Water Wave Scattering by an Elastic Thin Vertical Plate Submerged in Finite Depth Water

J. Marine Sci. Appl. (2013) 12: 393-399DOI: 10.1007/511804-013-1209-7Water Wave Scattering by an Elastic Thin Vertical PlateSubmerged in Finite Depth WaterRumpa Chakraborty and B. N. MandalPlhysics and Applied Mathematics Unit, Indian Statistical Institute, Kolkata 700108, IndiaAbstract: The problem of water wave sattrng by a thin vertical barrier, a bottom standing barrier, a barrier with a gap and aelastic plate submerged in uniform finite depth water is investigatedtotally submerged barrier. For each case they used anhere. The boundary condition on the elastic plate is derived fromapproximate method based on the Galerkin approximationthe Bernoulli Euler equation of motion satisfied by the plate. Usingand numerical estimates for the reflection and transmissionthe Green' s function technique, from this boundary condition, thecoefficients were obtained for different values of wavenormal velocity of the plate is expressed in terms of the differencebewen the vloiy plentiats (unknown) acroes the plate. The numbers. Also, earlier Goswami (1983) employed Geen'stwo ends of the platle are either clamped or fee. The relection and function technique to reduce the scattering problemtransmission coefficients are obtained in terms of the integrals' involving a fixed vertical rigid plate submerged in uniforminvolving combinations of the unknown velocityvelocty potenntial on the .finite depth water and solved it approximately by using atwo sides ofplate, whic1 satisfty three simutaneous integral perturbation method. Mandal and Dolai (1994) laterequations and are solved numerically. These coefficients areobtained very accurate upper and lower bounds for thecomputed numerically for various values of different parametersreflection and transmission cofficients for the problem ofand depicted graphically against the wave number in a number ofwater wave scattering by a thin vertical rigid barrier presentfigures.Keywords: thin vertical elastic plate; uniform finite depth water;in uniform finite depth water by using the Galerkinapproximation method.wave scattering; reflection and transmission cofficientsn the above mentioned problems, the barriers arArticle ID: 1671-9433(2013)04-0393-07assumed to be rigid so that the flexural response is notincluded. In many cases, a floating or submerged body is1 Introductioncapable of considerable flexure. Thus it is important to studywater wave scattering by a floating or submerged elasticA train of surface water waves when incident on anbody. The problem of water wave scattering involvingobstacle present in water, experiences partial reflection andelastic or viscoelastic bodies have been modeled as atrasmission by and over or below the obstacle. When the floating breakwater by Stoker (1957) and Taylor (1986), butobstacles are in the form of a thin rigid vertical barriersolutions to these problems were not given. Fox and Squirehaving some four basic gcometrical configurations present (1994) solved the problem of water wave sattering by a1 infinitely deep water or in finite depth water, thefloating ice-sheet modeled as an elastic beam, using thecorresponding wave scattering problems have been studiedlinear theory. More recently， water wave problemsin theliterature, assuming linear theory. Ursell (1947) involving elastic plates of different geomtrical shapes wereemployed Havelock's expansion of water wave potentialstudied by a number of authors. For example, Sturova (2003)followed by the use of the Cauchy singular integral equationinvestigated the unsteady motion due to a circular elasticto solve explicitly the problem of water wave scattering by aplate floating on shallow water under the action of anthin vertical rigid plate prilly immersed in ifinitely deep extermal load. A foating retangular elastic plate simulatesawater or by a completely submerged thin vertical rigidbarrier extending infintely downwards. Evans (1970) used aperiodic surface