C E T A 81-17
Irregular Wave Runup on Smooth Slopes
by
John P. Ahrens
C O A S T A L E N G I N E E R I N G T E C H N I C A L A I D N O . 81-17
DECEMBER 1981
A p p r o v e d for p u b l i c r e l e a s e ;
distribution u n l i m i t e d .
U.S. ARMY, C O R P S O F E N G I N E E R S
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
K i n g m a n B u i l d i n g
Fort B e l v o i r , V a . 2 2 0 6 0
UNCLASSIFIED G5.J.
S E C U R I T Y C L A S S I F I C A T I O N O F T H I S P A G E (When Dala Entered)
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READ INSTRUCTIONS BEFORE COMPLETING FORM
1, R E P O R T N U M B E R 2. G O V T A C C E S S I O N NO.
CETA 81-17
3. R E C I P I E N T ' S C A T A L O G N U M B E R
4. T I T L E Cand Subei(/e;
IRREGULAR WAVE RUNUP ON SMOOTH SLOPES
5. T Y P E O F R E P O R T * P E R I O D C O V E R E D
C o a s t a l E n g i n e e r i n g T e c h n i c a l A i d
4. T I T L E Cand Subei(/e;
IRREGULAR WAVE RUNUP ON SMOOTH SLOPES
6. P E R F O R M I N G O R G . R E P O R T N U M B E R 7. A U T H O R f s ; John p. A h r e n s a. C O N T R A C T O R G R A N T N U M B E R f a ) 9. P E R F O R M I N G O R G A N I Z A T I O N NAME AND A D D R E S S Department of t h e Army 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 (CERRE-CS) Kingman B u i l d i n g , F o r t B e l v o i r , V i r g i n i a 22060 10. P R O G R A M E L E M E N T , P R O J E C T , T A S K A R E A a WORK U N I T N U M B E R S D31229 I I . C O N T R O L L I N G O F F I C E N A M E AND A D D R E S S Department o f t h e Army 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 Kingman B u i l d i n g , F o r t B e l v o i r , V i r g i n i a 22060 12. R E P O R T D A T E December 1981 I I . C O N T R O L L I N G O F F I C E N A M E AND A D D R E S S Department o f t h e Army 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 Kingman B u i l d i n g , F o r t B e l v o i r , V i r g i n i a 22060 13. N U M B E R O F P A G E S 26
M. M O N I T O R I N G A G E N C Y N A M E & ADORESS(i{ dltferent Irom Controlling OUice) 15. S E C U R I T Y C L A S S , (of thla report)
UNCLASSIFIED
M. M O N I T O R I N G A G E N C Y N A M E & ADORESS(i{ dltferent Irom Controlling OUice)
tSa. D E C L ASSI F I C A T I O N / D O W N G R A D I N G S C H E D U L E
16. D I S T R I B U T I O N S T A T E M E N T f o / ( h i s ReporO
Approved f o r p u b l i c r e l e a s e ; d i s t r i b u t i o n u n l i m i t e d .
17. D I S T R I B U T I O N S T A T E M E N T (ol the abelract entered In Block 20, / / dlllereni Irom Report)
18. S U P P L E M E N T A R Y N O T E S
19. K E Y WORDS (Continue on r e v e r s e side il neceaaary and Identity by block number)
I r r e g u l a r waves VJave rundown Smooth p l a n e s l o p e s Wave runup
20. A B S T R A C T (Continue on r e v e r s e aide If neceaaary and Identify by block number)
The r e s u l t s o f s e v e r a l l a b o r a t o r y s t u d i e s have b e e n u s e d t o d e v e l o p a method t o e s t i m a t e t h e wave runup and rundown on p l a n e , smooth s l o p e s c a u s e d by i r r e g u l a r wave a c t i o n . C u r v e s and e q u a t i o n s a r e p r e s e n t e d w h i c ) i c a n be used t o compute t h e 2 - p e r c e n t runup, s i g n i f i c a n t runup, mean r u n u p , and a p p r o x i m a t e l o w e r l i m i t o f rundown. A p r o c e d u r e i s s u g g e s t e d f o r a d a p t i n g the s m o o t h - s l o p e r e s u l t s t o wave runup on r o u g h and p o r o u s s l o p e s . E x a m p l e problems i l l u s t r a t e t h e u s e o f t h e m a t e r i a l p r e s e n t e d .
PREFACE
T h i s r e p o r t p r e s e n t s a method f o r e s t i m a t i n g t h e m a g n i t u d e and d i s t r i b u t i o n of wave runup and rundown on p l a n e , smooth s l o p e s c a u s e d by i r r e g u l a r wave a c t i o n . W i t h i n t h e method's r a n g e o f a p p l i c a b i l i t y i t s u p e r s e d e s S e c t i o n 7,212, " I r r e g u l a r Waves," o f t h e S h o r e P r o t e c t i o n M a n u a l ( U . S . Army, C o r p s o f E n g i n e e r s , 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 , 1977) ; CETA 77-2 " P r e d i c t i o n of I r r e g u l a r Wave Runup" by J o h n P. A h r e n s ; and CETA 78-2 " R e v i s e d Wave Runup C u r v e s f o r Smooth S l o p e s " by P h i l i p N. S t o a . I t a l s o s u p e r s e d e s t h e p a r t s o f CETA 79-1 "Wave Runup on Rough S l o p e s , " by P h i l i p N. S t o a , w h i c h e s t i m a t e wave runup on r o u g h and p o r o u s s l o p e s by a d j u s t i n g t h e r u n u p f o r s i m i l a r wave c o n -d i t i o n s on smooth s l o p e s u s i n g a r o u g h - s l o p e c o r r e c t i o n f a c t o r . T h i s r e p o r t was p r e p a r e d by J o h n P. A h r e n s , O c e a n o g r a p h e r , u n d e r t h e gen-e r a l s u p gen-e r v i s i o n o f D r . R.M. S o r gen-e n s gen-e n , C h i gen-e f , C o a s t a l P r o c gen-e s s gen-e s and S t r u c t u r gen-e s B r a n c h , R e s e a r c h D i v i s i o n . Comments on t h i s p u b l i c a t i o n a r e i n v i t e d . A p p r o v e d f o r p u b l i c a t i o n i n a c c o r d a n c e w i t h P u b l i c Law 1 6 6 , 7 9 t h C o n g r e s s , approved 31 J u l y 1 9 4 5 , a s s u p p l e m e n t e d by P u b l i c Law 1 7 2 , 8 8 t h C o n g r e s s , approved 7 November 1 9 6 3 . C o l o n e l , C o r p s o f E n g i n e e r s Commander and D i r e c t o r
3
CONTENTS
Page
CONVERSION FACTORS, U.S. CUSTOMARY TO METRIC ( S I ) 5
SYMBOLS AND D E F I N I T I O N S 6 I INTRODUCTION . . . 7
I I IRREGULAR WAVE RUNUP ON PLANE, SMOOTH SLOPES 7
I I I IRREGULAR WAVE RUNDOWN 13 IV APPLICATION OF RESULTS TO ROUGH AND POROUS SLOPES 14
V EXAMPLE PROBLEMS 15 V I SUMMARY ^ 18
L I T E R A T U R E C I T E D 19 APPENDIX
A RUNUP S C A L E - E F F E C T CORRECTION FACTOR, k, FOR SMOOTH SLOPES . 21 B RUNUP REDUCTION FACTOR, r , FOR VARIOUS T Y P E S OF ROUGH
