Wave interaction with a floating and submerged elastic ... - Springer Link

J Eng Math (2014) 87:47–71 DOI 10.1007/s10665-013-9659-0

Wave interaction with a floating and submerged elastic plate system S. C. Mohapatra · T. Sahoo

Received: 28 January 2013 / Accepted: 15 June 2013 / Published online: 10 September 2013 © Springer Science+Business Media Dordrecht 2013

Abstract Surface gravity wave interaction with a floating and submerged elastic plate system is analyzed under the assumption of small-amplitude surface water wave theory and structural response. The plane progressive wave solution associated with the plate system is analyzed to understand the characteristics of the flexural gravity waves in different modes. Further, linearized long-wave equations associated with the wave interaction with the elastic plate system are derived. The dispersion relations are derived based on small-amplitude wave theory and shallow-water approximation and are compared to ensure the correctness of the mathematical formulation. To deal with various types of problems associated with gravity wave interaction with a floating and submerged flexible plate system, Fourier-type expansion formulae are derived in the cases of water of both finite and infinite depths in two dimensions. Certain characteristics of the eigensystems of the developed expansion formulae are derived. Source potentials for surface wave interaction with a floating flexible structure in the presence of a submerged flexible structure are derived and used in Green’s identity to obtain the expansion formulae for flexural gravity wavemaker problems in the presence of submerged flexible plates. The utility of the expansion formulae and associated orthogonal modecoupling relations is demonstrated by investigating the diffraction of surface waves by floating and submerged flexible structures of two different configurations. The accuracy of the computational results is checked using appropriate energy relations. The present study is likely to provide fruitful solutions to problems associated with floating and submerged flexible plate systems of various configurations and geometries arising in ocean engineering and other branches of mathematical physics and engineering including acoustic structure interaction problems. Keywords Expansion formulae · Floating and submerged elastic plate system · Green’s function · Reflection coefficient · Surface waves 1 Introduction To create calm zones near harbors and protect coastal facilities such as inlets and beaches from wave action, often both submerged horizontal plate and submerged breakwater structures are used. These structures are advantageous compared with subaerial breakwaters as these types of structures do not obstruct the ocean view, which is essential for development of recreational and residential shores. Burke [1] analyzed the surface wave scattering by a submerged S. C. Mohapatra · T. Sahoo (B) Department of Ocean Engineering and Naval Architecture, Indian Institute of Technology, Kharagpur, Kharagpur 721 302, India e-mail: [email protected]; [email protected]; [email protected]

123

48

S. C. Mohapatra, T. Sahoo

horizontal plate based on the Wiener–Hopf technique. Using a matched asymptotic expansion method, Siew and Hurley [2] analyzed the wave scattering by a submerged horizontal plate in shallow water. Patarapanich [3] analyzed the reflection characteristics of a submerged horizontal plate from deep- to shallow-water limits. Liu and Li [4] studied the water-wave motion over an offshore submerged horizontal porous-plate breakwater under the assumption of linear potential theory using the matched eigenfunction expansion method. Evans and Peter [5] analyzed the gravity wave interaction with a submerged semi-infinite porous plate by the Wiener–Hopf technique and finite porous plate by using the residue calculus technique in water of finite depth under the assumption of linear waterwave theory. On the other hand, Cho and Kim [6] studied the wave diffraction by a submerged flexible membrane using the boundary integral equation method and mode expansion method. Recently, Hassan et al. [7] analyzed the surface wave interaction with submerged flexible plates of finite and semi-infinite length in the two- as well as three-dimensional problem involving a circular plate by the matching method. Recently, Williams and Meylan [8] analyzed the surface wave interaction with a semi-infinite submerged elastic plate using the Wiener–Hopf technique. All these studies are meant for gravity wave interaction with submerged structures. On the other hand, there has also been significant study of surface wave interaction with very large floating structures (VLFS) for ocean space utilization. An interesting aspect of this class of problems is to reduce the structural response of VLFS. The traditional way to reduce the incident wave impact on a large floating structure is by using vertical breakwaters. However, in many situations, these breakwaters become an obstacle to the facilities mounted on the floating structures. To overcome these difficulties, antimotion devices have been proposed as alternatives for reducing the effect of waves on VLFS where the wave dissipation effect of breakwaters is small or there are no breakwaters. These antimotion devices are structures attached near the edges of the main structure of the VLFS and often do not require any mooring system. Takagi et al. [9] proposed a theory for an antimotion device to reduce the structural response of a VLFS in oblique sea. Watanabe et al. [10] analyzed the hydrodynamic characteristics of a floating elastic plate to which a submerged horizontal plate is attached at the fore-end. Recently, Wang et al. [11] reviewed various approaches for mitigating the hydroelastic response of a VLFS under wave action. In contrast to the significant progress in the literature on the interaction of surface waves with flexible submerged structures of various geometries as alternatives to subaerial breakwaters, there has been negligible progress in the literature on analyzing the effect of submerged structures for possible reduction of the structural response of very large floating structures. The mathematical complexity associated with gravity wave interaction with the floating and submerged flexible structures discussed above is the existence of higher-order boundary conditions on the structural boundaries associated with the Laplace/Helmholtz equation as the governing equation. These problems are non-Sturm–Liouville type in nature. Korobkin et al. [12] reviewed the modeling challenges associated with problems of hydroelasticity in various areas of mathematical physics and engineering. Fox and Squire [13] analyzed oblique wave scattering by a semi-infinite floating ice sheet using the conjugate gradient method. Sahoo et al. [14] used the eigenfunction expansion method and a suitable orthogonal mode-coupling relation to study the scattering of surface gravity waves by a semi-infinite floating elastic plate in two dimensions. Evans and Porter [15] analyzed the oblique wave scattering due to a crack in a floating ice sheet in finite water depth with the suitable use of the orthogonal relation developed by Lawrie and Abraham [16] and the Green’s function technique. Manam et al. [17] derived expansion formulae for wave interaction with floating flexible structures in the cases of water of both finite and infinite depths, and the utility of the developed expansion formulae is demonstrated by analyzing the flexural gravity wave scattering due to a crack in a floating ice sheet in water of infinite depth. Kohout et al. [18] analyzed wave scattering by multiple floating elastic plates of variable material characteristics with the help of the eigenfunction expansion method. Lawrie [19] derived the general form of the expansion formulae in three dimensions along with several characteristics of the eigenfunctions to deal with a class of problems associated with the Helmholtz equation arising in acoustic wave interaction with flexible structures. Mondal et al. [20] derived the expansion formulae and the orthogonal mode-coupling relations associated with the eigenfunctions in three dimensions to deal with gravity wave interaction with a floating elastic ice plate in channels of finite and infinite widths and depths. Mondal and Sahoo [21] derived the expansion formulae associated with flexural gravity wave motion in a two-layer fluid having an interface in three dimensions and studied the characteristics of the associated eigensystems. Unlike wave

123

Wave interaction

49

interaction with rigid structures, in the case of wave interaction with flexible structures, the flexural modes of the flexible structure contribute additional complex flexural gravity wave modes apart from the real and imaginary wave modes in the eigenfunction expansion method. Another, important characteristic associated with the eigenfunction expansion method is that the eigenfunctions are linearly dependent. In order to apply the eigenfunction expansion method to this class of problems, certain orthogonal mode-coupling relations are used to obtain the unknown coefficients in the expansion formulae. Lawrie [22] reviewed the theoretical developments on the eigensystems associated with wave–structure interaction problems. Lawrie [23] generalized the expansion formulae associated with the Helmholtz equation having higher-order boundary conditions to study the acoustic wave propagation in three-dimensional ducts with flexible walls. A contemporary account of various mathematical methods to deal with wave interaction with flexible structures is given by Sahoo [24]. In the present study, a more general mathematical formulation is developed to analyze the interaction of surface gravity waves with a floating and submerged elastic plate system in the cases of both finite and infinite water depths. The dispersion relation associated with the complex wave–structure interaction problem is analyzed to understand the characteristics of the flexural gravity waves on the floating and submerged flexible plate modes generated due to the interaction of surface waves with the floating and submerged plate system. The nature and location of the roots of the dispersion relation are determined through contour plots. Expansion formulae for the velocity potentials are derived using a mixed-type Fourier transform and the eigenfunction expansion method in water of infinite and finite depths, respectively. Various characteristics of the eigensystem associated with the expansion formulae are derived. The symmetric source potentials associated with the flexural gravity waves in the presence of a submerged flexible structure are obtained using the expansion formulae. Using the source potentials along with Green’s identity, expansion formulae associated with flexural gravity wavemaker problems in the presence of a submerged flexible plate are derived in infinite and finite water depths. The utility of the expansion formulae are demonstrated by analyzing the (i) diffraction of flexural gravity waves by a finite submerged flexible membrane and (ii) gravity wave scattering by two semi-infinite floating and submerged elastic plate systems. Considering the geometrical symmetry of the physical problem, the velocity potentials associated with the physical problems are rewritten as the sum of the symmetric and antisymmetric potentials (as in [7]) for simplification. To study the effect of a submerged and floating flexible plate system on wave motion, reflection coefficients are computed and analyzed. The computational results are checked using the energy relations derived in terms of reflection and transmission coefficients. The derived expansion formulae in two dimensions are generalized in the Appendix to deal with boundary-value problems having general-type higher-order conditions in two dimensions.

2 Mathematical formulation Under the assumption of the linearized theory of water waves and small-amplitude structural response, the problem is considered in the two-dimensional Cartesian coordinate system with the x-axis being in the horizontal direction and the y-axis in the vertically downward positive direction. An infinitely extended thin elastic plate is floating at the mean free surface y = 0, and another infinitely extended submerged flexible plate is kept horizontally at y = h in an infinitely extended fluid domain. The fluid domain bounded between the floating elastic plate and the submerged plate occupies the region 0 < y < h, 0 < x < ∞ and is referred to as region 1. On the other hand, the region between the submerged elastic plate and the bottom bed is referred to as region 2. Thus, region 2 occupies the region h < y < H, 0 < x < ∞ in case of finite water depth (as in Fig. 1) and h < y < ∞, 0 < x < ∞ in case of infinite water depth. Hereafter, subscript j = 1 refers to the floating plate and the fluid in region 1 and j = 2 refers to the submerged plate and the fluid in region 2. Assuming that the fluid is inviscid, incompressible, and irrotational and simple harmonic in time with angular frequency ω, the fluid motion is described by the velocity potentials  j (x, y, t) = Re{φ j (x, y)e−iωt } for j = 1, 2. Further, it is assumed that the deflection of the floating and submerged plates are of the form ζ j = Re{ζ j (x)e−iωt }. The velocity potential  j (x, y, t) satisfies the Laplace equation as given by

123

50 Fig. 1 Schematic diagram of floating and submerged flexible plates in water of finite depth

S. C. Mohapatra, T. Sahoo y

1

(x, t)

Floating elastic plate y

0 x

h

Submerged flexible plate

y

2

(x, t) y

h

y

H h y

H

∇ 2  j = 0 in the respective fluid domain.

