Progress r e p o r t wave set-up i n v e s t i g a t i o n (SK 31)
P.J. Visser
D e l f t U n i v e r s i t y o f Technology, Department of C i v i l Engineering D e l f t , The Netherlands
CONTENTS page 1 I n t r o d u c t i o n 1 2 Theory 3 3 Experimental procedure • 7 4 The waves 9 5 Experimental r e s u l t s 10
6 Summary, d i s c u s s i o n and suggestions f o r f u r t h e r work 15
Appendix A: references 18 Appendix B: symbols 20 Appendix C:. tables 22 Appendix D: f i g u r e s 28
I INTRODUCTION
When surface v/aves approach a coast and enter shallow water, the mean water l e v e l decreases s l i g h t l y u n t i l the breaker l i n e . This
i s c a l l e d wave set-down (negative wave s e t - u p ) . On the shoreward side of the breaker l i n e , i n the s u r f zone, where the wave h e i g h t decreases, the mean water l e v e l r i s e s and comes above the s t i l l water l e v e l ( a p a r t from p o s s i b l e wind e f f e c t s ) . This i s c a l l e d wave
set-up. The magnitude of the wave set-up i s d e f i n e d as t h e v e r t i c a l d i s t a n c e between mean and s t i l l water l e v e l .
The phenomenon i s important i n r e l a t i o n t o
~ c u r r e n t s , both p a r a l l e l and p e r p e n d i c u l a r t o the coast ( l o n g -shore c u r r e n t s , set-up c u r r e n t s , r i p c u r r e n t s ) and thus also , f o r t r a n s p o r t of sediment i n and near the breaker zone,
- design and c o n t r o l of shore p r o t e c t i o n works as d i k e s , sea w a l l s , beaches and dunes,
The f i r s t model and f i e l d observations of wave set-up are dated to about 1960 (see references i n [ l , 2, 3 j ) . The f i r s t t h e o r e t i c a l expressions f o r wave set-up were d e r i v e d by s e v e r a l researchers i n the p e r i o d 1961-1964, a f t e r the i n t r o d u c t i o n of the concept of r a d i a t i o n s t r e s s .
I n 1976 an i n v e s t i g a t i o n i n t o wave set-up due t o o b l i q u e l y i n c i d e n t r e g u l a r waves has been s t a r t e d i n the Laboratory of F l u i d Mechanics of the D e l f t U n i v e r s i t y of Technology. This r e p o r t describes the f i r s t series of experiments, namely the experiments w i t h normally i n c i d e n t waves. The purpose of these experiments i s t o make a l i n k w i t h other wave set-up experiments, e s p e c i a l l y the measurements by Bowen, Inman and Simmons [1] i n a flume and one of the wave set-up experiments by the D e l f t H y d r a u l i c s Laboratory i n a flume, which were performed i n 1975 and 1976.
Chapter 2 contains a s h o r t resume of the theory of wave set-up i n normally i n c i d e n t waves. I n chapter 3 the experimental procedure
2
i n the wave basin simultaneously w i t h the d e s i r e d wave (and may h i n d e r the steady and r e g u l a r wave p a t t e r n ) . Chapter 5 presents the experimental r e s u l t s and the comparison w i t h
r e s u l t s by other i n v e s t i g a t o r s . Chapter 6 contains the summary, d i s c u s s i o n and suggestions f o r f u r t h e r work.
2 THEORY
This chapter contains a s h o r t t h e o r e t i c a l d e s c r i p t i o n on wave set-up i n waves of perpendicular i n c i d e n c e . Reference i s made to [ 4 ] f o r the r e s t r i c t i o n s and the assumptions vjhich have been made and f o r the d e r i v a t i o n of s e v e r a l equations.
The equation d e s c r i b i n g the p o s i t i o n of the mean water l e v e l reads _ .dS
where h = s t i l l water depth,
q = wave set-up ( d i f f e r e n c e between mean and s t i l l water l e v e l ) ,
X = h o r i z o n t a l c o o r d i n a t e , normal t o t h e shore, p o s i t i v e
shorewards,
S = component of r a d i a t i o n s t r e s s t e n s o r .
XX
Equation (2.1) f o l l o w s from conservation of x-momentum. The r a d i a t i o n s t r e s s i s d e f i n e d as t h e c o n t r i b u t i o n of t h e waves t o the momentum t r a n s p o r t tensor. The concept of r a d i a t i o n s t r e s s has been developed by Longuet-Higgins and Stewart [ 2 , 5, 6, 7, 8 ] and independently by D o r r e s t e i n [ 3 ] and Lundgren (see [ 1 , 9, 10] ) .
I f A = wave l e n g t h , k = 2 T T/ A = wave number and H = wave h e i g h t , then the x-component of the r a d i a t i o n s t r e s s through second order of wave amplitude i s g i v e n by
which i n shallow.water reduces t o
S ^ ^ = l j p g / ( 2 . 3 )
Expression (2.2) f o l l o w s from l i n e a r A i r y wave t h e o r y , i n which a small wave amplitude i s supposed.
Two d i f f e r e n t r e g i o n s , one seaward» and one shoreward^ o f the breaker
4
Zone seawardsf o f the breaker l i n e
Outside the s u r f zone the waves have an ordered character: the motion i s n e a r l y i r r o t a t i o n a l and contains l i t t l e turbulence. The d i s s i p a t i o n of energy i s r e l a t i v e l y s m a l l , so i t can be
assumed t h a t energy i s approximately conserved:
1
2-5- pg H nc = constant,' (2.4)
o
where c = phase v e l o c i t y of the waves, nc = group v e l o c i t y of the wave t r a i n .
I n t h i s r e g i o n the mean f r e e surface can be computed [ l , 4, 7, 9 ] to g i v e n =
-2
1 • kH
8 s i n h 2kh (2.5)I n shallow water t h i s expression becomes
T) =
-1
6
h (2.6)Combination of (2.4) and (2.5) y i e l d s
n = - H :1 „2 , coth kh k (2.7)
8 o o 2kh + s i n h 2kh
where = deep water wave h e i g h t , k^ = deep water wave number.