AND POROUS STRUCTURES 22 C RUNUP SCALE CORRECTION FACTOR, k, FOR VARIOUS TYPES OF
ROUGH AND POROUS STRUCTURES 26 TABLES 1 R e g r e s s i o n c o e f f i c i e n t s f o r r u n u p p a r a m e t e r s Ra/Hg, Rg/H-s, and R/Hs 11 2 V a l u e s o f t h e r u n u p p a r a m e t e r s f o r e x a m p l e p r o b l e m 1. . 16 F I G U R E S 1 I r r e g u l a r wave r u n u p p a r a m e t e r s v e r s u s wave s t e e p n e s s f o r a p l a n e , smooth s l o p e o f 1 on 1, dg/Hg > 3 . 8 2 I r r e g u l a r wave r u n u p p a r a m e t e r s v e r s u s wave s t e e p n e s s f o r a p l a n e , smooth s l o p e o f 1 on 1.5, dg/Hg > 3 9 3 I r r e g u l a r wave r u n u p p a r a m e t e r s v e r s u s wave s t e e p n e s s f o r a p l a n e , smooth s l o p e o f 1 on 2, dg/Hg > 3 9 4 I r r e g u l a r wave r u n u p p a r a m e t e r s v e r s u s wave s t e e p n e s s f o r a p l a n e , smooth s l o p e o f 1 on 2.5, dg/Hg > 3 10 5 I r r e g u l a r wave r u n u p p a r a m e t e r s v e r s u s wave s t e e p n e s s f o r a p l a n e , smooth s l o p e o f 1 on 3, dg/Hg > 3 10 6 I r r e g u l a r wave r u n u p p a r a m e t e r s v e r s u s wave s t e e p n e s s f o r a p l a n e , smooth s l o p e 1 on 4, dg/Hg > 3 11 7 Rs/Hg v e r s u s t h e s u r f p a r a m e t e r f o r 3 < ds/Hg 13
CONVERSION FACTORS, U.S. CUSTOMARY TO METRIC ( S I ) UNITS OF MEASUREMENT .S. customary u n i t s of measurement used i n t h i s r e p o r t c a n be c o n v e r t e d t o
" . e t r i c ( S I ) u n i t s a s f o l l o w s : M u l t i p l y by To o b t a i n inches square i n c h e s c u b i c i n c h e s 25.4 2.54 6.452 16.39 m i l l i m e t e r s c e n t i m e t e r s s q u a r e c e n t i m e t e r s c u b i c c e n t i m e t e r s f e e t square f e e t c u b i c f e e t 30.48 0.3048 0.0929 0.0283 c e n t i m e t e r s m e t e r s s q u a r e m e t e r s c u b i c m e t e r s yards square y a r d s c u b i c y a r d s 0.9144 0.836 0.7646 m e t e r s s q u a r e m e t e r s c u b i c m e t e r s railes square m i l e s 1.6093 259.0 k i l o m e t e r s h e c t a r e s knots 1.852 k i l o m e t e r s p e r hour a c r e s 0.4047 h e c t a r e s foot-pounds 1.3558 newton m e t e r s m i l l i b a r s 1.0197 X 10"^ k i l o g r a m s p e r s q u a r e c e n t i m e t e r ounces 28.35 grams pounds 453.6 0.4536 grams k i l o g r a m s t o n , l o n g 1.0160 m e t r i c t o n s t o n , s h o r t 0.9072 m e t r i c t o n s degrees ( a n g l e ) 0.01745 r a d i a n s F a h r e n h e i t d e g r e e s 5/9 C e l s i u s d e g r e e s o r K e l v i n s ^ ^To o b t a i n C e l s i u s ( C ) t e m p e r a t u r e r e a d i n g s use f o r m u l a : C = ( 5 / 9 ) ( F - 3 2 ) , To o b t a i n K e l v i n ( K ) r e a d i n g s , u s e f o r m u l a ; from F a h r e n h e i t ( F ) r e a d i n g s , K = ( 5 / 9 ) ( F - 3 2 ) + 2 7 3 . 1 5 . 5
SYMBOLS AND D E F I N I T I O N S w a t e r d e p t h a t t h e t o e o f t h e s l o p e o r s t r u c t u r e on w h i c h runup o c c u r s g a c c e l e r a t i o n o f g r a v i t y , 32.2 f e e t p e r s e c o n d s q u a r e d Kg s i g n i f i c a n t wave h e i g h t a t t h e t o e o f t h e s t r u c t u r e k runup c o r r e c t i o n f a c t o r f o r s c a l e e f f e c t s Lo d e e p w a t e r w a v e l e n g t h , Lq = gTp^/2Tr R mean runup Rg s i g n i f i c a n t r u n u p , i . e . , a v e r a g e runup o f t h e h i g h e s t o n e - t h i r d o f wave r u n u p s R2 2 - p e r c e n t r u n u p , i . e . , e l e v a t i o n a b o v e t h e S t i l l w a t e r l e v e l e x c e e d e d by 2 p e r c e n t o f t h e r u n u p s Rdgö 9 8 - p e r c e n t rundown, i . e . , d e p t h b e l o w t h e S t i l l w a t e r l e v e l t h a t i s j u s t g r e a t e r t h a n 98 p e r c e n t o f t h e rundowns r r o u g h - s l o p e r u n u p c o r r e c t i o n f a c t o r , r a t i o o f r o u g h - s l o p e r u n u p t o s m o o t h - s l o p e r u n u p , a l l o t h e r c o n d i t i o n s t h e same Tp p e r i o d o f p e a k e n e r g y d e n s i t y o f t h e wave s p e c t r u m Tg s i g n i f i c a n t wave p e r i o d , i . e . , a v e r a g e p e r i o d o f t h e h i g h e s t o n e - t h i r d o f w a v e s 6 a n g l e formed b e t w e e n t h e s l o p e o f t h e s t r u c t u r e and t h e h o r i z o n t a l ? s u r f p a r a m e t e r , ? = [(R^/Lo)^^^ c o t 6 ] - ^
IRREGULAR WAVE RUNUP ON SMOOTH SLOPES
hy
John P. Ahrens
1. INTRODUCTION
T h i s r e p o r t p r o v i d e s g u i d a n c e on t h e m a g n i t u d e and d i s t r i b u t i o n o f wave .-inup and rundown e l e v a t i o n s c a u s e d by i r r e g u l a r wave c o n d i t i o n s s i m i l a r t o •hose o c c u r r i n g i n n a t u r e . T h e r e s u l t s p r e s e n t e d a r e f o r p l a n e , smooth s t r u c •ures w i t h r e l a t i v e l y deep w a t e r a t t h e t o e o f t h e s t r u c t u r e . F o r t h e s e c o n
-• i c i o n s t h i s r e p o r t s u p e r s e d e s e a r l i e r g u i d a n c e i n S e c t i o n 7.212 o f t h e S h o r e :'rotection Manual (SPM) (U.S. Army, C o r p s o f E n g i n e e r s , C o a s t a l E n g i n e e r i n g Research C e n t e r , 1 9 7 7 ) and A h r e n s ( 1 9 7 7 ) w h i c h i n d i c a t e t h a t i r r e g u l a r wave runup has a R a y l e i g h d i s t r i b u t i o n . W i t h i n t h e r a n g e o f t e s t c o n d i t i o n s t h x s report a l s o s u p e r s e d e s S t o a ( 1 9 7 8 a ) and t h e p a r t s o f S t o a ( 1 9 7 9 ) w h i c h e s t i -ziace wave runup on r o u g h and p o r o u s s l o p e s by a d j u s t i n g t h e runup on a smooth slope by a c o r r e c t i o n f a c t o r . T h e r a n g e o f t e s t c o n d i t i o n s c o v e r e d i n t h x s report i s d i s c u s s e d i n t h e n e x t s e c t i o n .