(1)

The rigid bottom boundary conditions are given by ∂2 = 0 at y = H in case of finite water depth ∂y

(2)

and 2 , |∇2 | → 0 as y → ∞ in case of infinite water depth.

(3)

The linearized kinematic conditions at the mean surface of the floating and submerged structures are given by ∂ζ1 ∂1 = on y = 0, ∂t ∂y ∂2 ∂1 ∂ζ2 = = on y = h. ∂t ∂y ∂y The plate deflection ζ j in the presence of compressive force satisfies (as in [25,26])       ∂4 ∂2 ∂2   for j = 1, 2, E j I j 4 + Q j 2 + m pj 2 ζ j = − P j (x, y, t) + − P j−1 (x, y, t) − y=a y=a ∂x ∂x ∂t where a = 0, h, P j (x, y, t) is the linearized hydrodynamic pressure in the jth region and is given by   ∂ j − gy , P j = −ρ ∂t

(4) (5)

(6)

(7)

P0 (x, y, t) is the constant atmospheric pressure exerted on the floating elastic plate near the free surface, and g is the acceleration due to gravity. Further, in Eq. (6), E j is the Young’s modulus, Q j is the compressive force acting on the plate, m pj = ρ pj d j , ρ pj is the mass per unit length of the floating plate, d j is the thickness of the floating plate, I j = d 3j /(12(1 − ν 2 )) associated with the jth plate, and ν is the Poisson’s ratio of each of the plates. Assuming P0 as the constant atmospheric pressure and eliminating P1 and ζ1 , the linearized condition on the floating thin elastic plate at y = 0 is obtained as   2   ∂ 1 ∂4 ∂2 ∂ 2 ∂1 ∂1 =ρ for 0 < x < ∞. (8) − g E 1 I1 4 + Q 1 2 + m p1 2 ∂x ∂x ∂t ∂y ∂t 2 ∂y Eliminating ζ from Eqs. (4), (5), and (6), the condition on the submerged plate at y = h is obtained as     2 ∂ 2 ∂4 ∂2 ∂ 2 ∂2 ∂ 2 1 E 2 I2 4 + Q 2 2 + m p2 2 for < x < ∞. =ρ − ∂x ∂x ∂t ∂y ∂t 2 ∂t 2

(9)

Thus, the spatial velocity potential φ j satisfies Eq. (1) along with the bottom boundary conditions as in Eqs. (2) and (3). The linearized kinematic condition on the flexible submerged plate surface at y = h as in Eq. (5) yields ∂φ1  ∂φ2  . (10)  +=  ∂ y y=h ∂ y y=h −

123

Wave interaction

51

Assuming that the thickness of the elastic plate is small compared with the wavelength (i.e., ρi d  1), the term ρi d is neglected hereafter as in [27]. Thus, from Eq. (8), on the mean surface y = 0 of the floating flexible plate, φ1 satisfies D1

∂ 5 φ1 ∂ 3 φ1 ∂φ1 − N1 3 + + K φ1 = 0 on y = 0, 0 ≤ x < ∞, 5 ∂y ∂y ∂y

(11)

whilst, from Eq. (9) on the submerged flexible plate at y = h, φ1 and φ2 satisfy D2

∂ 5 φ2 ∂ 3 φ2 − N2 3 + K (φ2 − φ1 ) = 0 for y = h, 0 < x < ∞, 5 ∂y ∂y

(12)

where D j = E j I j /ρg, N j = Q j /ρg, and K = ω2 /g. As in the case of a two-layer fluid having a free surface and interface, in the case of wave interaction with a floating and submerged elastic plate system, two flexural gravity wave modes will exist, and thus the far-field radiation condition is assumed to be of the form φ(x, y) =

II 

An gn (y)ei pn x as x → ∞,

(13)

n=I

with the pn s being the progressive flexural gravity wave modes generated due to the interaction of the surface gravity waves with the floating and the submerged flexible plates, gn (y) being the vertical eigenfunctions, and An being associated with the unknown wave amplitudes at far field.

3 Plane wave solution For the aforementioned problem in two dimensions, assuming that the plate deflections are of the forms ζ1 = Re{ζ10 ei( px−ωt) } and ζ2 = Re{ζ20 ei( px−ωt) }, the velocity potential is obtained as ⎧ iω ⎪ ⎨ − (μ cosh py + sinh py)ζ1 for 0 < y < h, (14) (x, y, t) = iω pcosh p(H − y)ζ 2 ⎪ ⎩ for h < y < H, p sinh p(H − h) where p satisfies the dispersion relation G( p) ≡ K −

p(1 + D1 p 4 − N1 p 2 ) = 0, μ

with μ being given by μ=

K {1 + coth ph coth p(H − h)} − (D2 p 4 − N2 p 2 ) p coth ph . K {coth ph + coth p(H − h)} − p(D2 p 4 − N2 p 2 )

An equivalent form of the above-mentioned dispersion relation is given by Rω4 − Sω2 + T = 0,

(15)

with R = {1 + coth ph coth p(H − h)}, S = gp{(D2 p 4 − N2 p 2 ) coth ph + (1 + D1 p 4 − N1 p 2 )(coth ph + coth p(H − h))}, T = g 2 p 2 (1 + D1 p 4 − N1 p 2 )(D2 p 4 − N2 p 2 ). Equation (15) yields 2 ω± =

S ± (S 2 − 4RT )1/2 . 2R

(16)

123

52

S. C. Mohapatra, T. Sahoo

In Eq. (16), the subscripts ± correspond to the flexural gravity waves generated due to the floating and submerged flexible plates, respectively. In practice, Eq. (15) is solved for p for a specific value of the wave period T which is used to determine the wave frequency ω± in different modes. Further, the ratio of the amplitudes of the deflection of the floating to submerged plate is given by 

sinh ph K {coth ph + coth p(H − h)} − p D2 p 4 − N2 p 2 ζ10 . (17) = ζ20 K √ Further, it may be noted that, for p = N2 /D2 = pcr , Eq. (16) yields 2 2 ω+ = S/R, ω− = 0.

(18)

Equation (18) ensures that, for p = pcr , no wave will propagate at the interface and only the wave in the surface mode will propagate in the presence of the submerged flexible plate, irrespective of water depth. The above situation may be of immense help to the designer to fix the position of the floating and submerged plate system for useful applications in marine environment. Keeping the realistic nature of the physical problem, it is assumed that Eq. (15) has two positive real roots, eight complex roots, and infinitely many imaginary roots. This assumption is a generalization of the plate-covered dispersion relation in the absence of a submerged plate, where there exist one real root corresponding to the propagating mode, four complex roots corresponding to nonpropagating modes, and infinitely many imaginary roots corresponding to evanescent modes. A similar situation exists in the case of wave interaction with a submerged flexible horizontal plate, in which case there are two propagating modes corresponding to surface waves, and flexural gravity modes exist, apart from the nonpropagating and evanescent modes. Hereafter, all computations are carried out throughout the paper by considering ν = 0.3, g = 9.81 ms−2 , and water density ρ = 1,025 kgm−3 unless otherwise mentioned. The nature and location of the roots of the dispersion relation are shown in Fig. 2 for specific plates. From the contour plots in Fig. 2, it is obvious that there are two real positive roots, eight complex roots, and infinitely many imaginary roots. A few of the imaginary roots are shown in Fig. 2b. In particular, in the case of deep water, for which ph 1, p(H − h) 1, coth ph → 1, and coth p(H − h) → 1, Eq. (15) yields gp 3 (D2 p 2 − N2 ) . (19) 2 On the other hand, in the case of infinite water depth, the dispersion relation can be derived from Eq. (15) by considering H → ∞. Equation (19) ensures that, in the case of deep water, there exist two plane waves due to the interaction of gravity waves with the two-plate system. In particular, for D1 = 0, N1 = 0, the wave number p has to be derived from Eq. (15) for the case of wave interaction with a submerged floating plate and then used in Eqs. (17) and (19) to obtain the ratio of the surface wave amplitude to the submerged plate deflection and wave frequency in surface and flexural gravity wave modes, respectively. 2 2 = gp(1 + D1 p 4 − N1 p 2 ) and ω− = ω+

4 Shallow-water approximation Proceeding in a similar manner as in [28], from the equation of continuity, it is derived that, in case of linearized long waves, on y = 0, the deflection of the floating plate ζ1 and the submerged plate deflection ζ2 satisfy ∂ζ2 ∂ 2 1 ∂ζ1 − =h ∂t ∂t ∂x2 and   2 ∂ 4 ζ1 ∂ 6 ζ1 ∂ ζ1 ∂ 2 (ζ1 − ζ2 ) − gh + N1 4 + D1 6 = 0. ∂t 2 ∂x2 ∂x ∂x Further, at y = h, the continuity equation yields ∂ζ2 ∂ 2 2 = (H − h) . ∂t ∂x2

123

(20)

(21)

(22)

Wave interaction

x 10

3

4

2

2

1

Im (p)

Im (p)

(b)

−3

6

0

0

−2

−1

−4

−2

−6 −6

−4

−2

0

2

4

Re (p)

(c) 8.5 8

−6

−4

−2

0

2

4

Re (p)

−3

6 −3 x 10

x 10 −7.6 −7.8 −8 −8.2 −8.4 −8.6 −8.8

−6

−4

−2

0

Re (p)

2

4

6 −3

x 10

−1

0

1

2

3

−3

−3

1

Im (p)

Im (p)

(d)

x 10

7.5

−2

Re (p)

−3

9

Im (p)

−3 −3

6 −3 x 10

x 10

1

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

Im (p)

(a)

53

0

0

−0.2

−0.2

−0.4

−0.4

−0.6

−0.6

−0.8

−0.8

−1 −10

−9.5 Re (p)