From wave theory f o l l o w s t h a t k h ( t g h kh) = k h , so
n = -H^ k f (k h) ,
0 0 o
(2.8)
where f ( k h) = f u n c t i o n of k and the l o c a l s t i l l water depth h. o o
The wave set-down a t the breaker l i n e f o l l o w s from (2.6) and, because rii h, can be w r i t t e n as
'b 16 ^ \ ' (2.9)
H.
b
where y (2.10)
= breaker h e i g h t ,
h^+n^ = mean water depth a t the breaker l i n e .
I n f a c t two regions seaward of the breaker l i n e can be d i s t i n q u i s h e d , namely (1) w e l l o u t s i d e the s u r f zone where the l i n e a r theory i s
most a p p l i c a b l e and (2) near the breaker l i n e v/here the waves are too steep t o use the l i n e a r theory.
Surf zone
I n the breaker zone, where the waves are u n s t a b l e and the f l u i d motion tends t o lose i t s ordered c h a r a c t e r , wave energy i s d i s s i p a t e d mainly due t o the g e n e r a t i o n of t u r b u l e n c e . For t h i s reason p o t e n t i a l f l o w theory i s no longer v a l i d . Other a n a l y t i c a l d e s c r i p t i o n s f o r the waves i n the s u r f zone, however, are not a v a i l a b l e . Therefore e m p i r i c a l or semi-empirical approaches are necessary.
Using s i m i l a r i t y arguments, i t i s assumed t h a t the wave h e i g h t i s p r o p o r t i o n a l t o the l o c a l mean water depth (LonguetrHiggins and Stewart [ 8 ] , Bowen, Inman and Simmons [ 1 ] ) :
H(x) = Y [ h ( x ) + n ( x ) ; . (2.11)
The l a b o r a t o r y measurements by Bowen e t a l [ 1 ] show t h a t an assumption of s i m i l a r i t y i n the s u r f zone i s reasonable. F u r t h e r the r a d i a t i o n stress expression ( 2 . 3 ) , which f o l l o w e d from l i n e a r wave t h e o r y , i s maintained.
6
S u b s t i t u t i o n of (2.11) i n t o (2.3) gives
S = -7-7-3 P8Y (b + n) 2 - (2.12)
XX
S u b s t i t u t i n g (2.12) i n t o (2.1) gives the g r a d i e n t of t h e wave set-up
3 2
dn ^ _ 8 dh dx ,,3 2 dx
This equation i n d i c a t e s t h a t t h e g r a d i e n t of the wave set-up i n the surf zone i s p r o p o r t i o n a l t o the l o c a l bottom slope, Equation (2.13) can be i n t e g r a t e d between x = x^ (breaker l i n e ) and
X = X ( l i n e of maximum s e t - u p ) , see [ 9 ] , y i e l d i n g m 3 2 n. max - n 'b (n + h, ) • max b (2,14) S u b s t i t u t i o n of (2.9) i n t o (2.14) gives max 5_ T6 (2.15)
The experiment's were made m the 16.60 x 34.00 m wave b a s i n o f the Laboratory of F l u i d Mechanics of the D e l f t U n i v e r s i t y of Technology. The wave b a s i n arrangement i s shovm s c h e m a t i c a l l y i n f i g u r e 1, The snake-type wave generator has a 32.80 m long
f l e x i b l e wave board, which c o n s i s t s o f rubber panels, each 0.40 m wide. The wave generator can produce r e g u l a r long-crested waves w i t h a constant angle of incidence which can be v a r i e d . The s t r o k e of the wave board a t the bottom can be a d j u s t e d between zero
(pure r o t a t i o n ) and the s t r o k e a t the s t i l l water l e v e l (pure t r a n s l a t i o n ) . I n a l l experiments the combination o f t r a n s l a t i o n and r o t a t i o n was chosen such t h a t the amplitude o f secondary waves was expected t o be minimal. Opposite t o the wave board a 1:10
smooth concrete slope was b u i l t . The d i s t a n c e between the t o e of the slope and the wave board was 8.35 m. The water i n the constant depth p a r t o f the wave b a s i n was 0.40 m deep.
The wave set-up and set-down were measured w i t h tappings, mounted i n the concrete beach, f l u s h w i t h the slope, i n two rays o f each 30 tappings ( f i g . 1). The h o r i z o n t a l d i s t a n c e between 2 tappings was 0.20 m. The i n s i d e diameter of the tappings was 1.5 mm. The
tappings were connected w i t h maiiometer tubes, i n which the s t a t i c head was measured. The assumptions i n v o l v e d i n t r a n s l a t i n g such measurement i n t o mean water l e v e l were considered by Longuet-Higgins and Stewart [ 7 ] ; see also l i t . [ 4 ] , [ 9 ] or [ 2 1 ] . The most im-p o r t a n t assumim-ptions are a g e n t l y s l o im-p i n g bottom and a slow v a r i a t i o n of the waves i n h o r i z o n t a l d i r e c t i o n . The manometer tubes were
readed by photograph, a l l o w i n g an accuracy of about + 0 , 1 mm. Wave set-up and set-down measurements were made i n r a y 1, and also i n and near the surf zone of r a y 2.
Surface e l e v a t i o n s were measured w i t h a r e s i s t a n c e - t y p e wave gauge and analysed by a c r e s t - t r o u g h apparatus t o determine the mean wave h e i g h t . The a c t u a l wave and the mean wave h e i g h t were recorded on paper. Wave h e i g h t s were measured i n p o i n t s of r a y 1, s t a r t i n g 2 m from the wave board and as f a r as p o s s i b l e on the slope. I n s i d e the s u r f zone the response of the wave gauge was n o t l i n e a r due t o the small water depths here; t h i s was c o r r e c t e d . The h o r i z o n t a l d i s t a n c e between 2 p o i n t s where wave h e i g h t s were measured was 0.20 m. I n each p o i n t wave h e i g h t s were measured d u r i n g about 90 seconds t o give a mean value*
Measurements o f wave run-up and the p o s i t i o n o f the plunge p o i n t were made v i s u a l l y . E s p e c i a l l y an accurate d e t e r m i n a t i o n of the p o s i t i o n of the plunge p o i n t v^as very d i f f i c u l t ; observations could be made w i t h an accuracy o f about + 0.05 m.
