paper_wave reflection from natural beaches

8/12/2019 Paper_Wave Reflection From Natural Beaches

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Ocean Engng Vol. 19, No 3, pp. 239-258, 1992. 0029-8018/92 5.00 + .00Printed in Great Britain. Pergamon Press Ltd

W A V E R E F L E C T I O N F R O M N A T U R A L B E A C H E ST. L. WALTON JR

U.S. Army Corps of Engineers, Coastal Engineering Research Center, USAE Waterways ExperimentStation, Vicksburg, Mississippi, U.S.A.Abstract--The reflection of wave energy from natural beaches is a quantity that is rarelymeasured. This paper addresses the question as to whether the reflected wave energy from abeach is an importan t component of energy flux from an engineering point of view. Wave run-up and wave set-up are important nearshore processes in which wave reflection appears to playa major role and, consequently, the prediction of reflected wave energy from the offshoreforcing irregular wave train energy at natural beaches is a first step in a better understandingof these processes. The present paper reports on wave reflection measurements at two naturalbeaches with different characteristics. The issue of wave reflection is addressed via separationof input and output signals using coincident pressure and velocity measurement s. In the presentlimited analysis, reflected wave energy was not found to be significant.

I N T R O D U C T I O NW A V E REFLECTION f r o m b ea ch e s i s a s u b j e c t o f m a j o r i mp o r t an ce t o u n d e r s t an d i n g t h en ea r s h o r e zo n e an d t o i mp r o v i n g co as t a l s t r u c t u r e d e s i g n . Th e l ev e l o f en e r g y fl uxd i s s ip a t i o n t h a t o c cu r s o n a b eac h i s d e p en d en t o n t h e m ag n i t u d e o f t h e w a v e r e f lec t i o nf r o m t h e b eac h . T h u s , i n an i n d ir ec t ma n n e r , w a v e re f l ec ti o n in f l u en ces m an y co as t a lp r o ces s e s s u ch a s r u n - u p w h i ch , i n t u r n , d e t e r mi n es co as t a l d e s i g n c r i t e r i a s u ch a s t h eh e i g h t o f a s eaw a l l o r f l o o d p r o t ec t i o n d u n e .

M u c h i n f o r m a t i o n o n r e f l e c ti o n c o m e s f r o m l im i t e d l a b o r a t o r y a n d t h e o r e t ic a l s t ud i e sb a s e d o n m o n o c h r o m a t i c w a t e r w a v e t h e o r y . M i c h e ( 19 5 1 ) p r o p o s e d a s e m i - em p i r ic a lc r i te r i o n f o r p r e d i c t i o n o f w a v e r e f lec t io n f r o m s l o p ed s t r u c t u r e s f o r t h e ca s e o fm o n o ch r o m a t i c i n c i d en t w a v es i n w h i ch li m i t ed en e r g y d is s i p a ti o n o ccu r r ed . M i ch es '( 1 9 5 1 ) r e s u lt s h a v e b e e n s h o w n to b e a r e a s o n a b l e r o u g h a p p r o x i m a t i o n f o r m o n o -c h r o m a t i c w a v e s in c i d e n t o n s o m e s t r u c tu r e s l o p e s [U r s e l l e t a l ( 1 9 6 0 ) ; B a t t j e s ( 1 9 7 4 ) ;G u z a a n d B o w e n ( 1 9 7 7 ) ] . M o r a e s ( 1 9 7 0 ) p e r f o r m e d e x t e n s i v e s m a l l s c a l e l a b o r a t o r yt e s ts o n m o n o c h r o m a t i c w a v e r e fl e ct io n f ro m b o t h s m o o t h a n d r o u g h s l o p e s w i th t h emi l d es t s l o p e b e i n g 1 o n 1 0. B a t t j e s ( 1 9 7 4 ) l a t e r u s ed t h e M o r ae s ' ( 1 9 7 0 ) d a t a s e t t od e r i v e a s em i - emp i r i ca l ex p r e s s i o n f o r r e f lec t i o n co e f fi c ien t b a s ed o n M i ch es ' ( 1 9 5 1 )w o r k w i th a d d e d i n d e p e n d e n t a s s u m p t io n s . S e e li g a n d A h r e n s ( 1 9 8 1 ) f u r t h e r r e f in e dt h e B a t t j e s ( 1 9 7 4 ) r e f l ec ti o n co e f f i c ien t ex p r e s s i o n b y em p i r ica l cu r v e f it ti ng .

