R.Ashok and S.R.Manam

Received:29 September 2021/Accepted:19 February 2022

?Harbin Engineering University and Springer-Verlag GmbH Germany,part of Springer Nature 2022

Abstract Oblique surface waves incident on a fixed vertical porous membrane of various geometric configurations is analyzed here.The mixed boundary value problem is modified into easily resolvable problems by using a connection.These problems are reduced to that of solving a couple of integral equations.These integral equations are solved by a one-term or a two-term Galerkin method.The method involves a basis functions consists of simple polynomials multiplied with a suitable weight functions induced by the barrier. Coefficient of reflection and total wave energy are numerically evaluated and analyzed against various wave parameters.Enhanced reflection is found for all the four barrier configurations.

Keywords Free surface gravity waves; Reflection coefficient; Singular integral equation; Galerkin approximation; Linear waves;Vertical porous membrane barrier

1 Introduction

Free surface water wave scattering problems involving various kind of structures play an important role in science and engineering. Thin vertical barriers are one such struc‐ture that attracted lot of attention due to their potential in developing rich mathematical solution methods to the asso‐ciated boundary value problems.Dean(1945)derived ana‐lytical solution of the two-dimensional normally incident water wave problem with a submerged thin vertical rigid barrier by using the complex variable technique.Then, us‐ing integral equation procedure, Ursell (1947) studied wa‐ter wave scattering problem involving a surface piercing rigid barrier. Scattering of gravity waves by a vertical bar‐rier with finite number of gaps in it was discussed by Lewin (1963), Mei (1966) and Evans (1970). Lewin(1963) and Mei (1966) solved the problem by converting the original problem into a Riemann-Hilbert problem. Ev‐ans(1970)used the method of complex variables to obtain the solution of the problem. Mandal and Chakrabarti(2000)proposed various solution methods to obtain the so‐lutions of the wave scattering problems involving different rigid barrier configurations.

Later, porous barriers gained attention due to its reflec‐tion and dissipative characteristics. Chwang (1983) intro‐duced porous wave maker theory by involving resistance effects.Then, the condition on the porous barrier has been modified by including inertial effects as well (Yu and Chwang 1994). Further, flexible breakwaters attracted lot more attention due to their structural flexibility. These kind of breakwaters are used in deep sea activities. Inci‐dent wave interaction on a flexible barrier that is hinged at the bottom of the sea and moored at the surface was ana‐lyzed by Lee and Chen (1990). Also, Williams et al.(1991) investigated flexible floating breakwater that was anchored to the sea floor with a buoy at the free surface.

Kim and Kee (1996) and Cho and Kim (1998) have studied wave interaction with a horizontal tensioned flexi‐ble membrane by utilizing the eigenfunction expansion method. Selvan and Behera (2020) have used the same method to handle the effect of a floating circular porous elastic membrane on surface gravity waves. Later, Cho and Kim (2000) discussed interaction of incident waves with a horizontal porous flexible membrane by using the multi-domain boundary element method. Yip et al. (2001)have investigated the oblique wave scattering problem in‐volving finite number of floating membranes in finite wa‐ter depth.Then, Suresh Kumar et al. (2007) have analysed wave scattering by a vertical flexible porous membrane in a two-layer fluid of finite depth,using an orthogonality re‐lation along with the least-square approximation. Karmak‐ar and Guedes Soares (2012) have analyzed the perfor‐mance of the multiple vertical surface-piercing porous membrane barriers in water of finite depth by applying the least-square approximation method. Mandal et al. (2016)studied oblique wave scattering problem involving the multiple flexible porous barriers in a two-layer fluid of fi‐nite depth. Using eigenfunction expansion method, Koley and Sahoo(2017)have investigated the oblique wave scat‐tering problem involving different configurations of a ver‐tical flexible porous membrane.

A single term Galerkin's method has been used by Evans and Morris (1972) to handle obliquely incident waves on a surface piercing rigid barrier and they found closed upper and lower bounds for the scattering quantities.Also,scatter‐ing of oblique incident surface waves by a finite length thin submerged vertical plate as well as a complete barrier with a gap in it has been handled by Mandal and Das(1996)and Das et al.(2018a,2019).They used one or two term Galer‐kin method to find the scattering quantities analytically.Fur‐ther, Galerkin method with many terms is utilized to solve oblique scattering problem involving a partially immersed and a submerged solid barrier by Das et al.(2018b).