4 THE WAVES
I n a d d i t i o n t o the primary wave, the p r o g r e s s i v e s i n u s o i d a l wave w i t h the p e r i o d T of the motion of the wave board and w i t h a c r e s t which i s p a r a l l e l t o the wave board, other undesired waves could be generated'simultaneously i n the wave b a s i n , v i z . ;
a r e f l e c t e d component of t h e primary wave ( r e f l e c t i o n on the beach), b r e f l e c t e d component of a ( r e f l e c t i o n against the wave b o a r d ) ,
c secondary waves (importance depends on U r s e l l parameter, see [11] ) , d subharmonic standing waves between wave board and beach (have
an unknown o r i g i n ) .
A l l these waves have c r e s t s which are p a r a l l e l t o the -wave board. Other d i s t u r b i n g waves are:
e standing waves between the s i d e - w a l l s ( p e r i o d T ) ,
g higher order components of e ( i n general n o t i m p o r t a n t ) , h standing edge waves between the s i d e - w a l l s w i t h p e r i o d T or 2T
(see f o r instance [12, 13, 14] ) ,
i standing cross-waves between the s i d e - w a l l s ( p e r i o d 2T, see [ 1 5 ] ) . These waves have c r e s t s p e r p e n d i c u l a r t o the wave board.
The occurrence of these d i s t u r b i n g waves and t h e i r magnitude depend on v a r i a b l e s and q u a n t i t i e s as the wave p e r i o d , t h e wave h e i g h t , the slope of t h e beach, the water depth, t h e d i s t a n c e between the wave board and the toe of the slope and the motion of t h e wave board. The waves c, d, e and h can d i s t u r b the wave p a t t e r n i n an u n d e s i r a b l e way. Moreover s e v e r a l kinds of c u r r e n t s may occur. E s p e c i a l l y r i p c u r r e n t s [ 1 6 ]
can hinder a steady wave p a t t e r n , because t h e i r p o s i t i o n s are n o t steady i n general.
Preceding t o the experiments described i n t h i s r e p o r t , an i n v e s t i g a t i o n was c a r r i e d out t o minimize the d i s t u r b i n g i n f l u e n c e s . The most r e g u l a r waves'were selected t o go on. Nevertheless these waves were n o t e x a c t l y r e p r o d u c i b l e ; the wave h e i g h t , f o r i n s t a n c e , measured at a c e r t a i n place
o u t s i d e the s u r f zone and averaged over 90 seconds, v a r i e d about + 4%. The f i g u r e s 2, 3 and 4 show p r o f i l e s of the s e l e c t e d waves.
10
5 EXPERIMENTAL RESULTS •
The experimental data of the o b s e r v a t i o n s i n r a y 1 are g i v e n i n t a b l e 1 The wave h e i g h t i n the constant depth p a r t of the wave b a s i n i s o b t a i n e d by averaging the mean wave h e i g h t s which were m.easured i n 33 p o i n t s . The s u r f s i m i l a r i t y parameter B,^, d e f i n e d as
r - t g g
o o
where = deep water wave l e n g t h ,
i s a v e r y i m p o r t a n t parameter: s e v e r a l s u r f zone p r o p e r t i e s and q u a n t i -t i e s can be expressed as f u n c -t i o n s o f 5^ -t 9 ] . The breaker -type c l a s s i f i c a t i o n r e s u l t i n g from the experiments by G a l v i n [ 1 7 ] can be w r i t t e n as (5.1) s u r g i n g or c o l l a p s i n g i f . p l u n g i n g s p i l l i n g i f 5, > 3.33, i f 0.46 < 5 < 3.33, o 5 < 0.46 o > ( 5 . 2 ) The U r s e l l parameter, d e f i n e d as Ur = A^H (5.3)
can be considered as a measure f o r the e f f e c t of secondary waves (see [11] ) . I n t h e experiments Ur < 13 ( i n the constant depth p a r t of the wave b a s i n ) , so the i n f l u e n c e of secondary waves can be expected t o be n e g l i g i b l e .
The measured breaker h e i g h t - t o - d e p t h r a t i o s H^/b^ are i n agreement w i t h measurements by B a t t j e s [ 9 ] , I v e r s e n [ 1 8 ] and Goda [19] .
The maximum set-up 1 ^ ^ ^ i s obtained by e x t r a p o l a t i n g the curve drawn through the measured and p l o t t e d set-up v a l u e s , w i t h the exception o f the measurement near the plunge p o i n t . Near t h i s p o i n t the mean
pressure a t the bottom was i n f l u e n c e d s t r o n g l y by the v e r t i c a l wave impact. T r a n s l a t i n g the mean pressure measurement i n t o mean water l e v e l i s n o t p o s s i b l e by n e g l e c t i n g t h i s i n f l u e n c e .
The experimental r e s u l t s are shown i n the f i g u r e s 2, 3 and 4, except the r e s u l t s of the wave set-up measurements i n ray 2, which are shown i n subsequent f i g u r e s . The break p o i n t i s defined as the p o i n t of the maximum V7ave h e i g h t . The h o r i z o n t a l d i s t a n c e between
2 p o i n t s i n the s u r f zone where v/ave h e i g h t measurements were made was 0.20 m except f o r experiment 31-4, f o r which t h i s d i s t a n c e
was 0.10 m. As can be seen from f i g u r e s 5, 6 and 7, a mutual d i s t a n c e of 0.20 m i s l i k e l y too l a r g e t o r e v e a l always the plunge p o i n t i n the wave h e i g h t graph. To v e r i f y the s i m i l a r i t y arguments on which equation (2.11) i s based, the measured values of y i n the s u r f zone are g i v e n i n t a b l e 2 and the nondimensional wave h e i g h t
H / H i s p l o t t e d versus the nondimensional water depth ( h + q ) / ( h + q, ) b b b i n f i g u r e 8. I n agreement w i t h the experimental r e s u l t s by Bowen,
Inman and Simmons [ 1"] the assumption of constant y i s reasonable. Compared t o the experiments 31-2 and 31-4, the s p a t i a l v a r i a t i o n of measured wave h e i g h t i s r a t h e r l a r g e f o r 31-3. This i s caused by more r e f l e c t i o n and more i n f l u e n c e of secondary waves (higher U r s e l l number).
