Monochromatic Wave over One and Two Bars

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1 Applied Mahemaical Sciences, Vol. 8, 204, no. 6, HIKARI Ld, hp://dx.doi.og/0.2988/ams Monochomaic Wave ove One and Two Bas L.H. Wiyano Faculy of Mahemaics and Naual Sciences, Bandung Insiue of Technology, Jalan Ganesha 0 Bandung, Indonesia. M. Jamhui Depamen of Mahemaics, Maulana Malik Ibahim Sae Islamic Univesiy, Jalan Gajayana 50 Malang, Indonesia. Copyigh 204 L. H. Wiyano and M. Jamhui. This is an open access aicle disibued unde he Ceaive Commons Aibuion License, which pemis unesiced use, disibuion, and epoducion in any medium, povided he oiginal wok is popely cied. Absac Based on small ampliude wae heoy we deemine an exac soluion fo he ampliude of he efleced and ansmied waves poduced when a monochomaic wave passes one and wo submeged bas, a esul of impoance in coasal engineeing. The opimal widh and he disance beween wo bas ae deemined, in ode o obain he smalles ampliude of he ansmied wave. Keywods: Monochomaic wave; poenial funcion; efleced wave; submeged ba; ansmied wave.. Inoducion In his pape, we examine he popagaion of monochomaic wave ove one and wo submeged bas. In paicula we deemine he eflecion and ansmission coefficien as a funcion of he deph and widh of he ba, and he disance beween wo bas. The effec of he bas is o educe he ampliude of he inciden wave, so ha such bas, eihe aificial o naual, ae of impoance in coasal engineeing. The change of he wae deph, fom deep o shallowe, affecs he waves by inceasing is ampliude, and he ohe way aound. A ba, submeged beneah wae suface, is ypical change of deph in a channel. I is ofen seen as pacical applicaion in coasal and ocean engineeing, such as beach poeco fom waves, since i can educe he wave ampliude. The incoming wave is paly efleced when hey pass ove he ba. The pecenages of he efleced and ansmied waves

2 308 L. H. Wiyano and M. Jamhui depend on he dimension of he ba. Moeove, anohe ba, behind he fis ba, can be consideed o fuhe educe he ampliude of he wave. The disance beween hose wo bas is anohe paamee ha is involved in ou calculaion, such ha we obain he minimum ampliude of he ansmied wave. In ode o answe he objecive, we use he poenial funcion and he dispesion elaion deived by Wiyano [5], fom a bounday value poblem of Laplace equaion, and i is solved by vaiable sepaaion mehod. Based on he deivaion, he opimal widh of a ba, giving he minimum ampliude of he ansmied wave, is obained numeically. This esul confims he analiical esul in [2], and also confims he esul obained by Pudjapaseya and Chenda [4], who calculae he opimal widh numeically fom shallow wae equaions. Since he equaions coespond o he lineaized dispesion elaion, he opimal widh in [4] is less accuae and slighly difeen wih he esul in [5], using he exac dispesion elaion. Anohe appoach in evaluaing he ansmied waves was povided by Goesen and Andonowai [], who show similaiy beween opic and wae waves based on he same model. Meanwhile, Yu and Mei [6] sudied waves ove a ba by including he effec of he eflecion fom beach. Ohe efeences can be found by eading he eview pape [3]. In impoving he esul fom one ba, we popose in his pape by adding anohe ba, and we deemine he opimal disance beween hose wo bas, so ha he ansmied waves have minimum ampliude. The poenial funcion and dispesion elaion deived by Wiyano [5] ae fis eviewed, and we ecalculae he ansmied waves fo one ba. These waves can be consideed as incoming waves of anohe ba, and afe passing ha ba, based on he esul in Wiyano [5], we shall obain he ansmied waves wih ampliude ha is squae of he ampliude of he pevious ansmied waves. This idea was also used in [4], o pedic he ampliude of he ansmied waves fo wo bas. When we calculae he ansmied ampliude by involving he disance beween wo bas, as a paamee, we obain he opimal disance, giving he minimum ansmied ampliude. This value is diffeen wih ou pevious calculaion, simila o he esul in [4]. 2. Monochomaic wave The poblem fomulaed hee is popagaion of wae wave. We use Caesian as he coodinaes, he x axis is chosen along he undisubed level of he wae suface, and he y axis is pependicula o he x axis. The boom channel is fla, wih wae deph h. Monochomaic wave popagaes on he suface of he wae in fom of η ( x, ) ikx ( w) = ae () Whee a is ampliude, k is wavenumbe, w is fequency, and i =. The wave avels o he igh.