−9 −3

x 10

x 10

−1

9

9.5 Re (p)

10 −3

x 10

Fig. 2 Contour plots showing the position of the roots of the dispersion relation (15)√with h/H = 0.3, plate rigidity E 1 I1 = 60 × √ 1011 Nm2 , E 2 I2 = 47 × 1011 Nm2 , compressive force Q 1 = 0.3 E 1 I1 ρg, Q 2 = 0.2 E 2 I2 ρg, and time period T = 5 s

From Eqs. (5), (21), and (22), the long-wave equation in terms of ζ2 is obtained as     ∂ 2 ζ2 ∂6 ∂4 1 1 ∂ 2 ζ1 ∂ 2 ζ2 D 2 6 + N 2 4 ζ2 = (23) − − 2 . ∂x ∂x g(H − h) ∂t 2 gh ∂t 2 ∂t Proceeding in a similar manner as discussed in the previous section, in case of a plane wave, ζ1 and ζ2 yield

2  ω − ghp 2 (D1 p 4 − N1 p 2 + 1) ζ10 − ω2 ζ20 = 0, (24)   2   2 2 4 2 (25) ζ20 = 0. ω (H − h)ζ10 − (H − h) + h ω − gh(H − h) p D2 p − N2 p From Eqs. (24) and (25), it can be easily derived that the dispersion relation derived here will be the same as Eq. (15) in case of shallow water. 5 Fourier-type expansion formulae In this section, various expansion formulae and characteristics of the eigensystem associated with the wave interaction with a floating and submerged elastic plate system are derived in the cases of water of both finite and infinite

123

54

S. C. Mohapatra, T. Sahoo

depths. As an application of the expansion formulae, source potentials associated with flexural gravity waves in the presence of a submerged flexible plate are derived in water of infinite and finite depths. The form of the orthogonal mode-coupling relation derived and used here to determine the unknown coefficients will be clear from the integral representation of the velocity potential derived based on Green’s identity discussed in a subsequent subsection.

5.1 Expansion formulae in two dimensions The spatial velocity potential φ(x, y) satisfying Eq. (1) along with the boundary conditions Eqs. (3), (10), (11), and (12) in the case of infinite water depth is given by φ(x, y) =

X 

∞ An Fn (y)e

iμn x

n=I

where



Fn (y) =

+

A(ξ )L(ξ, y)e−ξ x dξ for x > 0,

(26)

0

f n (y) −K e−μn (y−h)

for 0 < y < h, for h < y < ∞,

 n = I, II;

L(ξ, y) =

L 1 (ξ, y) L 2 (ξ, y)

for 0 < y < h, for h < y < ∞,

with

 f n (y) = (D2 μ4n − N2 μ2n )μn − K L 1 (iμn , y)/L 1 (iμn , h), L 1 (ξ, y) = K {T1 (ξ )ξ cos ξ y − K sin ξ y}, W (ξ ) = −ξ(D2 ξ 4 + N2 ξ 2 )D(ξ, h),

L 2 (ξ, y) = L 1 (ξ, y) − W (ξ ) cos ξ(y − h),

(26a)

D(ξ, h) = T1 (ξ )ξ sin ξ h + K cos ξ h,

T1 (ξ ) = D1 ξ + N1 ξ + 1. 4

2

The eigenvalues μn s in Eq. (26) satisfy the dispersion relation G(μn ) ≡ K −

μn T1 (iμn ){K (coth μn h + 1) − μn (D2 μ4n − N2 μ2n )} = 0. K (1 + coth μn h) − (D2 μ4n − N2 μ2n )μn coth μn h

(27)

Keeping the realistic nature of the physical problem, it is assumed that the dispersion relation (27) has two distinct real positive roots μI and μII and eight complex roots μIII , μIV , . . . , μX . It may be noted that, out of the eight complex roots, two pairs of roots are of the forms a ± ib and another two pairs of roots are of the forms −c ± id (as in [7,29]). However, the behavior of the roots of Eq. (27) associated with progressive waves is discussed briefly in Sect. 3. The far-field behavior of the velocity potential in Eq. (13) yields AVII = · · · = AX = 0 in Eq. (26) for bounded solution of the velocity potential. Further, it is easily derived that the eigenfunctions Fn s satisfy the orthogonal mode-coupling relation given by

Fm , Fn  = Fm , Fn 1 + Fm , Fn 2 = E n δmn for all m, n = I, . . . , VI,

Fn , L(ξ, y) = 0 for all ξ > 0, n = I, . . . , VI, with h

Fm , Fn 1 =

Fm (y)Fn (y) dy −

 N1 D1  Fm (0)Fn (0) + Fm (0)Fn (0) + Fm (0)Fn (0) , K K

0

∞

Fm , Fn 2 = lim

→0 h

123

e−y Fm (y)Fn (y) dy −

 N2 D2  Fm (h)Fn (h) + Fm (h)Fn (h) + Fm (h)Fn (h) , K K

(28)

Wave interaction

55

E n = −D(iμn , h) sinh μn h[Fn (0)]2 G (μn )/2K 2 μ2n , and D(iμn , h) = 0. Using the orthogonal mode-coupling relation, the unknowns in the expansion formulae in Eq. (26) are obtained as

φ(x, y), Fn (y) , En

An eiμn x =

n = I, . . . , VI,

A(ξ )e−ξ x =

2 φ(x, y), L(ξ, y) , π (ξ )

where (ξ ) = W 2 (ξ ) − 2K W (ξ )L 1 (ξ, h) + {ξ 2 T12 (ξ, 2) + K 2 }K 2 . Thus, Eq. (26) along with Eqs. (27) and (28) give the expansion formula for the velocity potential in case of infinite water depth completely. Next, the expansion formula for the velocity potential will be derived in case of finite water depth. Using the eigenfunction expansion method, the velocity potential φ(x, y) satisfying Eq. (1) along with the boundary conditions in Eqs. (2), (10)–(12) in finite water depth is written in the form ∞ 

φ(x, y) =

Bn ψn (y)ei pn x for x > 0,

(29)

n=I,...,X,1

where ψn (y) =

⎧ ⎨ ⎩

(D2 pn4 − N2 pn2 ) pn tanh pn (H − h) − K

−K cosh pn (H − y)/ cosh pn (H − h)

 L 1 (i pn , y) L 1 (i pn , h)

for 0 < y < h, for h < y < H,

(30)

and the Bn s are unknowns to be determined. The eigenvalues pn , n = I, II, . . . , IX, X in Eq. (29) satisfy the dispersion relation in p   p 1 + D1 p 4 − N 1 p 2 G( p) ≡ K − = 0, (31) μ with μ being given by 

  K 1 + coth ph coth p(H − h) − D2 p 4 − N2 p 2 p coth ph 

  μ= . K coth ph + coth p(H − h) − p D2 p 4 − N2 p 2 It is easy to check that the form of the dispersion relation in Eq. (15) is the same as in Eq. (31), which for H → ∞ reduces to the relation in Eq. (27). Keeping the realistic nature of the physical problem and considering the discussion on the root behavior in Sect. 3, it is assumed that the dispersion relation in Eq. (31) has two distinct positive real roots pn , n = I, II, eight complex roots pn , n = III, . . . , X, and an infinite number of purely imaginary roots pn , n = 1, 2, . . . of the form pn = iνn . It may be noted that, out of the eight complex roots, two pairs of roots are of the forms a ± ib and another two pairs of roots are of the forms −c ± id. Sample cases of the root behavior were illustrated by contour plot in the previous section. The boundedness characteristic of the far-field behavior of the velocity potential in Eq. (13) yields BVII = · · · = BX = 0 in Eq. (29). Further, it can be easily derived that the eigenfunctions ψn s satisfy the orthogonal mode-coupling relation as in Eq. (28) with ψm , ψn 2 and the E n s replaced by H

ψm , ψn 2 =

ψm (y)ψn (y) dy −

 N2 D2 ψm (h)ψn (h) + ψm (h)ψn (h) + ψm (h)ψn (h) , K K

(32)

h

En =

−D(i pn , h) sinh pn h[ψn (0)]2 G ( pn ) , 2K 2 pn2

where G( pn ) and D( pn , h) are the same as defined in Eqs. (31) and (26a), respectively. The unknown Bn s are given by

φ(x, y), ψn (y) . E n e i pn x Next, certain lemmas and a theorem are discussed with proof to highlight various characteristics of the eigenfunctions ψn (y)s as in Eq. (30). Bn =

123

56

S. C. Mohapatra, T. Sahoo

Lemma 5.1 The eigenfunctions ψn (y)s in Eq. (30) satisfy ∞ 

[ψn (0)]2 = 1 and En

n=I,...,VI,1

∞  n=I,...,VI,1

[ψn (h)]2 = 1. En

Proof Consider the integral    K 2 α2 1 1 − I1 = dα, 2π i α G(α) sinh αh αT1 (iα) sinh αh − K cosh αh

(33)

(34)

Γ

where the contour Γ consists of the semicircular arc γ of radius R 1 on the upper half-plane, the line segments (−R, pI − 1 ), a semicircle (γ 1 ) from −1 to 1 , the line segment ( pI + 1 , pII − 2 ), a semicircle (γ 2 ) from −2 to 2 , and the line segment ( pII + 2 , R) which contains all the poles in the upper half-plane (as in [21]). Using Jordan’s lemma and the Cauchy residue theorem, Eq. (34) yields I1 = 2

∞ 

K 2 pn2  − 1 = 0, G ( pn ) sinh pn h pn T1 (i pn ) sinh pn h − K cosh pn h n=I,...,VI,1

(35)

which after substituting for ψn (y) yields the first identity in Eq. (33). Proceeding in a similar manner, from the integral     2 α αT1 (iα) sinh αh − K cosh αh 1 1 − dα, (36) I2 = 2π i G(α) sinh αh α Γ

with Γ being the same as defined above, it can be easily derived that  ∞  pn2 pn T1 (i pn ) sinh pn h − K cosh pn h I2 = 2 − 1 = 0, G ( pn ) sinh pn h

(37)

n=I,...,VI,1

which yields the second identity in Eq. (33).