The f i g u r e s 9, 10 and 11 present a comparison between measured and t h e o r e t i c a l wave set-up and wave set-down i n r a y 1, and also the wave set-'up measured i n and near the s u r f zone of r a y 2. The d i f f e r e n c e between the wave set-up measured i n r a y 1 and r a y 2 i s r a t h e r s m a l l f o r 31-2 ( f i g . 9) and e s p e c i a l l y 31-4 ( f i g . 11). The d i f f e r e n c e s f o r 31-3 are g r e a t e r : i n r a y 2 the p o s i t i o n of the plunge p o i n t was more seaward and the g r a d i e n t of the wave set-up i n the s u r f zone was smaller than i n r a y 1. The maximum wave set-up, however, was almost the same. The t h e o r e t i c a l wave set-down i s obtained from ( 2 . 5 ) , u s i n g (2.4) and the measured value of H j . Well o u t s i d e the s u r f zone the d i f f e r e n c e between t h e o r e t i c a l and measured wave set-down i s s m a l l . The d i f f e r e n c e between t h e o r e t i c a l and observed wave set-down i s s i g n i f i c a n t near the breaker l i n e , where the waves were too steep f o r the l i n e a r theory t o \ remain v a l i d . Although the waves were higher than p r e d i c t e d by the
l i n e a r theory, the wave set-down was less than p r e d i c t e d by the same theory. This i s i n agreement w i t h the observations by Bowen e t a l [ 1 ] . The p o s i t i o n of the break p o i n t i n the t h e o r e t i c a l wave set-down
12
l i n e a r theory. I n the s u r f zone che t h e o r e t i c a l set-up i s obtained from (2.13), s u b s t i t u t i n g the measured value of y, the over the s u r f zone averaged value of y , and s t a r t i n g a t the computed )
break p o i n t . The f i g u r e s 9, 10 and 11 show a r a t h e r small d i f f e r e n c e between measured and t h e o r e t i c a l maximum wave set-up, b u t a l a r g e r d i f f e r e n c e between measured and t h e o r e t i c a l g r a d i e n t of wave s e t -up.
A comparison of wave set-up, vzave h e i g h t and wave run-up observations i n r a y 1 w i t h the theory and some e m p i r i c a l formulae i s given i n
t a b l e 3. Le Mehaute and Koh [ 2 0 ] d e r i v e d the f o l l o w i n g wave breaking c r i t e r i o n
^ = 0.76 ( t g a ) ' / ^ ( ^ ) - ' / ^ (5.4) o o
from several experimental i n v e s t i g a t i o n s i n two-dimensional wave tanks by other i n v e s t i g a t o r s . S u b s t i t u t i o n of (5.4) i n t o (2.15) gives an expression f o r the maximum value of the wave set-up
^max = ö-2^^«o^^g'^)'^' ^f^''^"" ^'-''^ o
A r e l i a b l e e m p i r i c a l formula f o r the wave run-up h e i g h t on a slope was given by Hunt (see reference i n [ 1 ] or [ 9 ] ) :
H
' o
where R = wave run-up h e i g h t (above S.W.L.), C = p o r o s i t y f a c t o r .
P
For a smooth slope eq. (5.6) can be w r i t t e n as
R = 5 H . o Q
The agreement between measured ïi^/li^ and the v a l u e p r e d i c t e d by Le Mehaute and Koh's formula (5.A) i s r a t h e r w e l l ( t a b l e 3 ) . As already appeared from the f i g u r e s 9 through 11 the d i f f e r e n c e between measured and t h e o r e t i c a l v/ave set-down near the break p o i n t i s l a r g e . This a p p l i e s also but t o a l e s s e x t e n t t o the g r a d i e n t of wave set-up i n the s u r f zone. The agreement between
measured n and the value computed from n = -tr" Y H, i s reasonable,
max ^ 'max 16 ' b * w h i l e the agreement between measured and t h e o r e t i c a l run-up i s
e x c e l l e n t . The d i f f e r e n c e between n and the v a l u e p r e d i c t e d max
by ( 5 . 5 ) , however, i s n o t small ( i n experiment 31-3). The experimental data of the experiments by Bowen e t a l [1 ]
and one of the experiments by the D e l f t H y d r a u l i c s Laboratory are given i n t a b l e 4. Bowen e t a l [ 1 ] do n o t r e p o r t on the breaker type; i n t a b l e 4 the breaker type i s obtained from Galvin's breaker type c l a s s i f i c a t i o n ( 5 . 2 ) . The experimental r e s u l t s by Bowen e t a l [ 1 ] are remarkable i n t h a t the measured values of H , /h, are l a r g e
b b
compared t o measurements by B a t t j e s [ 9 ] , I v e r s e n [ 1 8 ] and Goda [ 1 9 ] . The comparison of above measurements w i t h theory and some e m p i r i c a l formulae i s given i n t a b l e 5. The d i f f e r e n c e between measured
H ^ / H^ arid the value p r e d i c t e d by (5.4) i s about 10% a t most. The agreement between measured r\ and the value computed from
n = v r Y H, i s r a t h e r w e l l , w h i l e the agreement, between measured max 16 ' b
and t h e o r e t i c a l run-up i s e x c e l l e n t . The d i f f e r e n c e between measured n and the value computed from ( 5 . 5 ) , however, i s n o t small i n
max
some experiments.
Bowen e t a l [ 1 ] compared the t h e o r e t i c a l r a t i o of set-up slope t o beach slope, which i s given by
_ 3 2 1 dn _ 8 ^
(5.8) tga dx 3 2
' 8 ^
t o the measured r a t i o of set-up slope t o beach s l o p e , which was taken as
K = J k . (5.9)
tga X - X,
14
Thus, the measured r a t i o of set-up slope t o beach slope was averaged over the s u r f zone, g i v i n g a reasonably good agreement w i t h the t h e o r e t i c a l set-up slope-to-beach slope r a t i o as
expressed by ( 5 . 8 ) , but also n e g l e c t i n g the r e a l steeper s e t -up slope.