R e f l ec t i o n f r o m n a t u r a l ( m i l d , r o u g h , n o n - l i n ea r i n p r o f i le ) b eac h es u n d e r i r r eg u l a rw a v e ac t i o n i s co n s i d e r ab l y mo r e co mp l ex an d l it tl e is p r e s en t l y k n o w n co n c e r n i n g s u chp r o c e s s e s . T o d a t e , m e a s u r e m e n t s o f w a v e r e fl e ct io n c o e f fi c ie n ts f r o m b e a c h e s h a v eb een mad e p r i mar i l y v i a meas u r emen t s i n t h e n ea r s h o r e zo n e , t y p i ca l l y u s i n g e i t h e r ab es t fi t o f th e s t an d i n g w av e am p l i t u d e p a t t e r n t o p r o d u ce a co m p o s i t e r e f l ec ti o nco e f f i c ien t [ S u h ay d a ( 1 9 7 4 ) ] , o r u s in g li n ea r l o n g w a v e t h eo r y t o ca l cu l a t e th e co m p o s i t er e f l ec t i o n co e f f i c i en t [ Tak ezaw a e t a l ( 1 9 8 8 ) ] o r t o c a l c u l a t e t h e f r e q u e n c y d e p e n d e n t

239


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2 4 0 T L . W A L T O N J Rr e f l e c t i o n c o e f f i c i e n t [ T a t a v a r t i e t a l 1 9 8 8 ) ] . D e b a t e i n t h e l i t e r a t u r e h a s e x i s t e d ast o w h e t h e r t h e o b s e r v e d p h e n o m e n o n is t ru l y re f le c t io n i .e . s t a n d i n g w a v e s ) o r c ro s s-s h o r e s p a t ia l a m p l i t u d e v a r i a b i li t y d u e t o e d g e w a v e e f f e c t s [ G u z a 1 9 7 4 ) ] .

A n i m p o r t a n t q u e s t i o n t h a t r e m a i n s t o b e a n s w e r e d i s w h e t h e r t h e r e f l e c te d w a v ee n e r g y f r o m a b e a c h is a n i m p o r t a n t c o m p o n e n t o f e n e r g y f lu x f r o m a n e n g i n e e r i n gp o i n t o f v ie w . F o r e x a m p l e , i f n o n l i n e a r i n t e r a c ti o n w i th i n t h e s u r f z o n e c a s c a d e s e n e r g yd o w n t h e s p e c t r a t o w a r d l o w e r f r e q u e n c i e s a s s u g g e s t e d b y a r e d s h if t o f e n e r g y i n t h en e a r s h o r e s p e c tr a s e e n in m a n y st u d ie s [ G u z a a n d T h o r n t o n 1 9 8 9 ) ; E l g a r a n d G u z a1 9 8 5 ) ; F r e i li c h a n d G u z a 1 9 8 4 ) ] , t h e q u e s t i o n a r is e s a s t o w h e t h e r a s i g n if ic a n t

a m o u n t o f th i s e n e r g y e s c a p e s f r o m t h e s u r f z o n e i n a n o f f s h o r e d i r e c t io n . I f s i g n if ic a n ta m o u n t s o f e n e r g y a r e f o u n d t o b e e s c a p i n g f r o m t h e s u r f z o n e in a n o f f s h o r e d i re c t i o nt h e n a c t u a l e n e r g y a v a i l a b l e fo r s u r f z o n e p r o c e s s e s s u c h as s e t- u p a n d r u n - u p m i g h tb e c o n s i d e r a b l y l es s t h a n w h a t i s t y p i c a l ly a s s u m e d in e n g i n e e r i n g c a l c u l a t i o n s a s th ei n c i d e n t w a v e e n e r g y . T h e p r e s e n t p a p e r r e p o r t s o n a n a t t e m p t t o a d d r e s s t h i s i s s u ev ia s e p a r a t i o n o f th e m e a s u r e d w a v e f ie ld a t a n o f f s h o r e p r e s s u r e - v e l o c i t y s e n s o r g a g ei n t o i n c i d e n t a n d r e f l e c t e d w a v e e n e r g y a t t w o f i el d s it es .

S T U D Y L O C A T I O N ST h e t w o s it es a t w h ic h o f f s h o r e w a v e i n f o r m a t i o n a n d b e a c h p r o f i le i n f o r m a t i o n w e r e

c o l l e c t e d a r e s h o w n i n F i g. 1 . T h e f ir st s i te to b e a d d r e s s e d i s D u c k N C ) o n t h e O u t e rB a n k s o f th e N o r t h C a r o l i n a c o a s t li n e . T h e D u c k s it e is in th e m i d d l e o f a n a p p r o x i -m a t e l y 1 00 k m u n i n t e r r u p t e d s t r e tc h o f b a r r ie r i sl an d e x p o s e d t o t h e A t l a n t ic O c e a n .D u c k w a s th e s it e o f a m a j o r n e a r s h o r e h y d r o d y n a m i c s a n d s e d i m e n t t r a n s p o r t f ie ld

F IG . 1 . S i t e m a p o f s t u d y l o c a t i o n s .


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Wave reflection from natural beaches 24