 

Lemma 5.2 An equivalent form of the eigenfunctions ψn (y)s as in Eq. (30) is given by  ,y) i sinh pn (H − h) LD1(i(ippnn,h) for 0 < y < h, ψn (y) = −K cosh pn (H − y) for h < y < H. This form of ψn follows from the alternative form of the dispersion relation in Eq. (31) given by   D2 pn4 − N2 pn2 pn − K coth pn (H − h) 1 = . iL 1 (i pn , h) D(i pn , h) Theorem 5.1 The eigenfunctions ψn (y)s in Eq. (30) are linearly dependent. Proof Consider the integral  ψ(α, y) 1 dα, 0 < y < h, I = 2π tanh α(H − h)G(α)

(38)

Γ

with ψ(α, y) =

i tanh α(H − h)L 1 (iα, y) for 0 < y < h, K D(iα, h)

D(α, h), L 1 (α, y), and G(α) being the same as in Eq. (30) and (31), respectively. In Eq. (38), the contour Γ in the integral I consists of the semicircular arc γ of radius R 1 on the upper half-plane, the line segments (−R, pI −1 ), a semicircle (γ 1 ) from −1 to 1 , the line segment ( pI + 1 , pII − 2 ), a semicircle (γ 2 ) from −2 to 2 , and

123

Wave interaction

57

the line segment ( pII + 2 , R) which contains all the poles in the upper half-plane (as in [21]). Since the poles of the integrals in Eq. (38) are the same as the roots of the dispersion relation in Eq. (31), using the Cauchy residue theorem and considering the limiting cases as R → ∞ and  → 0, from the integral in Eq. (38), it can be easily derived that ∞  ∞  1 ψ( pn , y) ψ( p, y) d p ψ(α, y) dα + + = 0. (39) tanh pn (H − h)G ( pn ) tanh p(H − h)G( p) 2π tanh α(H − h)G(α) n=I,...,VI,1

−∞

γ

  As R → ∞, in Eq. (38), the second integral in Eq. (39) vanishes being an odd function and the third integral vanishes by Jordan’s lemma. Thus, Eq. (38) yields ∞  ψ( pn , y) = 0 for 0 < y < h. (40) tanh pn (H − h)G ( pn ) n=I,...,VI,1

Proceeding in a similar manner as in case of the integral I , applying the Cauchy residue theorem and Jordan’s lemma to the integral  1 α cosh α(H − h)ψ(α, y) dα for h < y < H, (41) 2π G(α) Γ

where ψ(α, y) = cosh α(H − y)/ cosh α(H − h) as in Eq. (30), it can be easily derived that ∞  pn cosh pn (H − h)ψ( pn , y) = 0 for h < y < H. G ( pn )

(42)

n=I,...,VI,1

Writing ψ( pn , y) = ψn (y), the relations in Eq. (40) and Eq. (42) ensure that the eigenfunctions ψn (y)s in Eq. (30) are linearly dependent. It may be noted that the relations for the unknowns Bn s, below Eq. (32), are expressed in terms of the velocity potential φ(x, y) and the eigenfunctions ψn s. Further, in Theorem 5.1, it has been proved that the eigenfunctions ψn s are linearly dependent. It is observed that the additional terms in the orthogonal mode-coupling relation make the Bn s dependent on the functional behavior at the end points. This dependency on end behavior is resolved by using the additional edge conditions required for specific physical problems in finite water depth (see [15,17,30]). Further, it can be seen from the expression for the An s and A(ξ ) below Eq. (28) that the orthogonal mode-coupling relation for the Fn s defined in Eq. (28) involves additional terms which include the end behavior and makes the unknowns dependent, which can be resolved to obtain the complete solution by using the required edge conditions for specific physical problems in infinite water depth. The same concept is also used in later sections while demonstrating the utility of the present method to solve specific physical problems. 5.2 Derivation of line source potentials Let G(x, y; x0 , y0 ) be the symmetric wave source potential (also referred to as the Green’s function) associated with the problem of interaction of gravity waves with a floating and submerged plate system which satisfies the Laplace equation in the fluid region except at the structural boundaries and at the source point (x0 , y0 ). Further, G(x, y; x0 , y0 ) satisfies the boundary conditions as in Eqs. (2–3) and (10–12). Near the source point (x0 , y0 ), the Green’s function behaves like  1 ln(r ) as r = (x − x0 )2 + (y − y0 )2 → 0. (43) G∼ 2π Assuming the symmetric property of the fundamental wave source potential about x = x0 , condition (43) yields (as in [28]) ∂G = δ(y − y0 )/2 on x = x0 , in the cases of both infinite and finite depths. (44) ∂x

123

58

S. C. Mohapatra, T. Sahoo

Using the generalized identity 

∞ δ(y − y0 )F(y) dy =

F(y0 ) F(y0 )/2

if y0 > 0, if y0 = 0,

(45)

0

and the expansion formula as discussed in the previous subsection, the source potential G(x, y; x0 , y0 ) in case of infinite depth is given by (proceeding in a similar manner as discussed in [17]) G(x, y; x0 , y0 ) =

VI 

An Fn (y)eiμn (x−x0 ) +

n=I

∞

A(ξ )L(ξ, y)e−ξ(x−x0 ) dξ for x > x0 .

(46)

0

Substituting for G from Eq. (46) in Eq. (44) and using the identity in Eq. (45) and comparing the obtained expression for δ(y − y0 ) with the expansion formulae in Eq. (26), the unknowns An s and A(ξ ) are obtained as ⎧ −iδ f (y ) 1 n 0 ⎪ for 0 ≤ y0 < h, ⎪ ⎪ ⎪ 2μ n En ⎪ ⎪ ⎪ ⎨ −i{ f n (h) + 1} for y0 = h, An = 2μn E n ⎪ ⎪ ⎪ ⎪ ⎪ −μn (y0 −h) ⎪ ⎪ ⎩ −iδ1 e for h < y0 < ∞ μn E n and

⎧ −δ1 K {ξ T1 (ξ ) cos ξ y0 − K sin ξ y0 } ⎪ ⎪ ⎪ ⎪ π ξ (ξ ) ⎪ ⎪ ⎪ ⎨ −[2K {ξ T1 (ξ ) cos ξ h − K sin ξ h} − W (ξ )] A(ξ ) = ⎪ π ξ (ξ ) ⎪ ⎪ ⎪ ⎪ ⎪ −[K {ξ T1 (ξ ) cos ξ y0 − K sin ξ y0 } − W (ξ ) cos ξ(y0 − h)] ⎪ ⎩ π ξ (ξ )

for 0 ≤ y0 < h, for y0 = h, for h < y0 < ∞,

with μn , Fn (y), L(ξ, y), f n (y), W (ξ ), and (ξ ) being the same as given in Sect. 5.1 and  1 for 0 < y0 < h, δ1 = 1/2 for y0 = 0. Proceeding in a similar manner as above, in case of finite water depth, the source potential G(x, y; x0 , y0 ) is obtained as G(x, y; x0 , y0 ) =

VI  n=I

An ψn (y)ei pn (x−x0 ) +

∞ 

Bn ψn (y)e−νn (x−x0 ) for x > x0 ,

(47)

n=1

where the An s are given by ⎧ −δ2 sinh pn (H − h)L 1 (i pn ; y0 ) ⎪ ⎪ ⎪ ⎪ 2 pn E n K D(i pn , h) ⎪ ⎪ ⎪ ⎨ − sinh pn (H − h)L 1 (i pn ; h) i cosh pn (H − h) − An = ⎪ 2 pn E n 2 pn E n ⎪ ⎪ ⎪ ⎪ ⎪ −iδ2 cosh pn (H − y0 ) ⎪ ⎩ 2 pn E n

for 0 ≤ y0 < h, for y0 = h, for h < y0 ≤ H,

and Bn , n = 1, 2, 3, . . . , are obtained from the expression for the An s with pn = iνn , where δ2 = 1 for y0 ∈ (0, h) ∪ (h, H ) and δ2 = 1/2 for y0 = 0, H and pn , νn , ψn (y), L 1 (i pn ; y0 ), and E n s being the same as in Sect. 5.1.

123

Wave interaction

59

5.3 Flexural gravity wavemaker problem In this subsection, using the source potential G(x, y, x0 , y0 ) derived in the previous subsection and Green’s identity, the expansion formulae for the flexural gravity wavemaker problem in the presence of a horizontal flexible plate are derived. In this case, the spatial velocity potential φ(x, y) satisfies the Laplace equation as in Eq. (1), along with the bottom boundary conditions as in Eqs. (2) and (3), and the boundary conditions on the floating and submerged flexible plates as in Eqs. (11) and (12). In addition, assuming that a vertical wavemaker oscillates with frequency ω and amplitude u(y) about its mean position at x = 0 on the wavemaker, the velocity potential satisfies ∂φ = u(y) on x = 0, (48) ∂x for 0 < y < H in finite water depth and 0 < y < ∞ in infinite water depth except at y = h. In order to derive an integral representation of the velocity potential in terms of the Green’s function G(x, y; x0 , y0 ) and u(y) as in Eq. (48), set G mod (x, y; x0 , y0 ) = G(x, y; x0 , y0 ) + G(−x, y; x0 , y0 ), with zero normal velocity on the wavemaker, i.e., potential φ(x0 , y0 ) is obtained as h φ(x0 , y0 ) =

∞ G mod 1 (0, y; x 0 , y0 )u(y) dy +

0

G mod x (0,

(49) y; x0 , y0 ) = 0. Using Green’s identity, the velocity

 mod  dx G mod 1y φ1 − φ1y G 1 y=0

0 H(∞)

G mod 2 (0,

+

∞ y; x0 , y0 )u(y) dy +

 mod  (φ2 − φ1 )G mod − G mod 2y − (G 2 1 )φ2y y=h dx,

(50)

0

h

where H (∞) = H in case of finite water depth and H (∞) = ∞ in case of infinite water depth, and the contribution from the bottom boundary is considered to be zero, with the subscripts “1” and “2” in the velocity potential φ and source potential G referring to fluid regions 1 and 2 as defined in Sect. 2. Using the fact that G(x, y, x0 , y0 ) is outgoing in nature at infinity and G mod (x, y; x0 , y0 ) satisfies G mod y (0, d; x 0 , y0 ) = 2G y (0, d; x 0 , y0 ),

G mod x y (0, d; x 0 , y0 ) = 0,

mod G mod yyy (0, d; x 0 , y0 ) = 2G yyy (0, d; x 0 , y0 ), G x x x y (0, d; x 0 , y0 ) = 0,

it can be easily derived that ∞ 

  2 mod mod  D1 G 1yyy φ1x y + G 1y φ1x yyy − N1 G 1y φ1x y  φ1 G 1y − φ1y G 1 dx = − y=0 K (x,y)=(0,0)

(51)