A comparison between the experimental r e s u l t s of T2 - 4B from the D e l f t H y d r a u l i c s Laboratory and experiment 31-4 i s sho^ra i n
f i g , 12. The s i m i l a r i t y i n beach slope and wave p e r i o d permits a comparison. The agreement i s good, e s p e c i a l l y f o r the g r a d i e n t of the set-up i n the s u r f zone, b u t also f o r the magnitudes
of set-up and set-down as appears from
31-4 - ^ ^ = 0 . 3 1 = -0,024
"b b n ri, T2-4B - 0,33 rr- = -0,025
b
The breaker depths are almost equal, i n s p i t e of h i g h e r waves f o r 31-4, Hence, the r a t i o of breaker h e i g h t t o mean water depth d i f f e r s s l i g h t l y : = 0,81 f o r T2-4B, y^^ = 0,95 f o r 31-4, The d i f f e r e n c e i n y i s s m a l l , however, which may be seen from f i g , 12.
6 SU1#IARY, DISCUSSION AND SUGGESTIONS FOR FURTHER WORK
The experiments described i n t h i s r e p o r t and a l s o the experiments by Bowen e t a l [ I J and the D e l f t H y d r a u l i c s Laboratory i n d i c a t e
t h a t the maximum value of the wave set-up on f l a t s o l i d beaches due t o normally incoming; waves may be approximated r a t h e r
a c c u r a t e l y by
n = ^ Y H~ (6.1)
max 1 6 ' b
where H^ = measured breaker h e i g h t .
Equation (6.1) expresses the maximum wave set-up i n terms of an inshore parameter (y) and an inshore q u a n t i t y (H^)• Equation (5.5)
o
gives the maximum wave set-up as a f u n c t i o n o f o f f s h o r e parameters and q u a n t i t i e s , the beach slope tga and s t i l l Y From the e x p e r i -mental r e s u l t s given by B a t t j e s [ 9 ] , I v e r s e n [ 1 8 ] and Goda [19]
i t i s p o s s i b l e , however, t o estimate Y from c,^ (see [ 9 ] , p. 21), The agreement of a l l measurements ( i n c l u d i n g the observations by Bowen e t a l t U and the one by the D e l f t H y d r a u l i c s Laboratory) w i t h the value p r e d i c t e d by (6.2) i s reasonable, b u t less than w i t h ( 6 . 1 ) . A l l experiments present an e x c e l l e n t agreement of measured run-up w i t h Hunt's formula.
As expected f o r plunging breakers the set-up does n o t s t a r t a t the break p o i n t but near the plunge p o i n t . Nevertheless the steeper set-up slope ( steeper than p r e d i c t e d by theory w i t h eq. (2.13) ) y i e l d s a maximun: set-up close t o the t h e o r e t i c a l value ( 6 . 1 ) . Near
the plunge p o i n t t r a n s l a t i o n o f mean pressure measurement i n t o mean water l e v e l f a i l s (does apply t o the experiments described i n t h i s r e p o r t , Bowen et a l [ 1 ] do n o t r e p o r t on t h i s ) . This i s caused by the plunge phenomenon (wave i m p a c t ) , N e g l e c t i n g the measurements near the plunge p o i n t s , the curves drawn through the other set-up measurement-points are p r a c t i c a l l y s t r a i g h t .
16
I n t h e s u r f zone the theory i s based on t h e assumption of
p r o p o r t i o n a l i t y of wave h e i g h t t o mean water depth (2.11). Also the experiments described i n t h i s r e p o r t show t h a t t h i s
assumption i s reasonable.
V/ell o u t s i d e the s u r f zone the d i f f e r e n c e between measured and t h e o r e t i c a l wave set-down i s small. Some s c a t t e r occurs due t o r e f l e c t i o n , secondary waves, measuring e r r o r s , e t c . I n a l l experiments the d i f f e r e n c e between observed and theo-r e t i c a l set-down i s s i g n i f i c a n t i n f theo-r o n t of the btheo-reaketheo-r l i n e
( t h i s d i f f e r e n c e w i l l increase when t h e t h e o r e t i c a l set-down i s computed from the measured wave h e i g h t s ) . A p o s s i b l e ex-p l a n a t i o n f o r t h i s ex-phenomenon may be t h a t n o t o n l y near the plunge p o i n t b u t also i n f r o n t of the break p o i n t the t r a n s -l a t i o n of mean pressure measurement i n t o mean water -l e v e -l f a i -l s . This idea i s elaborated below.
The mean pressure a t the bottom (p) can be w r i t t e n as (see [ 4 ] , [ 2 1 ] ) :
P = pg(h + n) + P
where pg(h + n) = mean h y d r o s t a t i c pressure a t t h e bottom, P = mean hydrodynamic pressure a t the bottom.
Equation (6.3) f o l l o w s from c o n s e r v a t i o n o f v e r t i c a l momentum. P can be w r i t t e n as ( [ 4 ] , [ 2 1 ] ) : dS p = — _ + small terms dx q where S = ƒ puwdz , -h n = e l e v a t i o n of f r e e surface above S.W.L., u T h o r i z o n t a l v e l o c i t y , w = v e r t i c a l v e l o c i t y ,
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 upwards from S.W. the overbar i n d i c a t e s a time average.
S can be considered as the (x,z)component o f the t h r e e
-dimensional r a d i a t i o n s t r e s s t e n s o r . S u b s t i t u t i o n of the l i n e a r wave theory expressions f o r r\, u and w i n t o (6.5) gives
S^^ = 0. Near break and-plunge p o i n t the v e l o c i t i e s due t o the waves do not behave s i n u s o i d a l l y , hovzever, b u t are s t r o n g l y
asymmetric. Hence, g e n e r a l l y S i s n o t zero here. Besides,
X Z
the growth and change of asymmetry occurs along a s h o r t d i s t a n c e ( i n p l u n g i n g b r e a k e r s ) , p o s s i b l y y i e l d i n g a c o n s i d e r a b l e g r a d i e n t of v e r t i c a l r a d i a t i o n s t r e s s . The term (6.5) w i l l have more i n f l u e n c e i n p l u n g i n g breakers: i n s p i l l i n g breakers the waves become a s y m m e t r i c a l l y t o o , but much less than i n p l u n g i n g breakers. Above mentioned e x p l a n a t i o n seems t o be confirmed by the experiments by Bowen e t a l [ 1 ] : the d i f f e r e n c e between t h e o r e t i c a l and measured
was less i n experiments w i t h breakers which were c a l c u l a t e d as s p i l l i n g . Some a d d i t i o n a l measurements were made t o check t h i s e x p l a n a t i o n . I n these experiments the mean water l e v e l was also measured w i t h a wave gauge. The s i g n a l o f t h e wave gauge was f e d
i n t o an e l e c t r o n i c f i l t e r t o damp the wave motion. The d i f f e r e n c e s between these set-down o b s e r v a t i o n s and t h e set-down observations w i t h manometers, however, were very s m a l l . Consequently, P have not i n f l u e n c e d t h e set-down measurements w i t h manoraeters i n f r o n t of the breaker l i n e .