A flexible porous tensioned membrane is considered here with all four type of configurations such as partially immersed, submerged barrier extending infinitely down‐wards, submerged barrier with a finite length and a com‐plete membrane with a gap in it. Free surface waves inci‐dent obliquely on the barrier from either side are consid‐ered. This defines two solution velocity potentials for the original problem. The upper-half plane problems for the two velocity potentials are reduced to problems in a quar‐ter-plane. These quarter plane problems are connected with auxiliary potentials through a couple of integral rela‐tions. Then, the auxiliary potentials are handled by the Galerkin's method with one or multi-term approximation.Further, lower and upper bounds for the scattering quanti‐ties are obtained. Formulation of the boundary value prob‐lem and its solution method is described in Section 2. In Section 3, numerical results are discussed for each of the barrier configuration by choosing a suitable weight func‐tion in the approximation. Finally, few conclusions are drawn in Section 4.

2 Mathematical formulation and method of solution

A mixed boundary value problem is considered in the Cartesian coordinate system(x,y,z)with the positivey-ax‐is measured vertically downwards and the horizontalxzplane is the undisturbed free surface.The fluid is occupied in the domainy>0 and ?∞＜x,z＜∞. The vertical porous membrane is placed in the fluid region atx=0,y∈Band?∞＜z＜∞in whichBtakes one of the following: (0,a), (b,∞), (0,a) ∪(b, ∞) and (a,b).The fluid is considered to be inviscid,incompressible and the flow is in irrotational mo‐tion. Incident surface waves come fromx=∞and make an angleβwith thexy- plane. Under the assumptions of the linearised water wave theory, time-harmonic fluid motion is described by the velocity potential Re{?1(x,y)eipze?iωt},wherep=Ksinβ,K=ω2/gin whichωis frequency of the in‐cident wave,gis the acceleration of gravity andtis time.The same wave motion is also described, when the inci‐dent waves come fromx=?∞and make an angleπ-βwith thexy-plane,by the velocity potential Re{?2(x,y)eipzeiωt}.Then,the spacial velocity potentials?j(x,y),j=1,2 satisfy

The free surface boundary condition is

In addition,?j(x,y),j=1,2 satisfy the conditions

and

whereris the local radius from the edge point of the barri‐er. The schematic diagram for the oblique wave scattering problem is shown in Figure 1.

The displacement of the horizontally oscillating mem‐brane is represented as

Figure 1 Free surface with submerged flexible porous membrane

whereνj(y)is the deflection amplitude of the membrane.

Assume that the absolute value of the deflection ampli‐tude is small as compared to the incident wave length.The boundary condition on the flexible porous membrane (Yu and Chwang 1994)is given by

whereΓ=Γ1+iΓ2is the non-dimensional complex porous effect parameter in whichΓ1is the resistance effect andΓ2is the inertial effect.

The fluid flow is continuous in the gapx=0,y∈Bˉ=(0,∞)B.That is,

The radiating conditions are described as

where?0(?x,y)=e?iK xcosβ?K yis the incident wave potential andR,Tare amplitudes of the reflected,transmitted waves.Also, the deflection amplitudeνj(y),j=1,2 of the mem‐brane satisfies the following condition

In the above,Tis tension in the membrane,ρis the density of water. The membrane frequency parameter may be de‐fined asα=ω ms/T, wherems=ρsdsis the uniform mass withρsanddsbeing the uniform mass density and the thickness of the membrane,respectively.

By utilizing the above equation in Equation (5), the boundary condition on the barrier Equation (5) is obtained as

2.1 Reduction to quarter-plan problems

In view of the continuity of the horizontal velocity of the fluid acrossx=0,the original velocity potentials?j,j=1,2 that are defined in the upper-half plane may be modified into a new potential functionsψj,j=1, 2 in the quarterplane.This is done by choosing(Lamb 1932)

The potential functionsψjdefined in the domain(?1)j+1x>0,j=1,2 satisfy Equations(1)?(4).By applying Equation(7) to the Equation (6), the boundary condition on the bar‐rier becomes

In addition,the boundary condition in the gap and the radi‐ation condition become

and

with a note thatT=1?R.

