10, w h i l e t h e maximum s t e e p -
ness a v a i l a b l e i n t h e experiments was
0.064.
These d i f f i c u l t i e s stem
from the d i f f i c u l t y o f g e n e r a t i n g steep waves i n s h a l l o w w a t e r w i t h
h i n g e d - f l a p wave g e n e r a t o r .
a
The Dana P o i n t data ( F i g . 1 1 ) , however,
r e p r e s e n t s p r o t o t y p e c o n d i t i o n s , and agrees w e l l w i t h t h e l a b o r a t o r y
d a t a , s u g g e s t i n g t h a t depth i s n o t a v e r y i m p o r t a n t f a c t o r , except poss i b l y f o r waves w h i c h break b e f o r e r e a c h i n g t h e s t r u c t u r e .
F i g u r e 11 p r o v i d e s a comparison
- 18 -
between t h e o r y and experiment f o r
a more r e a l i s t i c breakwater
form.
The e x p e r i m e n t a l
data
are from t h e U. S. Army Corps o f Engineers Report d e s c r i b i n g a
s e r i e s o f s c a l e model t e s t s f o r t h e design o f a h a r b o r a t Dana
Point, California.
As t h e Dana P o i n t breakwater
was a permeable
s t r u c t u r e , some wave energy was t r a n s m i t t e d even w i t h no overtopping ( k ^ < 0.1), and t h u s , t h e t r a n s m i t t e d wave h e i g h t s g i v e n
f o r s m a l l amounts o f o v e r t o p p i n g were n o t i n c l u d e d .
The runup
r a t i o s used f o r t h e Dana P o i n t data were R/H. = 1.0 and 1.1 f o r
p r o t o t y p e wave p e r i o d s o f 12 and 18 seconds r e s p e c t i v e l y . These
I
r a t i o s were e s t i m a t e d from t h e data f o r non o v e r t o p p i n g waves b u t
agree w e l l w i t h S a v i l l e ' s r e s u l t s .
The t h e o r e t i c a l p o i n t s
to t h e runup r a t i o s s t a t e d above, l o s s c o e f f i c i e n t k
correspond
= 0.8, and
i n t r i n s i c c o e f f i c i e n t o f r e f l e c t i o n p= 0.4.
R e f e r r i n g t o F i g , 11 i t i s e v i d e n t t h a t a s c a l e e f f e c t
exists.
The 1:50 s c a l e model y i e l d s l a r g e r t r a n s m i s s i o n c o e f f i c i e n t s
than t h e 1:5 s c a l e model.
T h i s i s p r o b a b l y a Reynolds e f f e c t
wherein
the r e - e n t r a n c e losses f o r t h e l a r g e r , more t u r b u l e n t model (and
t h e r e f o r e p r o t o t y p e ) a r e h i g h e r than i n t h e s m a l l e r model.
t h e o r e t i c a l s o l u t i o n l i e s between t h e two e x p e r i m e n t a l
The
results.
E x t r a p o l a t i n g these r e s u l t s t o p r o t o t y p e s c a l e , one expects
the theory
to give a c o n s e r v a t i v e l y high estimate f o r the transmission c o e f f i c i e n t .
For e n g i n e e r i n g purposes, however, t h i s i s a d e s i r a b l e c o n d i t i o n .
The t h e o r y does n o t account f o r d i r e c t t r a n s m i s s i o n through
a permeable breakwater.
The c o n t i n u i t y e q u a t i o n , Eq. ( 2 ) , n e g l e c t s
f l o w i n t o t h e pores o f t h e breakwater
o v e r t o p p i n g volume.
and thereby o v e r e s t i m a t e s t h e
Consequently, t h e t r a n s m i s s i o n c o e f f i c i e n t due
- 19 -
to pure o v e r t o p p i n g i s a l s o o v e r e s t i m a t e d .
compensating.
The
two e r r o r s are
However, the o v e r t o p p i n g r e g e n e r a t i o n process appears
to be more e f f i c i e n t than d i r e c t t r a n s m i s s i o n through
of low p e r m e a b i l i t y .
breakwaters
The net r e s u l t i s t h a t t h e t r a n s m i s s i o n co-
e f f i c i e n t i s s l i g h t l y o v e r e s t i m a t e d when t h e o v e r t o p p i n g t h e o r y i s
a p p l i e d t o r u b b l e mound breakwaters.
condition f o r engineering
Again,
this i s a desirable
estimates.
A l l o f the e x p e r i m e n t a l d a t a , i n c l u d i n g Dana P o i n t , are
presented
i n F i g . 12.
I t demonstrates t h e v a l i d i t y o f u s i n g H^/R
as the dimensionless
data.