0

and ∞  0

 mod mod  G mod {φ − φ } − {G − G }φ 2 1 2y  2y 2 1

dx y=h

  2 D2 G 2yyy φ2x y + G 2y φ2x yyy − N2 G 2y φ2x y  , =− K (x,y)=(0,h)

(52)

where d = 0, h. Using Eqs. (51) and (52) in Eq. (50), the velocity potential in terms of the Green’s function G as in Eq. (50) is obtained as     2 φ(x0 , y0 ) = − 2 G(0, y; x0 , y0 )u(y) dy + D1 G 1yyy φ1x y + G 1y φ1x yyy − N1 G 1y φ1x y  K (x,y)=(0,0) R     

 2 , (53) D2 G 2yyy φ2x y + G 2y φ2x yyy − N2 G 2y φ2x y  + K (x,y)=(0,h)

123

60

S. C. Mohapatra, T. Sahoo

where R is the semi-infinite interval (0, ∞) in case of infinite water depth and the finite interval (0, H ) in case of finite water depth. Next, substituting the explicit forms of the Green’s function from Eq. (46) into Eq. (53), the velocity potential in case of infinite water depth is obtained as ∞ VI  a(ξ )L(ξ, y)e−ξ x dξ iμn x an iδ1 Fn (y)e + , (54) φ(x, y) = ξ (ξ ) n=I

0

where an and a(ξ ) are given by ⎧∞    1 ⎨ μn an = D1 − μ2n φx y (0, 0) + φx yyy (0, 0) − N1 φx y (0, 0) Fn (y)u(y) dy + μn E n ⎩ D(iμn , h) 0 ⎫  ⎬ + μn D2 μ2n φx y (0, h) + φx yyy (0, h) − N2 φx y (0, h) for n = I, II, . . . , VI, ⎭ ⎧∞    2⎨ a(ξ ) = L(ξ, y)u(y) dy − K ξ D1 − ξ 2 φx y (0, 0) + φx yyy (0, 0) − N1 φx y (0, 0) ⎩ π 0 ⎫ ⎬   −ξ D(ξ, h) D2 − ξ 2 φx y (0, h) + φx yyy (0, h) − N2 φx y (0, h) , ⎭ with δ1 , μn , E n , Fn (y), (ξ ), and L(ξ, y) being the same as defined in Eq. (26). However, an and a(ξ ) are expressed in terms of the two unknowns φ y (0, 0) and φ yyy (0, 0), which have to be determined from appropriate edge conditions when dealing with a specific physical problem. Proceeding in a similar manner as in the case of infinite depth, the velocity potential φ(x, y) associated with the flexural gravity wavemaker problem in finite water depth is obtained as φ(x, y) =

VI 

An δ2 ψn (y)ei pn x +

n=I

∞ 

Bn δ2 ψn (y)e−νn x ,

(55)

n=1

where the An s are given by ⎡ H    1 ⎣ pn sinh pn (H − h) 2 D1 pn φx y (0, 0) + φx yyy (0, 0) − N1 φx y (0, 0) ψn (y)u(y) dy − An = pn E n K D(i pn , h) 0 ⎤   pn sinh pn (H − h) 2 − D2 pn φx y (0, h) + φx yyy (0, h) − N2 φx y (0, h) ⎦ , n = I, . . . , VI, K and Bn , n = 1, 2, 3, . . . , are obtained from the expression for the An s with pn = iνn . On the other hand, if the condition in Eq. (48) satisfied on the wavemaker is replaced by φ = u(y) on x = 0, then G

G mod

mod

(56)

is chosen as

(x, y; x0 , y0 ) = G(x, y; x0 , y0 ) − G(−x, y; x0 , y0 ), G mod (0,

(57)

y; x0 , y0 ) = 0. Proceeding in a similar manner as discussed above, the velocity potential which leads to φ(x, y) is obtained as    2 D1 G 1yyy φ1x y + G 1y φ1x yyy − N1 G 1y φ1x y  φ(x0 , y0 ) = 2 G x (0, y; x0 , y0 )u(y) dy + K (x,y)=(0,0) R    

 2 D2 G 2yyy φ2x y + G 2y φ2x yyy − N2 G 2y φ2x y  + , (58) K (x,y)=(0,h)

123

Wave interaction

61

Fig. 3 Schematic diagram of floating elastic plate and submerged finite membrane

y

Floating Elastic Plate y h

Submerged Membrane 2

0

x= -a

0 x

y x= a

H h y

H

x= 0

with R being the same as defined in Eq. (53). The velocity potential φ(x, y) in the cases of both infinite and finite depths can be easily obtained by proceeding in a similar manner as in the previous case. The integral forms of the velocity potentials in terms of the Green’s function in Eqs. (53) and (58) are similar to the form of the orthogonal mode-coupling relation as in Eq. (28). It may be noted that the present form of the Green’s function and the velocity potentials are derived based on the assumption that the roots of the dispersion relation are unique in nature. On the other hand, an alternative integral representation of the Green’s function has to be used in the Green’s identity to obtain the velocity potentials which will incorporate the occurrences of multiple roots arising in the dispersion relation while evaluating the complex integrals based on the Cauchy residue theorem. A similar approach was used in [31] while dealing with wave motion within porous structures to account for the situation when the roots of the dispersion relation coalesce.

6 Application of expansion formulae The utility of the expansion formulae and the flexural gravity wavemaker problems are demonstrated through two different physical problems in two subsections separately matching the continuity of pressure and velocity as in [7].

6.1 Diffraction of flexural gravity waves by a submerged membrane In this subsection, diffraction of flexural gravity waves by a submerged horizontal flexible membrane of finite length is analyzed in case of finite water depth in the two-dimensional Cartesian coordinate system. In this case, an infinitely flexible plate is floating at the free surface and a finite submerged flexible membrane is kept at y = h and is under uniform tension M2 . The submerged flexible membrane occupies the region y = h, −a < x < a and is assumed to be fixed at both ends at x = ±a, y = h. The fluid is assumed to be infinitely extended along the x-axis and occupies the region −∞ < x < ∞, 0 < y < H , except the submerged membrane, with H being water depth as in Fig. 3. The submerged membrane located at y = h divides the fluid domain into two regions. Region 1 occupies the fluid domain 0 < y < H , 0 < x < ∞, whilst the domain 0 < y < H , −∞ < x < 0 refers to region 2. In this case, the spatial velocity potential φ(x, y) will satisfy the governing equation (1) along with bottom boundary conditions as in Eq. (2) at y = H and the plate-covered boundary condition at y = 0 as in Eq. (11). The conditions on the submerged membrane are given by (similar to the case of a submerged plate as discussed in Sect. 2)   ∂φ  ∂φ  = for − a < x < a (59) ∂ y  y=h + ∂ y  y=h − and

 # "   ∂ 3 φ  M 3 + K φ  y=h + − φ  y=h − = 0 for − a < x < a, ∂ y y=h +

(60)

123

62

S. C. Mohapatra, T. Sahoo

where M = M2 /ρg. It may be noted that Eq. (60) is obtained from Eq. (12) by putting E 2 = 0, Q 2 = −M2 . Further, the assumption that the membrane is fixed at both ends yields φ y = 0, at x = ±a, y = h.

(61)

In order to solve the physical problem at hand, the deflection, slope of deflection, bending moment, and shear force acting on the floating elastic plate are assumed to be continuous at (0, 0) and (±a, 0) (as in [30]), which yields     ∂φ  ∂ 2 φ  ∂ 2 φ  ∂φ  = , = , (62) ∂ y (d + ,0) ∂ y (d − ,0) ∂ x∂ y (d + ,0) ∂ x∂ y (d − ,0)       ∂ 3 φ  ∂ 4φ ∂ 2φ ∂ 4φ ∂ 2φ ∂ 3 φ  = , D + N = D + N , (63) 1 1 1 1 ∂ y∂ x 2 (d + ,0) ∂ y∂ x 2 (d − ,0) ∂ y∂ x 3 ∂ y∂ x (d + ,0) ∂ y∂ x 3 ∂ y∂ x (d − ,0) for d = 0, ±a. Further, the membrane deflection and the slope of membrane deflection are assumed to be continuous at (±0, h), which yields     ∂φ  ∂φ  ∂ 2 φ  ∂ 2 φ  = , = . (64) ∂ y (0+ ,h) ∂ y (0− ,h) ∂ x∂ y (0+ ,h) ∂ x∂ y (0− ,h) The far-field radiation condition associated with the flexural gravity wave propagating below the floating plate in the presence of the submerged membrane is given by ⎧  cosh k0 (H − y) −ik0 x ⎪ ⎪ I0 e as x → ∞, + R0 eik0 x ⎨ cosh k0 H (65) φ(x, y) = ⎪ ⎪ ⎩ T0 cosh k0 (H − y) e−ik0 x as x → −∞, cosh k0 H where I0 is associated with the amplitude of the incident flexural gravity wave floating below the flexible floating plate assumed to be known, and R0 and T0 are unknown constants associated with the amplitude of the reflected and transmitted flexural gravity waves, respectively, and are to be determined. Further, k0 in Eq. (65) satisfies the dispersion relation in k as given by   (66) F(k) = k 1 − N1 k 2 + D1 k 4 tanh kh − K = 0. In addition, at the interfaces x = ±a, the continuity of pressure and velocity yield φ(±a + , y) = φ(±a − , y) and φx (±a + , y) = φx (±a − , y).