3 Monochomaic wave ove one and wo bas 309 The elaion beween k and w is obained by solving poenial funcion φ x, y, fom ( ) φ + φ = 0 (2) xx yy Subjec o he lineaized condiions η φ = 0 (3) y φ + g η = 0 (4) on he suface y = 0, and φ = 0 (5) y on he boom y = h. Hee g is he acceleaion of gaviy. Following Wiyano [5], vaiable sepaaion mehod is applied o he bounday value poblem (2)-(4), by expesing ig (6) φ( x, y, ) = η( x, ) F( y). w Theefoe, we obain 2 i w (7) φ( x, y, ) = η( x, ) cosh( ky) + sinh ( ky). w gk Condiion (5) hen gives 2 w = gk anh kh (8) ( ) as he dispesion elaion. The wave numbe k of he monochomaic wave is elaed o he wae deph. This quaniy changes when he wave avels passing ove a ba. This is followed by changing of he wave ampliude. We obseve he popagaion of he wave ove one and wo bas. Moeove, we can also calculae he flux of wae a he coss secion x 0, by inegaing he hoizonal velociy, defined by u = φx, giving iηx( x, o ) w (9) Q = k We need his fomula lae in calculaing he flux fo diffeen wae deph, a he edges of he ba he flux mus be coninue. 3. Opimal widh of a ba The efleced and ansmied waves fom an incoming wave ae obseved in his secion. A ba, having heigh d and widh L, is submeged beneah he wae

4 3020 L. H. Wiyano and M. Jamhui suface of wae deph h, illusaed in Figue. The incoming wave avels on he suface of deph h, having wave numbe k, saisfying (8), and ampliude a. Since he ba is on he boom of he channel a 0 < x < L, pa of he incoming wave is efleced wih ampliude a in diffeen diecion, expessed by giving diffeen sign a he exponen, so ha boh waves ineac a x < 0, having he same wave numbe. Similaly a he egion 0 < x < L, he ansmied wave wih ampliude b and wave numbe k 2, caused by he change of he deph a x = 0, ineacs wih he efleced wave fom he change of deph a x = L, having ampliude b and he same wave numbe k 2. Fo he egion x > L, hee is only ansmied wave wih ampliude c and wave numbe k 3, ha is he same as k if he wae has he same deph a he egion x < 0. η ( x, ) = ae ikx ( w) h η = ae i( kx w) h 2 η = ce ikx ( w) y = 0 d x = 0 x = L Figue Skech of wave popagaing ove a submeged ba. The above descipion is fomulaed in fom of he suface elevaion as ( ) ( ) ikx w i kx w ae + ae, x< 0 ikx ( 2 w) i( kx 2 w) η ( x, ) = b e + be, 0< x< L (0) ikx ( 3 w) ce, L< x Now, he poblem is o deemine he ampliude of each ype of waves. We can do ha by examining he coninuiy of he elevaion and flux a he edges of he ba, a x = 0 and x = L, giving a+ a = b + b he coninuiy of elevaion be be ce ik2l ik2l ik3l + = ( ) ( ) a+ a k2 = b + b k he coninuiy of flux ik3l be be k3 = ce k2 ik2l ik2l ( ) ()