The
parameter f o r p l o t t i n g o v e r t o p p i n g t r a n s m i s s i o n
envelope curve i s o f c o n s i d e r a b l e i n t e r e s t as i t appears
t o be a f a i r l y c o n s i s t e n t upper bound f o r t h e t r a n s m i s s i o n
coefficient,
of the form
k^ = 0.65
(1.10 - Hj^/R), f o r H^/R < 1.0
(12)
S a v i l l e (1955) has p u b l i s h e d data on f l o w r a t e s a s s o c i a t e d
w i t h t h e o v e r t o p p i n g o f v a r i o u s c o a s t a l s t r u c t u r e s . By m u l t i p l y i n g
the g i v e n discharges
( c f s / f t o f c r e s t w i d t h ) by t h e wave p e r i o d , an
o v e r t o p p i n g volume per wave i s o b t a i n e d .
w i t h Eq.
(9).
F i g . 13 shows Eq.
These r e s u l t s can be compared
(9) p l o t t e d dimensionlessly
as
2
V/(R
- Hj^)
(where V =
f o o t o f breakwater
A v , the d i m e n s i o n a l
c r e s t ) vs S a v i l l e ' s r e s u l t s .
f o r a s t r u c t u r e w i t h a smooth f a c e on a 1:1.5
of 4.5
and 9.0
t o 3.57.
The d a t a shown a r e
s l o p e , f o r t h e depths
f e e t a t the t o e , wave p e r i o d s o f 2.96
and f o r runup r a t i o s , based on n o n - o v e r t o p p i n g
2.54
o v e r t o p p i n g volume per
t o 6.4
seconds,
wave d a t a , r a n g i n g
from
B r e a k i n g waves and waves w i t h l i t t l e o v e r t o p p i n g have
- 20 -
been n e g l e c t e d .
For t h e l a t t e r t h e h e i g h t s and runups a r e
r e p o r t e d o n l y t o t h e n e a r e s t f o o t , and t h e s m a l l v a l u e s o f R - H^^
computed a r e n o t r e l i a b l e .
the given runup r a t i o s ,
The t h e o r e t i c a l p o i n t s solved f o r used
= 0,7 and p = 0.8. The c o r r e l a t i o n
between t h e o r y and experiment ( p e r f e c t c o r r e l a t i o n i s t h e 45° l i n e )
f u r t h e r supports
servatively
t h e a n a l y t i c a l assuinptions.
overestimates
A g a i n t h e t h e o r y con-
t h e volumes, a p r o b a b l y
consequence o f
the h i g h runup r a t i o s used,
CONCLUSIONS
The
t h e o r y presented
describes
the essential
features of
the process o f wave t r a n s m i s s i o n by o v e r t o p p i n g , f o r waves a r r i v i n g
at t h e b r e a k w a t e r w i t h o u t b r e a k i n g and a t normal i n c i d e n c e .
The
r e l a t i o n s h i p d e r i v e d i s dependent on t h e c o e f f i c i e n t s o f r e f l e c t i o n
and l o s s , and t h e runup r a t i o .
F u r t h e r i n v e s t i g a t i o n o f these quan-
t i t i e s would p e r m i t a r e f i n e m e n t o f t h e t h e o r y .
The
envelope curve may be used f o r p r e l i m i n a r y e s t i m a t e s
of t h e t r a n s m i s s i o n c o e f f i c i e n t .
For r u b b l e mound breakwaters t h i s
e s t i m a t e can be improved by u s i n g t h e t h e o r y a l o n g w i t h a runup r a t i o
R/H_j^
= 1.0, a l o s s c o e f f i c i e n t k^ = 0.8, and an i n t r i n s i c r e f l e c t i o n
c o e f f i c i e n t p = 0.4.
- 21 -
REFERENCES
U.
S. Army Corps of E n g i n e e r s ( 1 9 6 6 ) , "Shore P r o t e c t i o n , P l a n n i n g
and D e s i g n " , C o a s t a l E n g i n e e r i n g R e s e a r c h C e n t e r , T e c h n i c a l
R e p o r t No. 4, T h i r d E d i t i o n , U. S. Government P r i n t i n g O f f i c e .
Miche,
R., ( 1 9 5 3 ) , "The R e f l e c t i n g Power of M a r i t i m e Works Exposed
to A c t i o n o f t h e Waves", B u l l . Beach E r o s i o n B o a r d , U. S.
Army Corps o f E n g i n e e r s , V o l . 7, No. 2, A p r i l 1953, p. l-,7.
Sy, T.,
(1969), Personal
Communication
K i n g , N. , ( 1 9 7 0 ) , "Wave R e g e n e r a t i o n i n B r e a k w a t e r O v e r t o p p i n g " ,
S. M. and Ocean E n g i n e e r T h e s i s s u b m i t t e d to t h e Department
of N a v a l A r c h i t e c t u r e and M a r i n e E n g i n e e r i n g , M.I.T., Cambridge,
Mass., August 1970.
U.
