1 Introduction

The motion of a body of any geometrical configuration, floating on the surface of water, is investigated in the literature assuming linearized theory of water waves. The problem of heaving motion of a long, horizontal circular cylinder on the surface of water was investigated by Ursell(1949)using the method of multipole expansion of the time-harmonic stream function. The corresponding velocity potential also has a similar expansion. In fact, for an infinitely long horizontal cylinder of arbitrary cross section floating on the surface of water, the potential function in general can be expressed in terms of a regular wave, a wave source, a dipole and wave-free potentials(Ursell,1968; Athanassonlis,1984). The wave-free potentials are singular at some point and tend to zero rapidly at infinity. Obviously these satisfy the free-surface condition. Two and three dimensional problem of multipole expansions in the theory of surface waves in infinite deep water and also in water of uniform finite depth water has been given by Thorne(1953). Expansions in terms of the wave source and an infinite set of wave-free potentials were introduced for the three- dimensional problem involving a floating sphere half- immersed and making periodic heaving oscillations by Havelock(1955). Ursell(1961a),Ursell(1961b),Bolton and Ursell(1973), Mandal and Goswami(1984)considered problems where the potential functions is expansion in terms of wave sources and wave-free potentials. Taylor and Hu(1991)described expansion of the velocity potential for two and three dimensional wave diffraction and radiation problems. Linton and McIver(2001)briefly described the construction of wave free potentials in the case of water of infinite and finite depth water with a free surface.

There is a large class of problems in the field of fluid structure interaction where higher-order boundary conditions arise for a second-order partial differential equation. Various methods are being used to tackle these kind of mixed boundary-value problems(BVP)associated with the Laplace equation(or Helmholtz equation)arising in the study of waves propagating through solids or fluids. One of the widely used methods in wave structure interaction is the method multipole expansion. In most of the wave-structure interaction problems, the governing equation is either the Laplace or the Helmholtz equation and , thus, the features of the orthogonal relation mainly depend upon the nature of the bottom and upper surface boundary conditions. Higher-order boundary conditions occur frequently in fluid-structure interaction problems when we deal with very large floating structures(VLFS). Evans and Porter(2003)analysed the oblique wave scattering caused by a narrow crack in ice sheets floating on water of finite depth with the eigenfunction expansion method. Chakrabarti(2000)analysed the problem of scattering of surface water waves by the edge of an ice cover and obtained the explicit solution with a singular, Carleman-type integral equation. Das and M and al(2006),Das and M and al(2009)analysed the problem of water wave scattring by a circular cylinder with an ice-cover. Das and M and al(2008)also studied the water wave radiation by a sphere submerged in water with an ice-cover. Das and M and al(2010a),Mandal and Das(2010)presented in a systematic manner the construction of wave free potentials for two- dimensional deep water and finite depth water with free surface, or corresponding the effect of the surface tension at the free surface and also in water with an ice-cover. Dhillon and Mandal(2013)also presented the construction of a wave free potential for three dimensional deep water as well as finite depth water.

The multipoles and wave-free potentials have been described for single layer fluids, both in two and three dimensions and for infinite as well as finite depth water. More recently, however, interest has been extended to bodies which are floating, submerged or partially immersed in two-layer fluids, each fluid having a different density. The wave motion in a two-layer fluid has gained importance due to plans to construct under water pipe bridge across the Norwegian fjords. A fjord consists of a layer of fresh water on the top of a deep layer of salt water. For tow-dimension motions,Kassem(1982)presented the multipole expansions for two superposed fluids, each of finite depth. Linton and McIver(1995),Linton and Cadby(2002)have constructed a set of multipoles in the theory of surface waves in two-layer fluid with free surface for normal and oblique incident wave trains. Three-dimensional multipoles in two-layer fluid with free surface have been given by Cadby and Linton(2000). Das and M and al(2007);Das and M and al(2010b)constructed the multipole potentials in their problems for two-dimension and three-dimension in two-layer fluid with an ice-cover.

In the literature, two-layer fluid water wave problems have not been studied for wave-free potential function construction. However, for the various classes of water wave problems in two-layer fluid many researchers may use the wave-free potentials in the mathematical analysis. For circular cylinder of arbitrary cross section floating on the surface of a two-layer fluid or half-immersed circular cylinder in a two-layer fluid, the potential function in general can be expressed in terms of a regular wave, wave-free potential etc. Also in winter, a fjord is covered by a layer of ice, so that we have a cylindrical pipe bridge submerged below an ice-cover. Now if we consider the problem of partially or half-immersed circular cylinder in a two-layer fluid with an ice-cover, then potential functions may be expressed in terms of the regular potential as well as wave-free potential function. Thus it will study the problem of construction of wave-free potential in a two-layer fluid. Also in these problems(both single layer and two-layer)the higher-order boundary condition involves third order partial derivative(surface tension) and fifth order partial derivation(ice-cover). However, the boundary value problem involving higher-order boundary conditions more than fifth order partial derivative(Manam et al., 2006; Das et al., 2008)have not been extensively studied with a view to establish the multipole potentials and also wave-free potentials.