2.2 Connection between wave potentials

A well defined connection, similar to the one in Ashok et al.(2020),between the flexible porous wave potentialψj(x,y),j=1,2 and the auxiliary potentialχj(x,y),j=1,2 is in‐troduced as

The auxiliary potential functionsχj(x,y),j=1, 2 satisfy Equations(1)?(4).By differentiating the connection Equa‐tion(10),one obtains

where the variable suffix denotes derivative.By settingx=0 in Equation(11)and by applying the equation Equation(8),it may be obtained that

Also, by settingx=0 and by using the equation Equation (9)in the connection Equation(10),one finds the condition as given by

Further, the radiation condition forχj(x,y),j=1,2 may be specified as

whereRjandTjare constants.By applying the far-field be‐haviour in each term of the differential form of the connec‐tion Equation(11),a pair of relations is obtained as

and

Thus, the problem for the wave potentialψj(x,y) withj=1,2 has been decomposed into two solvable auxiliary poten‐tial problems with the aid of the connection Equation(10).Hence, the solution of the auxiliary potentialψj(x,y) may be found in terms of the wave potentialsχj(x,y) as in Ashok et al.(2020).The approximate solution of the wave potentialsχj(x,y)j=1,2 is obtained by the Galekin approxi‐mation method which in turn helps to find the estimated value of the reflection coefficient|R|.

2.3 Galerkin method

The general solution of the potential functionχ1(x,y)sat‐isfying Equations (1)?(3) and Equation (14) is given by(Manam et al.2006)

whereR1,T1are unknown constants,A(γ) is an unknown function,B(γ,y) =γcosγy?Ksinγyandγ1=Let

By utilizing the general solution Equation (17) in the above functionsg(y)andh(y),it may be obtained that

and

Then, the Havelock’s inversion theorem (Havelock 1929)is applied to the above pair of equations to get

and

By substitutingA(γ)from Equations(21)and(23)in the equations Equations (19) and (18) and then using the con‐ditions(12)and(13),it may be obtained that

and

where

and

Now,by defining the functions

and then,by making use ofG(u)andH(u)in Equation(24)and Equation(25),one obtains that

and

Then,the Equations(20)and(22)are modified as

and

where

The Equation(15)and Equation(16)are used in the above to find

In a similar manner, the same estimation of the constantFin terms of the reflection amplitudeRcan be obtained for the problem of the auxiliary potential functionχ2(x,y),by the aid of the Equations(15)and(16).

2.4 Bounds for F

Multi-term Galerkin approximation method is used to find the functionsG(u)andH(u)in the Equations(26)and(27) (Das et al. 2018b). We first define the symmetric and linear inner products as

Then,we define a pair of operatorsCandDas

Note that the operatorsCandDare linear, positive semidefinite and self-adjoint. To obtain the functionG(u) in Equation(26),we take the multi-term Galerkin approxima‐tion as given by

wheregn(y),n=0,1,…,Nmay be chosen appropriately for a particular barrier position.

By making use of Equation (31) in Equation (26), and multiplying the functiongm(y) before integrating overBˉ,we get the linear system of equations

where

The constantsan,n=0,1,…,Nare found by solving the above system of equations. By following Evans and Mor‐ris(1972),it is seen that

By substituting the equation Equation(28)in the above in‐equality,one obtains that

where

Further, the functionH(u) in Equation (27) can be found by considering the multi-term Galerkin approximation

wherehn(y),n=0,1,…,Nis chosen appropriately for a par‐ticular barrier problem. By substituting Equation (34) in Equation(27)and multiplying the functionhm(y)before in‐tegrating overB, we obtain the linear system of equations for the unknownsbn,n=0,1,…,Nas given by

where

Again, by following Evans and Morris (1972), it is seen that

By utilizing the Equation(29)in the above inequality,with the use of the far-field Equations (15) and (16), it is ob‐tained that

where

By modifying the relation Equation (30), the reflection coefficient is obtained as

whereFl≤F≤Fu.The upper and lower bounds for the reflec‐tion coefficient may now be found as where

and

The upper boundRuis computed first and then use it in the Equation (35) to estimate the lower boundFuwhich is matched with the lower boundFl. In the computation, it is found that these bounds forFare numerically matched up to few decimals.