experiment in October 1986 [Birkemeier e t a l . (1989)]. The primary extent of theexperiment was within the property limits of an Army Corps of Engineers FieldResearch Facility at Duck. The beach at this site has a steep foreshore with an averageslope of 1:10 while the offshore contains a bar feature with varying amplitude andposition on the nearshore profile depending on the wave climate. The grain sizecharacteristics of the beach typically exhibit bimodality with sands predominantly rang-ing from 0.15 to 0.30 mm in size and coarse shell fractions ranging predominantly from0.5 to 3 mm in size. Numerous oceanographic instruments were deployed offshore inan attempt to obtain a quality data set for better understanding of nearshore processesand hydrodynamics. In particular, the present analysis is based on data from a pressuresensor and bidirectional orthogonal axis electromagnetic current meter that was con-tained on a tripod located on the sea bed in approximately 6.5 m of water depth about500 m south of a research pier and approximately 500 m offshore. The pressure sensorwas located on the tripod approximately 1.45 m above the sea bed while the currentmeter sensor ball was located on the tripod 1.71 m above the sea bed. One channel ofthe bidirectional current meter was directed in the onshore-offshore direction whilethe other channel of the current meter was orthogonal to the first channel and orientedin the longshore direction. The instrument collected continuous data (pressure = p,velocity = u , v at sampling increments of 0.5 sec for 238 min, four times per day fora two week period during October 1986. The p u v records were further broken downinto 34 min records for time series processing, although two continuous records of 136min in length were also investigated. Further details on the p u v instrument, the datacollection, the instrument location, and beach characteristics at the site are given inBirkemeier e t a l . (1989). The primary records analyzed for the present study consistof data from a storm event with offshore significant wave heights exceeding 3 m thatoccurred during the period 10-14 October 1986. Wave height, period, and directionduring the time period of interest are given in Table 1 adopted from Birkemeier e t a l .(1989). The method of wave direction analysis for this gage is discussed in Grosskopf(1981). The azimuth angle of the offshore perpendicular to the beach is approximately70 true north, the same angle as the wave direction approach during the latter portionof the storm; thus, the predominant wave direction was approximately perpendicularto the beach during the period of interest. The predominance of waves in a cross-shoredirection is confirmed by the comparison of cross-shore velocity variance to the varianceof the alongshore velocity component at the gage site. In all cases analyzed, the ratioof alongshore velocity variance to cross-shore velocity variance was less than 15 andmean alongshore current was less than 0.1 m/sec.

The second site to be addressed is Ocean City (MD) where the first phase of a majorbeach nourishment project was completed in October 1988. Beach characteristics ofthe pre- and post-fill project are discussed in Hansen (1989). During the study periodthe beach directly landward of the offshore sensor site had a relatively steep nearshoreslope of approximately 1:15 out to about 5 m of water depth and a mild offshore slopeapproximately 1:150 in the deeper water offshore. The grain size characteristics of thebeach show predominantly quartz sands ranging from 0.15 to 0.3 in grain size [seeHansen (1989)]. During the pre- and post-nourishment phases of this beach nourishmentproject a bottom-resting tripod containing a pressure sensor and a bidirectional ortho-gonal axis electromagnetic current meter was collecting wave and current data at


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242 T . L . WAI.TON JR

TABLE 1. WAVE CLIMATOLOGY DucK N C )D a t e / t i m e H ..... m ) T ;, s e e ) W a v c d i r e c t i o nd e g r e e s )1 0 O ct o b er 1 9 8 6 1 2 4 6 2 .5 7 .5 5111 0 O c t o b e r 1 9 8 6 1 3 2[ ) 2 . 6 7 . 3 5 11 0 O c t o b e r 1 9 8 6 1 3 5 4 2 . 6 7 . 1 5 11 0 O c t o b e r 1 9 8 6 1 71 10 2 . 8 7 . 3 5 41 0 O c t o b e r 1 9 8 6 1 7 3 4 2 . 9 8 .[ ) 5 31 0 O c t o b e r 1 9 8 6 1 8 0 8 3 . 0 7 . 1 5 21 0 O c t o b e r 1 9 8 6 1 8 4 2 3 .1 7 . 8 5 61 0 O c t o b e r 1 9 8 6 1 9 1 6 3 . 0 7 . 8 5 31 0 O c t o b e r 1 9 8 6 1 9 5 0 3 . 1 8 . 3 5 81 0 O c t o b e r 1 9 86 2 0 2 4 3 . 1 7 , 5 6 01 0 O c t o b e r 1 9 86 2 3 2 0 3 . 3 8 . 0 5 81 0 O c t o b e r 1 9 8 6 2 3 5 4 3 . 2 9 . 1 6 411 O c t o b e r 1 9 8 6 0 0 2 8 3 . 3 8 . 3 6 11 1 O ct o b er 1 9 8 6 0 11 12 3 .4 7 .5 6 21 1 O c t o b e r 1 9 8 6 0 1 3 6 3 . 1 8 . 5 6 3l l O c t o b e r 1 9 8 6 0 21 11 3 . 2 7 . 8 6 311 O c t o b e r 1 9 8 6 1 12 44 3 . 1 8 . 8 6 511 O c t o b e r 1 9 8 6 0 5 4 0 3 . 2 9 . 1 6 21 1 O c t o b e r 1 9 8 6 0 6 1 4 3 . 1 8 . 5 6 311 O c t o b e r 1 9 86 0 6 4 8 3 . 4 9 . 5 6 61 1 O c t o b e r 1 9 8 6 ] 72 2 3 , 2 9 . 1 6 911 O c t o b e r 1 9 8 6 0 7 5 6 3 . 2 9 . 1 6 71 1 O c t o b e r 1 9 8 6 0 83 11 3 , 2 9 . 5 6 51 1 O c t o b e r 1 9 8 6 0 9 0 4 3 , 2 9 . l 6 21 1 O c t o b e r 1 9 8 6 1 2 0 0 3 , 2 9 . 9 6 91 1 O c t o b e r 1 9 8 6 1 2 3 4 3 . 2 1 1. 1 7 51 1 O c t o b e r 1 9 8 6 1 31 t8 3 . 2 1 0 . 7 7 21 1 O c t o b e r 1 9 8 6 1 3 4 2 3 . 3 1 1 , 6 7 31 1 O c t o b e r 1 9 8 6 1 4 1 6 3 . 2 1 1. 1 7 21 1 O c t o b e r 1 9 8 6 1 4 5 0 3 . 0 1 1 . 6 7 41 1 O c t o b e r 1 9 8 6 1 5 2 4 3 . 3 1 1. 1 7 41 1 O c t o b e r 1 9 8 6 1 8 2 0 3 , 0 1 1 . 6 7 41 1 O c t o b e r 1 9 8 6 1 8 5 4 2 . 9 1 2 . 2 7 61 1 O c t o b e r 1 9 8 6 1 9 2 8 2 . 8 1 2 . 2 7 31 1 O c t o b e r 1 9 8 6 2 0 0 2 2 . 8 1 1. 1 7 411 O c t o b e r 1 9 8 6 2 0 3 6 2 . 9 1 1. 1 7 31 1 O c t o b e r 1 9 8 6 2 1 1 0 2 . 9 1 2 . 2 7 31 1 O c t o b e r 1 9 8 6 2 1 4 4 2 . 8 1 1. 1 7 31 2 O c t o b e r 1 98 6 0 0 4 0 2 , 7 1 2 . 2 7 51 2 O c t o b e r 1 9 8 6 0 1 1 4 2 . 8 1 2 .8 7 61 2 O c t o b e r 1 9 8 6 0 1 4 8 2 . 7 1 2 .2 7 41 2 O c t o b e r 1 9 86 0 2 2 2 2 . 7 1 1 .6 7 61 2 O c t o b e r 1 9 86 0 2 5 6 2 . 6 1 0 .7 7 41 2 O c t o b e r 1 9 86 0 3 3 0 2 . 7 1 2 .2 7 31 2 O c t o b e r 1 9 8 6 0 4 0 4 2 . 5 9 . 9 7 51 2 O c t o b e r 1 9 8 6 0 6 4 5 2 . 3 1 1 ,6 7 51 2 O c t o b e r 1 9 8 6 0 7 1 9 2 . 3 l 1 ,1 7 71 2 O c t o b e r 1 9 8 6 0 7 5 3 2 . 4 1 1 ,6 7 3