The number of experiments, described i n t h i s r e p o r t , i s l i m i t e d to t h r e e . The agreement between these experiments and the 11 experiments by Bowen e t a l [ 1 ] and t h e experiment by the D e l f t H y d r a u l i c s Laboratory i s s a t i s f a c t o r y and t h e r e f o r e c o n t i n u a t i o n of the experiments w i t h o b l i q u e l y i n c i d e n t waves i s j u s t i f i e d . Resul of these experiments w i l l be r e p o r t e d i n t h e next progress report.'
18
APPENDIX A: REFERENCES
1. Bowen, A.J., Inman, D.L. and Sinmons, V.P., Wave "set-do\m" and set-up, J. Geoph. Res., v o l . 73, 1968, p. 2569-2577.
2. Longuet-Higgins, M.S. and Stewart, R.W., A note on wave set-up, J. Mar. Res., v o l , 21, 1963, p. 4-10.
3. D o r r e s t e i n , R,, Wave set-up on a beach, Proc, Second Techn, Conf. on H u r r i c a n e s , June 1961, Miami Beach; Washington, D.C,
1962, p. 230-241.
4. V i s s e r , P.J., Wave set-up, 1: l i t e r a t u r e and t h e o r e t i c a l i n v e s t i g a t i o n , i n p r e s s , 1977.
5. Longuet-Higgins, M.S. and Stewart, R.W., Changes i n the form of s h o r t g r a v i t y waves on long waves and t i d a l c u r r e n t s , J . F l u i d Mech., v o l . 8, 1960, p. 565-583.
6. Longuet-Higgins , M.S. and Stewart, R.W., The changes i n
amplitude of s h o r t g r a v i t y waves on steady non-uniform c u r r e n t s , J. F l u i d Mech,, v o l , 10, 1961, p, 529-549.
7. Longuet-Higgins> M.S. and Stewart, R.W., R a d i a t i o n s t r e s s and
mass t r a n s p o r t i n g r a v i t y waves, w i t h a p p l i c a t i o n t o " s u r f - b e a t s ' J. ..Fluid Mech.., v o l . 13, 1962, p. 481-504.
8. Longuet-Higgins, M.S. and Stewart, R.W., R a d i a t i o n stresses i n water waves; a p h y s i c a l , d i s c u s s i o n , w i t h a p p l i c a t i o n s , Deep-Sea Res., v o l . 11, 1964, p. 529-562.
9. B a t t j e s , J.A., Computation o f set-up, longshore c u r r e n t s , run-up and o v e r t o p p i n g due t o wind-generated waves. Communications on H y d r a u l i c s , Department o f C i v i l E n g i n e e r i n g , D e l f t U n i v e r s i t y of Technology, Report no. 74-2, 1974.
10. B a t t j e s , J.A., Set-up due t o i r r e g u l a r waves, Proc. 13th.Conf. Coastal Eng., Vancouver, B.C., 1972, p. 1993-2004.
11. Hulsbergen, C.H., O r i g i i i , e f f e c t and suppression of
secondary waves, Proc. 14th. Conf. Coastal Eng., Copenhagen, 1974, p.392-411.
12. Bowen, A.J, and Inman, D.L., Edge waves and c r e s c e n t i c b a r s , J. Geoph. Res., v o l . 76, 1971, p. 8662-8671.
13. Guza, R.Ï. and Davis, R.E., E x c i t a t i o n o f edge waves by waves i n c i d e n t on a beach, J. Geoph. Res., v o l . 79, 1974, p. 1285-1291.
14. Guza, R.T. and Inman, D.L,, Edge waves and beach cusps, J, Geoph, Res,, v o l , 80, 1975, p, 2997-3012
15. G a r r e t t , C.J.R., On cross-waves, J. F l u i d Mech. v o l . 4 1 , 1970, p. 837-849.
16. Bowen, A.J., Rip c u r r e n t s , 1. T h e o r e t i c a l I n v e s t i g a t i o n s , J. Geophys. Res., v o l . 74, 1969, p. 5467-5478.
17. G a l v i n , C.J. J r . , Breaker type c l a s s i f i c a t i o n on t h r e e l a b o r a t o r y beaches, J. Geoph. Res., v o l . 73, 1968, p. 3651-3659.
18. I v e r s e n , H.W. Laboratory study of b r e a k e r s , Nat. Bur. o f Standard C i r c u l a r 521, Washington, D.C, 1952, p. 9-32.
19. Goda, Y., A synthesis of breaker i n d i c e s . Trans. Jap. Soc. Civ. Eng., v o l . 2, 1970, p. 227-230.
20. Le Mehaute, B. and Koh, R.C.Y., On the b r e a k i n g of waves a r r i v i n g a t an angle to the shore, J.Hydr. Res., v o l . 5, p. 67-88.
21. L i u , P. L-F. and Mei, C.C., E f f e c t s o f a breakwater on nearshore c u r r e n t s due t o breaking waves. Tech. Mem. no. 57, C.E.R.C, 1975.