3 Numerical results for various barrier positions

3.1 Surface piercing finite membrane

In this case, the position and the gap are atB=(0,a) and=(a,∞), respectively. The functionG(y) has been taken(Das et al.2018b)as

wherean,n=0,1,…,Nare unknown constants and

The nature of the edge behaviour of the membrane is con‐sidered for taking the weight functiongn(y),n=0,1,…,N.Now, we calculate the analytic results for a two-term Galerkin approximation (N=1). Considerg0(y) andg1(y).Then,by solving the system Equation(32)to obtainFlas

where

andGi0=a Ki(2Ka),i=0, 1 in whichKi(2Ka),i=0, 1 are modified Bessel functions.It is observed thatB01=B10from symmetry of the kernel of the integral equations. Further,

the constantsA00,A01andA11are known as

and

where

Also,H(y)can be taken as

wherebn,n=0,1,…,Nare unknown constants. The weight functionshn(y),n=0,1,…,Nare given by

The same procedure is applied for the two-term Galer‐kin approximation, as mentioned above, to obtain the up‐per boundFu.It is numerically verified that both the upper and lower bounds are matching up to few decimals to esti‐mate the value of the reflection coefficient|R|.

The reflection coefficient |R| and the total energy |E|=|R|2+|T|2are calculated against the angle of incidenceβ. In Figures 2 and 3, estimation of the reflection coefficient |R|and the total energy|E|are depicted versus the angle of in‐cidenceβfor various values of the wavenumberKawith the membrane frequency parameterα=0.062 46 for both cases of porous as well as impermeable membranes. It may be observed, from Figures 2 and 3, that enhanced re‐flection and enhanced dissipation occur at smaller incident angles for shorter waves. The angle at which wave reflec‐tion gets its resonant peak increases with a decrease in the incident wave number. The reflection peak increases as length of the barrier increases, this is due to larger barrier reflect more waves. Also, complete reflection occurs at these incident angles in the case of the impermeable mem‐brane.

Reflection and energy curves are plotted against the an‐gle of incidence for various values of the membrane fre‐quency parameterαin Figures 4 and 5. For smaller values ofα, there is neither enhanced reflection nor enhanced en‐ergy loss for all angles of wave incidence. Reflection de‐creases gradually with an increase in the angle of inci‐dence for all values of the membrane frequency parameter.For smaller values of incidence,reflection is higher for the porous membranes with lesser tension. Also, mass of the membrane barrier increases when the reflection increases for shorter waves. This is because of barrier with higher mass reflect better.As expected,energy dissipation is high‐er for membranes with lower tension at smaller angles of wave incidence. However, enhanced reflection and total wave energy dissipation occur for porous membranes. Re‐flection or energy dissipation peaks happen at larger inci‐dent angles with a decrease in the tension of the porous membrane.

Figure 2 Graph of|R|and|E|versus β for different values of Ka with Γ=1 and α=0.062 46

Figure 3 Graph of|R|versus β for different values of Ka with Γ=0 and α=0.062 46

Reflection and energy curves are depicted against the angle of incidenceβfor different values of the porous effect parameterΓwhen the values ofα=0.062 46 andKa=0.4 are fixed in Figure 6.It is seen from Figure 6(a)that reflec‐tion in general is better for small incident angles when both the resistance and the inertial effects are present in the porous barrier. Further, all the reflection curves have peaks that are due to the flexible nature of the barrier and they occur within 50°to 75°.Also, enhanced reflection is evidently complete for the impermeable membrane. En‐hancement of the energy dissipation may be seen from Figure 6(b) around a particular angle of wave incidence for all the values of the porous effect parameter.In Figure 7,re‐flection and energy curves are plotted against the angle of incidenceβfor various values of the porous effect parame‐terΓwhenα=0 andKa=0.4.Overall reflection significant‐ly decreases again for the resistance as well as the inertial effects for small angles of wave incidence on the porous solid barrier. Figure 7(a) also shows that porosity alone will not cause enhanced reflection by the barrier since there is no evidence of the enhanced reflection in the absence of flexibility in the barrier.Also,as seen from Figure 7(b),en‐ergy dissipation is evidently caused mainly from the iner‐tial effects of the porous barrier. The smaller reflection is noted for the porous membrane barrier due to the fact that the porous barrier allows the water to pass through it.

