1 Introduction

Very large floating structures (VLFSs) refer to marine structures, the horizontal dimension of which is much greater than their thickness. VLFSs are usually constructed for offshore airports, docks, energy source production, storage facilities, military purposes, etc.^[1-2]. According to the characteristics of the structures, VLFSs are commonly classified into two groups, i.e., the pontoon type and the semi-submersible type. Pontoon-type VLFSs often serve in the offshore area against a small-amplitude wave environment. Due to the feature that the structure is very thin and large, the flexible deformation in response to the water waves is supposed to be considered, while the rigid-body motion is tiny enough to be neglected.

VLFSs are usually modelled as a thin elastic plate coupled with wave motion. The thin-plate model is also used for the ice sheets in the polar region^[3-4]. With the wave-plate interaction model, Fox and Squire^[3-4] investigated the reflection and transmission characteristics about the waves incident to the sea ices, and minimized an error term to obtain the relative coefficients with the method of least squares. They found that the evanescent modes, which were usually ignored in early studies, had remarkable impact on the reflection and transmission of the wave energy. Kashiwagi^[5] developed a time-domain calculation method to compute the hydroelastic responses of a pontoon-type VLFS subjected to arbitrary time-dependent external loads, and investigated the transient elastic deformation caused by the landing and take-off of an airplane. Sahoo et al.^[6] studied the surface wave scattering by a semi-infinite thin elastic plate, and introduced a new inner product with orthogonality by adding a differential term to dispose the matched eigenfunction expansions. Based on the dimensional analysis, Teng et al.^[7] improved the method of least squares in Fox and Squire^[3] by reducing the number of the Lagrange multipliers from 3 to 1, and performed a minor modification to the inner product method of Sahoo et al.^[6] by using successively the vertical eigenfunctions of the open-water and plate-covered regions. Xu and Lu^[8] found that, with a newly defined inner product with the vertical eigenfunction of the open-water region, the method of matched eigenfunction expansions would be apparently improved with much higher convergent results.

The density stratification effects of sea water are usually ignored in early studies, and an assumption of uniform density was often used for the fluid. Recently, the effects of density inhomogeneity have attracted more and more attention. Bhattacharjee and Sahoo^[9] and Xu and Lu^[10] used two-layer fluid models, which were the simplest assumption to formulate the sea water with a step-wise density variation along the depth. Bhattacharjee and Sahoo^[9] studied the flexural-gravity waves in two-layer fluids with finite and infinite water depths. Xu and Lu^[10] applied the optimized method of matched eigenfunction expansions proposed in Xu and Lu^[8] to a two-layer fluid problem, and analyzed the influence of the density ratio and interfacial position on the wave scattering.

In addition to the fluid stratification, more complicated models on other aspects were considered. Karmakar et al.^[11] investigated the flexural-gravity wave scattering by steps on a seabed, and improved the inner product method proposed by Sahoo et al.^[6] to the case involving the compressive forces within the elastic cover. Mohapatra et al.^[12] analyzed the effect of the compressive forces on the wave diffraction by a finite elastic plate. Sturova^[13] derived the velocity potential due to the unsteady transient sources in a three-dimensional fluid covered by a thin elastic plate with lateral stresses, and discussed the effect of the hydrodynamic loads on the ice sheets and broken floes. Lu^[14] modelled an elastic plate with lateral force floating on a two-layer fluid with a uniform current, and deduced analytically the corresponding dispersion relation. Mohanty et al.^[15] analyzed the coupled effects of the fluid current and plate compression on the time-dependent flexural-gravity wave motion in single-layer and two-layer fluids.

Recently, Mondal and Sahoo^[16] investigated the hydroelastic responses of a plate with lateral force on a three-layer fluid. The phase and group velocities are deduced from the dispersion relation for different water depths. A generalized expansion form was provided to calculate the wave scattering by a crack between two ice sheets via the method of matched eigenfunction functions.