I I . IRREGULAR WAVE RUNUP ON PLANE, SMOOTH SLOPES
T h r e e s o u r c e s o f d a t a w e r e u s e d i n e s t a b l i s h i n g t h e methods p r e s e n t e d i n t h i s r e p o r t : v a n O o r s c h o t and d'Angremond ( 1 9 6 8 ) , Kamphuis a n d Mohamed ( 1 9 7 8 ) , and A h r e n s ( 1 9 7 9 ) w h i c h d i s c u s s e d d a t a r e c e n t l y c o l l e c t e d a t t h e 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 ( C E R C ) . T h e c o -n d i t i o -n s c o -n s i d e r e d a r e a s t r u c t u r e with a p l a n e , smooth s l o p e f r o n t e d by a h o r i z o n t a l b o t t o m o f f s h o r e . T h e w a t e r depth a t t h e t o e o f t h e s t r u c t u r e i s r e l a t i v e l y d e e p , i . e . , 3 < dg/Hg < 1 2 , where dg i s t h e w a t e r d e p t h and Hg t h e s i g n i f i c a n t wave h e i g h t a t t h e t o e of the s t r u c t u r e . When t h e r e i s r e l a t i v e l y deep w a t e r a t t h e t o e o f t h e s t r u c t u r e t h e o f f s h o r e s l o p e o f t h e b o t t o m h a s l i t t l e i n f l u e n c e on t h e wave c o n d i -t i o n s and -t h e r e f o r e l i -t -t l e i n f l u e n c e on -t h e wave r u n u p s . T h i s l a c k o f i n f l u e n c e i n d i c a t e s t h a t t h e r u n u p r e s u l t s p r e s e n t e d c a n be a p p l i e d t o s i t u a t i o n s w h e r e there i s a n o f f s h o r e s l o p e . S i n c e t h e w a t e r d e p t h a l s o h a s l i t t l e i n f l u e n c e on wave runup f o r c o n d i t i o n s when dg/H^ > 8 ( S t o a , 1 9 7 8 a ) , w h e r e H^ i s t h e d e e p -water, u n r e f r a c t e d wave h e i g h t , S t o a ' s f i n d i n g s u g g e s t s t h a t t h e r e s u l t s o f t h i s study s h o u l d b e good f o r dg/Hg > 1 2 .
T h r e e r u n u p p a r a m e t e r s w e r e c h o s e n t o c h a r a c t e r i z e t h e r u n u p d i s t r i b u t i o n caused by i r r e g u l a r wave c o n d i t i o n s , i . e . , t h e mean r u n u p , R, t h e s i g n i f i c a n t runup, Rg, and t h e 2 - p e r c e n t r u n u p , Rz- T h e s i g n i f i c a n t runup i s t h e
aver-age runup o f t h e h i g h e s t o n e - t h i r d o f wave r u n u p s and t h e 2 - p e r c e n t r u n u p i s
the e l e v a t i o n exceeded by 2 p e r c e n t o f t h e wave r u n u p s .
F i g u r e 1 shows t r e n d - l i n e c u r v e s f o r Rz/Hg, Rg/Hg, a n d R/Hg f o r a p l a n e , smooth s l o p e o f 1 o n 1. T h e s e p a r a m e t e r s a r e p l o t t e d a s a f u n c t i o n o f t h e i r r e g u l a r wave s t e e p n e s s p a r a m e t e r , Hg/gTp^, w h e r e Tp i s t h e p e r i o d o f p e a k energy d e n s i t y o f t h e wave s p e c t r u m and g t h e a c c e l e r a t i o n o f g r a v i t y . T h e a p p r o x i m a t e r e l a t i o n s h i p b e t w e e n Tp and t h e a v e r a g e p e r i o d o f t h e s i g n i f i c a n t waves, T s , i s g i v e n by Goda ( 1 9 7 4 ) a s
Tp 1.05 Tg ( 1 )
5.0 0.0 L -3 4x 10 6« 10 8>I0 H s / q T p F i g u r e 1 . I r r e g u l a r w a v e r u n u p p a r a m e t e r s v e r s u s w a v e s t e e p n e s s f o r a p l a n e , s m o o t h s l o p e o f 1 o n 1 , dg/Hg > 3. F i g u r e s 2, 3, 4, 5, a n d 6, w h i c h a r e s i m i l a r t o F i g u r e 1 , show t r e n d l i n e s f o r s l o p e s o f 1 o n 1.5, 1 o n 2, 1 o n 2,5, 1 o n 3, a n d 1 o n 4, r e s p e c t i v e l y . The t r e n d l i n e s i n F i g u r e s 1 t o 5 a r e a l l o f t h e g e n e r a l f o r m
— = Cl + c
^s H. H. ^ " gT, 2 + Cc H. ( 2 ) .X r e p r e s e n t s R2, Rg, o r R, a n d C i , C2, a n d C3 a r e d i m e n s i o n l e s s r e -where R-g r e s s i o n c o e f f i c i e n t s . I n some c a s e s t r e n d l i n e i s s t r a i g h t . C2 o r C3 i s z e r o ; i f C3 i s z e r o t h e S i n c e a c a l c u l a t o r o r a c o m p u t e r may b e m o r e c o n v e n i e n t f o r c a l c u l a t i n g the r u n u p p a r a m e t e r s t h a n u s i n g t h e f i g u r e s . T a b l e 1 p r o v i d e s a t a b u l a t i o n o f the r e g r e s s i o n c o e f f i c i e n t s , a l o n g w i t h some s t a t i s t i c a l p a r a m e t e r s w h i c h c a n be u s e d t o e v a l u a t e how w e l l t h e c u r v e s f i t t h e d a t a . T h e s t a n d a r d d e v i a t i o n i s t h e s t a n d a r d d e v i a t i o n o f t h e d a t a a b o u t t h e t r e n d - l i n e c u r v e s a n d i s s h o w n i n F i g u r e s 1 t o 6 t o g i v e a n i n d i c a t i o n o f t h e m a g n i t u d e o f t h e s c a t t e r a b o u t the c u r v e s . T h e c o e f f i c i e n t o f v a r i a t i o n i s t h e s t a n d a r d d e v i a t i o n d i v i d e d b y the mean v a l u e o f Rx/Hg. U s i n g t h e c o e f f i c i e n t o f v a r i a t i o n t o d e t e r m i n e t h e p e r c e n t s c a t t e r i n d i c a t e s t h a t Rg/Hg c a n u s u a l l y b e e s t i m a t e d w i t h i n t h e r a n g e of ±5 t o 10 p e r c e n t a b o u t t h e t r e n d l i n e c u r v e s ; Ra/Hg a n d R/Hg c a n b e e s t i -mated w i t h i n t h e r a n g e o f ± 1 0 t o 15 p e r c e n t a b o u t t h e c u r v e s .0.01
2(10 4x10 6x10 - 3 8 XlO '
H s / q T p '
F i g u r e 2. I r r e g u l a r wave runup p a r a m e t e r s v e r s u s wave s t e e p n e s s f o r a p l a n e , smooth s l o p e o f 1 on 1.5, dg/Hg > 3, 0.5 2x10"' 4x10"' 6x10"' 8x10"' H s / g T p ^ F i g u r e 3. I r r e g u l a r wave r u n u p p a r a m e t e r s v e r s u s wave s t e e p n e s s f o r a p l a n e , smooth s l o p e o f 1 on 2, dg/Hg > 3. 9
5.0 4.5 4.0 3.5 3.0 • 2 . 5 2.0 1.5 1.0 0.5 0.0
1 1 1
f D e n o t e s + 1.0 s l d dev.ob out tre nd line
T 1 ~ — j p 2«I0 4x10 - 3 H s / g T p 6x10' 8x10'
F i g u r e 4. I r r e g u l a r wave runup p a r a m e t e r s v e r s u s wave s t e e p n e s s f o r a p l a n e , smooth s l o p e o f 1 on 2.5, dg/Hg > 3. O.ol I I I I I I I I 1 2x10"' 4x10"' 6x10"' 8x10"' H j / q T p F i g u r e 5. I r r e g u l a r wave r u n u p p a r a m e t e r s v e r s u s wave s t e e p n e s s f o r a p l a n e , smooth s l o p e o f 1 on 3, dg/Hg > 3.