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W a v e r e f l e c t i o n f r o m n a t u r a l b e a c h e s 243sampl ing interva l s o f 1 s ec wi th cont inuous data records o f 17 min every 1 -4 hr . Thet r ip o d l o c a t i o n w a s a p p r o x i m a t e ly 9 0 0 m o f f s h o r e o f t h e m i d p o r t io n o f O c e a n C i tyb e a c h 8 3 r d S t r e e t ) i n a p p r o x im a t e l y 1 0 . 8 m o f w a t e r . T h e p r e ss u r e se n s o r o n t h etr ipod was 0 .20 m above the s ea bed whi le the current s ensor ba l l on the tr ipod was0 . 4 6 m a b o v e t h e s e a b e d . O n e c h a n n e l o f t h e b i d ir e c ti o n a l c u r re n t m e t e r w a s d i re c t edi n t h e o n s h o r e - o f f s h o r e d i r e c t i o n w h i l e t h e o t h e r c h a n n e l o f t h e c u r r e n t m e t e r w a sor thogonal to the f i r s t channe l and or iented in the longshore d irec t ion . Dur ing thep e r i o d o f 2 1 - 2 3 M a y 1 9 9 0, a s tr o n g w e a t h e r s y s t em m o v e d i n o v e r t h e a re a o f c o n c er ncreat ing s igni f i cant wave he ights greater than 1 .6 m of f shore o f the beach. The data tobe d i scussed in the fo l low ing sec t ions are from this per iod in time . W ave he ight , per io d ,and direc t ion dur ing th i s s tudy per iod i s shown in Table 2 . The az imuth angle o f theof fshor e perpendicular to the beach i s approximate ly 90 true nor th . Th e m ethod of