3.2 Submerged membrane of infinite length

The membrane position and gap are atB=(b, ∞) andBˉ=(0,b)respectively.Again,the multi-term Galerkin approxi‐mation is used andG(y)(Das et al.2018b)is set as

wherean,n=0,1,…,Nare unknowns to be determined and the weight functionsgn(y),n=0,1,…,Nare

Again, the nature of the edge behaviour of the mem‐brane is considered for choosing the weight functiongn(y),n=0,1,…,N. Then, two-term Galerkin approximation (N=1) is used to solve the linear system Equations (32). The

Figure 4 Graph of|R|and|E|versus θ for different values of α with Γ=1 and Ka=0.4

Figure 5 Graph of|R|and|E|versus β for different values of α with Γ=1 and Ka=0.4

Figure 6 Graph of|R|and|E|versus β for different values of Γ with α=0.062 46 and Ka=0.4

Figure 7 Graph of|R|and|E|versus β for different values of Γ with α=0 and Ka=0.4

lower boundFlcan be obtained asFl=a0G00+a1G10,where

In the above,the constantsA00,A01andA11are

and

where

Now,H(y)can be taken as

wherebn,n=0,1,…,Nare unknown constants. The weight functionshn(y),n=0,1,…,Nare given by

Moreover, the upper boundFuis obtained by using the same procedure explained in the two-term Galerkin ap‐proximation method.

The reflection coefficient |R| and the total energy |E|=|R|2+|T|2are evaluated against the wavenumberKband the angle of incidenceβfor the submerged barrier extending infinitely downwards.

Figure 8 Graph of|R|and|E|versus β for different values of K b with Γ=1 and α=0.062 46

In Figure 8, numerical estimation of the reflection coef‐ficient|R|and the total energy|E|are plotted against the an‐gle of wave incidenceβfor different values of the wave‐numberKbwhenΓ=1 andα=0.062 46.These curves are al‐so shown in Figure 9 for the non porous membrane with the same flexibility. It is observed from Figure 8 that the reflection and the energy dissipation curves have resonant enhancement for all the incident wave numbers. Since the present barrier is the complementary nature of the previ‐ous case in the subsection 3.1,the resonant peaks at which enhanced reflection occurs move from lower to higher wave incident angles when the incident waves on the barri‐er become longer to shorter.This is exactly opposite to the results of the previous complementary case.Similar obser‐vation can be made for the energy dissipation curves with the change in the incident wavenumber. Flexibility alone causes the total reflection from the submerged non-porous membrane as seen from Figure 9. Thus, longer waves should incident the membrane by higher oblique angles to get better or complete reflection while shorter waves could incident the membrane by smaller oblique angles for a sim‐ilar enhanced reflection.

In Figure 10, reflection and energy curves are depicted against the angle of wave incidence for various membrane frequency parameter values when the wavenumberKb=0.4 andΓ=1 are fixed.The reflection peaks move towards high‐er oblique wave angles as the membrane frequency param‐eter increases. Similar behaviour is seen in the surface piercing membrane case as well.This shows that only flex‐ibility nature of the barrier,not a type of the barrier,causes the enhanced reflection. The reflection and energy curves are also plotted against the wavenumber for different mem‐brane frequency parameter values when the angle of oblique incidenceβ=30°andΓ=1 are fixed in Figure 11.However, reflection peaks move towards shorter incident waves as the membrane frequency parameter increases.This suggests longer waves must incident on a high ten‐sioned membrane and shorter waves must incident on a low tensioned membrane to get better resonant reflection. Re‐flection peak is attained for shorter and moderate waves for smaller mass of the membrane barrier. Similar observations can also be made from the energy curves in Figure 11.

