Scattering of gravity waves by multiple surface-piercing floating membrane

Applied Ocean Research 39 (2012) 40-52

C o n t e n t s l i s t s a v a i l a b l e a t S c i V e r s e S o l e n c e D i r e c t

Applied Ocean Research

j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o o a t e / a p o r

Scattering of gravity waves by multiple surface-piercing floating membrane

D. Karmakar, J. Bhattacharjee, C. Guedes Soares*

Centre for Marine Technology and Engineering (CENTEC), Instituto Superior Técnica, Technical University of Lisbon, Av. Rovisco Pais, 1049-001 Lisboa, Portugal

A R T I C L E I N F O A B S T R A C T

Article history: Received 21 April 2012

Received in revised form 22 July 2012 Accepted 2 October 2012

Keywords:

Surface-piercing breakwater Floating membrane Least-square approximation Surface gravity wave Wide-spacing approximation

Interaction of surface gravity w a v e s w i t h multiple vertically moored s u r f a c e - p i e r c i n g m e m b r a n e b r e a k w a t e r s in finite w a t e r depth is a n a l y z e d based on the linearized theory of w a t e r w a v e s . T h e study is carried out using least square a p p r o x i m a t i o n method to understand the effect of the vertical m e m b r a n e as effective b r e a k w a t e r Initially the p r o b l e m is studied for a single m e m b r a n e w a v e barrier but for the case of multiple m e m b r a n e breakwaters the study is carried out using the method of w i d e - s p a c i n g approximation. In the present study, it is observed that the deflection of the m e m b r a n e is reduced w i t h the increase in the stiffness parameter of the m o o r i n g lines attached to the membrane. In the case of single surface-piercing m e m b r a n e w i t h moored and fixed edge conditions, the reflection and transmission coefficients are compared and analyzed in detail. The resonating pattern in the reflection coefficients a r e also observed for multiple floating m e m b r a n e w h i c h can also be referred as Bragg's resonance. In the presence of the porosity constant the w a v e reflection is also observed to be decreasing a n d the change in the distance b e t w e e n the vertical floating b r e a k w a t e r s also helps in the attenuation of w a v e height. It is observed that the presence of multiple floating b r e a k w a t e r helps in the reduction of w a v e height in the transmitted region.

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1. Introduction

The p r o t e c t i o n o f harbors and other o f f s h o r e facilities near the coastline f r o m the i n c o m i n g w a v e attack has always been a concern to the ocean engineers and researchers. Suitable brealcwater systems are c o n s t m c t e d i n the ocean to p r o v i d e p r o t e c t i o n to the nearshore facilities b y r e d u c i n g t h e w a v e h e i g h t of the h i g h i n c o m i n g waves. Tra-d i t i o n a l l y , b o t t o m f i x e Tra-d r i g i Tra-d breakwaters are useTra-d f o r w a v e h e i g h t a t t e n u a t i o n . Poor b o t t o m f o u n d a t i o n , deep w a t e r regions, e n v i r o n -m e n t a l constraints -m a y o f t e n lead to the f a i l u r e o f these t r a d i t i o n a l breakwaters. I n this regard, f l o a t i n g f l e x i b l e breakwaters p r o v i d e an alternative t h a t can be used i n the coastal regions w i t h m i l d to m o d -erate w a v e c o n d i t i o n s . As a result, various researchers have proposed d i f f e r e n t types o f f l o a t i n g f l e x i b l e breakwater systems, the details o f w h i c h are described by McCartney and Bruce [1 ], Hales [2] and Oliver et al. [ 3 ] . These f l e x i b l e f l o a t i n g breakwaters are advantageous over the t r a d i t i o n a l b o t t o m m o u n t e d r i g i d ones as they are easy to carry, inexpensive, reusable, r a p i d l y deployable a n d removable. I n a d d i t i o n , the p a r t i a l f l e x i b l e b r e a k w a t e r s a l l o w the w a t e r c i r c u l a t i o n , the sedi-m e n t t r a n s p o r t and the safe passage of ocean life and t h e r e b y have less e n v i r o n m e n t a l i m p a c t s . However, the surface-piercing p a r t i a l break-w a t e r s m a y pose p r o b l e m s to the m a r i n e t r a n s p o r t and aesthetic c o n d i t i o n s . O n the o t h e r hand, h o r i z o n t a l m e m b r a n e s / e l a s t i c plates can also be used as brealcwater system t h a t allows t h e free m o v e -m e n t o f f i s h a n d f r e e passage o f seawater and s e d i -m e n t t r a n s p o r t .

* Corresponding author Tel.: +351 21 841 7957; fax: +351 21 847 4015. E-mail address: guedess@marist.utl.pt (C. Guedes Soares).

0141-1187/$ - see front matter ® 2012 Elsevier Ltd. All rights resei-ved. http://dx.d0i.0rg/l 0.1016/j.apor.2012.10.001

thus b e i n g f r i e n d l i e r to the e n v i r o n m e n t Several aspects o f w a v e i n -t e r a c -t i o n w i -t h f l o a -t i n g h o r i z o n -t a l m e m b r a n e s w e r e analyzed by Cho and K i m [ 4 - 6 ] , Yip et al. [ 7 ] , K a r m a k a r a n d Sahoo [ 8 ] , M i c h a i l i d e s and Angelides [ 9 ] , D i a m a n t o u l a l d and Angelides [10,11], Karmakar and Guedes Soares [12] a n d the l i t e r a t u r e c i t e d t h e r e i n .

In the past f e w decades, a l o t o f w o r k has been carried o u t to study the p e r f o r m a n c e o f v e r t i c a l f l e x i b l e b r e a k w a t e r f o r a t t e n u a t i o n of w a v e height. Lee and Chen [ 1 3 ] analyzed t h e w a v e i n t e r a c t i o n w i t h hinged flexible b r e a k w a t e r u s i n g e i g e n f u n c t i o n expansion m e t h o d . W i l l i a m s et al. [14] p e r f o r m e d studies o n the e f f i c i e n c y o f a v e r t i -cal elastic plate brealcwater c l a m p e d at the sea f l o o r . Isaacson [15] p r o v i d e d a detailed discussion o n the w a v e effects o n f l o a t i n g break-waters. A b u l - A z m [ 1 6 ] investigated the case o f dual h i n g e d b e a m and s h o w e d t h a t the e f f i c i e n c y o f the elastic-plate b r e a k w a t e r could be i m p r o v e d by t u n i n g t w o v e r t i c a l plates. M a h m o o d - u l - H a s s a n et al. [17] studied the w a v e i n t e r a c t i o n w i t h s u b m e r g e d elastic plate using e i g e n f u n c t i o n expansion m e t h o d . A t h r o u g h r e v i e w o f the d e v e l o p -m e n t s on wave i n t e r a c t i o n w i t h large f l o a t i n g f l e x i b l e elastic plates and ice sheets can be f o u n d i n W a t a n a b e et al. [ 1 8 ] and Squire [19] and the l i t e r a t u r e c i t e d t h e r e i n .

In order to m a k e the v e r t i c a l breakwaters m o r e e n v i r o n m e n t a l f r i e n d l y , the v e r t i c a l r i g i d b r e a k w a t e r can be replaced b y f l e x i b l e m e m b r a n e b r e a k w a t e r due to its added advantages i n a t t e n u a t i n g the w a v e h e i g h t The use o f v e r t i c a l f l e x i b l e m e m b r a n e as a p a r t i a l breakwater remains an i n t e r e s t i n g choice due to its l i g h t w e i g h t along w i t h the other advantages o f f l e x i b l e breakwaters. These m e m b r a n e breakwaters are made up o f s y n t h e t i c r u b b e r or a p o l y m e r i c m a t e r i a l w h i c h helps i n a b s o r b i n g the w a v e energy and t h e r e b y suppressing

D. Kannakaret al/Applied Ocean Researcl] 39 (2012)40-52 41

the s t r o n g vertical m o t i o n o f w a t e r particles to reduce the w a v e a m -p l i t u d e effectively. A -p a r t f r o m b e i n g used as -portable and sacrificial b r e a k w a t e r to protect h a r b o r entrances, a t t e n u a t i n g wave h e i g h t near the f l o a t i n g o f f s h o r e w i n d turbines, these f l o a t i n g flexible m e m b r a n e s can be used as o i l c o n t a i n m e n t b o o m to p r e v e n t oil spill i n the ocean, u n d e r w a t e r barrier to create a r t i f i c i a l fishing grounds f o r a q u a c u l t u r e activities and silt c u r t a i n to reduce the rate o f l i t t o r a l d r i f t [ 2 0 ] . These m e m b r a n e structures can be air d r o p p e d and self-erected at the site t h r o u g h a u x i l i a r y buoy. The shape and mass o f the system can be c o n t r o l l e d by filling i t w i t h w a t e r or air (see [ 2 1 , 2 2 ] ) .

Hence, f o r an e f f e c t i v e analysis o f v e r t i c a l m e m b r a n e as b r e a k w a -ter, T h o m p s o n et al. [ 2 3 ] i n v e s t i g a t e d the p e r f o r m a n c e o f a h i n g e d b u o y m e m b r a n e w a v e b a r r i e r a s s u m i n g constant added mass and w a v e d a m p i n g c o e f f i c i e n t o f the flexible m e m b r a n e . K i m and Kee [ 2 4 ] analyzed the i n t e r a c t i o n o f w a t e r waves w i t h a tensioned, u n -stretchable, vertical flexible m e m b r a n e e x t e n d e d to the sea bed us-ing b o t h e i g e n f u n c t i o n expansion m e t h o d a n d b o u n d a r y e l e m e n t m e t h o d . Further, Kee and K i m [ 2 5 ] s t u d i e d the w a v e i n t e r a c t i o n w i t h vertical membrane, h i n g e d at the sea floor and attached to a solid c y l i n d r i c a l b u o y at its t o p f o r b o t h submerged and s u r f a c e - p i e r c i n g b u o y / m e m b r a n e w a v e barriers using b o u n d a r y e l e m e n t m e t h o d .