(67)

Considering the geometrical symmetry of the physical problem about x = 0, the velocity potential associated with the physical problem mentioned above is rewritten as the sum of the symmetric and antisymmetric potentials as given by ϕ(x, y) = φ1 (x, y) − φ2 (−x, y) and ϒ(x, y) = φ1 (x, y) + φ2 (−x, y)

(68)

with φ1 and φ2 being the velocity potentials in regions 1 and 2, respectively. The reduced potentials ϕ(x, y) and ϒ(x, y) satisfy the governing Laplace equation in Eq. (1), and the surface and interface conditions in Eqs. (11), (59), and (60). The edge conditions (62) and (63) yield 3 ∂ y ϕ(d, 0) = 0, ∂ yx x ϕ(d, 0) = 0, 2 ∂ yx ϒ(d, 0)

= 0,

4 D1 ∂ yx x x ϒ(d, 0) +

(69a) 2 N1 ∂ yx ϒ(d, 0)

= 0,

(69b)

with d = 0, a. On the other hand, the continuity of velocity and pressure at the interface x = ±a in Eq. (67) yield that ϕ, ϒ, ϕx , and ϒx are continuous at x = a±. Further, on the submerged membrane, from Eq. (61), the edge conditions in terms of ϕ(x, y) and ϒ(x, y) yield that ϕ y and ϒ y vanish at (a, h). Further, from Eq. (64), the conditions on the membrane yield ϕ y = 0 and ϒx y = 0 at (0, h). Utilizing the expansion formulae, the reduced

123

Wave interaction

63

velocity potential ϕ(x, y) is expanded in the form ⎧ VIII ∞   ⎪ ⎪ ⎪ C sin p xψ (y) + Cn sin pn xψn (y) ⎪ n n n ⎨ n=I n=1 ϕ(x, y) = ∞  ⎪ ⎪ −ik0 (x−a) Y (y) + ⎪ I e An eikn (x−a) Yn (y) ⎪ 0 0 ⎩

for 0 < x < a, (70) for a < x < ∞,

n=0,I,...,IV,1

where the pn s and ψn (y)s are the same as defined in Sect. 5.1 with D2 = 0 and N2 = −M. Thus, the eigenvalues pn s in this case satisfy the dispersion relation given by     K coth ph + coth p(H − h) − M p 3 p 1 − N1 p 2 + D1 p 4

 C( p) ≡ K − = 0. (71) K 1 + coth ph coth p(H − h) − M p 3 coth ph On the other hand, in the floating elastic plate-covered region without the submerged membrane, the vertical eigenfunctions Yn (y)s are given by ⎧ cosh kn (H − y) ⎪ ⎪ for n = 0, I, II, . . . , IV, ⎨ cosh kn H (72) Yn (y) = ⎪ ⎪ ⎩ cos kn (H − y) for n = 1, 2, . . . , cos kn H where the kn s satisfy the dispersion relation in k as in Eq. (66), which has one real root k0 , four complex roots of the forms a ± ib, −c ± id, and infinitely many imaginary roots of the form ikn . In order to have bounded solution of ϕ(x, y) in the plate-covered region, the contribution from two of the complex roots is ignored, which yields AIII = AIV = 0. Further, the vertical eigenfunctions Yn (y)s satisfy the orthogonal mode-coupling relation H

Ym , Yn  = 0



 N  D1  1 Ym Yn + Ym Yn − Y Y  Ym (y)Yn (y) dy + = En δmn , K K m n  y=0

(73)

where En = F (kn )Yn (0)/2kn K with F(kn ) being the same as in Eq. (66). Using the continuity of velocity and pressure at x = a (as in Eq. 67) and the orthogonal mode-coupling relation as in Eq. (73), two systems of equations for the determination of the An s and Cn s are obtained as (hereafter, the series for velocity potentials are truncated after N terms) N 

Em Am =

Cn sin pn a X mn − I0 E0 δm0 , m = 0, I, II, 1, 2, . . . N ,

(74)

n=I,...,VIII,1

ikm Em Am =

N 

pn Cn cos pn a X mn + ik0 I0 E0 δm0 , m = 0, I, II, 1, 2, . . . N ,

(75)

n=I,...,VIII,1

where



X mn =

H

ψn (y)Ym (y) dy = − sinh pn (H − h)Umn + Vmn ,

0

Umn =

and Vmn

  1 k sinh k (H − h)[ p T (i p ) sinh p h + K cosh p h m n 1 n n n 2 − p 2 ) cosh k H [−D(i p , h)] m (km m n n + pn cosh km (H − h)[−D(i pn , h)]  − {[ pn T1 (i pn )km sinh km H + K pn cosh km H ]}

 − km sinh km (H − h) cosh pn (H − h) − pn cosh km (H − h) sinh pn (H − h) . = 2 − p 2 ) cosh k H (km m n

123

64

S. C. Mohapatra, T. Sahoo

Further, the fixed edge condition of the membrane yields N 

Cn pn sin pn a sinh pn (H − h) = 0,

(76)

n=I,...,VIII,1

where the infinite series is truncated after N terms. Thus, the system of equations (74–76) is solved simultaneously to determine the unknown constants associated with the reduced potential ϕ. It may be noted that the continuity conditions in Eqs. (62–64) in terms of ϕ are incorporated while deriving the system of equations in Eqs. (74) and (75) by using the orthogonal mode-coupling relation defined in Eq. (73). Proceeding in a similar way as in case of ϕ, the velocity potential ϒ(x, y) is obtained as ⎧ VIII ∞   ⎪ ⎪ ⎪ G cos p xψ (y) + G n cos pn xψn (y) for 0 < x < a, ⎪ n n n ⎨ n=I n=1 (77) ϒ(x, y) = ∞  ⎪ ⎪ ikn (x−a) −ik (x−a) ⎪ 0 Y0 (y) + Bn e Yn (y) for a < x < ∞, ⎪ ⎩ I0 e n=0,I,II,1

where the pn s and ψn (y)s are the same as in Eq. (70) with D2 = 0 and N2 = −M as defined in Eq. (70). Matching the velocity and pressure at x = a and proceeding in a similar manner as in case of ϕ, two systems of equations for the determination of the Bm s and G m s for m = 0, I, II, 1, 2, . . . , N are obtained as N 

Em Bm =

G n cos pn a X mn − I0 E0 δm0 ,

(78)

n=I,...,VIII,1

ikm Em Bm = −

N 

pn G n sin pn a X mn + ik0 I0 E0 δm0 .

(79)

n=I,...,VIII,1

Further, from the fixed edge of the membrane, the condition in terms of ϒ yields N 

G n pn cos pn a sinh pn (H − h) = 0,

(80)

n=I,...,VIII,1

where the infinite series is truncated after N terms. It may be noted that the required continuity conditions in Eqs. (62)–(64) in terms of ϒ are incorporated while deriving the system of equations in Eqs. (78) and (79) by using the orthogonal mode-coupling relation defined in Eq. (73). The system of equations in Eqs. (78)–(80) can be solved numerically to determine the unknown coefficients associated with the reduced potentials ϒ. Once the unknown coefficients associated with the reduced potentials ϕ and ϒ are obtained, the reflection and transmission coefficients K r , K t will be computed using the relations K r = |(A0 + B0 )/(2I0 )| and K t = |(A0 − B0 )/(2I0 )| which satisfy the energy relation K r2 + K t2 = 1. Using the orthogonal characteristics of the eigenfunctions ψn (y), an alternative system of equations for the determination of the unknowns associated with the velocity potentials ϕ and ϒ can be obtained, details of which are deferred here. In Fig. 4a, the reflection coefficient K r versus time period is plotted for different values of the tensile force parameter M2 /(ρg H 2 ). The reflection coefficient is resonating in nature for smaller values of wave period for fixed tensile force. The resonating pattern in the reflection coefficient may be due to the change of phase of the incident and reflected waves in the presence of the submerged flexible structure. As the wave period increases, wave reflection diminishes and the pattern is similar to that of Hassan et al. [7]. However, with an increase in tensile force, the general pattern in reflection coefficient is increasing in nature. For larger values of wave period T , irrespective of tensile force, wave reflection becomes negligible. In Fig. 4b, the reflection coefficient K r versus time period T is plotted for different values of nondimensional membrane length a/H . The general pattern of reflection coefficient is similar to that in Fig. 4a. However, it is observed that the reflection coefficient increases with an increase in a/H . However, optimum values of the reflection coefficient of up to 1 are observed for higher values of nondimensional membrane length a/H for certain values of the wave period T .

123

Wave interaction 1

(b)

2

M2/ρgH =0.08

1

0.9

M /ρgH =0.1

0.9

0.8

M2/ρgH2=0.15

0.8

2

a/H=3 a/H=5 a/H=6

2

0.7

0.7

0.6

0.6

Kr

Kr

(a)

65

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1 5

10

15

20

25

30

35

40

45

50

5

10

15

Wave period (T)

20

25

30

35

40

45

50

Wave period (T)

√ Fig. 4 Variation of reflection coefficient K r versus time period with E 1 I1 = .47 × 1011 Nm2 and Q 1 = 0.3 E 1 I1 ρg for various values of: a tensile force parameter M2 /(ρg H 2 ) with a/H = 3, b a/H with tensile force M2 /(ρg H 2 ) = 0.1 Fig. 5 Schematic diagram of semi-infinite floating and submerged elastic plate

Semi-infinite floating plate y

Semi-infinite submerged plate

y

h

0

Open water region

y

H

x 0

6.2 Diffraction of surface waves by a semi-infinite floating plate in the presence of a submerged flexible plate The diffraction of surface gravity waves by a semi-infinite floating plate in the presence of a semi-infinite submerged flexible plate is analyzed in case of finite water depth in the two-dimensional Cartesian coordinate system with the x-axis being horizontal and the y-axis being vertically downward positive. It is assumed that both plates have free edges at x = 0, y = 0, h. The fluid is assumed to be infinitely extended along the x-axis and occupies the region −∞ < x < ∞, 0 < y < H , except the floating and submerged plates, with H being water depth as in Fig. 5. The floating and submerged plates divide the fluid domain into two regions. Region 1 consists of the region occupying the fluid domain 0 < y < H and 0 < x < ∞, whilst region 2 occupies the fluid domain −∞ < x < 0, 0 < y < h, or h < y < H in case of finite water depth. The spatial velocity potentials φ j for j = 1, 2, satisfy Eq. (1), the linearized free surface condition in Eq. (11) with D1 = 0, N1 = 0, and the bottom boundary condition in Eq. (2) in the open water region. On the other hand, in the plate-covered region, the φ j s satisfy the condition in Eq. (11) on the plate-covered mean free surface y = 0, whilst, on the mean surface y = h of the submerged plate, the φ j (x, y)s satisfy the condition D2

∂ 5 φ2 ∂ 3 φ2 − N + K (φ2 − φ1 ) = 0 on y = h, −∞ < x < 0. 2 ∂ y3 ∂ y5

(81)

Assuming that both plates have a free edge at the end x = 0, the vanishing of the shear force and bending moments at x = 0 yield ∂ 4 φ2 (0, 0) ∂ 2 φ2 (0, 0) = 0, + N 1 ∂ y∂ x 3 ∂ y∂ x φ2x x y (0, 0) = 0, φ2x x y (0, h) = 0. D1

D2

∂ 4 φ2 (0, h) ∂ 2 φ2 (0, h) = 0, + N 2 ∂ y∂ x 3 ∂ y∂ x

(82) (83)

123

66

S. C. Mohapatra, T. Sahoo

In addition, at the interface x = 0, 0 < y < H , the continuity of pressure and velocity yield φ1 (0, y) = φ2 (0, y) and φ1x (0, y) = φ2x (0, y).