Figure 9 Graph of|R|versus β for different values of Kb with Γ=0 and α=0.062 46

Figure 10 Graph of|R|and|E|versus β for different values of α with Γ=1 and Kb=0.4

Figure 11 Graph of|R|and|E|versus Kb for different values of α with Γ=1 and β=30°

Reflection and energy curves are also depicted against the angle of incidence for various porous effect parameter values when the wavenumberKb=0.4 andα=0.062 46 are fixed in Figure 12. By comparing these curves with the ones in Figure 6, it may be concluded that the reflection and the energy dissipation is mostly caused by the flexible nature of the membrane. Enhanced reflection becomes a complete one for the impermeable barrier at a particular in‐cident wave angle as observed in the case of surface pierc‐ing membrane barrier.Figure 13 shows reflection and ener‐gy curves for the solid porous barrier.They are qualitative‐ly comparable with the ones in Figure 7.

3.3 Submerged membrane of finite length

In this case, the membrane position and the gap are atB=(a,b)andBˉ=(0,a)∪(b,∞)respectively.TheGalerkin ap‐proximationforG(y) is specified asG(y)≈∑an gn(y),wherean,n=0,1,…,Nareunknown constantsandgn(y),n=0,1,…,Nare theweightfunctions.Thenatureofthe edge conditions of the porous membrane is considered to setgn(y)as

Figure 12 Graph of|R|and|E|versus β for different values of Γ with α=0.062 46 and Kb=0.4

Figure 13 Graph of|R|and|E|versus β for different values of Γ with α=0 and Kb=0.4

Here, we obtain the analytic results by using a single term Galerkin approximation (N=0). Considerg0(y). Then,by solving the linear system Equation (32), we obtaina0=G00/A00.By usinga0into Equation(33)and getFl=G002/A00where

and

with

Likewise, a single term Galerkin approximation for the functionH(y)is specified as

Again, the edge behaviour of the membrane is considered to sethn(y),n=0,1,…,Nas

Then, the upper boundFuis obtained by using a single term Galerkin method. Thus, the reflection and the trans‐mission coefficients are attained between two close bounds and the numerical results are obtained for the scat‐tering quantities against various parameters.

The reflection coefficient |R| and the total energy |E|=|R|2+|T|2are calculated against the non-dimensional wave‐numberKbas well as the angle of incidenceβfor a sub‐merged barrier of finite length. Numerical estimation of the reflection and the energy curves are plotted against the wavenumberKbfor variousμ=values whenβ=30°andΓ=1 are fixed in Figure 14.For the fixed barrier length pa‐rameterμ, wave reflection and energy loss reach maxi‐mum for waves with a moderate wavelength.The measure of these quantities are remained uniform for moderate to short incident waves. For fixed larger wavenumber, the length of the barrier decreases as reflection decreases. It is due to shorter waves interaction with smaller barriers.

In Figures 15 and 16,the reflection coefficient and the to‐tal wave energy are depicted against the wavenumber for different values of the membrane frequency parameter when the oblique angle of incidenceβ=30°is fixed,in both cases of the porous and the non-porous membranes,respec‐tively. Reflection increases in general when the tension in the membrane increases for all membrane lengths. Further,enhanced reflection takes place for a specific smaller length of the membrane due to the flexible nature of the barrier.Both reflection and dissipation of wave energy maintain uni‐formly high for moderate to longer membranes.Similar ob‐servations can be made, from Figure 16, in the case of the impermeable tensioned membrane. However, impermeable membranes with higher tension cause significantly higher reflection in comparison with the porous ones.

In Figure 17,the reflection coefficient and the total ener‐gy are depicted against the incident wave angle for variousΓvalues whenKb=0.2 andα=0.062 46 are fixed. From Figure 17, it may be observed that the enhanced reflection and the enhanced energy loss occur in between 30°to 50°and the curves reach minimum in the vicinity of 90°.Again, this is due to the flexible nature of the submerged membrane.Complete reflection is possible at a specific in‐cident wave angle for the impermeable tensioned mem‐brane. This resonant reaction from the barrier is absent when one considers solid porous membrane, as shown in Figure 18. Again, reflection and energy curves for the solid membrane are qualitatively similar to the ones in Figures 7 and 13.