3,0
2.0
1.0
D e n o t e s ± 1.0 s t d . dev. about trend line
%s=l'.6l f '''Ws .-1.2 5 ^ '^Hs = 0 . 8 4 f 0.01 0.02 0.03 0.04 0.05 H . / q T p ^ 0.06 0.07 0.08 0.09 o . a o ^
F i g u r e 6. I r r e g u l a r wave runup p a r a m e t e r s v e r s u s V7ave s t e e p n e s s f o r a p l a n e , smooth s l o p e 1 on 4, dg/Hg > 3.
Table 1 . Regression c o e f f i c i e n t s f o r runup parameters Rj/Hs, Rs/tls, and R/llg (see e.q. 2 ) .
R e g r e s s i o n c o e f f i c i e n t s C o t 6 C l C2 C3 S t d , d e v . C o e f f . o f v a r i a t i o n Rz/Hs 1.0 2.32 7.15 X 10' 0 0.343 0.134 1.5 2.52 1.95 X 102 0 0.487 0.156 2.0 3.21 7.19 X l o i 0 0.421 0.123 2.5 3.39 1.29 X 102 -1.61 X 10"* 0.420 0.118 3.0 3.70 0 -1.70 X 10" 0.415 0.120 4.0 3.60 -2.22 X 102 0 0.330 0.117 R3/R s 1.0 1.34 6.61 X l O l 0 0.133 0.085 1.5 1.38 3,18 X 102 -1.97 X 10" 0.195 0.094 2.0 1.64 3.57 X 102 -3.09 X 10" 0.136 0.059 2.5 1.94 2.79 X 102 -3.21 X 10" 0.184 0.078 3.0 2.11 1.87 X 102 -2.67 X 10" 0.190 0.081 4.0 2.52 -7.94 X 10' 0 0.122 0.053 R/H: s l.Q 0.71 1.10 X 102 -8.07 X 10^ 0.150 0.157 1.5 0.75 1.97 X 102 -1.14 X 10" 0.143 0.119 2.0 0.93 2,42 X 102 -1.93 X 10" 0.142 0.101 2.5 1.00 2.78 X 102 -3.13 X 10" 0.141 0.099 3.0 1.19 2.09 X 102 -2.96 X 10" 0.181 0.123 4.0 1.47 7.25 X 10' -1.70 X 10" 0.127 0.085 v =
F i g u r e 6, f o r a s l o p e o f 1 on 4, i s somewhat d i f f e r e n t t h a n F i g u r e s 1 t o ^ f o r s t e e p e r s l o p e s . P l u n g i n g waves become t h e d o m i n a n t b r e a k e r t y p e o n t h e
: on 4 s l o p e , i n d i c a t i n g t h a t wave runup c a n be p r e d i c t e d u s i n g a t y p e of f o r
-- u l a s u g g e s t e d by Hunt ( 1 9 5 9 ) and u s e d b y v a n O o r s c h o t and d'Angremond ( 1 9 6 8 ) . f i g u r e 6 shows t r e n d - l i n e c u r v e s , u s i n g e q u a t i o n ( 2 ) , f o r t h e l e s s s t e e p wave c o n d i t i o n s , i . e . , Hs 0.005 < — V ^ 0-003 8Tp2 jnd a H u n t - t y p e f o r m u l a i s u s e d f o r t h e s t e e p e r wave c o n d i t i o n s , i . e . , Hs/gTp^ > 0.003 w h e r e p l u n g i n g w a v e s d o m i n a t e . The H u n t t y p e f o r m u l a s f o r F i g -ure 6 a r e g i v e n by t h e e q u a t i o n s ^ = 1 . 6 1 , ( 3 ) Hs = 1.25 C ( 4 ) — = 0.84 ? ( 5 ) Hs where t h e s u r f p a r a m e t e r , E,, i s g i v e n by C =
nr;:
o r t a n e ( H 3 / L o ) l / 2 c o t e (Hs/L„)l/^ Lq i s t h e d e e p w a t e r w a v e l e n g t h g i v e n b y and c o t 6 i s t h e c o t a n g e n t o f t h e a n g l e 6 b e t w e e n t h e s t r u c t u r e s l o p e and t h e h o r i z o n t a l . F i g u r e 7 p r o v i d e s a d i f f e r e n t p e r s p e c t i v e and a d d i t i o n a l i n s i g h t on t h e trends t o be e x p e c t e d f o r i r r e g u l a r wave r u n u p . T h e Rg/Hg c u r v e s f r o m F i g u r e s 1 to 6 h a v e b e e n t r a n s f e r r e d t o F i g u r e 7 and p l o t t e d v e r s u s t h e s u r f p a r a m e t e r , •) to show t h e i n f l u e n c e o f b r e a k e r c h a r a c t e r i s t i c s o n r u n u p . When ? < 2.0, nost of t h e l a r g e r w a v e s i n t h e i n c i d e n t wave t r a i n p l u n g e d i r e c t l y on t h e s t r u c t u r e and Rg/Hg d e c r e a s e s w i t h i n c r e a s i n g Hg/gTp2 a n d i n c r e a s i n g c o t Q. This p l u n g i n g wave r e g i o n i s w h e r e a H u n t - t y p e f o r m u l a ( H u n t , 1 9 5 9 ) s u c h a sequations ( 3 ) , ( 4 ) , and ( 5 ) i s v a l i d . When 5 > 3.5, no w a v e s p l u n g e on t h e s t r u c t u r e i n d i c a t i n g a s t a n d i n g wave c o n d i t i o n o r s u r g i n g wave r e g i o n . The i n f l u e n c e o f Hg/gTp2 and c o t 6 o n Rs/Hg i s r e v e r s e d f o r s u r g i n g w a v e s a s
1,0 2.0 3,0 4.0 S u r f P n r a m e t e r , ^ 5.0 6.0 7.0 F i g u r e 7. Rg/Hg v e r s u s t h e s u r f p a r a m e t e r f o r dg/Hg > 3, compared t o p l u n g i n g w a v e s ; i . e . , Rg/Hg i n c r e a s e s a s Hg/gTp^ i n c r e a s e s and c o t 9 i n c r e a s e s . The r e v e r s a l o f i n f l u e n c e c r e a t e s a t r a n s i t i o n r e g i o n , 2.0 < ; < 3.5, w h e r e t h e r e i s l i t t l e n e t i n f l u e n c e o f Hg/gTp^ and c o t 6 on Rg/Hg. I t i s i n t h i s t r a n s i t i o n r e g i o n t h a t t h e l a r g e s t v a l u e s of Rg/Hg o c c u r , p r o b -ably b e c a u s e t h e most n o n l i n e a r s u r g i n g w a v e s o c c u r i n t h i s r e g i o n . F i g u r e 7 i d e n t i f i e s t h e s e r e g i o n s and shows t h e r u n u p t r e n d s . E q u a t i o n s ( 3 ) , ( 4 ) , and (5) can be u s e d on s l o p e s f l a t t e r t h a n 1 on 4 a s l o n g a s p l u n g i n g w a v e s p r e -dominate, i . e . , 5 < 2.0.