(84)

The velocity potential in region 1 is given by ∞  φ1 (x, y) = A0 Y0 (y)e−ik0 x + Rn Yn (y)eikn x , x > 0,

(85)

n=0

where A0 and R0 are related to the amplitude of the incident and reflected waves, and Rn for n = 1, 2, . . . are unknown constants to be determined. The vertical eigenfunctions Yn (y)s in Eq. (85) are given by Yn (y) = √ cosh kn (H − y)/ Nn satisfying the orthonormal relation H Ym (y)Yn (y) dy = δmn , (86) 0

with Nn =

H 2

 1+

sinh 2kn H 2kn H

 , the kn s satisfy the dispersion relation

ω2 = gkn tanh kn H, for n = 0, 1, 2, . . ., and δmn is the Kronecker delta function. Proceeding in a similar manner as discussed in Sect. 5.1, the velocity potential in region 2 is expanded as ∞  φ2 (x, y) = Bn ψn ( pn , y)e−i pn x for x < 0, (87) n=I,...,VI,1

where the Bn s are unknown constants to be determined, and the eigenfunction ψn s are the same as defined in Sect. 5.1. However, the eigenvalues pn , n = I, . . . , VI, 1, 2, . . . satisfy the dispersion relation in the submerged plate region as given in Eq. (31). Keeping in mind the realistic nature of the physical problem, it is assumed that the dispersion relation (31) has two distinct positive real roots pn , n = I, II, eight complex roots pn , n = III, IV, . . . , X of the forms −a ± ib, and an infinite number of purely imaginary roots of the form pn = iμn , n = 1, 2, . . . (similar to the ones in Mohapatra et al. [28]). Further, the boundedness characteristics of the velocity potential yield that the four complex roots will not contribute to the expansion of the velocity potential, which yields BVII = BX = 0. From Eq. (85), it is clear that there is only one wave mode in the water region leading to incident and reflected waves associated with the surface gravity waves. On the other hand, due to the interaction of surface waves with the floating flexible structure and the submerged flexible plate, there exist two progressive wave modes as in Eq. (87). The continuity of pressure across x = 0 in Eq. (84) yields ∞ ∞   Rn Yn (y) = Bn ψn (y). (88) A0 Y0 (y) + n=0

n=I,...,VI,1

Using the orthonormal characteristics of the Yn s, it can be easily derived that ∞  Rm = −A0 δm0 + Bn X mn , n=I,...,VI,1

where H X mn =

ψn (y)Ym (y) dy = − sinh pn (H − h)Umn + Vmn , 0

Umn =

 1 k sinh km (H − h) [ pn T1 (i pn ) sinh pn h √ 2 − p 2 ) N [−D(i p , h)] m (km m n n + K cosh pn h] + pn cosh km (H − h) [−D(i pn , h)]

 − pn T1 (i pn )km sinh km H + K pn cosh km H ,

123

(89)

Wave interaction

67

and

 − km sinh km (H − h) cosh pn (H − h) − pn cosh km (H − h) sinh pn (H − h) . Vmn √ 2 − p2 ) N (km m n Further, applying the continuity of velocity across x = 0 as in Eq. (84) and proceeding in a similar manner as above, another system of equations is obtained as =

∞ 

A0 δm0 (km + k0 ) =

  Bn X mn km + pn .

(90)

n=I,...,VI,1

Using the edge condition as in Eqs. (82) and (83), the system of equations is obtained as N  n=I,...,VI,1 N  n=I,...,VI,1 N 

 Bn pn2 K  D1 pn2 − N1 sinh pn (H − h) = 0, D(i pn , h)

(91)

Bn pn3 K sinh pn (H − h) = 0, D(i pn , h)

(92)

  Bn pn2 D2 pn2 − N2 sinh pn (H − h) = 0,

(93)

Bn pn3 sinh pn (H − h) = 0.

(94)

n=I,...,VI,1

and N  n=I,...,VI,1

The system of equations (89–94) is to be solved to find the Bn s and Rn s. Thus, the reflection coefficient, in this case given by K r = R0 /A0 , can be obtained to understand the reflective characteristics of the submerged plate. Applying Green’s identity to the velocity potential along with its complex conjugate in the fluid domain (as in [7,29]), the energy relation in terms of the reflection and transmission coefficients is obtained as   k0 |A0 |2 − |R0 |2 + pI E I |BI |2 + pII E II |BII |2 = 0, (95) with A0 and R0 being the same as in Eq. (95), Bn for n = I, II being the same as defined in Eq. (87), and E n for n = I, II being the same as defined in Eq. (32). As mentioned in the previous subsection, the orthogonality of the

(a)

1

(b) 6

2

0.9

E2I2=47×10 Nm

0.8

E2I2=47×10

6.1

2

6.2

2

E2I2=47×10

0.7

Nm

0.8

Nm

h/H=0.2 h/H=0.3 h/H=0.5

0.7 0.6 r

0.6 0.5

K

Kr

1 0.9

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1

0

0 2

4

6

8

10

12

14

Wave period (T)

16

18

20

5

10

15

20

25

30

35

40

45

50

Wave period (T)

√ Fig.√6 Variation of reflection coefficient K r versus time period with E 1 I1 = 47 × 1011 Nm2 , Q 1 = 0.3 E 1 I1 ρg, and Q 2 = 0.2 E 2 I2 ρg for various values of: a submerged plate rigidity E 2 I2 with h/H = 0.3, b depth of submergence h/H with E 2 I2 = 47 × 109 Nm2

123

68

S. C. Mohapatra, T. Sahoo

eigenfunctions ψn (y)s can be used to derive a system of equations similar to the system of equations in (89–94) as in [14], and details are deferred here. In Fig. 6a, the reflection coefficient K r versus time period is plotted for different values of E 2 I2 . A certain resonating pattern is observed in the reflection coefficient for smaller values of wave period. However, for higher values of wave period, full reflection is observed. The resonating pattern in the wave reflection may be due to the interaction of the two progressive wave modes in the floating and submerged plates. The optimum values of the reflection coefficient may be due to the change of phase of the incident and reflected waves in the presence of the submerged flexible structure as discussed in Fig. 4a. On the other hand, the flexibility of the submerged structure has negligible effect on wave reflection. In Fig. 6b, the reflection coefficient K r versus time period T is plotted for different values of depth of submergence h/H . It is observed that the resonating pattern is more predominant when the plate is close to the free surface for higher values of wave period. However, for smaller values of wave period, irrespective of water depth, the resonating pattern in the reflection is similar to that of Fig. 6a. Thus, for all practical purposes, the flexible structure should not be kept close to the floating structure to avoid the resonance effect and achieve maximum wave reflection.

7 Conclusions In the present paper, surface wave interaction with a floating and submerged elastic plate system is investigated. In the presence of floating and submerged flexible structures in the fluid, surface gravity waves interact with the floating and submerged structures to generate flexural gravity waves in both the floating and submerged plate modes. For computational purposes, contour plots are used to locate the initial guess of the roots of the dispersion associated with the wave–structure interaction problem. Various expansion formulae and characteristics of the eigensystem derived in the present study will be useful to deal with various physical problems associated with wave–structure interaction problems in two dimensions. The utility of the expansion formulae is demonstrated by analyzing two physical problems associated with wave interaction with floating and submerged structures of specific configurations. For each physical problem, energy relations are used to check the accuracy of the computational results. From the two physical problems discussed in the manuscript, it is clear that the flexible submerged plate significantly increases the wave reflection of both surface gravity waves and flexural gravity waves. Further, wave reflection attained optimum values for certain intermediate wave periods, which may be due to phase change of incident and reflected waves leading to constructive/destructive interference of the waves in the presence of the submerged structure. Thus, suitable positioning and configuration of the floating and submerged plate systems are likely to be helpful for mitigating the structural response of large floating structures used for various infrastructural activities in the marine environment. A similar concept can be easily utilized to deal with wave–structure interaction problems in acoustics and related problems of mathematical physics and engineering. Acknowledgments Financial support received from the Naval Research Board, New Delhi, Government of India is gratefully acknowledged. The authors gratefully acknowledge the reviewers for their advice and suggestions.

Appendix Generalized expansion formulae Under the assumption of the linearized theory of water waves, the spatial velocity potential φ(x, y) satisfies the Laplace equation (1). On the surface at y = 0, it is assumed that the velocity potential φ satisfies a general type of boundary condition of the form     ∂ ∂φ ∂ L +M φ = 0, (96) ∂x ∂y ∂x

123

Wave interaction

69

where L and M are the linear differential operators of the form       l0 m0 ∂ 2n ∂ 2n ∂ ∂ = = cn 2n , M dn 2n , L ∂x ∂x ∂x ∂x n=0

(97)

n=0

with the cn s and dn s being known constants (as in [29]). It is assumed that l0 ≥ m 0 , so that l0 determines the highest-order derivative in the boundary condition at y = 0. The linearized condition on the flexible submerged plate at y = h and the bottom boundary condition are the same as defined in Sect. 2. However, in case of finite water depth, the pn s satisfy the dispersion relation in p given by



 G( p; l0 , m 0 ) ≡ K Q( p; m 0 ) 1 + coth p(H − h) coth ph − K T2 (i p) − 1 p coth ph



 − p P( p; l0 )Q( p; m 0 ) coth ph + coth p(H − h) + p 2 P( p; l0 ) T2 (i p) − 1 = 0, (98) where P( p; l0 ) =

l0  (−1)n cn p 2n ;

Q( p; m 0 ) =

n=0

m0  

− 1)n dn p 2n



(99)

n=0

are the characteristic polynomials associated with the linear differential operators L(∂/∂ x) and M(∂/∂ x), respectively. The general form of the velocity potential φ(x, y) satisfying Eq. (1) along with the boundary conditions in Eqs. (3), (10), (12), and (96) in case of infinite depth is of the form φ(x, y) =

2l 0 +2 n=I

∞ An Fn (y)e

iμn x

+

A(ξ )L(ξ, y)e−ξ x dξ for x > 0,

(100)

0

where the Fn (y)s and L(ξ, y) are given by ⎧ ⎨ − iL 1 (iμn , y; l0 , m 0 ) , 0 < y < h, K D(iμn , h; l0 , m 0 ) Fn (y) = ⎩ −μn (y−h) , h < y < ∞, e  L 1 (ξ, y; l0 , m 0 ) for 0 < y < h, L(ξ, y; l0 , m 0 ) = L 2 (ξ, y; l0 , m 0 ) for h < y < ∞, with