In this paper construction of wave-free potentials and multipoles are presented in a systematic manner. The cases of two-dimensional non-oblique and oblique waves in two-layer fluid with free surface condition with higher order partial derivative are considered. Also the cases of three-dimensional waves in two-layer fluid with free surface with higher order partial derivative are considered. When the higher order partial derivative reduces to first order(free surface)or fifth order partial derivative(ice-cover), multipoles exactly coincide with the multipoles for two-layer fluid with free surface(Linton and McIver, 1995; Linton and Cadby, 2002; Cadby and Linton, 2000), or for two-layer fluid with ice-cover(Das and Mandal, 2007; Das and Mandal, 2010b).

2 Multipoles and wave-free potentials for non-oblique waves

In a two-layer fluid, both the upper and lower fluids are assumed to be homogeneous, incompressible and inviscid. Let ρ^I be the density of the upper fluid and ρ^II(>ρ^I)be the same for the lower fluid. Let the lower fluid extend infinitely downwards while the upper one has a finite height h above the mean interface. Let y-axis points vertically upwards from the undisturbed interface y=0. Thus the upper layer occupies the region 0＜y＜h while the lower layer occupies the region y＜0. Under the usual assumption of linear theory and irrotational two-dimensional motion, velocity potentials ,ω being angular velocity, describing the fluid motion in the upper and lower layers exist. For a general BVP,ψ^{I, II}satisfy

On the upper surface having the mean position y=h,ψ^Isatisfies the free-surface condition with higher-order derivatives of the form( Landau LD and Lifshitz, 1959): If the free-surface has an ice-cover modelled as a thin elastic plate, where D=Eh³₀/12(1-v²)ρg),ε= ρ₀h₀/ρ,ρ₀ is the density of ice,ρ is density of water,h₀ is the small thickness of ice-cover,E,v are the Young’s modulus and Poission’s ratio of the ice and K=ω²/g,g being the acceleration due to gravity. A generalization of(2)for more higher-order derivatives has been introduced by Manam et al.(2006) and has the form: where $L$ is a linear differential operator of the form: where c_m(m=0,1,…,m₀)are known constants. Keeping in mind various physical problems involving fluid structure interaction, only the even order partial derivatives in x are considered in the differential operator $L$ .

The linearised boundary conditions at the interface y=0 are

where ρ=ρ^I/ρ^II(＜1) and the bottom condition is given by

2.1 Singularities in the lower layer

We first consider solutions of Laplace equation in two dimensions(x, y)which are singular at(0,f＜0). Polar co-ordinates(r,θ)are defined in the(x, y)-plane by

Now for the case of normal incidence, the solutions of Laplace’s equation singular at y=f＜0 are r^-ncosnθ and r^-nsinnθ,n≥, and these have the integral representations(Thorne,1953)

Let ψ^s_n and ψ^a_n denote the symmetric and antisymmetric multipoles satisfying(1),(2)except at(0,f)with boundary conditions(4)to(7) and Also they represent outgoing waves as |x|→∞ .

The mutipoles are constructed as Linton and McIver(1995)

where A(k),B(k),C(k) are functions of k to be found such that the integrals exist in some sense and satisfy the generalized boundary condition(4) and the interface conditions(5) and (6) and are of outgoing nature at infinity. All the conditions are satisfied if we choose A(k),B(k) and C(k) as where H(k)is given by

The path of the integration in the integrals in(14)to(17)is indented below the poles at k=k₁ and k=k₂ on the real k-axis to take care of their outgoing behaviour as |x|→∞ and k₁,k₂ are only two real positive roots of the equation H(k)=0(Das et al.,2008).

The far-field forms of the multipoles, in the lower fluid, is given by

where as x→±∞ . Here C^k₁,C^k₂ are the residues of C(k) at k=k₁ and k=k₂, given by Using(22) and (23), we find The combinations do not contribute anything as |x|→∞ so that they are wave-free. Now using the representations(25) and (26)it can be shown that and Letting f→0 in(27) and (28)we obtain the symmetric and antisymmetric wave-free potentials with singularity near the interface between two-layer and are given by and where C^*(k)is the limiting value of C(k) when f→0.

2.2 Singularities in the upper layer

To develop multipoles singular at y=f>0 and polar co-ordinates are again defined via(9). The solutions of Laplace’s equation singular at y=f>0 are r^-ncosnθ and r^-nsinnθ,n≥1, and let ψ^s_n and ψ^a_n denote the symmetric and antisymmetric multipoles satisfying(1),(2)except at(0,f)with boundary conditions(4)to(7) and

Also they represent outgoing waves as |x|→∞ .