20
APPENDIX B: SYMBOLS
The symbols used i n the t e x t are l i s t e d below. The remaining symbols used i n appendix C are l i s t e d i n t a b l e 3,
c = phase v e l o c i t y
g = g r a v i t a t i o n a l a c c e l e r a t i o n H V7ave h e i g h t
= deep water wave h e i g h t
Hj = mean wave h e i g h t i n constant depth p a r t of. vrave b a s i n = breaker h e i g h t
h - s t i l l water depth
h j = s t i l l V7ater depth i n constant depth p a r t of wave b a s i n h^ = s t i l l water depth a t breaker l o c a t i o n
K = over s u r f zone averaged (measured) r a t i o of set-up slope to beach slope
k = wave number
k = deep water wave number o
nc = group v e l o c i t y
P = mean hydrodyiiamic pressure a t the bottom p = mean pressure a t the bottom
R = wave run-up h e i g h t S = ( x , x)-component of r a d i a t i o n s t r e s s tensor XX S = ( x , z)-component of 3-dimensional r a d i a t i o n s t r e s s t e n s o r xz T = wave p e r i o d u = h o r i z o n t a l v e l o c i t y , 2 3 i Ur = A H/.h = U r s e l l parameter ' w = v e r t i c a l v e l o c i t y : X = h o r i z o n t a l c o o r d i n a t e , p o s i t i v e shorewards' z = v e r t i c a l c o o r d i n a t e , p o s i t i v e upwards from S.W.L.
a = slope angle w i t h r e s p e c t t o the h o r i z o n t a l
Y wave height-mean water depth r a t i o i n s u r f zone Y = over s u r f zone averaged value of y
= wave height-mean water depth r a t i o at breaker l o c a t i o n n = e l e v a t i o n of f r e e s u r f a c e above S.W.L.
n mean value of n: wave set-up or wave set-do^m wave set-down a t breaker l o c a t i o n
n . mm
minimum wave set-up (maximum wave set-doxm) n
max
= maximum wave set-up
X = wave l e n g t h
X
o
= deep water wave l e n g t h
X
o =
tga//H /A• o o
22
APPENDIX C: 5 TABLES
The symbols used i n t a b l e 5 correspond w i t h the symbols used i n t a b l e 3.
31-3 1.69 0'. 101 447 40.0 0.089 304 8.56 9.08 0.020 0.71 12.3 12.6 10.97 1.15 -0.33. p i . -0.51 1.14 . 3.90 6.6 31-4 I 1.30 0.101 264 . 40.1 0.152 217 7.70 8.42 0.032 0.56 5.6 10.3 11.10 0.93 . -0.25 p i . -0.34 0.97 3.22 4.7 1 ! 1
meas c a l c E e a s c a l c meas c a l c meas meas meas meas meas meas meas
the i n d e x 1 r e f e r s to a v a l u e i n c o n s t a n t depth p a r t o f wave b a s i n , p i . = p l u n g i n g b r e a k e r ,
Y = o v e r s u r f zone a v e r a g e d v a l u e o f y , R = wave run-up .
distance from breakpoint i n cm experiment 0 10 20 30 40 50 60 70 80 90 100 110 T . 31-2 1.24 1.17 1.03 1.10 1.03 1.11 31-3 1.18 1.22 1.19 0.98 1.22 1.15 1.04 1.14 31-4 0.95 l.OI ].03 1.09 1 .06 0.77 1.02 1.01 0.81 0.99 0.93 0.91. 0.97
3 1 - 3 1 . 3 9 1-.21 1 . 4 6 - 0 . 3 3 - 0 . 9 0 ' - 0. 5 1 3 . 9 0 • 4 , 4 8 4 . 7 1 0 . 4 1 0 . 3 3 0 . 2 8 6 . 6 6 . 4 5 3 1 - 4 1 . 2 2 1 . 0 8 1-.29 - 0 . 2 5 - 0 . 6 2 - 0 . 3 4 3 . 2 2 3 . 1 1 3 . 3 1 0 . 3 6 0 . 2 6 0 . 2 4 4 . 7 4 . 7 2 •
meas meas meas meas meas , meas
1 K c a l c u l a t e d w i t h l i n e a r t h e o r y . max 1 6 H - = 0 . 7 6 itga)'/' (^)-^/^ o 0 • 0 b = - T6 ^ \ 1 dq o o K K ' = K"= tga dx 3 - 2 1 + ^ Y
= measured set-up slope-to-beach slope r a t i o .
= t h e o r e t i c a l set-up slope-to-beach slope r a t i o .
n - Tl
max b ^S^-^^max"
over s u r f zone averaged set-up slope-to-beach slope r a t i o .
expe-r i m e n t T s e c t g a X 0 cm cm
\
X 0 cm «1 cm H 0 cm H 0 X 0 ^ 0 I Ur^ cm\
cm\
\
\
cm b r e a -k e r t y p e \ i i n y n max cm R cm 71/3 0.82 0.082 105 3.60 0.034 0.45 4.40 4. 15 1.06 -0.17 s p . 0.90 1.48 1.70 71/4 0.82 0.082 105 5.15 0.049 0.38 5.90 5.5 1.07 -0.19 s p . 0.88 1.60 1.84 51/4 . 1. 14 0.082 202 4.20 0.021 0.58 6.60 5.0 1.32 -0.19 p i . 1.1! 2.07 2.32 51/6 1 . 14 0.082 202 6.45 0.032 0.47 8.55 6.8 1.26 -0.32 p i . 1.15 2.95 3.25 51/8 1 . 14 0.082 202 9.00 0.045 0.40 10.60 9.7 1.09 -0.47 sp 1.00 3.30 3.70 35/7 1.65 0.082 424 4.25 0.010 0.83 7.75 5.9 1.31 -0.18 p i . 1.22 3.37 3.66 35/10 1.65 0.082 424 5.85 0.014 0.71 9.65 6.8 1.42 -0.25 p i . . 1.19 3.70 4.23 35/12 1.65 0.082 424 7.10 0.017 0.65 11.45 9.5 1.21 -0.26 p i . 1.17 • 4. 15 4.75 35/15 1.65 0.082 424 8.90 0.021 0.58 13.00 9.7 1.34 -0.43 p i . 1.17 4.65 5.20 24/17 2.37 • 0.082 876 6.20 0.0071 0.99 11.80 8.8 1.34 -0.30 p i . 1.24 4.50 6.17 24/20 2.37- 0.082 876 7.50 0.0086 0.90 12.70 9.2 1.38 -0.38 p i . 1.28 5.28 6.60 T2-4B 1.28 0.10 j 255 29.8 0.117 1.92 6.21 6.71 0.026 0.62 8.6 8.62 10.8 0.80 -0.22 p i . -0.33 0.94 2.8671/3 through 24/20 : e x p e r i m e n t s by Bowen, Inman and Simmons T2-4B : one o f t h e e x p e r i m e n t s by t h e D e l f t H y d r a u l i c s L a b o r a t o r y