5 Monochomaic wave ove one and wo bas 302 This sysem of equaions is hen solved o obain a, b, b, c, which ae all elaive o he ampliude of he incoming wave a. The wave numbes k, k 2, k 3 ae calculaed fom he dispesion elaion (8), by conseving he fequency w fom he incoming wave. Since he ampliudes ae obained in complex numbe, we calculae each ampliude by aking he modulus of he numbe. We pefom some compuaions of he above heoy fo g = 0, w= 2. The wae deph is h = 3.5 ; and he dimension of he ba is heigh d =.5 ( h 2 = 2 ) and widh L =.5. Figue 2 Plo of he ansmied ampluude c / a vesus he widh of he ba L, fo g = 0, w= 2, wae deph h = 3.5 ; and he ba heigh d =.5 ( h 2 = ). The opimal widh is L op = 3.040, coesponding o c / a= Afe calculaing he wave numbe fo each egion, he sysem of equaion () is solved, and we obain he ansmied wave wih ampliude c = a, and he efleced ampliude a = 0.33 a. We hen calculae he ampliudes fo vaious numbes of L, and we obain ha he ampliude of he ansmied wave eaches minimum value c / a= 0.987, a L = This minimum ampliude can be eached fo ohe widh of ba peiodically. Plo of c / a vesus L is shown in Figue 2. We hen call he fis value of L as he opimal widh, denoed by L op. The pofile of he plo confims o he esul in [2] and [4]. Fo anohe ba heigh, we calculae he ampliudes using d = 2.5 ( h 2 = ), simila plo is obained, bu lowe. The opimal widh is L op = 2.320, coesponding o he ansmied wave wih ampliude c / a=

6 3022 L. H. Wiyano and M. Jamhui 4. Opimal disance beween wo bas When wo bas ae on he boom of a channel, he ansmied wave passing ove hose bas can be calculaed similaly as pesened in he pevious secion. The opimal disance beween wo bas ae obseved, so ha we obain he minimum ampliude of he ansmied wave. If he wo bas ae placed side by side wihou disance, he ansmied wave has ampliude which is no smalles, as we can see fom he plo in Figue 2. On he ohe hand, we assume he disance beween hose wo bas is elaively fa. The ansmied wave of he fis ba has ampliude c, popoional o he incoming ampliude a, namely c = α a. This wave can be consideed as he incoming wave fo he second ba, so ha afe passing hough he second ba, he wave has ampliude α 2 a, fo he same widh of he fis ba. Since α is less han, wo bas can moe educe he ampliude. Howeve, he effec of he disance beween wo bas is ou concen in his secion. η ( x, ) = ae ikx ( w) η = ee ikx ( w) 5 h h 2 h h 4 3 h5 S x = 0 x = L L Figue 3 Skech of wave popagaing ove wo submeged bas. In descibing he poblem, we divide he wae egion ino 5 sub egions, and he suface elevaion is wien as ikx ( w) i( kx w ) ae + ae x< 0 ikx ( 2 w) i( kx 2 w) be + be 0 < x< L ikx ( 3 w) i( kx 3 w η ) ( x, ) = ce + ce L< x< L+ S (2) ikx ( 4 w) i( kx 4 w) d e + de L+ S < x< 2L+ S ikx ( 5 w) ee 2L+ S< x The fis ba of widh L is placed a 0 < x < L, and he second ba wih he same widh is a L+ S < x< 2L+ S. The disance beween hose wo bas is S. Each