 L 1 (ξ, y; l0 , m 0 ) = K ξ P(iξ ; l0 ) cos ξ y − Q(iξ ; m 0 ) sin ξ y ,

L 2 (ξ, y; l0 , m 0 ) = L 1 (ξ, y; l0 , m 0 ) − W (ξ, h; l0 , m 0 ) cos ξ(y − h), W (ξ, h; l0 , m 0 ) = −ξ {T2 (ξ ) − 1}D(ξ, h; l0 , m 0 ), T2 (ξ ) = D2 ξ 4 + N2 ξ 2 + 1, D(ξ, h; l0 , m 0 ) = ξ P(iξ ; l0 ) sin ξ h + Q(iξ ; m 0 ) cos ξ h. The eigenfunctions μn s satisfy the dispersion relation in μ given by  P(μ; l0 ) μQ(μ; m 0 )(coth μh + 1) − μ2 {T2 (iμ) − 1}

  G(μ; l0 , m 0 ) ≡ − K = 0. Q(μ; m 0 ) 1 + coth μh − T2 (iμ) − 1 μ coth μh

(101)

It is assumed that the dispersion relation in Eq. (101) has two distinct real positive roots and 2l0 number of complex roots of the forms a ± ib and −c ± id, with μI and μII being the real roots and μIII , μIV , . . . , μ2l0 +2 being the complex roots. However, out of the 2l0 + 2 roots of the dispersion relation, only roots leading to boundedness of the velocity potential will be considered in the expansion formulae in Eq. (100). It may be noted that D(iμn , h) = 0 for all 0 < h < ∞. Further, the eigenfunctions Fn (y)s and L(ξ, y; l0 , m 0 ) satisfy the generalized orthogonal mode-coupling relation as given by

Fm , Fn  = Fm , Fn 1 + Fm , Fn 2 = E n δmn for m, n = I, II, . . . , 2l0 + 2

(102)

123

70

S. C. Mohapatra, T. Sahoo

and

Fm , L(ξ, y; l0 , m 0 ) = 0 for m = I, II, . . . , 2l0 + 2, ξ > 0, where δmn is the Kronecker delta function, and the E n s are given by

(103)



En = −

sinh μn h[ψn (0)]2 G (μn ; l0 , m 0 )D(iμn , h; l0 , m 0 ) , 2K 2 μ2n h

Fm , Fn 1 =

Fm (y)Fn (y) dy +

l0  (−1) j j=1

0

+

m0  (−1) j+1 j=1

 cj 2 j−(2k−1) Fm2k−1 (0)Fn (0) Q(μn ; m 0 ) j

k=1

 dj 2 j−2k Fm2k−2 (0)Fn (0), P(μn ; l0 ) j

k=1

∞

Fm , Fn 2 = lim

e

→0

−y

  D2 N2 Fm (h)Fn (h) + Fm (h)Fn (h) − F (h)Fn (h). Fm (y)Fn (y) dy + K K m

h

Using Eqs. (102) and (103), the unknowns An s and A(ξ ) in Eq. (100) are obtained as

φ(x, y), Fn (y) 2 φ(x, y), L(ξ, y; l0 , m 0 ) , (104) , A(ξ ) = An = En π (ξ, h; l0 , m 0 ) with

(ξ, h; l0 , m 0 ) = W 2 (ξ, h; l0 , m 0 ) − 2W (ξ, h; l0 , m 0 )L 1 (ξ, h; l0 , m 0 ) 

+K 2 ξ 2 P 2 (iξ ; l0 ) + Q 2 (iξ ; m 0 ) . This completes the derivation of the expansion formulae in case of infinite water depth. On the other hand, the general form of the velocity potential φ(x, y) satisfying Eq. (1) along with the boundary conditions (2), (10), (12), and (96) in case of finite depth is φ(x, y) =

2l 0 +2 n=I

An ψn (y)e

i pn x

+

∞ 

An ψn (y)e− pn x for x > 0,

(105)

n=1

where the eigenfunction ψn s can be obtained as ⎧ i tanh pn (H − h)L 1 (i pn , y; l0 , m 0 ) ⎪ ⎪ 0 < y < h, ⎨− K D(i pn , h; l0 , m 0 ) (106) ψn (y) = cosh pn (H − y) ⎪ ⎪ ⎩ h < y < H, cosh pn (H − h) with the eigenvalues pn s satisfying the dispersion relation in Eq. (98). Similar to the case of infinite water depth, in this case also, it is assumed that the dispersion relation in Eq. (98) has two distinct positive real roots pI , pII , pn = a ± ib, −c ± id for n = III, . . . , 2l0 + 2, and an infinite number of purely imaginary roots pn , n = 1, 2, . . . of the form pn = iνn . In the expansion formulae, terms leading to unboundedness of the velocity potential will not be included. In this case, the eigenfunctions ψn (y)s satisfy the orthogonal mode-coupling relation in Eq. (102) with

ψm , ψn 2 and E n s being replaced by H  N D2  2

ψm , ψn 2 = ψm (y)ψn (y) dy + ψm (h)ψn (h) + ψm (h)ψn (h) − ψ (h)ψn (h), K K m h

sinh pn h[ψn (0)]2 G ( pn ; l0 , m 0 )D(i pn , h; l0 , m 0 ) En = − . 2K 2 pn2 Thus, the unknowns An s in Eq. (105) are obtained as

φ(x, y), ψn (y) An = for n = I, . . . , 2l0 + 2, 1, 2, . . . . En Thus, the generalized expansion formulae in case of finite water depth are derived completely.

123

Wave interaction

71

References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31.

Burke JE (1964) Scattering of surface waves on an infinitely deep fluid. J Math Phys 5:805–819 Siew PF, Hurley DG (1977) Long surface waves incident on a submerged horizontal plane. J Fluid Mech 83:141–151 Patarapanich M (1984) Maximum and zero reflection from submerged plate. J Waterw Port Coast Ocean Eng ASCE 110:171–181 Liu Y, Li Y (2011) An alternate analytical solution for water-wave motion over a submerged horizontal porous plate. J Eng Math 69(4):385–400 Evans DV, Peter MA (2011) Asymptotic reflection of linear water waves by submerged horizontal porous plate. J Eng Math 69(2–3):135–154 Cho IH, Kim MH (1998) Interaction of a horizontal flexible membrane with oblique incident waves. J Fluid Mech 367:139–161 Hassan ULM, Meylan MH, Peter MA (2009) Water wave scattering by submerged elastic plates. Q J Mech Appl Math 62:245–253 Williams TD, Meylan MH (2012) The Wiener–Hopf and residue calculus solutions for a submerged semi-infinite elastic plate. J Eng Math 75:81–106 Takagi K, Shimada K, Ikebuchi T (2000) An anti-motion device for a very large floating structure. Mar Struct 13:421–436 Watanabe E, Utsunomiya T, Kuramoto M (2003) Wave response analysis with an attached submerged plate. Int J Offshore Polar Eng 13:1053–5381 Wang CM, Tay ZY, Takagi K, Utsunomiya T (2010) Literature review of methods for mitigating hydroelastic response of VLFS under wave action. Appl Mech Rev 63(3):030802 Korobkin A, Parau EI, Vanden Broeck J-M (2011) The mathematical challenges and modelling of hydroelasticity. Philos Trans R Soc A 369:2803–2812 Fox C, Squire VA (1994) On the oblique reflection and transmission of ocean waves at shore fast sea ice. Philos Trans R Soc Lond A 347:185–218 Sahoo T, Yip TL, Chwang AT (2001) Scattering of surface waves by a semi-infinite floating elastic plate. Phys Fluids 13(11):3215– 3222 Evans DV, Porter R (2003) Wave scattering by narrow cracks in ice sheets floating on water of finite depth. J Fluid Mech 484:143–165 Lawrie JB, Abraham ID (1999) An orthogonality relation for a class of problems with higher order boundary conditions; applications in sound–structure interaction. Q J Mech Appl Math 52:161–181 Manam SR, Bhattacharjee J, Sahoo T (2006) Expansion formulae in wave structure interaction problems. Proc R Soc Lond A 462:263–287 Kohout AL, Meylan MH, Sakai S, Hanai K, Leman P, Brossard D (2007) Linear water wave propagation through multiple floating elastic plates of variable properties. J Fluids Struct 23(4):649–663 Lawrie JB (2009) Orthogonality relations for fluid-structural waves in a three-dimensional, rectangular duct with flexible walls. Proc R Soc Lond A 465:2347–2367 Mondal R, Mohanty SK, Sahoo T (2013) Expansion formulae for wave structure interaction problems in three dimensions. IMA J Appl Math 78:181–205 Mondal R, Sahoo T (2012) Wave structure interaction problems for two-layer fluids in three dimensions. Wave Motion 49:501–524 Lawrie JB (2012) Comments on a class of orthogonality relations relevant to fluid–structure interaction. Meccanica 47(3):783–788 Lawrie JB (2012) On acoustic propagation in three-dimensional rectangular ducts with flexible walls and porous linings. J Acoust Soc Am 131:1890–1901 Sahoo T (2012) Mathematical techniques for wave interaction with flexible structures. CRC Press, Boca Raton Magrab EB (1979) Vibrations of elastic structural members. SIJTHOFF and NOORDHOFF Alphen aan den Rijn, The Netherlands Mohapatra SC, Ghoshal R, Sahoo T (2013) Effect of compression on wave diffraction by a floating elastic plate. J Fluids Struct 36:124–135 Schulkes RMSM, Hosking RJ, Sneyd AD (1987) Waves due to a steadily moving source on a floating ice plate. Part 2. J Fluid Mech 180:297–318 Mohapatra SC, Karmakar D, Sahoo T (2011) On capillary-gravity wave motion in two-layer fluids. J Eng Math 71(3):253–277 Bhattacharjee J, Sahoo T (2008) Flexural gravity wave problems in two-layer fluids. Wave Motion 48:133–153 Karmakar D, Bhattacharjee J, Sahoo T (2010) Oblique flexural gravity-wave scattering due to changes in bottom topography. J Eng Math 66:325–341 Dalrymple RA, Losada MA, Martin PA (1991) Reflection and transmission from porous structures under oblique wave attack. J Fluid Mech 224:625–644

123