A l l t h e r e s u l t s i n t h i s r e p o r t w e r e o b t a i n e d i n r e l a t i v e l y s m a l l - s c a l e l a b o r a t o r y s t u d i e s and must be c o r r e c t e d f o r s c a l e e f f e c t s ( S t o a , 1 9 7 8 a ) . The c o r r e c t i o n f o r s c a l e e f f e c t s of wave runup on smooth s l o p e s c a n be f o u n d i n S t o a ( 1 9 7 8 b ) (shown i n App. A ) . E x a m p l e p r o b l e m 1 i n S e c t i o n V i l l u s t r a t e s the method o f a p p l y i n g t h i s c o r r e c t i o n . The r e s u l t s i n F i g u r e s 1 t o 7 a r e a l l p r e s e n t e d i n t e r m s o f t h e s i g n i f i c a n t wave h e i g h t a t t h e t o e of t h e s t r u c t u r e , Hg, r a t h e r t h a n t h e d e e p w a t e r , u n -r e f -r a c t e d wave h e i g h t , H^. I f i t i s d e s i -r e d to c o n v e -r t t h e -r e s u l t s o f t h i s study t o d e e p w a t e r c o n d i t i o n s , Hg s h o u l d be m u l t i p l i e d by t h e s h o a l i n g c o e f -f i c i e n t , g i v e n i n A p p e n d i x C o-f t h e SPM (U.S. Army, C o r p s o -f E n g i n e e r s , 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 , 1 9 7 7 ) , c a l c u l a t e d u s i n g dg and Tp t o o b t a i n an e s t i m a t e o f t h e d e e p w a t e r , u n r e f r a c t e d s i g n i f i c a n t wave h e i g h t .
I I I . IRREGULAR WAVE RUNDOWN
I r r e g u l a r wave rundown i s c h a r a c t e r i z e d by t h e 98 p e r c e n t i l e rundown, Rdge, i . e . , t h e rundown d e p t h b e l o w t h e S t i l l w a t e r l e v e l w h i c h i s g r e a t e r t h a n 98 p e r c e n t o f t h e wave r u n d o w n s . The i r r e g u l a r wave rundown p a r a m e t e r , Rdge i s analogous t o t h e r u n u p p a r a m e t e r , R2, s i n c e o n l y 2 p e r c e n t o f t h e rundowns are l o w e r t h a n Rdgg. F i g u r e 8 shows t h e t r e n d o f t h e r e l a t i v e rundown, Rdge/Hg a s a f u n c t i o n o f t h e s u r f p a r a m e t e r , and t h e a p p r o x i m a t e u p p e r
F i g u r e 8. Rdgs/Hg v e r s u s t h e s u r f p a r a m e t e r .
and lower l i m i t s o f d a t a s c a t t e r a b o u t t h e t r e n d - l i n e c u r v e . The t r e n d - l i n e curve f o r r e l a t i v e rundown i s g i v e n by t h e e q u a t i o n Rdge -2.46/5 = - 2 . 3 2 e ( 6 ) The a b s o l u t e v a l u e o f r e l a t i v e rundown i s s m a l l f o r s m a l l v a l u e s o f t h e s u r f parameter s i n c e t h e p l u n g i n g w a v e s w h i c h d o m i n a t e t h e s e c o n d i t i o n s c a u s e c o n -s i d e r a b l e wave -s e t u p . A-s t h e -s u r f p a r a m e t e r i n c r e a -s e -s a -s t a n d i n g wave d e v e l o p -s a g a i n s t t h e s t r u c t u r e and t h e r e l a t i v e rundown a p p r o a c h e s - 1 . 7 5 , a l t h o u g h v a l u e s o c c a s i o n a l l y a s low a s -2.25 w e r e o b s e r v e d . E q u a t i o n ( 6 ) p r o v i d e s a s i m p l e way to e s t i m a t e t h e a p p r o x i m a t e l o w e r l i m i t of rundown. T h e r e i s no s c a l e - e f f e c t c o r r e c t i o n f a c t o r s p e c i f i c a l l y d e v e l o p e d f o r wave rundown, so i t i s recommended t h a t t h e c o r r e c t i o n f a c t o r f o r wave r u n u p be applied to rundown a s i l l u s t r a t e d i n e x a m p l e p r o b l e m 2 i n S e c t i o n V.
I V . A P P L I C A T I O N OF RESULTS TO ROUGH AND POROUS SLOPES
The r e s u l t s g i v e n i n t h i s r e p o r t c a n be a p p l i e d t o p l a n e , r o u g h - and p o r o u s - s l o p e s t r u c t u r e s , i f t h e r e i s . r e l a t i v e l y deep w a t e r a t t h e t o e of t h e s t r u c t u r e ( a s d i s c u s s e d p r e v i o u s l y i n S e c . I I ) . To a p p l y t h e s e r e s u l t s i t i s n e c e s s a r y t o h a v e a r e l i a b l e e s t i m a t e o f t h e r o u g h s l o p e r u n u p c o r r e c t i o n f a c -l^or, r , w h i c h i s t h e r a t i o o f wave r u n u p on a r o u g h o r p o r o u s s l o p e t o t h e
runup on a smooth s l o p e , a l l o t h e r c o n d i t i o n s b e i n g t h e same ( S t o a , 1 9 7 8 a ) . :;ormally, r i s d e t e r m i n e d i n l a b o r a t o r y e x p e r i m e n t s u s i n g m o n o c h r o m a t i c wave c o n d i t i o n s b u t i t a p p e a r s t h a t r f a c t o r s d e t e r m i n e d i n t h i s manner c a n a l s o be a p p l i e d t o i r r e g u l a r wave c o n d i t i o n s ( B a t t j e s , 1 9 7 4 ) . V a l u e s of r f o r v a r i o u s t y p e s of r o u g h and p o r o u s s l o p e s a r e g i v e n by S t o a ( 1 9 7 9 ) (shown i n App. B) .
O f t e n wave r u n u p on r o u g h s l o p e s must be c o r r e c t e d fox s c a l e e f f e c t s and
the c o r r e c t i o n f a c t o r s a r e g i v e n i n S t o a ( 1 9 7 9 ) (shown i n App. C ) . E x a m p l e
problem 3 i l l u s t r a t e s how t h e r e s u l t s p r e s e n t e d i n t h i s r e p o r t c a n be a p p l i e d
to a r o u g h and p o r o u s s l o p e and t h e method o f a p p l y i n g t h e r o u g h s l o p e s c a l e
-e f f -e c t c o r r -e c t i o n f a c t o r .
V. EXAMPLE PROBLEMS
A * * * * * * * * * * * * * * EXAMPLE PROBLEM 1 * * * * * * * * * * * * * * *
T h i s e x a m p l e i l l u s t r a t e s t h e u s e o f t h e runup e q u a t i o n . F i g u r e s 1 t o 6, and the recommended method o f i n t e r p o l a t i o n b e t w e e n s l o p e s .
GIVEN: A p l a n e , smooth s l o p e o f 1 on 2.75 i s s u b j e c t e d to i r r e g u l a r wave a c t i o n . The s i g n i f i c a n t wave h e i g h t , s i g n i f i c a n t wave p e r i o d , and w a t e r d e p t h a t t h e t o e o f t h e s t r u c t u r e a r e 6.0 f e e t (1.83 m e t e r s ) , 7.0 s e c o n d s , and 24.0 f e e t ( 7 . 3 m e t e r s ) , r e s p e c t i v e l y .