I n a d d i t i o n to the flexible nature o f the brealcwater, porous structures are o f t e n suggested f o r b r e a k w a t e r systems as i t is m o r e e f f i -c i e n t i n dissipating the i n -c i d e n t w a v e energy and thus r e d u -c i n g the w a v e load o n the barrier [ 2 6 , 2 7 ] . Porous flexible structures are ef-f e c t i v e i n c o n t r o l l i n g the occurrence o ef-f resonance, t y p i c a l l y inside the s e m i - c o n f i n e d harbor regions. M o r e o v e r the p o r o s i t y o f the bar-rier enables the passage o f the undervvater stream. W a n g and Ren [ 2 8 ] presented a theoretical s t u d y o n the scattering o f s m a l l a m p l i -tude waves by a flexible, porous and t h i n b e a m - l i k e b r e a k w a t e r h e l d fixed i n the seabed. The i n t e r a c t i o n o f linear w a t e r w a v e i n a chan-nel o f constant d e p t h i m p i n g i n g on a v e r t i c a l t h i n porous b r e a k w a t e r w i t h a semi-submerged a n d fixed rectangular obstacle i n f r o n t o f i t was investigated by Yang et al. [ 2 9 ) . W u et al. [ 3 0 ] investigated the p h e n o m e n o n o f w a v e r e f l e c t i o n by a v e r t i c a l w a l l w i t h a h o r i z o n t a l submerged porous plate i n finite w a t e r d e p t h using e i g e n f u n c t i o n expansion m e t h o d . They observed t h a t the occurrence o f w a v e t r a p -p i n g near to the w a l l takes -place due to the -presence o f submerged p o r o u s plate. Liu and Li [ 3 1 ] presented an a l t e r n a t i v e analytical s o l u -t i o n f o r w a -t e r w a v e m o -t i o n over a s u b m e r g e d h o r i z o n -t a l porous pla-te u s i n g m a t c h e d e i g e n f u n c t i o n e x p a n s i o n approach. Recentiy, Evans and Peter [ 3 2 ] studied t h e linear w a t e r - w a v e r e f l e c t i o n by a sub-m e r g e d porous plate f o r b o t h the s e sub-m i - i n f i n i t e plate and the finite plate backed b y a solid w a l l using W i e n e r - H o p f m e t h o d and residue calculus technique.

W a v e i n t e r a c t i o n w i t h v e r t i c a l barriers has d r a w n s i g n i f i c a n t att e n att i o n o f atthe scienattific c o m m u n i att y f o r atthe p r o att e c att i o n of o f f s h o r e f a c i l -i t y -i n the recent decades. Several a n a l y t -i c a l and n u m e r -i c a l techn-iques are developed over the years to tackle this class o f p r o b l e m s . Lee and C h w a n g [ 3 3 ] used the least square m e t h o d to study the scattering and g e n e r a t i o n o f w a t e r waves b y v e r t i c a l permeable barriers. Sahoo et al. [ 3 4 ] studied the t r a p p i n g and g e n e r a t i o n o f surface waves by s u b m e r g e d vertical permeable barriers at one e n d o f a s e m i - i n f i n i t e l y l o n g channel o f finite w a t e r d e p t h . They analyzed d i f f e r e n t fixed bar-rier c o n f i g u r a t i o n s and the e f f e c t o f p e r m e a b i l i t y on the g r a v i t y w a v e p r o p a g a t i o n . Y i p et al. [ 3 5 ] e x t e n d e d t h e s t u d y o f Sahoo et al. [ 3 4 ] to i n c l u d e the flexible barriers used the e i g e n f u n c t i o n expansion a l o n g w i t h the o r t h o g o n a l m o d e - c o u p l i n g relations t o solve the b o u n d a r y value p r o b l e m . They m o d e l e d the b a r r i e r as a porous flexible elastic plate and applied the E u l e r - B e r n o u l l i b e a m e q u a t i o n a l o n g w i t h the porous b o u n d a r y c o n d i t i o n to describe the b a r r i e r e f f e c t M a n a m and Sahoo [ 3 6 ] developed a n a l y t i c a l s o l u t i o n s based o n the Havelock's t y p e expansion f o r m u l a e f o r r a d i a t i o n or scattering o f o b l i q u e w a t e r waves by a f u l l y extended porous barriers i n case o f t w o layer flu-ids. K u m a r and Sahoo [ 3 7 ] i n v e s t i g a t e d the p e r f o r m a n c e o f a v e r t i c a l porous flexible plate b r e a k w a t e r i n t w o layer fluids o f finite depths.

The b r e a k w a t e r was assumed to be e x t e n d e d over the e n t i r e w a t e r d e p t h . The o r t h o g o n a l m o d e c o u p l i n g relations f o r t w o layer fluids a l o n g w i t h the least square m e t h o d w e r e u t i l i z e d to solve the b o u n d -ary value p r o b l e m . K u m a r et al. [ 3 8 ] e x t e n d e d the s t u d y to analyze v e r t i c a l m e m b r a n e barrier i n t w o layer fluids.

The effectiveness o f v e r t i c a l flexible barriers o f various c o n f l g -urations is s t u d i e d using d i f f e r e n t m a t h e m a t i c a l m e t h o d s by ana-l y z i n g the m u t u a ana-l response o f the s t r u c t u r e and the surface waves. W i l l i a m s [ 3 9 ] t h e o r e t i c a l l y investigated the h y d r o d y n a m i c properties o f a flexible floating brealcwater c o n s i s t i n g o f a m e m b r a n e s t r u c t u r e attached to a s m a l l float restrained by m o o r i n g s u s i n g b o u n d a r y i n t e -gral e q u a t i o n m e t h o d . The r e f l e c t i o n and t r a n s m i s s i o n characteristics in t h i s case are observed to be sensitive to the spacing b e t w e e n the m e m b r a n e and the t e n s i o n parameter. Cho et al. [ 4 0 ] i n v e s t i g a t e d the o b l i q u e w a v e i n t e r a c t i o n w i t h a d u a l v e r t i c a l flexible m e m b r a n e w a v e barrier h i n g e d at the sea floor u s i n g b o u n d a r y e l e m e n t m e t h o d . It is observed t h a t the w a v e b l o c k i n g e f f i c i e n c y s i g n i f i c a n t l y increases in n o r m a l and o b l i q u e i n c i d e n t waves w i t h the use o f d u a l flexible m e m b r a n e . Lo [ 4 1 , 4 2 ] analyzed the p e r f o r m a n c e o f v e r t i c a l flexible m e m b r a n e w a v e barrier o f a finite e x t e n t using e i g e n f u n c t i o n e x p a n -sion m e t h o d . Lee a n d Lo [ 4 3 ] s t u d i e d the p e r f o r m a n c e o f m u l t i p l e surface p e n e t r a t i n g flexible w a v e barriers o f finite d r a f t u n d e r the as-s u m p t i o n o f l i n e a r i z e d w a v e t h e o r y and as-small m e m b r a n e reas-sponas-se. Here, b o t h theoretical and e x p e r i m e n t a l studies are p e r f o r m e d f o r the m u l t i p l e barriers and i t is observed t h a t increasing t h e t e n s i o n o f the m e m b r a n e decreases the w a v e transmission. Liu and Li [ 4 4 ] e x a m i n e d the h y d r o d y n a m i c p e r f o r m a n c e o f a w a v e a b s o r b i n g d o u b l e c u r t a i n -w a l l p a r t i a l b r e a k -w a t e r t h a t consists o f a sea-ward p e r f o r a t e d -w a l l and a s h o r e w a r d i m p e r m e a b l e w a l l . Liu a n d Li [ 4 5 ] o b t a i n e d the s o l u t i o n o f w a t e r w a v e i n t e r a c t i o n w i t h submerged p e r f o r a t e d s e m i - c i r c u l a r breakwaters by u s i n g the m u l t i p o l e m e t h o d .

A l t h o u g h t h e scattering o f w a t e r waves by obstacles has been w e l l d o c u m e n t e d i n the l i t e r a t u r e , the scattering o f w a t e r waves b y m o o r e d vertical s u r f a c e - p i e r c i n g porous m e m b r a n e b r e a k w a t e r has n o t received m u c h a t t e n t i o n to the best o f the k n o w l e d g e o f the authors. Therefore, i n the present study, m u l t i p l e s u r f a c e - p i e r c i n g m o o r e d porous b r e a k w a t e r s are analyzed i n d e t a i l f o r various m o o r e d and fixed edge c o n d i t i o n s . The analysis is c a r r i e d o u t to under-stand the effectiveness o f the m o o r i n g present i n the s u r f a c e - p i e r c i n g b r e a k w a t e r . N u m e r i c a l c o m p u t a t i o n s are p e r f o r m e d o n w a v e i n t e r -a c t i o n w i t h m u l t i p l e flo-ating vertic-al m e m b r -a n e s u s i n g w i d e - s p -a c i n g a p p r o x i m a t i o n f o r various porosity constants a n d t h e distance bet w e e n bethe membranes. I n bethe n e x bet secbetion, bethe m a bet h e m a bet i c a l f o r m u l a t i o n o f the b o u n d a r y value p r o b l e m (BVP) f o r g r a v i t y w a v e i n t e r -a c t i o n w i t h m u l t i p l e surf-ace-piercing v e r t i c -a l p o r o u s m e m b r -a n e s i n w a t e r o f finite d e p t h is described i n d e t a i l .