In this paper, we consider two-dimensional models of a three-layer fluid covered by thin elastic plates with lateral stresses. The method of matched eigenfunction expansions is employed, and a new inner product for the three-layer fluid is proposed. In Section 2, assuming that the length of the plate is semi-infinite, we first derive the dispersion relation, and then use the inner product to deduce a solvable set of simultaneous equations. In Section 3, we study the wave scattering by a finite elastic plate under different edge conditions, which can be deduced via the same method, and perform the calculation simply by a symmetric decomposition. In Section 5, the reflection and transmission of the waves and the deflection and inner forces of the plate are discussed for different frequencies and lateral stresses, where the correctness and accuracy are validated by an energy conservation equation.

2 Semi-infinite plate model 2.1 Mathematical formulation

A mathematical model is formulated to describe two-dimensional hydroelastic responses of a semi-infinite thin elastic plate floating on a three-layer fluid. The fluid region is vertically divided into three layers due to the density difference in the real ocean (see Fig. 1) . The horizontal orientation of the fluid domain is assumed to be infinite. We use the subscript m=1, 2, 3 to represent the upper, middle, and lower layers, respectively. The densities of the three layers are denoted by ρ_m with ρ₁ ＜ ρ₂ ＜ ρ₃, and the thicknesses are denoted by h_m~(m=1, 2, 3). A Cartesian coordinate system (x, z) is placed in such a way that the undisturbed positions of the surface, the upper interface, the lower interface, and the rigid bottom are on z=0, z=-H₁, z=-H₂, and z=-H₃, respectively, where .

The fluid is assumed to be inviscid and incompressible, and the motion is assumed to be irrotational. Then, the potential flow theory can be used. Under these assumptions, the potential function, denoted by Φ_m(x, z, t), is governed by the Laplace equation

(1)

In the framework of the small-amplitude wave theory, the boundary conditions can be linearized analytically. We introduce ζ_m(x, t) to denote the displacements of the surface (m=1) , the upper interface (m=2), and the lower interface (m=3) . On the top surface (z=0) , the kinematic and dynamic conditions are given by

(2)

(3)

where g is the acceleration due to gravitation, and P is the relative pressure. For the free-surface region (-∞＜x< 0, z=0) , P=0. For the plate-covered region (0＜x＜+∞, z=0) , with the assumption that the plate is homogeneous and there is no cavitation between the plate and the fluid, the balance among the elastic force, the lateral force, the inertial force, gravity, and the relative pressure can be expressed by

(4)

where D = Ed³/(12(1 − ν²)) is related to the bending rigidity of the plate, Q is related to the lateral stress of the plate (with compression at Q>0 and stretch at Q< 0) ^[13], and M=ρ_ed. E, d, υ, and ρ_e are Young's modulus, the thickness, Poisson's ratio, and the density of the semi-infinite thin elastic plate, respectively. The condition is required to ensure the stability of the floating elastic plate^[13]. Combining Eqs. (2) , (3) , and (4) , an associated boundary condition in the plate-covered region is derived as follows:

(5)

where 0＜x＜+∞, and z=0. Obviously, Eq. (5) reduces to the free-surface boundary condition as D, Q, and M tend to zero.

On the middle (z = −H²) and lower (z = −H³) interfaces, the kinematic and dynamic conditions are

(6)

(7)

respectively, where m=1, 2. Then, the combined boundary condition on the interfaces is derived as

(8)

where γ_m=ρ_m/ρ_m+1, and m=1, 2.

On the rigid bottom of the fluid, it is assumed that the fluid particles do not permeate across the physical boundary. Then, the bottom boundary condition reads

(9)

At the left end of the fluid-plate interface, we assume that the plate floats freely on the fluid. Thus, the bending moment and the shear force should vanish at the left edge, i.e.,

(10)

(11)

Considering the continuities of the pressure and the mass flux across the interface (x=0) between the free-surface and plate-covered regions, we have the matching conditions as follows:

(12)

(13)

where m=1, 2, 3, and H₀=0.