TABLE 2. WA VE CLIMATOLOGY--OCEAN CITY M D )D a t e / t i m e Hm m ) Tp s e c ) W a v e d i r e ct i o nd e g r e e s )21 M ay 1990 1600 0 .97 4 .3 6621 M ay 1990 1700 1 .16 4 .5 6921 M ay 1990 1800 1.33 5.2 7321 M ay 1990 1900 1 .25 6 .6 8421 M ay 1990 2000 1 .38 6 .0 7921 M ay 1990 2100 1 .53 4 .5 6921 M ay 1990 2200 1 .54 6 .0 6821 Ma y 1990 2300 1 .50 6 .0 6522 Ma y 1990 0000 1 .52 5 .2 6522 Ma y 1990 0100 1 .54 7 .3 7222 M ay 1990 0200 1 .61 6 .0 6122 M ay 1990 0300 1 .34 5 .2 6822 M ay 1990 0400 1 .24 7 .3 8122 M ay 1990 0500 1 .20 8 .3 8622 M ay 1990 0600 1 .35 7 .8 8122 M ay 1990 0700 1 .37 8 .3 8222 M ay 1990 0800 1 .29 7 .8 8522 M ay 1990 0900 1 .14 8 .8 8122 M ay 1990 1000 1 .16 8 .3 8322 M ay 1990 1100 1 .22 8 .3 8122 M ay 1990 1200 1 .13 8 .8 7822 M ay 1990 1300 1.27 8.8 6522 M ay 1990 1400 1 .09 8 .8 8022 M ay 1990 1500 1 .10 9 .5 9822 M ay 1990 1600 1 .21 9 .5 9522 M ay 1990 1700 1 .18 4 .2 7322 M ay 1990 1800 1 .22 7 .8 8522 M ay 1990 1900 1 .20 7 .8 8322 M ay 1990 2000 1 .13 8 .8 7922 M ay 1990 2100 1 .15 6 .9 8722 M ay 1990 2200 1 .21 5 .2 8522 M ay 1990 2300 1 .05 8 .8 8423 M ay 1990 0000 0 .97 8 .8 8323 M ay 1990 0100 1 .15 9 .5 7523 M ay 1990 0200 0 .97 9 .5 9123 M ay 1990 0300 0 .78 9 .5 93


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244 T. L . ~ ALION J~

w a v e d i r e c t i o n a n a l y s i s f o r th i s s it e is d i s c u ss e d i n G r o s s k o p f (1 9 8 1 ) . T h e p r e d o m i n a n tw a v e d i r e c t io n f o r t h e r e c o r d s s u b s e q u e n t ly a n a l y z e d w a s a p p r o x i m a t e l y p e r p e n d i c u l a rt o t h e b e a c h . T h e p r e d o m i n a n c e o f w a v e s in a c r o s s - s h o r e d i re c t i o n w a s c o n f i r m e d b yc o m p a r i s o n o f t h e a l o n g s h o r e v e l o c i ty v a r i a n c e t o t h e c r o s s - s h o r e v e l o c i ty v a r i a n c e . I na ll c a s e s a n a l y z e d , t h e r a t i o o f th e a l o n g s h o r e v e l o c i ty v a r i a n c e t o t h e c r o s s - s h o r ev a r i a n c e w a s le s s t h a n 3 0 % a n d m e a n a l o n g s h o r e d i r e c t e d v e l o c i ty a t t h e g a g e s it e w a sl e s s t h a n 0 . 2 m / s e c .

M E T H O D O L O G YI n th e p r e s e n t c a s e , l o n g c r e s t e d w a v e s a r e a s s u m e d t r a v e l i n g in a n o n s h o r e d i r e c t io n

( i .e . t o w a r d t h e b e a c h ) a s s h o w n i n F i g. 2 . T h e s u r f a c e e le v a t i o n f o r s tr i ct ly o n s h o r ed i r e c t e d w a v e s c a n b e d e s c r i b e d a s a s u m m a t i o n o f w a v e s o f v a ri o u s i n c id e n t a m p l i t u d e sa n d p h a s e s w h e r e o n e s u ch f r e q u e n c y w a v e c o m p o n e n t is d e f in e d a t l o c a ti o n x to b eo f t h e f o r m :

i , c c rm n A t ) 1 )q i , n A t ) = a , , o s ( k , , , x -w h e r e ~ , , = 2 ~ r f m = r a d i a l fr e q u e n c y , f , , = w a v e f r e q u e n c y f o r g i v e n f r e q u e n c y c o m -p o n e n t m , k m = w a v e n u m b e r , a n d a ~ = i n ci d en t w a v e a m p l i t u d e . T h e s u p e r s c r ip t

i n r e f er s to th e w a v e a p p r o a c h i n g t o w a r d t h e b e a c h ( s e e Fig . 2 ) . T h e i n c o m i n g p h a s ea t x = 0 , t = 0 h a s b e e n a s s u m e d e q u a l t o z e r o w i t h o u t l o s s o f g e n e r a l i t y .

A t t h e b e a c h a p o r t i o n o f t h e w a v e e n e r g y is r e f l e c t e d r e s u l ti n g i n a n o u t g o i n gw a v e tr a i n s u r fa c e e l e v a t i o n d e f i n e d f o r o n e f r e q u e n c y c o m p o n e n t a s :

INCIDENTW VE

Fl6. 2. Coordinate system.

i ~ ) : BE CH


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Wave reflection from natural beaches 245~ o ut ( n A t ) = a ~ t cos(k,,, x + ~ r m n A t + ~ , ~ ) (2)

where a phase change ~,,, between incident and reflected waves has been assumedfor generality. In the present situation the term reflected is used in a general mannerto specify the existence of a partially standing wave field offshore. The outgoing wavecomponent at a given frequency may be either a direct reflection of the incoming wavetrain from the beach at the input frequency, or, a free wave created within the surfzone via non-linear interaction of incident waves and then reflected from the beach ordirectly generated in an outgoing direction by surf zone forcing in some manner [see,for example, Symonds e t a l . (1982)]. The reflection coefficient can be defined forthis one component (one frequency) wave system as:

aO,,r m = i- -~ 3 )

a mThe water surface elevation for a composite sum of waves at harmonic frequenciescan be defined in terms of a finite Fourier series as:

N - - 1q. = rl(nA/) = ~] (a~ cos (k . .x - or,.n a t )m = O~- a u t c o s k m . x - -~ f f m n a t + 6.,)) (4)

where a term for the mean water surface (m = 0 component) has been included.Utilizing the notation for a lag domain discrete Fourier transform series (DFT) as perBorgman (1973):

N - l [ i 2 7 r m n \x . = x ( n A t ) -- .,=o] X,. e xp ~ ) Af (5)with

X m = X ( m A f ) (6)and the frequency domain discrete Fourier transform series (DFT) as:

N - 1 { - - i 2 t r m n ~X m = n = o E xnexp~ ~ ] At (7)

the water level surface elevation for the composite incident and reflected wave trainseries can be written as:N - - 1 [ . [ t 2 r r m n \ { i 2 1 r m n ~0 . = ~ ( n A t ) : ,,,= o \ ~ A ~ ex p~ ]~ / + a~,, t ex p~ )) AT (8)

where A~ and A u ' are the complex amplitudes of the incident and reflected waves atfrequency component m.The bottom pressure as recorded by a subsurface pressure sensor for one wavefrequency component (incident and reflected) is given by [Dean and Dalrymple (1984)]a s : p ( n h t ) = p g K pm ~ ( n a t ) (9)with


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246 T .L . WALTON Rco sh (I k , , [ .s ),) (10)

K , , , . c s h ( l k m l h )w h e r e p = d e n s i t y o f s e a w a t e r , g = a c c e l e r a t i o n o f g r a v i t y , h = w a t e r d e p t h ,Sp = h e i g h t o f p r e s s u r e s e n s o r a b o v e t h e s e a b e d , a n d K p i s o f t e n r e f e r r e d t o a s t h ed y n a m i c p r e s su r e r e s p o n se f a c t o r f o r a gi v en f r e q u e n c y c o m p o n e n t m .

F o r a w a v e t r a in m a d e u p o f a n u m b e r o f w a v e f r e q u e n c y h a r m o n i c c o m p o n e n t s ,t h e p r e s s u r e t i m e s e r i es c a n a l s o b e w r i t t e n i n a si m i l ar m a n n e r t o t h a t o f su r f a c ee l e v a t i o n a s f o l l o w s :

N - , [ [ i 2 r r m n \ { i 2 r r m n ) t A fp , , = p ( n A t ) = ~ p gK p ,, , I A Z e x p k - ~ ) + m ' / ' e x p , - - N - / ] ( 1 1 )m ~

T h e w a t e r w a v e p a r t i c le v e l o c it y i n t h e o n s h o r e - o f f s h o r e d i r e c t i o n a s r e c o r d e d b y as u b s u r f a c e c u r r e n t m e t e r f o r o n e w a v e f r e q u e n c y c o m p o n e n t i s g i v e n b y [ D e a n a n dD a l r y m p l e ( 1 9 8 4 ) ] :

u ( n A t ) = ~ , , K , ,, , cos0m (a~ c os (k , , , x - o r,,,n a t )- a',~{ co s( ki n, x + ~ m n A t + ~m)) (12)

with m b e i n g t h e w a v e d i r e c t i o n f o r a g i v e n f r e q u e n c y c o m p o n e n t a s d e f i n e d i n F i g .2 , a n d w h e r e :

cosh Ikm l s ,, ) 13)K , , , - s i n h ( l k , - I h )

w i t h sv = h e i g h t o f v e l o c i t y s e n s o r a b o v e t h e s e a b e d .F o r a w a v e tr a i n m a d e u p o f a n u m b e r o f w a v e f r e q u e n c y c o m p o n e n t s t h e

o n s h o r e - o f f s h o r e v e l o c i ty t i m e s e r ie s c a n a ls o b e w r i t t e n in a s i m i la r m a n n e r t o t h a to f s u r f a c e e l e v a t i o n a s f o l l o w s :

N ~ ( . [ i 2 r r m n ~ { i 2 ~ r m n ~ ' ~u , , = u ( n A t ) = ~ '~ c r,~ Kv ,, cosO,, , \A ~ ex p l ~ - - ) - A t e x p t - ~ - N - - ) ] A f .t T l ~ ( 1 4 )

I n a s i m i l a r m a n n e r t h e l a g d o m a i n d i s c r e t e F o u r i e r t r a n s f o r m s e ri e s o f t h e m e a s u r e dp r e s s u r e a n d v e l o c i t y v a l u e s c a n b e d e f i n e d a s:

N I / i 2 ~ r m n \p , , = p ( n A t ) = ~ , P ~ e x p ~ - - ~ ) A f (1 5 )a n d

N -- L [ i 2 u m n ~u ,, = u ( n b t ) = ~ .. U m e x p l ~ ) A f . ( 16 )m=O

E q u a t i n g f r e q u e n c y c o m p o n e n t t e r m s o f t h e la g d o m a i n D F T s e ri es , th e f o l lo w i n ge q u a t i o n s f o r th e c o m p l e x a m p l i t u d e p r e s s u r e a n d v e l o c i t y t e r m s c a n b e f o u n d a s:P m = 9 g K p , , ( A ~ , + A ' ) ( 1 7 )a n d