2. Mathematical formulation

The BVP associated w i t h w a v e scattering b y m u l t i p l e surface-p i e r c i n g surface-porous m e m b r a n e s i n w a t e r o f finite d e surface-p t h h is f o r m u l a t e d u n d e r the assumptions o f linear wave t h e o r y . The p r o b l e m is analyzed in the t h r e e - d i m e n s i o n a l Cartesian c o - o r d i n a t e s y s t e m w i t h x-z b e i n g the h o r i z o n t a l plane a n d y - a x i s being v e r t i c a l l y d o w n w a r d positive. The fluid occupies the regions -oo < x, z < oo and 0<y <h w i t h y = 0 a n d - o o < x,z < CO b e i n g the u n d i s t u r b e d w a t e r surface. It is assumed t h a t N n u m b e r s o f t h i n porous m e m b r a n e s o f equal l e n g t h d{d < h) are placed v e r t i c a l l y w i t h ends either fixed or m o o r e d as s h o w n i n Fig. 1. The first m e m b r a n e is placed at x = - Q ] , z = 0 a n d a l l the f o l l o w i n g m e m b r a n e s are placed i n l i n e w i t h t h e first one b u t are n o t equally spaced. The fluid d o m a i n is d i v i d e d i n t o N +1 s u b - d o m a i n s , n a m e l y

D l = ( - O l < x < o o , 0 < y <h, , - oo < z < o o ) , D j + i = ( - O j + i < X < - Q j , 0 < y < h, - o o < z < o o ) ,

i = 1 , 2 , . . . N - l

42 D. Kannakar et al./Applied Ocean Research 39(2012) 40-52

Trnnsmiitcd wïivc ,x-=-3-j _{=-a; x=-ai riidik-iU wave}

Ar

i'oroi!^ Membniiie

NTooring üiu's

>• X

Fig. 1. Schematic diagram for multiple surface-piercing membrane.

as s h o w n i n Fig. 1. Hereafter, the subscripts w i l l denote the para-m e t r i c values i n the respective f l u i d regions.

A m o n o c h r o m a t i c surface g r a v i t y w a v e is o b l i q u e l y i n c i d e n t at an angle 9 o n t h e m e m b r a n e at x = - O i , y = /i, w h i c h t h e n propagates t h r o u g h the m e m b r a n e s at x = Oj, y = / i f o r j = 2, 3 N and f i -nally t r a n s m i t s the energy i n the s e m i - i n f i n i t e region D/v+i. A s s u m i n g t h a t the f l u i d is i n v i s c i d as w e l l as incompressible and the m o t i o n is i r r o t a t i o n a l and s i m p l e h a r m o n i c i n t i m e and a l o n g the z-axis, the v e l o c i t y p o t e n t i a l * j (x,y,z,t) and the associated free surface e l e v a t i o n X j ix,z,t) are expressed i n the f o r m * j (x,y,z,t) = Re {4>j{x,y)e''^'''^'} and X] (^,z,f) = Re { f j ( x ) e ' ' ^ - ' ^ ' } w h e r e Re denotes the real part, co is the angular f r e q u e n c y o f i n c i d e n t wave, and (is the c o m p o n e n t o f the w a v e n u m b e r along z-axis. The spatial v e l o c i t y p o t e n t i a l (pj{x,y) f o r j = 1,2 N + 1 satisfies the g o v e r n i n g p a r t i a l d i f f e r e n t i a l e q u a t i o n g i v e n by

(•^iy ^ </>; = 0 o n - oo < X < oo, 0 < y < h. (1)

w h e r e V^^ = {iP/dx^ + i^'^/ay'^). I n the o p e n w a t e r r e g i o n , the f r e e surface b o u n d a r y c o n d i t i o n is g i v e n b y

+ K<pj=0, o n y = 0, x e ( - o o , - a w )

(2) u . . . ( - a j . - a j _ , ) . - - u ( - a i , o o ) .

w h e r e K = co'^/g. The n o f l o w c o n d i t i o n at the r i g i d u n i f o r m b o t t o m b o u n d a r y is g i v e n by

-P-=0

on y = h,

i)y

oo < X < oo. (3)

The m e m b r a n e is assumed to be porous and is d e f l e c t e d h o r i z o n -t a l l y w i -t h -the d i s p l a c e m e n -t gj{y,z,-t) = R e { ^ j ( y ) } e ' ' ^ - ' ' ^ f , w h e r e (y) is the c o m p l e x d i s p l a c e m e n t a m p l i t u d e o f the m e m b r a n e . Thus the b o u n d a r y c o n d i t i o n o n the porous m e m b r a n e barriers a t x = -aj,j = 1 , 2 , . . . , N based on the Darcy's l a w is g i v e n by

^ = l y o G - 0 j ) - ico^j o n y s (0, d) f o r j = 1,2 N ( 4 a ) w h e r e G = Gr + iG, is the c o m p l e x porous e f f e c t parameter as d e f i n e d by Y u and C h w a n g [46] a n d is g i v e n by

r _

nf

+ 'S)

K o d ( / 2 + S2)' (4b)

i n w h i c h y is the p o r o s i t y constant, ƒ is the resistance force c o e f f i c i e n t , S is the i n e r t i a l force c o e f f i c i e n t , d is the thickness o f the porous m e d i u m and yo is the w a v e n u m b e r o f the i n c i d e n t wave. The real p a r t Gr represents the resistance e f f e c t o f the porous m a t e r i a l against the seepage flow w h i l e t h e i m a g i n a r y part G; denotes the inertia e f f e c t of the fluid inside the porous m a t e r i a l .

The c o n t i n u i t y o f v e l o c i t y and pressure across the free flowing interfaces y e (d, h) b e l o w the m e m b r a n e s at x = ~aj,j = 1,2 N yields

(Pi=^j+-\ o n y e ( d , h ) , x = - a j .

C o m b i n i n g Eq. (4a) and the second c o n d i t i o n o f (4c), the b o u n d a r y c o n d i t i o n on all a l o n g the m e m b r a n e b a r r i e r and the gap is g i v e n by

{(pj+i - <Pj) il<oC dx 0

i + icü^j f o r 0 < y < d, f o r d < y < /!.

(5)

The e q u a t i o n o f m o t i o n f o r the flexible m e m b r a n e b a r r i e r acted u p o n b y the fluid pressure and a p o r t i o n o f w h i c h is i n t o u c h w i t h the atmosphere is g i v e n b y (see [ 4 7 ] ) :

(6) w h e r e T is the t e n s i o n a l o n g the m e m b r a n e , iris is the mass o f the m e m b r a n e per u n i t area and p is the d e n s i t y o f w a t e r . For the u n i q u e s o l u t i o n o f the BVP, edge c o n d i t i o n s are to be prescribed at the end o f the m e m b r a n e . Physically, at least one end o f t h e m e m b r a n e is to be k e p t fixed or m o o r e d to other fixed or floating s t r u c t u r e i n order to keep the m e m b r a n e at a desired p o s i t i o n o f interest. In the present study, t h r e e d i f f e r e n t types o f edge b e h a v i o r f o r t h e surface-piercing m e m b r a n e s are prescribed, w h i c h are as f o l l o w s :

(i) Fixed-moored. I n this case the m e m b r a n e is assumed t o be fixed at the u p p e r end at y = 0 and m o o r e d t h r o u g h a s p r i n g at the l o w e r end a t y = d. Therefore, the edge c o n d i t i o n s are g i v e n by

ay

where/<d = 2/cj sin^ o; is the s p r i n g constant, o; is the m o o r i n g angle and l(] is the s p r i n g stiffness.

(ii) ivioored-fixed. I n this case the m e m b r a n e is assumed to be m o o r e d a t y = 0 a n d fixed a t y = d. Hence, the edge c o n d i t i o n s are g i v e n by

( y ) = 0 at y = 0 and l<d^j a t y = d. (7a)

ay l<d^j at y = 0 and ( y ) = 0 at y = d. (7b)

( i i i ) Moored-moored. In this case the m e m b r a n e is assumed to be m o o r e d t h r o u g h springs at b o t h ends a t y = 0 a n d y = d. Hence, the edge c o n d i t i o n s are g i v e n by

^ = / < d , ^ j a t y = 0 and 'Él = i ^ j j y = d. (7c)

(8) w h e r e and i<d^ are the s p r i n g c o n s t a n t at u p p e r and l o w e r ends, respectively. Further, the v e l o c i t y p o t e n t i a l s s a t i s f y t h e far field r a d i -a t i o n c o n d i t i o n g i v e n by

w h e r e RQ and TQ are the c o m p l e x a m p l i t u d e s o f the r e f l e c t e d and t r a n s m i t t e d waves, respectively and yo is the w a v e n u m b e r o f the i n c i d e n t w a v e s a t i s f y i n g the dispersion r e l a t i o n :

Yo tanh Yoh ~ K = 0, ( 9 ) w i t h / = y o sin 9, = /<QF!^. In the n e x t section, the s o l u t i o n p r o

-cedure f o r the o b l i q u e w a v e scattering by m u l t i p l e s u r f a c e - p i e r c i n g v e r t i c a l m e m b r a n e w i l l be described i n d e t a i l .

N.B: The s i n g u l a r i t y o f the v e l o c i t y near the tip o f the m e m b r a n e is observed i n the s t u d y o f vertical m e m b r a n e as b r e a k w a t e r (see [48]). The s i n g u l a r i t y i n the velocity field is g o v e r n e d b y the r e l a t i o n :

V(/) = 0 ( r - ' / 2 ) 0. (10)

w h e r e r denotes the distance o f an a r b i t r a r y p o i n t f r o m the edge o f the b a r r i e r . I t m a y be noted t h a t the m a t c h e d e i g e n f u n c t i o n expansion m e t h o d does n o t account f o r the above c o n d i t i o n e x p l i c i t i y . H o w e v e r , the specific edge c o n d i t i o n s prescribed at the m e m b r a n e edges do

D. Karmakaret al./Applied Ocean Research 39 (2012) 40-52 43 Traiisniitled wave

A

r

x - 0 Incident wave Porous Membrane

Ar

M o o r i n g linos y=d y=h Fig. 2. Scliemadc diagram for single surface-piercing membrane.

tal<e care o f tlie s i n g u l a r i t y b e h a v i o r i n t h e s o l u t i o n o f the physical p r o b l e m .