2.2 Dispersion relation

We assume that the motion is time-harmonic and the time variable can be separated from the full potential function Φ_m(x, z, t). Hereinafter, we introduce to represent the spatial potential function, i.e., , where ω is the angular frequency. According to the general solution of the Laplace equation, we assume that

(14)

where k refers to the wave number, and C_m1 and C_m2 are coefficients to be determined.

Substituting the general solution to the Laplace equation into Eqs. (5) , (6) , (8) , and (9) , we can derive the dispersion relation as follows:

(15)

where

(16)

(17)

(18)

(19)

(20)

(21)

(22)

(23)

(24)

(25)

We have Γ=0, λ=0, and σ=0 for the open-water region while , and for the plate-covered region. Three explicit solutions to the dispersion equation (15) with respect to ω² are given by

(26)

where

(27)

(28)

When γ₁=1 or γ₂=1, the dispersion relation (26) will reduce to the case for a two-layer fluid, which can be validated by comparing with that obtained by Xu and Lu^[10] for Q=0. For a given frequency ω, the wave numbers can be determined numerically with Eq. (26) . The wave numbers of the flexural-gravity waves were explained in detail by Fox and Squire ^[3] for a single-layer fluid. For the case of a three-layer fluid, we can find three positive real roots κ_0m representing three incident wave modes coming from -∞ and three negative real roots -κ_0m representing the corresponding wave modes reflecting off x=0. The wave numbers of the waves transmitted into the plate-covered region are represented by κ_0m＞0. There also exist infinite evanescent modes, which should vanish at ± ∞ due to the far-field radiation conditions in both the open water region and the plate-covered region. We have the modes represented by for the open water region and by for the plate-covered region. There are also two damping progressive modes in the plate-covered region, represented by , and , where α＞, and β＞0.

2.3 Velocity potential functions

For the sake of conciseness, we introduce a new symbol $\phi$ to represent uniformly the stratified spatial potential functions as follows:

(29)

Then, we use the superscripts L and R for the potential function to refer to the open-water region and the plate-covered region, respectively. Based on the physical understanding and the mathematical analysis on the wave numbers, the spatial potential function for both regions can be written as

(30)

(31)

where R_0m, R_i, T_0m, and T_j are coefficients to be determined, m=1, 2, 3, i=1, 2, 3, …, and j=I, II, 1, 2, 3, …. Z_0m, Z_i, , and are related to the vertical eigenfunctions defined by

(32)

where

(33)

(34)

(35)

and K=gk/ω². The coefficient for the incident wave in the corresponding layer I_0m can be represented by the amplitude ξ_m of the incident wave with

(36)

According to Eq. (2) , ζ₁ in Eqs. (10) and (11) can be expressed in terms of the potential function $\phi $₁. Substituting $\phi $^R(x, z) in Eq. (31) into the edge conditions Eqs. (10) and (11) yields

(37)

(38)

Substituting $\phi $^L(x, z) in Eq. (30) and $\phi $^R(x, z) in Eq. (31) into Eqs. (12) and (13) , we have the matching conditions as follows:

(39)

(40)

To calculate the reflection and transmission coefficients, an inner product method is usually employed to dispose the matching relations. Subsequently, the calculation may be simplified with the aid of the orthogonality of the eigenfunctions. Sahoo et al.^[6] employed the vertical eigenfunctions with respect to the plate-covered region to define an inner product, in which the eigenfunctions are orthogonal with an additional differential term. Teng et al.^[7] improved the method of Sahoo et al.^[6] by using the vertical eigenfunctions of the open-water and plate-covered regions. For a simpler calculation, Xu and Lu^[8] defined an inner product for a two-layer fluid system, which involved the vertical eigenfunction of the open-water region only. Following the idea of Xu and Lu^[8], we introduce here a new orthogonal inner product for the three-layer fluid as follows:

(41)

where l, n=0₁, 0₂, 0₃, 1, 2, 3, …. One can check the orthogonality of Eq. (41) that P_ln=0 as l ≠ n while P_ln≠q 0 as l = n. By performing the inner product operation on the two sides of Eqs. (39) and (40) with Z_n(z), we have

(42)

(43)

(44)

(45)

where

(46)

Equations (39) , (40) , (42) , (43) , (44) , and (45) form a set of equations for the unknown coefficients. By truncating the infinite summation with N terms, a closed system of 2N+8 equations for 2N+8 unknown coefficients is established, which can be solved numerically.