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Wave reflection from natural beaches 247

Urn = tr,,, Kv,, COS0m (m/m A ') (1 8)U s i n g t h e a b o v e t w o e q u a t i o n s , t h e c o m p l e x w a v e a m p l i t u d e s A ~ o f t h e i n c o m i n gw a v e tr a in a n d ~n t o f t h e o u t g o i n g w a v e t r a i n c a n b e f o u n d a s :

A ~ - P m + Um ( 1 9)pgKpm ffmKvm os0ma n d

A~, '- Pm Um ( 2 0 )pgKpm tYmKvm OS0mI n t h e a b o v e e q u a t i o n s t h e s e n si ti v it y o f t h e i n c o m i n g a n d o u t g o i n g w a v e t ra i n

c o m p l e x a m p l i t u d e m o d u l u s t o t h e d i r e c t io n o f t h e i n c om i n g a n d o u t g o i n g w a v e t r a ini s s een t o b e r e l a t i v e ly s m a l l b y v ir t u e o f t h e C OS0m t e r m in t h e d en o m in a to r o f t h ev e l o c i ty f r e q u e n c y d o m a i n t e r m . I t is a ls o n o t e d t h a t i f t h e w a v e v e l o c it y c o m p o n e n ti s a s s u m e d t o b e i n t h e o n s h o r e - o f f s h o r e d i r e c t i o n , t h e c o m p l e x i n b o u n d a m p l i t u d e i sm in im ized w h i l e t h e o u tg o in g w av e t ra in co m p lex am p l i t u d e co e f f i c i en t i s m a x im ized ,th u s m ax im iz in g co m p u ted r e f l e c t i o n .I n t h e p r e s en t co n t ex t , t h e s p ec t r a l en e r g y co n ten t o f a s i g n a l c an b e d e f in ed a s t h em o d u l u s s q u a r e d d i v i d e d b y th e r e c o r d l e n g th [ B o r g m a n ( 1 9 7 3 )] ; h e n c e t h e i n ci d e n tman d r e f l e c t ed s p ec t r a l en e r g y fo r a g iv en f r eq u e n cy co m p o n en t f = mar = N A t a r egiven as :

$ 2 - l A b i aNAt ( 2 1 )a n d

S o . . _ LA 'I2NAt ( 2 2 )Th e f r eq u e n cy d ep e n d en t r e f l e c t io n co e f f ic i en t c an b e d e f in ed a s:

I A o - Ir m - I A ~ I ( 2 3 )t h e r a t io o f t h e o u t g o i n g w a v e t ra i n c o m p l e x a m p l i t u d e m o d u l u s t o t h e i n c o m i n g w a v et ra i n c o m p l e x a m p l i t u d e m o d u l u s i n t h e s it u a ti o n w h e r e i n c i d e n t e n e r g y is g r e a t e r t h a nr e f l ec t ed en e r g y , o r , i n an ad hoc m a n n e r a s t h e i n v e rs e o f E q u a t i o n ( 2 3 ) w i t h an e g a t i v e s ig n i n t h e s it u a t io n w h e r e t h e r e is m o r e o u t g o in g r e f l e c te d w a v e e n e r g yt h a n i n c o m i n g i n c i d e n t w a v e e n e r g y .

A s a c h e c k o f th e r e a s o n a b l e n e s s o f a li n e a r w a v e t h e o r y r e la t i o n sh i p b e t w e e n t h ep r e s s u r e a n d v e l o c i t y s ig n a ls , t h e c o h e r e n c y s q u a r e d c a n b e c o m p a r e d w i t h a v a l u e o f1 .0 s u g g es t i n g a p e r f ec t l i n ea r r e l a t i o n s h ip in t h e ca s e o f s m o o th e d co h e r en cy e s t i-m a t e s [ B e n d a t a n d P i e r so l ( 1 9 8 6 )] . T h e c o h e r e n c y s q u a r e d f o r th e p r e s s u r e a n d v e l o c i t ys ig n a l s i s d e f in ed a s [ B en d a t an d P i e r s o l ( 1 9 8 6 ) ] :


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2 4 8 T L . ~ ' A I T O N J R

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I s . m l eY P S,,,,,,,S/,t,,, 24 )

I u I = (25)t h e s p e c t r a o f t h e o n s h o r e - o f f s h o r e v e lo c i ty c o m p o n e n t s ig n al , a n d

Iv 2S p m = N ~ t 26 )t h e s p ec t r a o f t h e p re s s u re s i g n a l , an d

U * P mS P m - N A t 27 )t h e c ro s s - s p ec t r a o f t h e v e l o c i t y an d p re s s u re s i gn a ls , wh e re t h e q u an t i t y ( ) r ep re s e n t st h e c o m p l e x c o n j u g a t e o f t h e a r g u m e n t i n p a r e n t h e s i s .