3. Method of solution

I n i t i a l l y the s o l u t i o n procedure is described f o r the case o f single v e r t i c a l porous m e m b r a n e , i.e. f o r N = 1 f o r better u n d e r s t a n d i n g o f the m e t h o d . Under the assumptions o f linear t h e o r y o f w a t e r waves, the v e l o c i t y potentials i n each r e g i o n can be expanded i n t e r m s o f ap-p r o ap-p r i a t e e i g e n f u n c t i o n s w i t h u n k n o w n coefficients by the m e t h o d of e i g e n f u n c t i o n expansion. The u n k n o w n c o e f f i c i e n t s can be d e t e r -m i n e d either by t h e -m e t h o d o f least square a p p r o x i -m a t i o n or b y a p p l y i n g the o r t h o g o n a l properties o f t:he e i g e n f u n c t i o n s . Both the approaches w i l l lead to a l i n e a r system o f algebraic equations to be solved n u m e n c a l l y to d e t e r m i n e the u n k n o w n s . Here, i n the present study, t h e m e t h o d o f least square a p p r o x i m a t i o n is applied to o b t a i n the f u l l s o l u t i o n o f the BVP. The same approach can be adhered to solve the p r o b l e m o f m u l t i p l e membranes. A l t h o u g h , i t m a y be n o t e d t h a t increasing the n u m b e r o f membranes w i l l drastically increase the t o t a l n u m b e r o f u n k n o w n s and thus the s o l u t i o n procedure w i l l become cumbersome. Therefore, w a v e scattering b y m u l t i p l e surfacep i e r c i n g m e m b r a n e s is solved b y the m e t h o d o f w i d e s surfacep a c i n g a surfacep surfacep r o x -i m a t -i o n , w h -i c h assumes t h a t t h e d-istance b e t w e e n t w o consecut-ive m e m b r a n e s is large enough to neglect the local effects near one m e m -brane due to the presence o f the o t h e r one. This approach d r a s t i c a l l y reduces the n u m b e r o f u n k n o w n s to be d e t e r m i n e d and hence saves c o m p u t e r m e m o r y a n d time f o r n u m e r i c a l evaluation. In the n e x t sub-section, the e i g e n f u n c t i o n expansion c o m b i n e d w i t h the least square a p p r o x i m a t i o n is described f o r single m e m b r a n e .

3. J. Single vertical porous tnembrane

A single v e r t i c a l porous m e m b r a n e is considered to be placed at the o r i g i n x = 0 as s h o w n i n Fig. 2. Therefore, the w h o l e d o m a i n o f consideration is d i v i d e d i n t o t w o regions as s h o w n i n Fig. 2 and the BVP described i n Section 2 is a c c o r d i n g l y m o d i f i e d f o r N = 1. I t m a y be n o t e d t h a t the m e m b r a n e can be placed at any distance f r o m the o r i g i n and a s i m p l e t r a n s l a t i o n o f the axis w i l l lead to a m e m b r a n e placed at o r i g i n . Hence, the m e m b r a n e is placed at the o r i g i n f o r the sake o f n u m e r i c a l convenience w i t h o u t any loss o f g e n e r a l i t y . The v e l o c i t y p o t e n t i a l ^ j ( x , y ) , j = 1,2 satisfies the H e l m h o l t z equation i n the fluid d o m a i n as i n Eq. (1) a l o n g w i t h the b o u n d a r y c o n d i t i o n s (2) and ( 3 ) .

Using the expansion f o r m u l a e , the v e l o c i t y potentials i n each o f the t w o regions are g i v e n by

0 1 (X, y ) = (e-'**'^ + Roe''^") fo ( y ) + £ R„e-'"''fn ( y ) f o r x > 0, (11) 0 2 ( x , y ) = Toe-''^'Vo ( y ) + J2 Tne'"'fn ( y ) f o r x < 0.

w h e r e R„, T,,, n = 0,1,2... are the u n k n o w n constants to be d e t e r m i n e d . The e i g e n f u n c t i o n s / n ( y ) ' s are g i v e n b y r , , c o s h Y n l h ~ y ) ^ fn(y) = — . . . . . . ' f o r n = 0. and / n ( y ) = cosh Ynh cos Sn{h-y)

t o r n = 1,2,

(12) cos Snh

w h e r e Yn satisfies the dispersion r e l a t i o n

yn t a n h / n h - / c = 0. f o r n = 0. ( 1 3 ) w i t h Yn = iSn f o r n = 1,2... The d i s p e r s i o n r e l a t i o n i n Eq. (13) has

t w o real roots ± yo and i n f i n i t e n u m b e r s o f p u r e l y i m a g i n a r y roots y „ = ± !(5nwith52 = A : 2 - ; 2 f o r n = 1 , 2 . . . . I n the present study, o n l y the p o s i t i v e roots are considered f o r the sake o f boundedness o f the s o l u t i o n .

The c o n t i n u i t y o f h o r i z o n t a l v e l o c i t y across the porous m e m b r a n e as w e l l as the gap holds good and t h a t y i e l d s

^ 902 ax ax a t x = 0, y e ( 0 , / I ) . S u b s t i t u t i n g ^ ] and 0 2 h o r n E q . ( 1 1 ) i n E q . ( 1 4 ) g i v e s Ro + To = ^. i?„ + r „ = 0, f o r n = l , 2 . . . . (14) (15) U p o n s u b s t i t u t i n g Eq. (15) i n the p o t e n t i a l f u n c t i o n 0 j ( x , y ) , j = 1,2 w e have

M (X, y ) = {e-"^" + Roe'i^') fo (y) - F J2 Rm-""'fn (y) for x > 0, n=l

02 ( X , y) = (1 - J?o) e-'*^Vo (y) - E i^ne^"Vn (y) for x < 0. (16)

A l o n g the line x = 0, w e d e f i n e the pressure p o t e n t i a l 0p, as

OC

0p = ( 0 i - 0 2 ) = 2 / ? o / o ( y ) + 2 E i ? n / n ( y ) a t x = 0. • ( 1 7 ) n=l

S u b s t i t u t i n g 0p i n Eq. (6) and t h e n s o l v i n g the d i f f e r e n t i a l equa-tion, the m e m b r a n e d e f l e c t i o n ^ i ( y ) is o b t a i n e d as

^ , ( 3 , ) = ; , ^ + P ^ i " ^ o y 2ipw

cos Sod sin Sgd

/ o ( y ) + f : - • / n ( y ) ,

(18)

w h e r e A, B are u n k n o w n constants associated w i t h t h e m e m b r a n e d e f l e c t i o n .

S u b s t i t u t i n g 0 j , J = 1,2 f r o m Eq. ( 1 6 ) and § , ( y ) f r o m Eq. (18) i n the first c o n d i t i o n o f Eq. (5) f o r 0 < y < d, w e o b t a i n

Ro\il<o + 2iYoC

f^RA-kn

+ liyoG + 2pco^

My)-fn ( y ) - i/<o/o ( y )

(19)

M r o A ' ^ ^ + i c o B ' - ^ m ^ O f o r y e ( 0 , d ) .

cos Sod sin Sod

A p p l y i n g the c o n t i n u i t y o f pressure across the gap y e (d, /i) as i n the second c o n d i t i o n o f Eq. (5), w e have

J ? o / o ( y ) + Ë i ? n / n ( y ) = 0 for y e ( d . / I ) (20)

Next, the edge conditions as i n Eqs. ( 7 a ) - ( 7 c ) are used f o r the d e t e r m i n a t i o n o f the u n k n o w n constants A and B i n t e r m s o f the u n k n o w n s Rn, n = 0 , 1 , 2

4 4

( i ) Fixed-moored. In this case the m e m b r a n e is assumed to be fixed at y = 0 a n d m o o r e d at y = d. Hence, using the a p p r o p r i a t e edge c o n d i t i o n t h e values o f A a n d B are g i v e n by

D. Kannakaret al./Applied Ocean Research 39(2012) 40-52

f u n c t i o n H ( y ] can be d e f i n e d as

A = 2ipco cos Sod

2ipco R„ Ro ( 2 1 a ) r ( r s o cos s o d - / < d ) ( 2 1 b ) [Tf'o (d) - l<dfo ( d ) + cos s o d ( r s o tan sod + l<d)\

+ y ^

ti{-yn'+s'o)

{T ƒ ' „ (d) -ƒ<<,ƒ„ ( d ) + cos S o d ( r s o tan Sod + l<d)}].

(ii) Moored-flxed. I n this case the m e m b r a n e is assumed t o be m o o r e d at y = 0 a n d fixed at y = d. Hence, using the a p p r o p r i a t e edge c o n d i t i o n the values o f A and B are g i v e n b y

2ipcü ( r f o i O ) -Ro l<d + foid) cos Sod | / „ ( d ) - - l ( r / ' , ( 0 ) - / < d + \ 1 ^ _Rn COS Sod t l { - Y n ' + S ^ ) n ( d ) ) (22a) 2ipco

T P 7

Th'{0)-l<d + kd cos Sgd / o ( d )

+ i : ^ ^ ^ j ^ / n ' ( 0 )

-„=1 ( - y „ 2 + s2)

I

i<d + kd COS Sod fn{d) (22b)

w h e r e Po = l(/Cd/cos Sod) + {Tso/sin Sod)}.