FIND: R, Rg, and R2 f o r t h e g i v e n c o n d i t i o n s . Would t h e r e be s u b s t a n t i a l wave o v e r t o p p i n g i f t h e f r e e b o a r d o f t h e s t r u c t u r e w e r e 20.0 f e e t ( 6 . 1 0 m e t e r s ) ? SOLUTION: S i n c e t h e r e i s no f i g u r e o r s e t o f c o e f f i c i e n t s f o r t h e r u n u p
e q u a t i o n ( e q . 2) f o r a s l o p e o f 1 on 2.75 i t i s n e c e s s a r y t o compute R, Rg, and R2 f o r s l o p e s o f 1 on 2.50 and 1 on 3.00 and i n t e r p o l a t e b e t w e e n them. To s t a r t , c a l c u l a t e t h e p e r i o d of p e a k (maximum) e n e r g y d e n s i t y , T^,, u s i n g e q u a t i o n ( 1 ) . F ' Tp = 1.05 Tg = 1.05 ( 7 . 0 ) = 7.35 s e c o n d s Then compute t h e s t e e p n e s s p a r a m e t e r , Hg/gTp^ Hs 6.0 gTp2 3 2 . 2 ( 7 . 3 5 ) 2 = 0.00345 U s i n g t h e above v a l u e o f s t e e p n e s s i n e q u a t i o n ( 2 ) w i t h t h e c o e f f i c i e n t g i v e n i n T a b l e 1 a l l o w s t h e c o m p u t a t i o n of Rx/Hg. F o r e x a m p l e , t o c a l c u -l a t e Ra/Hg f o r a 1 on 2.5 s -l o p e R2 ^ = 3.39 + [ 1 2 9 . 0 ( 0 . 0 0 3 4 5 ) ] + [ - 1 6 , 1 0 0 ( 0 . 0 0 3 4 5 ) 2 ] = 3.64 s The a b o v e v a l u e o f R2/HS c a n be c o n f i r m e d , u s i n g F i g u r e 4. T h e r e f o r e , R2 = 3 . 6 4 ( H g ) = 3 . 6 4 ( 6 . 0 ) = 21.8 f e e t (6.64 m e t e r s )
The o t h e r runup p a r a m e t e r s Rg and R c a n be c a l c u l a t e d i n a s i m i l a r manner, t h e n u s e d f o r i n t e r p o l a t i o n t o g i v e t h e v a l u e s of t h e r u n u p p a r a m e t e r s f o r t h e 1 on 2.75 s l o p e a s shown i n T a b l e 2.
T a b l e 2. V a l u e s of t h e runup p a r a m e t e r s f o r e x a m p l e p r o b l e m 1. c o t e R2/Hs ^2 Rs/Hg Rg R/Hg R ( f t ) ( f t ) ( f t ) 2.50 3.64 21.8 2.52 15.1 1.58 9.5 3.00 3.49 21.0 2.43 14.6 1.56 9.4 2.75 21.4^ 14.9^ 9.4^ ^ I n t e r p o l a t e d v a l u e . The i n t e r p o l a t e d v a l u e s i n T a b l e 2 s h o u l d be c o r r e c t e d f o r s c a l e e f f e c t s t o y i e l d t h e r e q u i r e d a n s w e r . The s c a l e c o r r e c t i o n f a c t o r f o r a s l o p e o f 1 on 2.75 i s 1.125 ( s e e App. A ) ; t h e r e f o r e , R2 = 21.4 ( 1 . 1 2 5 ) = 24.1 f e e t ( 7 . 3 5 m e t e r s ) Rg = 14.9 ( 1 . 1 2 5 ) = 16.8 f e e t ( 5 . 1 2 m e t e r s ) R = 9.4 ( 1 . 1 2 5 ) = 10.6 f e e t ( 3 . 2 8 m e t e r s ) A f r e e b a o r d of 20.0 f e e t f a l l s b e t w e e n R2 and Rg, so t h e s t r u c t u r e c r e s t would n o t be o v e r t o p p e d f r e q u e n t l y , p r o b a b l y by l e s s t h a n 10 p e r c e n t o f the w a v e s . I t i s , t h e r e f o r e , e x p e c t e d t h a t t h e v o l u m e o f o v e r t o p p i n g would n o t be g r e a t . I t i s d i f f i c u l t t o d e t e r m i n e how h i g h a smooth s t r u c t u r e w o u l d h a v e t o be to p r e v e n t a l l wave o v e r t o p p i n g b u t a r e a s o n a b l e e s t i m a t e w o u l d be Rmax - R2 + Hg
where Rmax i s t h e e l e v a t i o n o f t h e maximum r u n u p .
* * * * * * * * * * * * * * * EXAMPLE PROBLEM 2 * * * * * * * * * * * * * * *
T h i s e x a m p l e i l l u s t r a t e s how t o c a l c u l a t e t h e a p p r o x i m a t e l o w e r l i m i t o f rundown.
GIVEN: A p l a n e , smooth 1 on 2.50 s l o p e i s s u b j e c t e d t o i r r e g u l a r wave a c t i o n . The s i g n i f i c a n t wave h e i g h t , s i g n i f i c a n t wave p e r i o d , and w a t e r d e p t h a t t h e toe of t h e s t r u c t u r e a r e 7.0 f e e t ( 2 . 1 3 m e t e r s ) , 8.0 s e c o n d s , and 30.0 f e e t (9.14 m e t e r s ) , r e s p e c t i v e l y . FIND: Rdge f o r t h e a b o v e c o n d i t i o n s ; t h i s i s t h e a p p r o x i m a t e l o w e r l i m i t o f wave rundown. SOLUTION: The p e r i o d o f p e a k e n e r g y d e n s i t y i s Tp = 1 . 0 5 ( T s ) = 1.05 X 8.0 = 8.40 s e c o n d s and t h e s u r f p a r a m e t e r i s ^ ^ 1 ^ 1 ( H s / L o ) ^ / 2 Q { 7 . 0 / [ 3 2 . 2 x (8 .4) 2 ] / 2 t t } ( 2 . 5 )
•sir.g t h i s v a l u e o f C i n e q u a t i o n ( 6 ) p i v e s t h e r e l a t i v e rundown, i . e . , -2.46/C = - 2 . 3 2 e = -0.99 w h i c h c a n be c o n f i r m e d i n F i g u r e 8. Then Rdge = ( 7 . 0 ) ( - 0 . 9 9 ) = -6.9 f e e t (-2.10 m e t e r s ) and u s i n g A p p e n d i x A t o c o r r e c t t h i s rundown f o r s c a l e e f f e c t s g i v e s Rdge ( c o r r e c t e d ) = - 6 . 9 ( 1 . 1 2 8 ) = -7.8 f e e t (-2.38 m e t e r s )
The same s c a l e c o r r e c t i o n f a c t o r u s e d f o r runup i s u s e d f o r rundown.