7 Monochomaic wave ove one and wo bas 3023 egion conains efleced and ansmied waves, wih ampliude denoed by subscip fo eflecion and fo ansmiion, and diffeen wavenumbe k j, given by index j =, 2, L,5, elaing o he wae deph, saisfying (8). Afe passing he second ba, hee is only ansmied wave. The wave ampliude is hen deemined by examining he coniuiy of he elevaion and flux a he edges of he ba, a x = 0, LL, + S, 2L+ S, giving a+ a = b + b ik2l ik2l ik3l ik3l be + be = ce + ce 3( ) 3( ) 4( ) 4( ) he coninuiy of elevaion ik L+ S ik L+ S ik L+ S ik L+ S ce + ce = de + de ik4( 2L+ S ) ik4( 2L+ S ) ik4( 2L+ S ) de + de = ee ( a+ a) k2 = ( b + b) k ik2l ik2l ik3l ikl 3 ( be be ) k3 = ( ce ce ) k2 ik3( L+ S ) ik3( L+ S ) ik4( L+ S ) ik4( L+ S ) he coninuiy of flux ( ce ce ) k4 = ( de de ) k3 ik4( 2L+ S ) ik4( 2L+ S ) ik4( 2L+ S ) ( de de ) k5 = ee k4 () This sysem of equaions is hen solved o obain a, b, b, c, c, d, d, e, which ae all elaive o he ampliude of he incoming wave a. We examine he above fomulaion o obseve he effec of he disance beween wo bas o he ansmied wave. Fom ou pevious calulaion, fo g = 0, w= 2, wae deph h = 3.5 (also h 3, h 5 ), and ba heigh d = 2.0 ( h2 = h4 =.5 ), we obain he opimal widh L op = We hen exend he model fo wo bas wih ha widh, bu vaious disance S. Fo S = 2, he wave ampliude afe passing wo bas is e = a. Hee, we compae he ampliude wih espec o he incoming ampliude a. Fo smalle disance S, we obain lage aio numbe e / a, and he ansmied wave has he same ampliude o he incoming ampliude a S = 0. This agees wih ou esul fo one ba, wih ba widh L = 2L op. Fo lage disance fom S = 2, he aio numbe of he ampliude deceases by inceasing he disance, unil a S = 3.58 coesponding o e = a, as he minimum ampliude of he ansmied wave. This numbe is smalle han ou calculaion using assumpion vey fa disance, giving e = a. Plo beween S and e / a is shown in Figue 4, which indicaes he peiodiciy he cuve. The nex opimal disance can be obained a S = 0.72.

8 3024 L. H. Wiyano and M. Jamhui Figue 4 Plo of he ansmied ampluude e / a vesus he disance beween wo bas S, fo g = 0, w= 2, wae deph h = 3.5 (also h 3, h 5 ); and he ba heigh d =.5 ( h 2 = h 4 =.5 ). Fo he ba widh L op = 3.040, he opimal disance is S op = 3.58 (also S op = 0.72 ), boh coesponding o e / a= Conclusions An analyical soluion of wave popagaion passing ove one and wo bas has been obained using vaiable sepaaion of poenial funcion fo monochomaic ype wave. The opimal dimension of he ba and he opimal disance beween wo bas, ha gives he minimum ampliude of he ansmied wave, is obained numeically as he wave numbe has o be calculaed fom nonlinea dispesion elaion, and o educe he complexiy of he fomula. Acknowledgemens. The auhos ae gaeful o Bandung Insiue of Technology and Diecoae Geneal of Highe Educaion, Minisy of Educaion and Culue-Indonesia fo suppoing he eseach of his pape unde Reseach-Innovaion and Decenalizaion Gans, 204. The auhos also hank D Pudjapaseya, I. Magdalena and Sugi S.T. fo useful discussions duing solving he poblem. Refeences [] E. Goesen, Andonowai, Similaiies beween opical and suface wae waves, J. Indones. Mah. Soc., 8 (2002), -8.

9 Monochomaic wave ove one and wo bas 3025 [2] C.C. Mei, The applied dynamics of ocean suface waves, Wold Scienific, 989. [3] C.C. Mei, P.L.F. Liu, Suface waves and coasal dynamics, Annual Rev. Fluid Mech., 25 (993), [4] S.R. Pudjapaseya, H.D. Chenda, An opimal dimension of submeged paallel bas as a wave efleco, Bull. Malay.Mah. Sci. Soc., 32 (2009), [5] L.H. Wiyano, Wave popagaion ove a submeged ba, ITB J. Sci., 42A (200), [6] J. Yu, C.C. Mei, Do Longshoe bas shele he shoe?, J. Fluid Mech., 404 (2000), Received: Apil 7, 204

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