( i i i ) Moored-moored. I n this case the m e m b r a n e is assumed t o be m o o r e d at b o t h ends a t y = 0 and y = d. Thus, the edge c o n d i t i o n s y i e l d 2ipco 2ipa> Ro Rn Po (23a) Ro

' o ( o ) - f e d , ) ^ l + E

Qnfkd^ , Po U s o (23b) + t a n s o d + ( r / ' o ( 0 ) - / i d , sin Sod

w h e r e PQ = {/sü, - kd^ - T tan sod - ( / C d , k d ^ / T s o ) tan sod), Qo =

{Tf'o id) - kd, fo (d)) - (Tso cot sod - kd,) {Tf'o (0) - kd,) ,

(T ƒ ' „ ( d ) - l<d, fn ( d ) ) - ( r s o cot Sod - l<d,) {Tf'n (0) - l<d,) s i n Sod rso

It is e v i d e n t f r o m Eqs. (19) and ( 2 0 ) t h a t the o r t h o g o n a l properties o f the e i g e n f u n c t i o n s cannot be used i n the usual m a n n e r to o b t a i n a linear system o f equations. H o w e v e r , f r o m Eqs. ( 1 9 ) and (20), a n e w

H [ y ] =

Ro i/<o + 2iyoG + _{H y )}

t{ I T{-Y^+sl)

-ik,S,(y) + ia,A'^^+icB'^'^y

cos Sod ' • " sin

sod

RoSo{y) +

Y,Rnfn(y)

= o

11=1 w h i c h can be f u r t h e r r e d e f i n e d as. w [ y ] = R o / o ! y l + E ^ " / " W + M 3 ' l . n=l w h e r e = 0 for y e (0. d ) , for y e (d. / i ) , (24) / o [ y l ^ fn[y\ i7(b + 2 i y o G + -/ o ( y ) , lin + 2iYoG + / n ( y ) . 2püy-2pui^ / o ( y ) f o r y s ( 0 , d ) . f o r y s (d, h), f o r y e ( 0 , d ) , f o r y s ( d , / i ) , (25a) (25b) (25c) 0, f o r y e ( d . h ) .

N o w o n a p p l i c a t i o n o f the least square a p p r o x i m a t i o n m e t h o d w e have,

fh

I IH [ y ] | dy = m i n i m u m . ( 2 6 )

J

0

M i n i m i z i n g the above i n t e g r a l w i t h respect to each i?„, leads to

/ H * [ y ] W , n [ y ] d y = 0 f o r m = 0 , 1 . 2 N , ( 2 7 ) J 0

w h e r e H''[y] = R'^f^[y\ + Y.Z,RtJ*[y\ + h'\y\ and H ^ y l = •dH\y\/dRm w i t h * denotes the c o m p l e x c o n j u g a t e . Thus, i n the b a r r i e r region, the respective derivatives to p e r f o r m the i n t e g r a t i o n are g i v e n b y

3R~o

3H

as,,

i/Co + 2 i y o G + - k n + 2 i y o G + 2pa)2 r ( y o ' + % ' ) 2pu)

fo{y)

f o r y e ( O . d ) , (28a) T{-y^+si)^

and the derivatives i n the gap r e g i o n are g i v e n by

= / o ( y )

f o r y 6 ( d . ; ! ) . (28b)

S u b s t i t u t i n g H*[y] and Hm[y] i n Eq. ( 2 7 ) a n d u p o n t r u n c a t i o n o f the i n f i n i t e series sums up to a finite n u m b e r o f t e r m s M (say), w e d e r i v e

M

E ' ^ n ' ^ ' n n = Öm f o r m = 0. 1, 2 . . . . M , ( 2 9 ) n=0

w h i c h is a system o f ( M + 1) x ( M + 1) linear equations f o r the d e t e r m i n a t i o n ofR*n w i t h X „ , n and fa,„ g i v e n b y

Xmn = j Jn\y]fm\y]dy and bm= / h*[y\f„,[y\dy f o r m, n = 0 , 1

./ 0

D. Kannakaret al./Applied Ocean Research 39 (2012)40-52 45

The r e f l e c t i o n and t r a n s m i s s i o n c o e f f i c i e n t s Kr and Kt are d e f i n e d as

Kr = \Ro\ and fC, = ITQI = |1 - RQ! (31)

In the n e x t subsection, the study o f w a v e scattering by a s i n -gle surface-piercing m e m b r a n e is f u r t h e r extended to study w a v e s c a t t e r i n g b y m u l t i p l e s u r f a c e - p i e r c i n g m e m b r a n e s . I t m a y be n o t e d t h a t the same matched e i g e n f u n c t i o n expansion and the least square a p p r o x i m a t i o n can be applied to o b t a i n t h e a n a l y t i c a l s o l u t i o n f o r m u l t i p l e membranes. H o w e v e r , t h e n u m b e r o f u n k n o w n s w i i l be i n -creased drastically w i t h increasing n u m b e r o f m e m b r a n e s . Therefore, the m a t h e m a t i c a l calculations w i l l become c u m b e r s o m e that w i l l lead to m o r e c o m p u t a d o n m e m o r y and t i m e . In t u r n the m e t h o d becomes i n e f f i c i e n t . Hence, the m e t h o d o f w i d e spacing a p p r o x i m a d o n is ap-p l i e d under a ap-p ap-p r o ap-p r i a t e c o n d i t i o n s to o b t a i n the s o l u t i o n f o r m u l t i ap-p l e membranes, the details o f w h i c h are described i n the n e x t subsection.

3.2. Multiple vertical porous membranes

The m e t h o d o f w i d e spacing a p p r o x i m a t i o n is based o n the as-s u m p t i o n t h a t the dias-stance b e t w e e n the t w o conas-secutive m e m b r a n e as-s is m u c h larger t h a n the w a v e l e n g t h o f the i n c i d e n t plane progres-sive w a v e w h i c h ensures t h a t the local e f f e c t is negligible and the evanescent modes do n o t c o n t r i b u t e to t h e s o l u t i o n (as i n [ 4 9 , 5 0 ] ) . A s s u m i n g t h a t N n u m b e r o f floating s u r f a c e - p i e r c i n g m e m b r a n e s are placed w i d e l y apart, the a s y m p t o t i c f o r m o f the v e l o c i t y p o t e n d a l 0 j ( x ) f a r a w a y f r o m the m e m b r a n e s i n the respective regions are g i v e n by /

0, = e-''*" + RNe"^)\ - a , < x < o c .

-. AjC-'V j = 1.2 N - l . (32)

U s i n g the m a t c h i n g c o n d i t i o n s across the v e r t i c a l m e m b r a n e , a s y s t e m o f 2 N linear equations associated w i t h 2N u n k n o w n s K N , TN, A j , Bj, j = 1 , 2 , . . . , N - 1 are o b t a i n e d as g i v e n b y Rwe-"^"! = n e ' ' * " ' i - i - B i f , e - ' ' * ' " i . A j - e - " ^ " / = f l Aj_,e''*)''J + B j r i e - ' ' ^ ° J , B j . e - ' ' ^ " j + i = A j r i e ' ' ^ ' ' J + i + ti B j + i e-"'o''j+i, r ^ e ' M N = A w - i t i e " * " " , (33)

w h e r e n and t i correspond to the a m p l i t u d e o f the reflected and t r a n s m i t t e d waves f o r single s u r f a c e - p i e r c i n g m e m b r a n e i n i s o l a t i o n w i t h Ao = 1 and Bu = 0. Solving the above s y s t e m o f equations, u n k n o w n s Rfi and TN are o b t a i n e d a n d i n t u r n | R N I and [1^1 w i l l c o r r e -s p o n d to the r e f l e c t i o n and t r a n -s m i -s -s i o n coefficient-s, re-spectively.

I t is n o t e w o r t h y to m e n t i o n t h a t M a r c h e n k o and V o l i a k [51 ] and Porter and Evans [ 5 2 ] had separately o b t a i n e d the r e l a t i o n f o r r e f l e c -t i o n a n d -transmission c o e f f i c i e n -t s using w i d e - s p a c i n g a p p r o x i m a -t i o n for m u l t i p l e cracks i n floating ice-sheet. M a r c h e n k o and Voliak [ 5 1 ] d e r i v e d the e x p l i c i t f o r m u l a e to d e t e r m i n e the r e f e c t i o n and trans-m i s s i o n coefflcients f o r N n u trans-m b e r o f cracks as g i v e n b y p n , ( r w _ i ) ^ e 2 " ^ ° j i ? i = l ^ R i i ^ N . i e ^ ' M ; ^ " d ^ ' ^ r w - i T i e ' ^ ' - J R i A-R-iRN-ie^''^'! (34)

On the o t h e r hand. Porter and Evans [ 5 2 ] p r o v i d e d the f o r m u l a e f o r the r e f l e c t i o n and t r a n s m i s s i o n c o e f f i c i e n t s i n case o f N n u m b e r o f cracks as g i v e n by

R and T N = (35)

CTN-ATICTN-I """" A f ^ - i ( o - N - A T i a w - i ;

w h e r e X = e'**"J w i t h R-i and Ti are the r e f l e c t i o n a n d transmission c o e f f i c i e n t s f o r single crack a t x = 0 w i t h

s m ( N Q ! ) , cos(arg(ri)-I-<na;

aw = — and cos

a

= v

B\

^ u

sma ITil (36)

The n u m e r i c a l c o m p u t a t i o n shows t h a t the results o b t a i n e d by the present f o r m u l a e as described i n Eq. ( 3 3 ) are i d e n t i c a l w i t h t h e results o b t a i n e d b y the f o r m u l a e g i v e n by M a r c h e n k o a n d V o l i a k [ 5 1 ] and Porter a n d Evans [ 5 2 ] . However, the f o r m u l a e g i v e n by M a r c h e n k o and V o l i a k 151] r e q u i r e the results f o r single and ( N 1) n u m b e r o f cracks i n o r d e r to o b t a i n the results f o r N n u m b e r o f cracks.