* * * * * * * * * * * * * * * EXAMPLE PROBLEM 3 * * * * * * * * * * * * * * * T h i s e x a m p l e i l l u s t r a t e s how t h e r e s u l t s o f t e s t s w i t h i r r e g u l a r w a v e s on smooth s l o p e s c a n be a p p l i e d t o s i t u a t i o n s w h e r e t h e s t r u c t u r e i s r o u g h and porous. GIVEN: A rubble-mound b r e a k w a t e r i s t o be b u i l t w i t h a s l o p e on t h e s e a w a r d f a c e of 1 on 2 w h i c h w i l l be o v e r t o p p e d by wave a c t i o n o n l y o c c a s i o n a l l y under t h e d e s i g n c o n d i t i o n s . The d e s i g n c o n d i t i o n s i n c l u d e a s i g n i f i c a n t wave h e i g h t , s i g n i f i c a n t wave p e r i o d , and w a t e r d e p t h a t t h e t o e o f t h e s t r u c t u r e o f 15.0 f e e t ( 4 . 5 7 m e t e r s ) , 12.0 s e c o n d s , and 45.0 f e e t ( 1 3 . 7 2 m e t e r s ) , r e s p e c t i v e l y . The c o r e o f t h e b r e a k w a t e r w i l l be s l i g h t l y a b o v e t h e d e s i g n w a t e r l e v e l , i . e . , a h i g h c o r e b r e a k w a t e r . FIND: The h e i g h t a t w h i c h t h e b r e a k w a t e r w i l l o n l y o c c a s i o n a l l y be o v e r t o p p e d d u r i n g t h e d e s i g n c o n d i t i o n s . SOLUTION; The p e r i o d o f p e a k e n e r g y d e n s i t y i s Using e q u a t i o n ( 2 ) w i t h t h e c o e f f i c i e n t s i n T a b l e 1 f o r a p l a n e , smooth s l o p e of 1 on 2 and RaH/g g i v e s Tp = 1 . 0 5 ( T s ) = 1.05 ( 1 2 . 0 ) = 12.6 s e c o n d s and t h e s t e e p n e s s p a r a m e t e r i s 3 2 . 2 ( 1 2 . 6 ) 2 15.0 = 0.00293 — = 3.2083 + 71.879 ( 0 . 0 0 2 9 3 ) = 3.42 s ( t h i s v a l u e c a n be c h e c k e d i n F i g . 3) and R2 = 3 . 4 2 ( 1 5 . 0 ) = 51.3 f e e t ( 1 5 . 6 4 m e t e r s )
17
:he runup r e d u c t i o n f a c t o r , r , f o r rubble-mound b r e a k w a t e r s w i t h h i g h cores i s Q.52 ( s e e App. B) and t h e s c a l e - e f f e c t c o r r e c t i o n f a c t o r i s 1.06 (see App. C) so R2 f o r t h e b r e a k w a t e r i s
R2 ( b r e a k w a t e r ) = 5 1 . 3 ( 0 . 5 2 ) 1.06 = 28.3 f e e t ( 8 . 6 3 m e t e r s )
Rg and R a r e found i n a s i m i l a r manner t o be
Rg ( b r e a k w a t e r ) = 2 0 . 0 f e e t ( 6 . 1 0 m e t e r s ) R ( b r e a k w a t e r ) = 1 2 . 2 f e e t ( 3 . 7 2 m e t e r s ) These c a l c u l a t i o n s i n d i c a t e t h a t i f t h e f r e e b o a r d w e r e 28.3 f e e t o n l y 2 p e r -cent of t h e w a v e s w i t h a Hg = 15 f e e t and Tg = 12 s e c o n d s s p e c t r u m w o u l d overtop t h e s t r u c t u r e w h i l e a f r e e b o a r d of 12.2 f e e t w o u l d a l l o w a b o u t h a l f t h e waves t o o v e r t o p . A f r e e b o a r d e q u a l t o Rg, i . e . , 20 f e e t , w i l l s a t i s f y t h e c o n d i t i o n o f o n l y o c c a s i o n a l wave o v e r t o p p i n g s i n c e about 13 p e r c e n t o f t h e waves w o u l d be e x p e c t e d t o o v e r t o p t h e b r e a k w a t e r . * * * * * * A * * A A A A f t A * A * * * * * A * A A A A A * * * A * A * * * * V I . SUMMARY
E q u a t i o n s and c u r v e s a r e p r e s e n t e d f o r c o m p u t i n g t h r e e runup p a r a m e t e r s and one rundown p a r a m e t e r f o r p l a n e , smooth s l o p e s e x p o s e d t o i r r e g u l a r wave c o n d i -tions where dg/Hg > 3. T h e s e p a r a m e t e r s a r e R2, t h e e l e v a t i o n e x c e e d e d by only 2 p e r c e n t o f t h e r u n u p s ; Rg, t h e a v e r a g e runup o f t h e h i g h e s t o n e - t h i r d of the wave r u n u p s ; R, t h e mean runup of a l l t h e r u n u p s ; and Rdge, t h e depth below t h e S t i l l w a t e r l e v e l w h i c h i s j u s t g r e a t e r t h a n 98 p e r c e n t o f t h e rundown. E x a m p l e p r o b l e m 1 i l l u s t r a t e s t h e u s e o f e q u a t i o n ( 2 ) i n c o m p u t i n g the rundowns, p a r a m e t e r s , and t h e method o f i n t e r p o l a t i o n f o r r u n u p on s l o p e s not s p e c i f i c a l l y c o v e r e d i n t h i s r e p o r t . E x a m p l e p r o b l e m 2 i l l u s t r a t e s t h e method of c o m p u t i n g rundown. E x a m p l e 3 i l l u s t r a t e s how t h e s t u d y r e s u l t s f o r smooth s l o p e s c a n be a p p l i e d t o r o u g h and p o r o u s s l o p e s , i n t h i s c a s e t o com-pute the d e s i r e d f r e e b o a r d f o r a rubble-mound b r e a k w a t e r .
LITERATURE C I T E D
AHRENS, J . P . , " P r e d i c t i o n o f I r r e g u l a r Wave Runup," CETA 7 7 - 2 , U.S. Army, C o r p s of E n g i n e e r s , 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 , F o r t B e l v o i r , V a . , J u l y 1977 .
AHRENS, J . P . , " I r r e g u l a r Wave Runup," Proaeedings of the Conference on Coastal
Structures '79, A m e r i c a n S o c i e t y o f C i v i l E n g i n e e r s , V o l . I I , 1 9 7 9 , pp.
998-1019.
BATTJES, J.A., "Wave Runup and O v e r t o p p i n g , " T e c h n i c a l A d v i s o r y C o m m i t t e e on P r o t e c t i o n A g a i n s t I n u n d a t i o n , R i j k s w a t e r s t a a t , T h e Hague, N e t h e r l a n d s , 1 9 7 4 . GODA, U., " E s t i m a t i o n o f Wave S t a t i s t i c s f r o m S p e c t r a l I n f o r m a t i o n , "
Pro-aeedings of the Symposium on Ocean Wave Measurement and Analysis, V o l . I ,
1974, pp. 3 2 0 - 3 3 7 .
HUNT, I . A . , " D e s i g n o f S e a w a l l s and B r e a k w a t e r s , " Joumal of the Waterways and
Harbors Division, V o l . 8 5 , No. WW3, S e p t . 1 9 5 9 , pp. 1 2 3 - 1 5 2 .
KAMPHUIS, J.W., a n d MOHAMED, N., "Runup on I r r e g u l a r Waves on P l a n e , Smooth S l o p e , " Journal of the Waterway, Port, Coastal, and Ocean Division, V o l . 104, No. WW2, May 1 9 7 8 .
STOA, P.N., " R e a n a l y s i s o f Wave Runup on S t r u c t u r e s and B e a c h e s , " TP 7 8 - 2 , U.S. Army, C o r p s o f E n g i n e e r s , 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 , F o r t B e l v o i r , V a . , Mar. 1 9 7 8 a .
STOA, P.N., " R e v i s e d Wave Runup C u r v e s f o r Smooth S l o p e s , " CETA 7 8 - 2 , U.S. Army, C o r p s o f E n g i n e e r s , 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 , F o r t B e l v o i r , Va., J u l y 1 9 7 8 b .
STOA, P.N., "Wave Runup on Rough S l o p e s , " CETA 7 9 1 , U.S. Army, C o r p s o f E n g i -n e e r s , 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 , F o r t B e l v o i r , V a . , J u l y 1 9 7 9 . U.S. ARMY, CORPS OF ENGINEERS, COASTAL ENGINEERING RESEARCH CENTER, Shore
Protection Manual, 3d e d . . V o l s . I , I I , ' and I I I , S t o c k No. 0 0 8 - 0 2 2 - 0 0 1 1 3 - 1 ,
U.S. Government P r i n t i n g O f f i c e , W a s h i n g t o n , D . C , 1 9 7 7 , 1,262 pp.
VAN OORSCHOT, J.H., and D'ANGREMOND, K., "The E f f e c t o f Wave E n e r g y S p e c t r a on Wave Runup," Proceedings of the 11th Conference on Coastal Engineering, A m e r i c a n S o c i e t y o f C i v i l E n g i n e e r s , 1 9 6 8 , pp. 8 8 8 - 9 0 0 .
APPENDIX A
APPENDIX B
RUNUP REDUCTION FACTOR, r , FOR VARIOUS TYPES OF ROUGH AND POROUS STRUCTURES ( S t o a , 1 9 7 9 )
I . VALUE OF r FOR QUARRYSTONE RUBBLE-MOUND STRUCTURE (HIGH CORE)
r = 0.52
Q u a r r y s t o n e ^ormor l a y e r
I I . VALUES OF r FOR CONCRETE ARMOR UNITS Embankment. a. Gobi B l o c k s . r 0 . 9 3 f o r H ^ k ^ o r H/k^ « 6 ( u s e H' when d /H' > 3 a n d H when d „ / H ' < 3 ) O 8 0- S O b. S t e p p e d S l o p e s . V a l u e s o f r f o r s t e p p e d s l o p e s . T y p e o f s t e p S l o p e ( c o t 9 ) r ^ V e r t i c a l r i s e r s 1 . 5 0 . 7 5 2 , 0 0 . 7 5 3 . 0 0 . 7 0 Rounded e d g e s 3 . 0 0 . 8 6 h < H^/k^ < 12 w h e r e i s t h e h e i g h t o f t h e r i s e r .