Figure 14 Graph of|R|and|E|versus Kb for different values of μ with β=30°and Γ=1

Figure 15 Graph of|R|and|E|versus Kb for different values of α with μ=0.05,β=30°and Γ=1

Figure 16 Graph of|R|versus Kb for different values of α with μ=0.05,β=30°and Γ=0

3.4 Complete membrane with a gap

In this case,the membrane position and the gap are atB=(0,a)∪(b,∞) andBˉ=(a,b), respectively. The multi-term Galerkin approximation for the functionG(y) can be taken by

wherean,n=0,1,…,Nare unknown constants. The edge behaviour of the membrane is considered to set the weight functionsgn(y),n=0,1,…,Nas

Likewise, considerH(y)≈∑bn hn(y), wherebn,n=0,1,…,Nare unknown constants and the edge behaviour of the membrane is considered to choosehn(y),n=0, 1,…,Nas

Figure 17 Graph of|R|and|E|versus β for different values of Γ with μ=0.5,Kb=0.2 and α=0.062 46

Figure 18 Graph of|R|and|E|versus β for different values of Γ with μ=0.5,Kb=0.2 and α=0

The same procedure is used to obtain the bounds for the scattering quantities as given in the previous case and the details are not included here.Graphs of the reflection coef‐ficient and the total energy are depicted by considering pa‐rameter values as mentioned in the case of the submerged membrane with a finite length.

In Figure 19, estimated values of the reflection coefficient|R| and the total energy |E| are computed against the wave‐numberKbfor differentμ=values whenβ=30°andΓ= 1 are fixed. When a particular frequency of waves incident on the tensioned or the non-tensioned porous membrane with a gap in it, reflection and energy dissipation curves get reso‐nant enhancement for a specific length of the gap.

Graphs of the coefficient of reflection and the total ener‐gy are given in Figures 20 and 21 against the wavenumber for variousαvalues when the porous and the impermeable barrier, respectively, are obliquely incident by waves that make an angleβ=30°.They reveal that these curves attain resonant peaks when waves of certain frequency incident on the tensioned membrane barrier with a particular gap length. These peaks move towards higher values ofKbwith a decrease in the tension value of the membrane.It is remarked that resonant reflections become a complete one for the impermeable membrane with a gap.

Figure 19 Graph of|R|and|E|versus Kb for different values of μ with β=30°and Γ=1

Figure 20 Graph of|R|and|E|versus Kb for different values of α with μ=0.05,β=30°and Γ=1

Figure 21 Graph of |R| and |E| versus Kb for different values of α with μ=0.05,β=30°and Γ=0

Figure 22 Graph of|R|and|E|versus β for different values of Γ with μ=0.5,Kb=0.2 and α=0.062 46

Finally, reflection and total energy curves are plotted in Figure 22 against the incident angle for variousΓvalues whenμ=0.5,Kb=0.2 andα=0.062 46 are fixed. They re‐veal that the resonant peaks of these curves are attained at high incident angles as compared to the ones in Figure 17 for the case of the submerged membrane of finite length.Again, enhanced reflection is almost complete when the membrane becomes non-porous. In addition, solid porous barrier will not have any resonant reactions from the inci‐dent waves. Also, the reflection and the energy curves in Figure 23 for the solid porous barrier with a gap are com‐parable qualitatively with those in other barrier configura‐tions.

Figure 23 Graph of|R|and|E|versus β for different values of Γ with μ=0.5,Kb=0.2 and α=0

4 Conclusions

The present work is concerned with the scattering prob‐lem of water waves obliquely incident on four types of the vertical porous membranes such as surface piercing, sub‐merged but finite length,submerged but infinite length and complete one with a finite gap.The original problem is de‐composed into a pair of resolvable problems for the auxil‐iary potentials.These problems are solved by the Galerkin method with a one term or a two term approximation.Closed upper and lower bounds are obtained for the scat‐tering quantities by choosing a suitable weight function in all the four barrier configurations. Then, the graphs of the reflection and the total energy are plotted against various parameters such as the non-dimensional wavenumber and the angle of wave incidence. Enhanced reflection and en‐hanced energy dissipation are found in all the four barrier configurations and they are sensitive to the incident wave angles as well as the membrane parameters. This is useful in the design of removable breakwaters such as vertical membranes by adjusting the length of the membrane suit‐ably chosen so as to attain maximum reflection in order to mitigate harmful effects of the incident waves.

Founding InformationSupported by DST with Grant No. MTR/2019/000561