4. Results and discussion

The i n t e r a c t i o n o f oblique surface g r a v i t y waves w i t h single and m u l t i p l e s u r f a c e - p i e r c i n g v e r t i c a l porous m e m b r a n e s i n w a t e r o f fi-n i t e d e p t h is s t u d i e d by a fi-n a l y z i fi-n g the r e f l e c t i o fi-n c o e f f i c i e fi-n t /Cr, trafi-ns- trans-m i s s i o n c o e f f i c i e n t Kt, d e f l e c t i o n o f the trans-m e trans-m b r a n e ^ j ( y ) , the free sur-face w a v e elevation ^ j ( x ) and energy d i s s i p a t i o n c o e f f i c i e n t Ke. In case o f m u l t i p l e membranes, the n u m e r i c a l results are c o m p u t e d f o r u n i -f o r m l y spaced m e m b r a n e s . Hence, i t gives rise to a p e r i o d i c g e o m e t r y that m a y lead to Bragg scattering o f ocean waves. The s t u d y is car-r i e d o u t f o car-r d i f f e car-r e n t values o f w a t e car-r d e p t h h, distance b e t w e e n the m e m b r a n e L, n o n - d i m e n s i o n a l tension p a r a m e t e r T/pgh^. p o r o s i t y c o n s t a n t G, angle o f incidence 6 and n o n d i m e n s i o n a l w a v e n u m -ber Yoh. The values o f the physical parameters, w h i c h are k e p t fixed t h r o u g h o u t the n u m e r i c a l c o m p u t a t i o n s are p = 1025 k g m ^ ^ , g = 9.8 m s~2 and n o n - d i m e n s i o n a l mass per u n i t area o f the m e m b r a n e m = ms/ph = 0 . 0 1 . The energy r e l a t i o n o b t a i n e d t h e o r e t i c a l l y i n the p r e s e n t s t u d y provides a n u m e r i c a l check f o r t h e results o b t a i n e d t h r o u g h n u m e r i c a l c o m p u t a t i o n . In order to p e r f o r m the n u m e r i c a l c o m p u t a t i o n , the s y s t e m o f e q u a t i o n is t r u n c a t e d u p to a finite n u m -ber o f N t e r m s . The convergence c r i t e r i o n is checked f o r N = 50, and i t is observed t h a t Kr a n d Kt satisfy the conservation o f energy r e l a t i o n - t Kf = 1 f o r G = 0. This shows t h a t there is no e n e r g y loss due to the b a r r i e r w h e n the m e m b r a n e is i m p e r m e a b l e . O n t h e o t h e r hand f o r G > 0, some o f the w a v e energy gets dissipated b y the b a r r i e r due to the porous n a t u r e o f the m e m b r a n e and as a r e s u l t Kf + Kf < 1 f o r all G > 0.

4.1. Reflection and transmission coefficient

The r e f l e c t i o n and transmission c o e f f i c i e n t s are p l o t t e d f o r b o t h single and m u l t i p l e s u r f a c e - p i e r c i n g m e m b r a n e w i t h fixed edge a t y = 0 and m o o r e d edge a t y = d. A b r i e f c o m p a r i s o n o f the results f o r all the t h r e e edge c o n d i t i o n s is also presented.

4.J.J. Single membrane

In Fig. 3, the r e f l e c t i o n and t r a n s m i s s i o n c o e f f l c i e n t s Kr a n d Kt are p l o t t e d versus n o n - d i m e n s i o n a l w a v e n u m b e r Yo^ f o r v a r i o u s values o f m e m b r a n e l e n g t h d/h c o n s i d e r i n g 6 = 30=, T/pgh'^ = 1.0 and G = 0 . 0 1 . I t is observed t h a t the w a v e r e f l e c t i o n keeps o n i n c r e a s i n g w i t h the increase i n w a v e n u m b e r whereas the t r a n s m i s s i o n c o e f f i c i e n t decreases w i t h the increase i n w a v e n u m b e r . The r e f l e c t i o n c o e f f i c i e n t is also observed to be higher w i t h the increase i n the m e m b r a n e l e n g t h d/h.

In Fig. 4, the r e f l e c t i o n and t r a n s m i s s i o n c o e f f l c i e n t s , Kr a n d Kt are p l o t t e d versus n o n - d i m e n s i o n a l wave n u m b e r Yoh f o r v a r i o u s values o f angle o f incidence 9 considering d/j, = 0.5, T/ pg\? = 1.0 a n d G = 0.01. I n t h i s case i t is f o u n d t h a t the r e f l e c t i o n c o e f f i c i e n t keeps o n decreasing w i t h the increase i n the angle o f i n c i d e n c e 9 w h e r e a s i n the case o f t r a n s m i s s i o n c o e f f i c i e n t an opposite p a t t e r n is observed.

In Fig. 5, the r e f l e c t i o n and t r a n s m i s s i o n c o e f f i c i e n t s , Kr and Kt are p l o t t e d versus n o n - d i m e n s i o n a l w a v e n u m b e r y^h f o r various values o f t e n s i o n o f the m e m b r a n e T/ p g / i ^ c o n s i d e r i n g A/h = 0.5, 9 = 30= a n d G = 0 . 0 1 . I t is observed t h a t w i t h the increase i n the t e n s i o n o f t h e m e m b r a n e the r e f l e c t i o n c o e f f i c i e n t increases w h e r e a s t h e t r a n s m i s s i o n c o e f f i c i e n t decreases.

In Fig. 6, the r e f l e c t i o n and t r a n s m i s s i o n c o e f f i c i e n t s , Kr a n d Kt are p l o t t e d versus n o n - d i m e n s i o n a l w a v e n u m b e r yoh f o r v a r i o u s values

4 6 D. Kannakaret al./Applied Ocean Researcli 39 (2012) 40-52 1.0 ^ 0.6 H 0.4 0.2 K, -jv^^" y^.".

\

\ --•' \ / - "* •' ' \ - - ~ d/h = 0.2 d/h = 0.5 - d/h = 0.8 " - d/h = 0.4 d/h = 0.6

-

/ •' •' \ -. , / / ' 1 5 3 1 1 4 5

Fig. 3. Kr and /(, versus y o'i for different values of d/h considering f/ = 30 , T/pgh^ •¬ 1.0 and G = 0.01. 0.6 4 0.4 4 0.0 0 = 0° 0 = 30" 0 = 4 5 ° •• 0 = 6 0 " 2 3 1.0-, 0,8 / V.' T/pgh^ = 0.01 • T/pgh' = 0.1 T/pgh' = 0.05 T/pgh' = 0.5 T/pgh^ = 1.0 (b) 0.4 4 0.0 T/pgh' = 0.01 T/pgh' = 0.05 T/pgh' = 0.1 T/pgh' = 0.5 — T/pgh' = 1.0 1 2 3 4 5

Fig. 5. Kr and Kf versus yoh for diffei-ent values of T/pgh^ considering d/h = 0.5,0 = 30 and G = 0.01.

Fig. 4. Kr and K, versus Yoh for different values of O considering d/h = 0.5, T/pglv' = 1.0 and C = 0.01.

of porosity constant G considering d/h = 0.5, 9 = 30= and T/ pgh'^ = 1.0. I t is observed ttiat w i t h the increase i n the porosity constant the r e f l e c t i o n c o e f f l c i e n t decreases. This may be due to the fact the as the porosity constant increases m o r e waves get t r a n s m i t t e d t h r o u g h the m e m b r a n e a n d as a r e s u l t the r e f l e c t i o n c o e f f l c i e n t decreases.

I n Fig. 7, t h e r e f l e c t i o n and transmission coefficients, Kr and Kt are p l o t t e d versus n o n - d i m e n s i o n a l w a v e n u m b e r yoh f o r various edge c o n d i t i o n c o n s i d e r i n g d/h = 0.5, Ö = 30=, /C^ = 10^ N m - i , G = 0.01 and T/ pgh'^ = 1.0. It is observed t h a t w a v e r e f l e c t i o n is higher w h e n u p p e r edge is fixed and l o w e r edge is m o o r e d whereas the w a v e r e f l e c t i o n is l o w e r i n t h e case o f the m e m b r a n e w i t h b o t h ends b e i n g m o o r e d . This may be due to t h e fact t h a t f o r b o t h the edges being m o o r e d , m o r e waves get t r a n s m i t t e d t h r o u g h the m e m b r a n e due to the high flexibility at the edges and as a result the w a v e r e f l e c t i o n is l o w as compared to fixed edge c o n d i t i o n .

4.J.2. Multiple membranes

The r e f l e c t i o n a n d t r a n s m i s s i o n coefficients are p l o t t e d f o r a d o u -ble surface-piercing m e m b r a n e w i t h fixed edge a t y = 0 and m o o r e d edge a t y = d. It is considered t h a t the t w o membranes are separated by distance L so that a, = L and Uj = 2L (see Fig. 1). Therefore, t h e ge-o m e t r y becge-omes perige-odic i n nature. I n Fig. 8, the r e f l e c t i ge-o n and trans-mission coefficients, Kr and Kt are p l o t t e d versus n o n - d i m e n s i o n a l w a v e n u m b e r yoh f o r various values of L/h considering 8 = 30=, T/ p g / i ^ = 1.0, d/h = 0.5 and G = 0 . 0 1 . The resonating p a t t e r n i n the

Fig. 6. Kr and K, versus yo'i for different values of C considering d/h/h/h = 0.5,0 = 30 andT/pgh^ = 1.0.

r e f l e c t i o n coefficient is observed to be h i g h e r as the distance between the t w o membranes is increased. This m a y be due to the t r a p p i n g o f m o r e waves w i t h the increase i n t h e distance o f the membranes and as a result the resonating p a t t e r n increases. These resonation patterns i n the r e f l e c t i o n c o e f f i c i e n t m a y also be t e r m e d as Bragg's resonance as described i n Bennetts et al. [53). The resonating p a t t e r n generally occurs i n w a t e r w a v e p r o b l e m s h a v i n g periodic structures.

D. Karmakar et al./Applied Ocean Research 39 (2012)40-52 47 1.0 0.6 0.4 0.2 4

\ \ / '

- — Fixed-Moored \ >v'' \ . ' . . .

A

Moored-Fixed \ >v'' \ . ' . . .

A

Moored-Moored /.' \

/ \

/ \

/

/

I ' l ' I ' l ' (a) 1.0n 0.8 4 0.2 4 2 a Ólh = 0.2 d/h = 0.4 d/h = 0.5 d/h = 0.6 d/h = 0.8

Fig. 7. Kr and K, versus yoh for different values of edge condition considering d//? = 0.5.61=30 , C = 0.01 andr//)gh2 = 1.0. (a) ^ 0 1.0 T-0.2 4 0.0 •'/••! •' i •7 ;''. ' •7 \ • L / h = 1.0 U h = 2.0 U h = 3.0 L/h = 4.0 U h = 5.0 0.8 0.2 0.0-- 0.0-- U h = 1.0 - U h = 2.0 • U h = 3.0 - L/h = 4.0 • U h = 5.0 1 2 3 4 5

Fig. 8. Kr and K, versus yoh for different values of L/h considering d/h = 0.5,0 = 30 C = 0.01 andr//5gh2 = i.o.