R E S U L T SA s h o r t s e g m e n t o f t h e m e a s u r e d 34 m i n r e c o r d s o f p r e s s u r e h e a d ( m e t e r s o f w a t e r )a n d o n s h o r e - o f f s h o r e v e lo c i ty c o m p o n e n t ( m / s e c ) a r e s h o w n in F ig s 3 a a n d 7 a f o r th e

t . Sta0a t

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11/20

W a v e r e f le c t io n f r o m n a t u r a l b e a c h e s 2 4 9

D u c k ( N C ) s i t e f o r t h e r e c o r d p e r i o d s b e g i n n i n g 0 0 2 8 1 1 O c t o b e r 1 9 8 6 a n d 0 6 1 4 1 1O c t o b e r 1 9 8 6 . C o h e r e n c e e s t i m a t e s b e t w e e n t h e p r e s su r e a n d v e l o c i t y si g n a ls fo r t h es a m e p e r i o d s ( 3 4 m i n r e c o r d s) a s c o m p u t e d v i a E q u a t i o n ( 2 4 ) w i th s m o o t h e d s p e c tr a le s t i m a t o r s a r e s h o w n i n F i g s 3 b a n d 7 b , A s e x p e c t e d , t h e r e is a h i g h d e g r e e o f c o h e r e n c ef o r th e t w o s ig n a l s. S m o o t h e d s p e c t ra l e n e r g y o f m e a s u r e d p r e s s u r e h e a d Spp a n do n s h o r e - o f f s h o r e v e l o c i t y c o m p o n e n t S , , f o r t h e t w o t i m e p e r i o d s a r e s h o w n i n F i g s4 a n d 8 . S m o o t h e d s p e ct r al e n e r g y c o n t e n t o f i n c id e n t w a v e t r a in w a t e r s u r fa c e e l e v a t i o nS i a n d o f r e f le c t ed w a v e t r ai n w a t e r s u r f a c e e l e v a t i o n S t a s c o m p u t e d v i a E q u a t i o n s( 2 1 ) a n d ( 2 2 ) a r e s h o w n i n F i g s 5 a n d 9 . R e f l e c t i o n c o e f f i c i e n t c o m p u t e d f r o m t h es m o o t h e d i n g o i n g a n d o u t g o i n g w a v e t r a i n i s s h o w n a s a f u n c t i o n o f f r e q u e n c y i n F i g s6 and 10 .

T h e f r e q u e n c y p l o t s h a v e b e e n s m o o t h e d b y d i v i d in g r e c o r d le n g t h s i n t o 8 s u b r e c o r d s ,r e m o v i n g t h e m e a n o f t h e s u b r e c o r d s , c a l c u l a t i n g s p e c t r a o f t h e s u b r e c o r d s , a n d t h e na v e r a g i n g e n e r g y c o n t e n t w i t h i n f r e q u e n c y b i n s , t h e r e f o r e p r o v i d i n g a n e q u i v a l e n t o f1 6 d e g r e e - o f - f r e e d o m s m o o t h e d s p e ct r al e st i m a t o r s [ B e n d a t a n d P i er s o l ( 1 9 8 6 ) ] . I n t h ec o m p u t a t i o n o f t h e i n c i d e n t a n d r e f l e c t e d w a v e t r a i n s a h i g h f r e q u e n c y c u t - o f f o f0 . 2 5 H z w a s u t il iz e d t o p r e v e n t a m p l i fi c at io n o f n o i s e d u e t o t h e h y d r o d y n a m i c d a m p i n gf a c to r s in t h e d e n o m i n a t o r s o f E q u a t i o n s ( 1 9 ) a n d ( 2 0 ) . P l o t s o f b o t h S i a n d S , t h es p e c t r a l e n e r g y e s t i m a t e s o f t h e i n c i d e n t a n d r e f l e c t e d w a v e t r a i n e l e v a t i o n s , s h o wd a s h e d c u r v e s a b o v e a n d b e l o w t h e a v e r a g e f r e q u e n c y c o n t e n t o f th e s i g n a ls r e p r es e n t in go n e r o o t m e a n s q u a r e e r r o r f r o m t h e a v e r a g e a s d e fi n e d b y B e n d a t a n d P i e r so l [ ( 1 9 8 6 ) ,c h . 9 , p . 2 9 5 ] . M a x i m u m r e f le c t io n o v e r f r e q u e n c y b a n d s w i t h s i g n if ic a n t e n e r g y c o n t e n t

8

L,II.N 4( It8 . Ze,.

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t..N,(

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F I G . 4 .

D U C K 0 0 2 8 , I I OC T OB E R 1 9 8 6 (a)

8 8 . 8 5 8 . 1 8 . 1 5 8 . 2 8 . 2 5F r e q u e n c y ( h e r t z ) (b)

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G

t

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8 8 8 . 8 5 8 . 1 8 . 1 5 m . 2 8 . 2 5Fn,e q , , e n c y ( h e r t z )

( a ) V a r i a n c e s p e c t r u m o f p r e s su r e . ( b ) V a r i a n c e s p e c t r u m o f o n s h o r e - o f f s h o r e v e l o c it y .


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2 5 0 T . L . W A L T O N J R

LJ~N

paper_wave reflection from natural beaches

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