3 Finite plate model 3.1 Mathematical formulation

The semi-infinite plate model is usually used for a large floating ice sheet while the finite plate model is usually used for an artificial VLFS. Recently, Xu and Lu^[10] investigated a finite plate floating on a two-layer fluid with symmetric free-edge conditions. Maiti and Mandal^[17] studied the interaction between the wave motion and a finite elastic plate with the symmetric edges on a single-layer fluid bounded over a porous bed. In this section, we will employ the finite-plate model to study the hydroelastic response of the plate with different edge conditions on a three-layer fluid. The length of the plate is denoted by 2L. The z-axis is put at the midpoint of the plate (see Fig. 2) . The fluid region is horizontally separated by the plate into three regions, i.e., the open-water region on the left side (-∞＜x＜-L), the plate-covered region in the middle (-L＜x＜L), and the open-water region on the right side (L＜x＜+∞).

Under the same hypotheses as those in Section 2, the governing equation, the kinematic condition, the dynamic condition, and the matching conditions will take similar forms in the corresponding regions. For the edge conditions of a finite plate, we can consider the following three cases: (i) a free edge where both the bending moment and the shear force vanish, (ii) a simply-supported edge where both the deflection and the bending moment equal zero, and (iii) a built-in edge where both the deflection and the rotation angle equal zero.

3.2 Velocity potential function

We use the superscripts L, M, and R to refer to the left, middle, and right regions, respectively. Based on the analysis on the wave numbers in Section 2 and the physical understanding on the wave scattering by a finite plate, the spatial velocity potentials are expanded as follows:

(47)

(48)

(49)

where R_0m^L, R_i^L , T_0m^M, R_0m^M, T_j^M , R_j^M , T_0m^R, and T_i^M(i=1, 2, 3, …; j=I, II, 1, 2, 3, …) are the coefficients to be determined, I_0m^L is given by Eq. (36) , and the vertical eigenfunctions Z_0m, Z_i, , and are given by Eqs. (32) -(35) .

For the case with the symmetric boundary conditions, Mei and Black^[18] and Wu et al.^[19] divided the spatial potential function into two parts, i.e., a symmetric part $\phi $^s(x, z) and an antisymmetric part $\phi $^a(x, z). Each part satisfies the governing equation and the boundary conditions, and can readily be solved with the method in Section 2. We hereinafter employ this technique to study the wave scattering by a finite plate floating on a three-layer fluid. Then, the potential function is rewritten as follows:

(50)

where

(51)

(52)

To obtain the full spatial potential function, we solve the symmetric potential and the antisymmetric potential separately.

According to Eqs. (47) -(49) , the symmetric potential expansion is written by

(53)

(54)

where

(55)

Substituting $\phi $^s(x, z) into Eqs. (12) and (13) and using Z_n(z) to make an inner product over the equations, we obtain

(56)

(57)

(58)

(59)

For the antisymmetric potential functions, the series solutions are given by

(60)

(61)

where

(62)

By the approach similar to that for the symmetric part, we have

(63)

(64)

(65)

(66)

By truncating the infinite summation in Eqs. (56) -(59) and (63) -(66) with N terms and adding the edge conditions, we have a set of 2N+8 simultaneous equations, from which the 2N+8 unknown coefficients can be solved numerically. Then, the symmetric and antisymmetric potentials are determined.