The Bragg's resonances f o r the periodic f l o a t i n g structures are w e l l described i n Karmakar et al. [ 5 4 ] .

I n Fig. 9, the r e f l e c t i o n and transmission coefficients, Kr and Kt are p l o t t e d versus n o n - d i m e n s i o n a l wave n u m b e r yoh f o r various values o f d/h considering 0 = 30=, T/ pgh'^ = 1.0, L/h = 2.0 and G = 0.01. I t shows t h a t m u l t i p l e zeros i n the r e f l e c d o n c o e f f i c i e n t occur w i t h i n 0 < y o ' i < 2 f o r membranes o f smaller lengths. This i m p l i e s t h a t m e m b r a n e s w i t h smaller lengths ( a p p r o x i m a t e l y d / / i < 0.3) yields

d/h = 0.2 d/h = 0.4 d/h = 0.5 d/h = 0.6 d/h = 0.8

I \ J \

A .

Fig. 9. Kr and K, versus Yoh for different values of d/h considering L/h = 3.0.fi = 30 , C = 0.01 and T/pgh2 = 1.0.

f u l l r e f l e c t i o n f o r i n c i d e n t waves w i t h 0 < y o'' < 2.

I n Fig. 10, the r e f l e c t i o n and t r a n s m i s s i o n coefficients, Kr and Kf are p l o t t e d versus n o n - d i m e n s i o n a l w a v e n u m b e r y o h f o r various values o f T/ pgh'^ c o n s i d e r i n g L/h = 3.0, 9 = 30=, G = 0.01 a n d d/h = 0.5. In this case i t is observed t h a t as the t e n s i o n parameter keeps o n increasing the w a v e r e f l e c t i o n also increases. The m e m b r a n e s become stiffer as the t e n s i o n increases and t h a t perhaps leads to m o r e w a v e r e f l e c t i o n . I t is also f o u n d t h a t the resonating p a t t e r n increases f o r higher values o f t h e tension o f the m e m b r a n e . This m a y be due to the decrease i n the flexibility o f the m e m b r a n e w h i c h results i n the increase i n resonating p a t t e r n . A n opposite p a t t e r n is observed i n the case o f transmission c o e f f i c i e n t .

4.2. Membrane deflection

The d e f l e c t i o n o f the m e m b r a n e is p l o t t e d f o r fixed edge at y = 0 and m o o r e d edge a t y = d as given i n Eq. (18) f o r single surface-piercing floating m e m b r a n e .

In Fig. 11(a), the m e m b r a n e d e f l e c t i o n is p l o t t e d versus distance X f o r various values o f porous parameter G c o n s i d e r i n g yoh = 2.5, T/ pgh^ = 0 . 0 1 , 0 = 3 0 = and d/h = 0.5. The d e f l e c t i o n o f t h e m e m b r a n e is observed to be decreasing w i t h the increase i n the p o r o s i t y constant o f the m e m b r a n e . This suggests t h a t as the p o r o s i t y c o n s t a n t increases m o r e waves get t r a n s m i t t e d t h r o u g h the m e m b r a n e a n d as a result the d e f l e c t i o n o f t h e m e m b r a n e decreases. On the o t h e r hand i n Fig. 11(b), the m e m b r a n e d e f l e c t i o n is p l o t t e d versus distance x f o r various values o f n o n - d i m e n s i o n a l w a v e nurrtber yoh c o n s i d e r i n g G = 1.0, d / h = 0.5, T/ pgh'^ = 0.01, 9 = 30 and d/h = 0.5. The d e f l e c t i o n o f

48 D. Kannakaret al./Applied Ocean Research 39 (2012) 40-52 (a) 0.4 H 0.2 4

\

- T/pgh^ = 0.01 - T/pgh' = 0.05 - - T / p g h ' = 0.5 T/pgh' = 0.1 • T/pgh' = 0.05 - - T / p g h ' = 0.5 - - T / p g h ' = 1.0 (a) G = 0.0 G = 1.0 G = 1.0+0.5i G = 2.0 G = 1.0+2.0i G = 5.0 — I — 0.02 0.04 1.0 0,0-T/pgh' = 0.01 - - • 0,0-T/pgh' = 0.05 T/pgh' = 0.1 • T/pgh' = 0.5 T/pgh' = 1.0 4

Fig. 10. Kr and K, versus yoh for different values otT/pgh^ considering L/h = 3.0.9 = 30 , C = 0.01 and d/h = 0.5. ( b ) •ê -10 O CU •25

./ :

0.000 0.025 X

Fig. 11. Membrane deflection f ; versus distance x for different values of (a) porous parameter G with yoh = 2.5 and (b) rion-dimensional wave number yoh with C = 1.0 considering T/pgh^ = 0.01. = 30^ and d/h = 0.5.

the m e m b r a n e is observed to be increasing w i t h the increase i n the n o n - d i m e n s i o n a l w a v e n u m b e r

J/Q/I.

This shows t h a t as yoh increases the w a v e l e n g t h o f the i n c i d e n t w a v e decreases w h i c h results i n the increase i n the m e m b r a n e d e f l e c t i o n .

I n Fig. 12(a), the m e m b r a n e d e f l e c t i o n is p l o t t e d versus distance X f o r various values o f m e m b r a n e l e n g t h d/h c o n s i d e r i n g T/ pgh^ = 0.01, yoh = 2.5, 9 = 3 0 ° and G = 1.0. The d e f l e c t i o n o f the m e m -brane decreases as the l e n g t h o f the m e m b r a n e decreases. This shows t h a t w h e n m e m b r a n e is o f small l e n g t h , then the w a v e gets trans-m i t t e d b e l o w the trans-m e trans-m b r a n e and as a result the trans-m e trans-m b r a n e d e f l e c t i o n reduces.

I n Fig. 12(b), the m e m b r a n e d e f l e c t i o n is p l o t t e d versus distance X f o r various values o f t e n s i o n parameter T/ p g h ^ c o n s i d e r i n g yoh = 2.5, d/h = 0.5, e = 3 0 ° and C = 1.0. I t is observed t h a t as the tension o f the m e m b r a n e increases, the d e f l e c t i o n o f the m e m b r a n e decreases. This is due to the fact t h a t w i t h the increase i n the tension p a r a m e t e r the f l e x i b i l i t y o f the m e m b r a n e decreases and as a result the d e f l e c t i o n o f the m e m b r a n e decreases.

I n Fig. 13(a) the m e m b r a n e d e f l e c t i o n is p l o t t e d versus distance X f o r v a r i o u s values o f m o o r i n g angle a c o n s i d e r i n g yoh = 2.5, 9 = 3 0 ° , T/ pgh'^ = 0.01, d/h = 0.5 and G = 1.0. I t is observed t h a t the change i n the m o o r i n g angle has v e r y l i t t i e e f f e c t o n the d e f l e c t i o n o f the m e m b r a n e . For higher values o f the m o o r i n g angle the m e m b r a n e d e f l e c t i o n is observed to be almost same f o r a l l values o f o;.

I n Fig. 13(b), the m e m b r a n e d e f l e c t i o n is p l o t t e d versus distance x f o r v a r i o u s values o f m o o r i n g stiffness IQ c o n s i d e r i n g yoh = 2.5, 9 = 3 0 ° , T/ pgh'^ = 0.01, d/h = 0.5 and G = 1.0. Here, i t is observed t h a t as the stiffness o f the m o o r i n g decreases, the m e m b r a n e becomes

f l e x i b l e and as a result the d e f l e c t i o n o f the m e m b r a n e increases at y = d. H o w e v e r , as the m o o r i n g stiffness increases, the a m o u n t o f displacement o f the m e m b r a n e edge decreases.

In Fig. 14, the m e m b r a n e d e f l e c t i o n is p l o t t e d versus distance x f o r various values o f angle o f incidence 0 c o n s i d e r i n g yoh = 2.5, T/ pgh'^ = 0.01, d / h = 0.5 a n d G = 1.0. The d e f l e c t i o n o f the m e m b r a n e is observed to be decreasing as the angle o f incidence o f t h e i n c o m -i n g w a v e -increases. Th-is m a y be due to the f a c t t h a t as the angle o f incidence increases the e f f e c t o f i n c o m i n g w a v e o n the m e m b r a n e reduces.

4.3. Wave elevation

The free surface g r a v i t y w a v e elevations i n the i n c i d e n t a n d trans-m i t t e d w a v e regions are o b t a i n e d f r o trans-m the r e l a t i o n :

- i c w f j = (j)jy. o n y = 0, j = 1,2, w h i c h can be expressed as ( 3 7 ) (e-''*'" -F Roe"*") / ' o (0) + Ë R n e - ' ' " V „ (0) f o r x > 0, ( 3 8 ) Toe-''* V o ( 0 ) + E Tne^"" ƒ ' „ ( 0 ) f o r x < 0.

In Fig. 15(a), the w a v e elevations are p l o t t e d versus distance x f o r various values o f porous parameter G c o n s i d e r i n g yoh = 5.0, 9 = 3 0 ° , T/ pgh^ = 1.0 and d/h = 0.5. It is observed t h a t the w a v e elevation i n the t r a n s m i t t e d region is reduced as c o m p a r e d to the

D. Karmakaret al./Applied Ocean Research 39 (2012) 40-52 49 (a) -40 d/h = 0.2 - d/h = 0.4 d/h = 0.5 d/h = 0.6 - d/h = 0.8 •0.01 0.00 0.01 0.05 0.02 0.06 O-, -20 T/pgh' = 0 . 0 0 5 T/pgh' = 0.01 T/pgh' = 0.025 ' - T/pgh' = 0.05 - T/pgh^ = 0.1 0.00

Fig. 12. Membrane deflection versus distance x for different values of (a) membrane length d/h with T/pg/i^ = 0.01 and (b) tension parameter T/pgh'^ with d/h = 0.5 considering yoh = 2.5, fl = 30 and G = 1.0.