51/4 1.57 1.18 1.40 -0.19 -0.46 2.07 2.29 2.04 0.32 0.32 2.32 2.42 51/6' 1.33 1.11 1.26 -0.32 -0.61 2.95 3.07 2.91 0.33 0.34 -3.25 3.00 51/8 1.18 1.04 1.16 -0.47 -0.66 3.30 3.31 3.25 0.27 0.30 3.70 3.54 35/7 1.82 1 .34 1.69 , -0.18 -0.59 3.37 2.95 2.74 0.34 0.39 3.66 3 . 5 3 35/10 1.65 1 .20 1.55 -0.25 -0.72 3.70 3.59 3.37 0.34 0.38 4.23 4.1 1 3 5 / 1 2 1.61 1.20 1.48 -0.26 -0.84 4.15 4.19 3.85 • 0.34 0.31 4.75 4.55 35/15 1.46 1.20 1.40 . -0.43 -0.95 4.65 4.75 4.55 0.34 0.37 5.20 5 J 0 24/17 1.90 1.45 .1.84 -0.30 -0.91 • 4'. 50 4.57 4.42 0.37 0.36 6.17 6.1 0 24/20 1.69 1.42 1.75 -0.38 -1.02 5.28 5.08' 5.26 0.38 0.40 6.60 6.7 3 X2-4B 1.28 1.06 1.36 -0.22 -0.51 -0.33 2.86 2.53 2.68 0.36 0.25 0.22 4.17
71/3 through 24/20: e x p e r i m e n t s by Bowen, Inman and Simmons
T2-4B : one o f t h e e x p e r i m e n t s by t h e D e l f t H y d r a u l i c s L a b o r a t o r y
D I S T A N C E F R O M WAVE B O A R D = 3./.5 m D I S T A N C E FROM WAVE B O A R D = 4.45 m D I S T A N C E F R O M WAVE B O A R D 2 5.45 m D I S T A N C E FROM V/AVE B O A R D = 6.A5 m D I S T A N C E FROM WAVE B O A R D = 7.45 m 1 I 1 ~ i 1 1 1 1 —; >- J I M E IN S E C O N D S E X P . 3 1 - 2 T s l . 1 7 s e c h , = 4 0 . 0 c m H , ^ 8 . S 5 c m . . . r a y 1 F I G . 2 ; W A V E P R O F I L E S
E u D I S T A N C E FROM W A V E B O A R D = 4.45 m 1 0 1 2 3 A 5 6 I , , . 1 H 1 — T I M E IN S E C O N D S E X P . 3 1 - 3 T : : ; 1 . 6 9 s e c h, = 4 0 . 0 c m H, = 9.56 cm r a y 1 F I G . 3 : W A V E P R O F I L E S
D I S T A N C E FROM V M V E B O A R D = 3./15 m D I S T A N C E F R O M WAVE B O A R D = 4.45 m D I S T A N C E F R O M WAVE B O A R D = 5.45 m D I S T A N C E FROM WAVE B O A R D = 6.45 m D I S T A N C E F R O M WAVE B O A R D = 7.45 m 1 ^ ^ 1 ^—) . 1 &. T I M E IN S E C O N D S I E X R 3 1 - 4 T = 1.30 s e c h, = 4 0 . 1 c m H, = 7 . 7 0 c m r a y 1 F I G . 4 : V/AVE P R O F I L E S
B R E A K O B S E R V E D A H in cm P O I N T I ' P L U N G E POINT 11 s • 8 • 7 • 6-5 4 4-3 21 -I . 42 S.W.L. T O E O F S L O P E B E A C H ƒ max 4-0.5 D I S T A N C E F R O M 10 11 12 V/AVE B O A R D IN M E T E R S E X P . 3 1 - 3 T = 1.69 s e c h,= 40.0 c m H,= 8.56 c m FIG.6 : W A V E H E I G H T S A N D S E T - U P M E A S U R E D ! N RAY 1 .
8 • 7 • 6-2 . . X X X X X X X X " X X X X V W " » X X X T O E O F S L O P E 2 in c m i, • A . | 2 3 4-2 S.W.L E X P . 3 1 - 4 T = 1.30 s e c h,= 40.1 c m - 0 . 5 D I S T A N C E F R O M 10 11 . . 12 WAVE B O A R D IN M E T E R S H, =7.70 c m F i G . 7 : V/AVE H E I G H T S A N D S E T - U P M E A S U R E D I N R A Y 1 .
Hb 0.4 4-X E X P . 31 - 2 + E X P . 3 1 - 3 . E X P . 3 1 - / i 0.2 0.4 0.6 0.8 ( h . 1.0 -F I G . 8 NONDIMENSIONAL WAVE H E I G H T A S F U N C T I O N OF NONDIMENSIONAL M E A N W A T E R D E P T H
- r T H E O R E T I C A L S E T - U P RAY 1 B R E A K P O I N T P L U N G E POINT , , ( R A Y 1 ) , , + 3 42 S.W.L T O E O F S L O P E E X P . 3 1 - 2 T = 1.17 s e c 10 hjr^O.O c m 11 12 •0.5 ^ D I S T A N C E FROM. WAVE B O A R D IN M E T E R S F I G . 9 : M E A S U R E D V/AVE S E T - U P IN R A Y S 1 A N D 2 A N D T H E O R E T I C A L S E T - U P IN RAY 1
12 in c m . T O E O F S L O P E - - • , - i , - • • . • , : , • D I S T A N C E F R O M 8 9 10 11 12 W A V E B O A R D - I N M E T E R S E X P . 3 1 - 3 T = 1.69 s e c h^ = AÖ.0 cm
TOE O F S L O P E S.W.L + 2 4-1 <• + i- + _ © 10 11 12 - 0 . 5 DISTANCE FROM V M V E B O A R D . IN M E T E R S E.X?. 3 1 - - T= 1.30 s e c h, =40.1 cm F I G . 1 1 : M E A S U . R E D W A V E S E T - U P IN R A Y S 1 A N D 2 AND T H E O R E T I C A L S E T - U P IN RAY 1
H in c m
4
E X P . 3 1 - A : T= 1.30 s e c , h , = ^ 0 . 1 c m , H, = 7.70 cm , T G oc = 0.10 , WAVE T A N K . E X P . T 2 - 4 B : T=1.28 s e c , h, = 2 S . 8 c m , H, = 6.21 c m , T G o c = 0.10 , F L U M E .