2. Embankment and R u b b l e Mound. V a l u e s o f r f o r c o n c r e t e armor u n i t s . Armor u n i t a n d p l a c e m e n t method L e n g t h d i m e n s i o n , k
r
A r m o r - l a y e r t h i c k n e s s (No. o f u n i t s ) V a l u e s o f r S l o p e s ( c o t 6 ) T e t r a p o d Random U n i f o r m \ 2 2 0.45 0.51 1.3 t o 3.0 1.3 t o 3.0 Q u a d r i p o d Random U n i f o r m 2 2 0.51 0.51 1.3 t o 3.0 1.3 t o 3.0 T r i b a r Random U n i f o r mI
2 1 0.45 0.50 1.3 t o 3.0 1.3 t o 3.0 M o d i f i e d c u b e Random U n i f o r m U n i f o r m U n i f o r m k, z\/...
r---\
1 \
1
2 1 1 1 0.48 0.62 0.73 0.55 1.3 t o 3.0 1.5 2.0 3.0 2 4I I I . VALUES OF r FOR QUARRYSTONE EMBANKMENT S l o p e ( c o t 9 ) H / k
V
r 1 . 5 3 t o 4 0 . 6 0 2 . 5 3 t o 4 0 . 6 3 3 . 5 3 t o 4 0 . 6 0 5 . 0 3 0 . 6 0 5 . 0 4 0 . 6 8 5 . 0 5 0 . 7 2APPENDIX C
RUNUP SCALE CORRECTION FACTOR, k, FOR VARIOUS TYPES OF ROUGH AND POROUS STRUCTURES ( S t o a , 1 9 7 9 )
S t r u c t u r e T y p e k Q u a r r y s t o n e , rubble-mound b r e a k w a t e r 1.06 Q u a r r y s t o n e , r i p r a p r e v e t m e n t 1.00 C o n c r e t e armor u n i t s , r u b b l e mound o r r e v e t m e n t 1.03 2 6
Ahrens, John P.
I r r e g u l a r wave runup on smooth s l o p e s / by John P. A h r e n s . — F o r t B e l v o i r , Va. : U.S. Army C o a s t a l E n g i n e e r i n g Research Center ; S p r i n g f i e l d , Va. : a v a i l a b l e f r o m NTIS, 1981. [26] p. : i l l . ; 28 c m . — ( C o a s t a l e n g i n e e r i n g t e c h n i c a l a i d ; no. 81-17) Cover t i t l e . "December 1981." B i b l i o g r a p h y : p. 19.
The r e s u l t s o f s e v e r a l l a b o r a t o r y s t u d i e s have been used t o d e v e l o p a method t o e s t i m a t e the wave runup and rundown on p l a n e , smooth s l o p e s caused by i r r e g u l a r wave a c t i o n . Curves and e q u a t i o n s a r e p r e s e n t e d which can be used t o compute t h e 2 p e r c e n t r u n u p , s i g n i f i -c a n t r u n u p , mean r u n u p , and a p p r o x i m a t e l o w e r l i m i t o f rundown. A p r o c e d u r e i s suggested f o r a d a p t i n g the smooth-slope r e s u l t s t o wave runup on rough and porous s l o p e s - Example problems i l l u s t r a t e t h e use o f the m a t e r i a l p r e s e n t e d .
1. Wave runup. 2. Water waves. I . T i t l e . I I . S e r i e s .
TC203 .0581ta no. 81-17 627 A h r e n s , J o h n P. I r r e g u l a r wave r u n u p on s m o o t h s l o p e s / by J o h n P. A h r e n s . — F o r t B e l v o i r , V a . : U.S. Army 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 ; S p r i n g f i e l d , V a . : a v a i l a b l e f r o m N T I S , 1981. [ 2 6 ] p. : i l l . ; 28 c m . — ( C o a s t a l e n g i n e e r i n g t e c h n i c a l a i d ; n o . 81-17) C o v e r t i t l e . "December 1981." B i b l i o g r a p h y : p. 19. The r e s u l t s o f s e v e r a l l a b o r a t o r y s t u d i e s h a v e b e e n u s e d t o d e v e l o p a method t o e s t i m a t e t h e wave r u n u p a n d rundown on p l a n e , s m o o t h
s l o p e s c a u s e d by i r r e g u l a r wave a c t i o n . C u r v e s a n d e q u a t i o n s a r e p r e s e n t e d w h i c h c a n be u s e d to c o m p u t e t h e 2 p e r c e n t r u n u p , s i g n i f i -c a n t r u n u p , mean r u n u p , a n d a p p r o x i m a t e l o w e r l i m i t o f r u n d o w n . A p r o c e d u r e I s s u g g e s t e d f o r a d a p t i n g t h e s m o o t h - s l o p e r e s u l t s t o wave
Ahrens, John P.
I r r e g u l a r wave runup on smooth s l o p e s / by John P. A h r e n s . — F o r t B e l v o i r , Va. : U.S. Army C o a s t a l E n g i n e e r i n g Research Center ; S p r i n g f i e l d , Va. : a v a i l a b l e f r o m NTIS, 1981. [26] p. : i l l . ; 28 c m . — ( C o a s t a l e n g i n e e r i n g t e c h n i c a l a i d ; no. 81-17) Cover t i t l e . "December 1981." B i b l i o g r a p h y : p. 19.
The r e s u l t s o f s e v e r a l l a b o r a t o r y s t u d i e s have been used t o d e v e l o p a method t o e s t i m a t e t h e wave runup and rundown on p l a n e , smooth s l o p e s caused by I r r e g u l a r wave a c t i o n . Curves and e q u a t i o n s a r e p r e s e n t e d w h i c h can be used t o compute t h e 2 p e r c e n t r u n u p , s i g n i f i -c a n t r u n u p , mean r u n u p , and a p p r o x i m a t e l o w e r l i m i t o f rundown. A p r o c e d u r e i s suggested f o r a d a p t i n g t h e smooth-slope r e s u l t s t o wave runup on rough and porous s l o p e s . Example problems i l l u s t r a t e t h e use o f the m a t e r i a l p r e s e n t e d .
1. Wave runup. 2. Water waves. I . T i t l e . I I . S e r i e s .
TC203 .U581ta no. 81-17 627
Ahrens, John P.
I r r e g u l a r wave runup on smooth s l o p e s / by John P. A h r e n s . — F o r t B e l v o i r , Va. : U.S. Army C o a s t a l E n g i n e e r i n g Research Center ; S p r i n g f i e l d , Va. : a v a i l a b l e from'NTIS, 1981. [26] p. : i l l . ; 28 c m . — ( C o a s t a l e n g i n e e r i n g t e c h n i c a l a i d ; no. 81-17) Cover t i t l e . "December 1981." B i b l i o g r a p h y : p. 19.
The r e s u l t s o f s e v e r a l l a b o r a t o r y s t u d i e s have been used t o d ev e lo p a method t o e s t i m a t e t h e wave runup and rundown on p l a n e , smooth s l o p e s caused by i r r e g u l a r wave a c t i o n . Curves and e q u a t i o n s a r e p r e s e n t e d which can be used t o compute t h e 2 p e r c e n t r u n u p , s i g n i f i -c a n t r u n u p , mean r u n u p , and a p p r o x i m a t e l o w e r l i m i t o f rundown. A