pressure on a rectangular floating elasticmethod based on the theory of complex variable to solve theplate，which is important to engineers engaged in theproblem of wave scattering by a thin vertical rigid plate ofconstruction of these types of airports. Sturova (2006 a, b)finite vertical height submerged in deep water. Also Porterstudied such problems for shallow water and for infinitelyand Evans (1995) considered oblique wave scattering by adeep water. Wave scattering by a semi-infinite elastic platethin vertical rigid brier in unifom finite depth water floaing on ifinitely deep water was studied by Chakrabartihaving four basic configurations namely, a surface piercing(2000) reducing th| singularintegral equation中国煤化工。sar erecenreacatei. 2012.019al.(2001) considereMYHCNM HG water andused the eigen function expansion for the solution method.*Corresponding author Email: biren@ isicalac.inGayen and Mandal (2009) studied water wave scattering by◎Hartbin Engineering University and Springer- Verlag Berlin Heidelberg 201339Rumpa Chakraborty, et al. Water Wave Scatering by an Elastic Thin Vertical Plate Submerged in Finite Depth Watera thin elastic plate of arbitrary width floating in deep waterFree sufaceafter reducing the problem to solving singular integralequations of the Carleman type. All these problems involve((x.y)T.(x.2) .horizontal elastic plates, floating or submerged in deep orR,(-x.))finite depth water. A water wave problem involving avertical elastic plate was studied for the first time by Meylanb .(x, y)-incident wave(1995) who generalized the problem of water wavesattering by a surface piercing thin vertical rigid plateoriginally investigated by Ursell (1947) in infinitely deepwater, as mentioned earlier, to a thin vertical elastic plateFig.1 Schematic diagram of the problemsubmerged in water of uniform finite depth. In the presentpaper wave scattering by a thin vertical elastic plate2 Mathematical formulationsubmerged in water of uniform finite depth is studied. ThisWe consider two-dimensional motion of surface waves undermay be regarded as a generalization of the rigid platethe action of gravity only. Water is assumed to be an inviscid,problem considered by Porter and Evans (1995)， as alsomentioned earlier, to an elastic plate. The problem isuniform finite depth h. A rectangular Cartesian coordinateformulated in terms of a boundary value problem for thevelocity potential function where the boundary conditionsystem (x, y, z)is chosen wherein the y-axis is taken verticallyover the plate is derived from the Bernoulli-Euler equationdownwards into the fluid region and the plane y = 0 representsof motion satisfied by the elastic plate. By using the Green'sthe undisturbed free surface. The motion is independent of thefunction technique, from the Bernoulli-Euler equation ofco-ordinate z. The position of the submerged thin elasticmotition, the normal velocity (unknown) of the plate isvertical plate is given by: x= 0,a8mn to be continuous at 17=y,25)2φ, 2= 0 in the fluid region, (12)8mpn(y+0,y)- 8nn(y -0,y)while=-1,Jax2 ay2and g is to be symmetric in y and n. Then g (n, y)is foundthe free surface conditiontobe.Kφ+9P= 0 on y=0 (13)Aeiah+Beiah+Ceah +Derah, a0) in the region9.(0,y)=分. 8(7>)(Q()- a())dn, a0 (36)coshkohEq. (36) gives a representation of 0(x, y) for x >0, andandG(5，7;x, y)(x <0)in the region bounded externallyhence the reflection coefficient T is obtained by makingby the lines 7= 0,-X≤ξ≤0 ; ξ=0， 0≤η≤h;x→∞in Eq. (36) and comparing with Eq. (22), we findη=h,-X≤ξ≤0;ξ=-X ,0≤η≤h and intermally by a circlex=:+x()mro,(-o)osh,an(37)of very small radius 8 with center at (x, y) and ultimately2k。h + sinh 2k。hmaking X -→∞andδ -→0, we obtainAgain, if we choose x = +0, then in the region used in theGreen's integral theorem we have to take a half circle ofφ(x,y)=e"coshkg(h-y).small radius δ with the center at(+0, y), so that instead wecoshk.hobtainf.2G(.7;x,y)4(M)- G0.,:x:>(4,()) for x<0(31)a:()=-=f(0.,0,).()- G(0.7:0, y)0.(7)]dn,where G(5, n:x,y)=0