(b) 0-1 K^= 1 0 ' N m ' K^ = 10° Nm'' K^ = 10'Nm-* K = 1 0 ' Nm'' 0.00 001 0,02 0.03 X

Fig. 13. Membrane deflection versus distance x for different values of (a) mooring angle a and (b) mooring stiffness /Cj considering yoh = 2.5. fl = 30 , = 0.01, d//) = 0.5 and G = 1.0.

i n c i d e n t region. I t m a y also be n o t e d t h a t w i t h the increase i n the p o r o s i t y constant the w a v e e l e v a t i o n i n the i n c i d e n t region decreases and the w a v e e l e v a t i o n i n the t r a n s m i t t e d region increases. Hence, membranes w i t h less p o r o s i t y w i l l be m o r e effective i n r e d u c i n g the i n c o m i n g w a v e height. However, the w a v e l e n g t h remains the same f o r d i f f e r e n t values o f t h e p o r o s i t y constant. On the other hand i n Fig. 15(b), the w a v e elevations are p l o t t e d versus distance x f o r various values o f tension p a r a m e t e r T/ p g / i ^ considering yoh = 5.0, 0 = 30=, d/h = 0.5 and G = 1.0. It is observed t h a t as the tension o f the m e m b r a n e increases t h e w a v e e l e v a t i o n i n the t r a n s m i t t e d r e g i o n decreases. This is i n agreement w i t h Fig. 10, w h i c h shows t h a t w a v e r e f l e c t i o n increases w i t h increase i n the m e m b r a n e tension. Thus, the a t t e n u a t i o n o f w a v e h e i g h t takes place w i t h the increase i n the tension o f the m e m b r a n e .

I n Fig. 16(a), the w a v e elevations are p l o t t e d versus distance x f o r various values o f angle o f incidence 6 considering yoh = 5.0, d/h = 0.5, T/ pgh^ = 1.0 and G = 1.0. The change i n the phase o f the w a v e elevation i n the i n c i d e n t and t r a n s m i t t e d region is observed w i t h the change i n the angle o f incidence. However, the w a v e elevations i n the i n c i d e n t and t r a n s m i t t e d regions are observed to be a l m o s t same f o r all i n c i d e n t w a v e angles. It shows t h a t the change i n the i n c i d e n t w a v e angle does n o t a f f e c t the w a v e h e i g h t a t t e n u a t i o n . H o w e v e r , i t alters the w a v e l e n g t h i n b o t h the regions.

I n Fig. 16(b), the w a v e elevations are p l o t t e d versus x f o r various values o f n o n - d i m e n s i o n a l w a v e n u m b e r y^h considering 0 = 30=, d/h = 0.5, r / pgh'^ = 1.0 and G = 1.0. I t is observed t h a t w i t h the increase i n the n o n - d i m e n s i o n a l w a v e number, the w a v e elevations

-0,01 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 X

Fig. 14. Membrane deflection versus distancex for different values of angle of incidence fl considering yah = 2.5, T/pgli^ = 0.01, d/Zi = 0.5 and C = 1.0.

i n the i n c i d e n t and t r a n s m i t t e d regions increase. This is due to the f a c t t h a t as the w a v e n u m b e r increases the w a v e l e n g t h o f the i n c i d e n t w a v e decreases and thus the w a v e h e i g h t increases.

50 D. Karmakaret al./Applied Ocean Research 39 (2012) 40-52 G = 1 . 0 G = 1.0+0.51 G = 2.0 G = 1.0+2.0i G = 5.0 —1— -150 -100

(a)

T/pgh' = 0.01 -

• llpgh' = 0.05

T/pgh' = 0.1 T/pgh' = 0.5 T/pgh'= 1.0

Fig. 15. Vertical deflection f j versus distances for different values of (a) porous param-eter Gwitfi T/pgh^ = 1.0 and (b) tension paramparam-eter 7/pgh^ with G = 1.0 considering yo/i = 5.0, fl = 30 and d/h = 0.5. ••e •c 0.15-1 0.10 0.05 0.00 -0.05 -0.10¬ -0.15-0 = -0.15-0" - • 0 = 30° 0 = 45" - 0 = 60°

vïïm

KIAMM

.200 -150 -100 -50 50 100 150 200 ( b ) 0.05 4 0.00-^ -015

Fig. 16. Vertical deflection versus distance x for different values of (a) angle of incidence fl with yoh = 5.0 and (b) non-dimensional wave number yoh with 0 = 30-considering d/h = 0.5, T/pgli' = 1.0 and C = 1.0.

4.3. Energy dissipation coefficient

The energy d i s s i p a t i o n c o e f f l c i e n t Kg f o r the floating s t r u c t u r e as suggested by C h w a n g and Chan [26] is g i v e n b y

Ke = \-l<.j -l<}. ( 3 9 )

I n Fig. 17(a), the energy dissipation c o e f f i c i e n t f o r single surfacep i e r c i n g m e m b r a n e m o o r e d at the b o t t o m is surfacep l o t t e d versus n o n -d i m e n s i o n a l w a v e n u m b e r y o ' i c o n s i -d e r i n g 9 = 3 0 ° , -d/h = 0.5 an-d T/ pgh^ = 1.0. I t is observed t h a t the energy d i s s i p a t i o n c o e f f i c i e n t decreases w i t h the increase i n the resistance e f f e c t t e r m o f the porous e f f e c t parameter. On the o t h e r hand w h e n b o t h the resistance e f f e c t t e i m and the i n e r t i a e f f e c t t e r m o f the porous e f f e c t p a r a m e t e r are present t h e n the energy d i s s i p a t i o n decreases. A s i m i l a r o b s e r v a t i o n can be f o u n d i n C h w a n g a n d Chan [ 2 6 ] .

I n Fig. 17(b), the energy d i s s i p a t i o n c o e f f i c i e n t f o r d o u b l e surfacep i e r c i n g m e m b r a n e m o o r e d at the b o t t o m is surfacep l o t t e d versus n o n -d i m e n s i o n a l w a v e n u m b e r yoh c o n s i -d e r i n g L/h = 3 . 0 , 9 = 3 0 ° , -d/h = 0.5 and T/ pgh^ = 1.0. As observed i n Fig. 17(a), the energy d i s s i p a t i o n c o e f f i c i e n t decreases w i t h the increase i n the resistance e f f e c t t e r m o f the porous e f f e c t p a r a m e t e r and w h e n b o t h the resistance e f f e c t t e r m and the i n e r t i a e f f e c t t e r m are present t h e n the energy dissipa-tion c o e f f i c i e n t is observed to be decreasing. I t m a y be n o t e d t h a t i n the case o f d o u b l e s u r f a c e - p i e r c i n g m e m b r a n e the energy d i s s i p a t i o n c o e f f i c i e n t is observed to behave as w a v e p a t t e r n w i t h i n the region 1.5 < y o ' i < 5. Further, c o m p a r i s o n o f Fig. 17(a) and (b) reveals t h a t the d i s s i p a t i o n o f energy is m o r e i n case o f d o u b l e s u r f a c e - p i e r c i n g

m e m b r a n e t h a n the single s u r f a c e - p i e r c i n g m e m b r a n e . Therefore, i t m a y be c o n c l u d e d t h a t m u l t i p l e s u r f a c e - p i e r c i n g m e m b r a n e s w o r k b e t t e r as a brealcwater system i n c e r t a i n c o n d i t i o n s .

5. Conclusion

The p e r f o r m a n c e o f v e r t i c a l s u r f a c e - p i e r c i n g porous m e m b r a n e b r e a k w a t e r i n w a t e r o f finite d e p t h is analyzed based o n the l i n earized t h e o r y o f w a t e r waves. The associated BVP is solved by a p p l y -i n g t h e m e t h o d o f e -i g e n f u n c t -i o n e x p a n s -i o n a l o n g w -i t h the m e t h o d o f least square a p p r o x i m a t i o n . The s t u d y is first c a r r i e d o u t f o r single m e m b r a n e and t h e results are f u r t h e r e x t e n d e d f o r m u l t i p l e floating m e m b r a n e s u s i n g w i d e - s p a c i n g a p p r o x i m a t i o n .

N u m e r i c a l results are c o m p u t e d f o r r e f l e c t i o n and t r a n s m i s s i o n coefflcients, m e m b r a n e d e f l e c t i o n , f r e e surface w a v e elevations and energy d i s s i p a t i o n c o e f f l c i e n t The correctness o f the n u m e r i c a l results is checked b y s a t i s f y i n g the r e q u i r e d energy r e l a t i o n . The c o n -vergence o f the i n f l n i t e series sums is v e r i f i e d a n d i t is observed t h a t the desired results are o b t a i n e d f o r M = 50 n u m b e r o f evanescent modes. I t shows t h a t the m e t h o d is e f f i c i e n t i n t e r m s o f c o m p u t e r m e m o r y and time.

The c o m p a r i s o n o f the results f o r various fixed and m o o r e d edge c o n d i t i o n s is analyzed f o r r e f l e c t i o n and t r a n s m i s s i o n c o e f f l c i e n t s . The p o r o s i t y o f the m e m b r a n e is also added to analyze the e f f e c t o f porous p a r a m e t e r o n the p e r f o r m a n c e o f the v e r t i c a l m e m b r a n e as an e f f e c t i v e brealcwater. I n the case o f single s u r f a c e - p i e r c i n g m e m b r a n e , w i t h the increase i n the l e n g t h o f the m e m b r a n e a n d t e n s i o n o f the m e m b r a n e the w a v e r e f l e c t i o n increases. I t is observed t h a t as the