S, Army Corps o f E n g i n e e r s ( 1 9 6 5 ) , " G e n e r a l D e s i g n f o r Dana P o i n t
H a r b o r , Dana P o i n t , C a l i f o r n i a " , September 1965.
S a v i l l e , T., J r . , ( 1 9 5 5 ) , " L a b o r a t o r y D a t a on Wave Runup and O v e r t o p p i n g on S h o r e S t r u c t u r e s " , U. S. Army Corps o f E n g i n e e r s ,
Beach E r o s i o n B o a r d T e c h n i c a l Memorandum No. 64.
L a m a r r e , P., ( 1 9 6 7 ) , "Water Wave T r a n s m i s s i o n by O v e r t o p p i n g o f an
Impermeable B r e a k w a t e r " , S. M. T h e s i s s u b m i t t e d to t h e D e p a r t ment o f C i v i l E n g i n e e r i n g a t M.I.T., Cambridge, Mass.,
September 1967.
- 2 2 -
24
25
C o n t r o l Volume
Figure 4
C o n t r o l Volume, Leeward Face
Figure 5
vs K^/R ,
T = 1.5 Sec
28
O
29
T
1
T—^
1
1—
-T
T = 0.8 sec
Slopes
Rough
Smooth
r
•
1
1
1•
i
1
Dana P o i n t
Model Data
0.6
O
1:50 Model
A
1:5 Model
0.5
Theory
0.4
•^•^^
•
•
A
0.3
a
A.
»
0
A
•
A
A
i«%
0.2 -
• '
0
A
0.1
0
1
•
0.1
0.2
•
•
0.3
0.4
F i g u r e 11
0.5
k
0.6
1
I
1
0.7
0.8
0.9
H^/R
vs KL /R - Dana P o i n t Model Data and Theory
1.
T
0.6
i
ï
Lab, Smooth Slope
O
OO
Lab, Rough Slope
O*
0.4
•
Dana P t . 1:50 Scale
A
Dana P t . 1:5 Scale
0.65 (1.10 - H^/R)
II
0.3
0.4
0.8
0.2
0.1
0.1
0.2
0.3
F i g u r e 12
0.4
0.5
^076
0 ~
o.8„ ,^ 0.9
H^/R
vs H^/R - A l l Lab Data and Dana P o i n t Model Data
1.0
0.4
Experimental
gure 13
Comparison o f T h e o r e t i c a l O v e r t o p p i n g Volumes w i t h
S a v i l l e ' s Data
35
LIST OF SYMBOLS
A
p a r t i a l s t a n d i n g wave a m p l i t u d e
A.
i n c i d e n t wave a m p l i t u d e
X
A
r e f l e c t e d wave a m p l i t u d e
r
\
t r a n s m i t t e d wave a m p l i t u d e
C
wave c e l e r i t y
c
energy p r o p a g a t i o n r a t e (group v e l o c i t y )
g
E
run down energy
E.
i n c i d e n t wave energy d e n s i t y
X
E
overtopping
energy
o
e
d i m e n s i o n l e s s o v e r t o p p i n g energy
o
E
r e f l e c t e d wave energy d e n s i t y
r
\
t r a n s m i t t e d wave energy d e n s i t y
H
s t i l l water
h
dimensionless s t i l l water
\
b r e a k w a t e r c r e s t e l e v a t i o n above SWL
\
dimensionless breakwater c r e s t
H.
i n c i d e n t wave h e i g h t
depth
depth
elevation
X
h
rundown l o s s c o e f f i c i e n t
k
Miche's r e f l e c t i o n c o e f f i c i e n t
m
k
breakwater r e f l e c t i o n c o e f f i c i e n t
r
\
transmission c o e f f i c i e n t
L
i n c i d e n t , r e f l e c t e d wave l e n g t h
h o r i z o n t a l d i s t a n c e from f i r s t
maximum runup
36
trough t o point of
LIST OF SYMBOLS
(continued)
t r a n s m i t t e d wave l e n g t h
constant c o e f f i c i e n t i n parabola equation
exponent i n p a r a b o l a
equation
p o t e n t i a l energy o f runup wedge
d i m e n s i o n l e s s p o t e n t i a l energy o f runup wedge
runup h e i g h t
d i m e n s i o n l e s s runup h e i g h t
b r e a k w a t e r s l o p e , seaward
face
" d i m e n s i o n l e s s " b r e a k w a t e r s l o p e , SA/L„
R
wave p e r i o d
overtopping
volume
dimensionless overtopping
horizontal
volume
coordinate
h o r i z o n t a l d i s t a n c e t o Y = 0, H^, r e s p e c t i v e l y
d i m e n s i o n l e s s , X, X^, X2
vertical
coordinate
dimensionless v e r t i c a l coordinate
s p e c i f i c weight
Miche's i n t r i n s i c r e f l e c t i o n c o e f f i c i e n t