4 Energy conservation

For the harmonic wave motion in a uniform fluid, the average energy flux over a period across the arbitrary closed curve in the fluid must be equal to zero due to the periodic energy variation of the fluid, such that an identity can be given as follows:

(67)

where E_f stands for the average energy flux over a period, C is an arbitrary closed curve in the fluid with n for its outward-pointing normal vector hereinafter, and denotes the directional derivative about n. Substituting into Eq. (67) , the expression of energy conservation is deduced as follows:

(68)

where refers to the imaginary part of one complex variable, and the star symbol * refers to the conjugate quantity of the related functions. Obviously, Eq. (68) can also be corroborated via validating Green's theorem in the fluid domain. From Eq. (68) , we can deduce that, for a given frequency, the energy flux across the certain boundary of the fluid can be measured by a path integral and the fluid density. Thus, for the consideration of energy conservation of the three-layer case, we denote a concise form by to define the energy flux across the boundary C along the direction n as follows:

(69)

Taking the fluid domain (-X＜x＜X, -H₃＜z< 0) and considering the out-pointing normal direction of the boundary, we have

(70)

where C_m^± are related to the boundary (x = ±X, −H_m＜z＜−H_m−1).C_S and C_B refer to the surface boundary and the bottom boundary, respectively. i and k are the unit vectors for the x-axis and the z-axis, respectively. Associating Eqs. (2) , (3) , and (9) , it is conspicuous that the energy flux across the free surface and the bottom boundary will vanish. Therefore, will only keep the remain of the plate-covered part (denoted by C_P). Subsequently, we obtain the energy conservation relation for the three-layer fluid covered by an elastic plate as follows:

(71)

The numerical results for the semi-infinite plate model or the finite plate model can be checked via validating Eq. (71) .

In particular, for the finite plate model, a further and more simplified derivation can be obtained. By letting X tend to ∞ to eliminate the evanescent modes and applying the orthogonal relation of the vertical eigenfunction for the free-surface region to Eq. (71) , the energy conservation relation can be simplified as follows:

(72)

where

(73)

By applying the dynamic boundary condition Eq. (5) and integrating by parts, we can simplify as follows:

(74)

where

(75)

Then, the energy conservation relation for the finite plate model can be written as

(76)

According to the above analysis, we can deduce that, the physical meaning of △_m is related to the energy change of the incident wave of the mode, and k_0m and △_e represent the part of the wave energy transferred to the elastic plate due to the lateral stress Q. Three incident waves and the elastic plate form a conservative system.

5 Results and discussion

To have graphical representation for the wave scattering and the hydroelastic response of the plate, numerical calculations are performed, which is validated via the law of energy conservation. We choose H=H₃, g, and ρ₁ as the characteristic quantities to the non-dimensionalized variables:

(77)

To keep clarity, the hat marks over the non-dimensional variables are omitted hereinafter.

In the following computational processing, we take

Figures 3(a) and 3(b) show that the frequency ω(k) and the phase velocity c(k) vary with the wave number k, which are determined by Eq. (26) . When the lateral force approaches a critical value of compression, i.e., Q=Q_cr, the frequency and the phase speed at the lower interface (m=3) will become zero at a peculiar wave number. With a supercritical compression force Q＞Q_cr, the wave propagation is impossible.

Figure 4 shows the flexural-gravity wave number κ₀₁ of the surface mode in the plate-covered region as a function of the lateral stress Q with different D and ω. The surface mode for the plate-covered region is induced from the synthetic action of gravity with the elastic restoring force of the plate. We observe that the surface mode κ₀₁ is positively correlated with the increase in Q from stretching to compression. For a given Q, a larger value of κ₀₁ appears for a smaller D or a larger ω.

Let A₁, A₂, and A₃ represent the wave amplitudes of the surface, the upper interface, and the lower interface, respectively, i.e.,

(78)

With the free-edge conditions at the two edges (±L, 0), Figs. 5, 6, and 7 show the effects of ω on the amplitudes of the surface, the upper interface, and the lower interface with D=0.05 and Q=0.4. At a low frequency, e.g., ω=0.25, the positions of the crests and the troughs for the surface, the upper interface, and the lower interface are not identical for different plate models, but the magnitudes are almost of the same order. Figures 5, 6, and 7 show the wave profiles for different incident frequencies, which suggest that, when the frequency increases, the maximal wave amplitudes at the open-water and plate-covered regions increase while the wave lengths decrease. The wave reflections on the surface, the upper interface, and the lower interface will be strengthened while the transmissions will be

weakened with an increasing frequency. The curves for the upper and lower interfaces have similar profiles with those on the surface, which means that the surface wave mode is the predominant component of the incident waves. These results are consistent with those obtained by Xu and Lu^[10] for the case of a two-layer fluid.

Let M_B(x) and F_S(x) represent the bending moment and the shear force of the plate, respectively, which are mathematically given by

(79)

Figure 8 shows the effects of the lateral stress Q on the surface fluctuation, the bending moment, and the shear force with D=0.1 and ω=1 for a finite plate with free edges at (±L, 0).. In this case, we have Q_cr ≈ 0.63. Figure 8(a) shows that the fluctuation of the plate deflection (−1 ＜ x/L＜1) increases when |Q| becomes larger. Different from the changing rule of deflection, the bending moment and the shear force are always enhanced when Q increases (see Figs. 8(b) and 8(c)).

Figures 9 and 10 show the effects of different edge conditions on the variations of the deflection, the bending moment, and the shear force of a finite plate, where the parameters are D=0.1 and ω=1. For the free-edge condition, the curves for the deflection, the bending moment, and the shear force have the smallest difference between the crusts and the troughs. For the built-in condition, the smallest values occur at Q = -0.4 (see Figs. 9(a), 9(b), and 9(c)), while the largest ones appear at Q = 0.4 (see Figs. 10(a), 10(b), and 10(c)). It suggests that the elastic plate with built-in edges has more sensitive responses when the lateral force changes from stretching to compression.

6 Conclusions

The hydroelastic responses of a thin elastic plate of either semi-infinite or finite length floating on a three-layer fluid are studied with the consideration of the internal lateral stress of the plate. Within the framework of the linear potential flow theory, the velocity potential of wave motion is deduced with the method of matched eigenfunction expansions. A valid approach, making an inner product with orthogonality for a three-layer fluid, is newly proposed to improve the calculation. The inner product is defined only by taking the vertical eigenfunctions at the open-water region, which is automatically orthogonal. For the plate-covered region, the inner product can be calculated from a differential term in Eq. (46) , which is related to the properties of the plate, i.e., D and Q. The convergence of the series is guaranteed with this definition. In the derivation for a finite plate model, the potential is decomposed into a symmetric part and an antisymmetric part, which is helpful to simplify the computation.

Based on the mathematical model and method of solution, some physical characteristics of the hydroelastic responses and wave scattering are discussed. An energy conservation relation is derived to check the correctness and accuracy of the numerical solution for the expansion coefficients. It is noticed that there is a term related to the lateral stress under the free-edge condition, indicating that some energy is transferred to the elastic plate due to the lateral force. The energy conservation relation implies that the far-field energy fluxes for the three incident wave modes are conserved with the elastic plate. The numerical results show that, when the incident frequency increases, the wave reflection of each layer will be strengthened and the transmission will be weakened. The surface wave mode is predominant during the scattering process.

The effects of the lateral stress of the plate are considered. For the stretching status of the elastic plate with free edges, a part of energy is absorbed by the elastic plate over all frequencies. For a compression status, an elastic plate will release the energy into the whole fluid domain over a specific frequency interval. The bending moment and shear force are always enhanced when the lateral stress increases. A comparison on the hydroelastic responses due to different edge conditions is also studied. It is shown that the deflection and inner forces of the plate with a built-in condition are more sensitive than those with other edge conditions when the lateral stress changes. All the results reported for a floating plate without the lateral stress on a two-layer fluid are extended.

The hydroelastic interaction between a floating plate and the incident waves in a fluid of arbitrary layer will be of interest and can be considered in the future work.