The mutipoles are constructed as(Linton and McIver, 1995)

where

The path of the integration in the integrals in(33)to(36)is indented below the poles at k=k₁ and k=k₂ on the real k-axis to take care of their outgoing behaviour as |x|→∞ .

The far-field forms of the multipoles, in the upper fluid, is given by

where as |x|→∞ Here A⁽ⁱ⁾_n(k₁),A⁽ⁱ⁾_n(k₂), and B⁽ⁱ⁾_n(k₁),B⁽ⁱ⁾_n(k₂)are the residues of A⁽ⁱ⁾_n(k) and B⁽ⁱ⁾_n(k)respectively at k=k₁ and k=k₂, given by where

Using(40) and (41), we find Now using the representations(44) and (45)it can be shown that and where

The last contour integrals in(46) and (47)can be written in the form:

and the integrals being in the sense of Cauchy principal value.

Thus(46) and (47)reduce to

and the last integrals of(50) and (51)being in the sense of Cauchy principal value. Letting f→h in(50) and (51)we obtain the symmetric and antisymmetric wave-free potentials with singularity in the upper surface of the two-layer and are given by and where $M_j^*{\text{(}}k{\text{)}}$,$N_j^*{\text{(}}k{\text{)}}$,j=1,2 are the limiting values of $M_j^{}{\text{(}}k{\text{)}}$,$N_j^{}{\text{(}}k{\text{)}}$, j=1,2 respectively when f→h.

In particular, choose c₀=1,c_i=0,i=1,2,…,m₀, the BVP becomes the BVP for two-layer fluid with free surface(Linton and McIver, 1995) and the multipoles exactly coincide with those for the case of two-layer fluid with free surface(Linton and McIver, 1995) and the wave-free potentials become the wave-free potentials for two-layer fluid with free surface. Similarly, if choose c₀=1−εK,c₁=0,c₂=D,c_i=0,i=3,4,…,m₀, then the BVP becomes the BVP for two layer fluid with ice-cover boundary condition(3)( Das and M and al, 2007) and obtain the corresponding multipoles( M and al, 2007) and wave-free potentials. If we let c₀=1,c_i=0,i=1,2,…,m₀ and ρ→0 in this problem then it can be shown that the multipoles and wave-free potential functions go over to the single layer multipoles evaluated by Thorne(1953) and wave-free potential evaluated by Das and M and al(2010a). Thus by letting ρ→0 in the above analysis we recover the results for the single layer fluid.

3 Multipoles and wave-free potentials for oblique waves

Under the usual assumption of linear theory and irrotational two-dimensional motion, velocity potentials ,γ is the wave number component along the z-direction, describing the fluid motion in the upper and lower layers exist. For a general BVP,ψ^{I, II} satisfy

Eqs.(6) and (7)represent the linearized boundary conditions at the interface y=0, while the free-surface condition with higher-order derivatives at y=h is where the differential operator In(57),c_m(m=0,1,…,m₀)are known constants. Keeping in mind various physical problems involving fluid structure interaction, only the even order partial derivatives in x are considered in the differential operator $M$ and (8)is the bottom condition of the lower layer.

3.1 Singularities in the lower layer

Let ψ^s_n and ψ^a_n denote the symmetric and antisymmetric multipoles satisfying(54) and (55)for upper and lower fluid except at(0,f)with boundary conditions(56) and (6)to(8) and represent an outgoing waves at infinity. Near the point(0,f), the behaviours of ψ^{IIs, a}_n are given by

where K_n(z)denotes the modified Bessel function of second kind.

The multipoles are constructed as(Linton and Cadby, 2002)

where ν=γcoshk and A(k),B(k),C(k) are functions of k to be found such that the integrals exist in some sense. Also we have(Thorne,1953)

The functions ψ^s_n and ψ^a_n are singular solutions of the modified Helmholtz equation and satisfy the generalized free surface condition(56) and the interface conditions(6) and (7) and are of outgoing nature at infinity. Then A(k),B(k) and C(k) have the forms:

The path of the integration in the integrals in(60)to(63)is indented below the poles at k=μ₁ and k=μ₂, where

k₁,k₂ are only two real positive roots of the equation H(v)=0.

The far-field forms of the multipoles, in the lower fluid, is given by

where as |x|→∞, where Cμ₁ and Cμ₂ are given by where ${\beta _j} = {{\text{(}}k_j^2 - {\gamma ^2}{\text{)}}^{1/2}}$.

Using(70) and (71), we find

Thus and

Letting f→0 in(75) and (76)we obtain the symmetric and antisymmetric wave-free potentials with singularity near the interface between two layers and are given by

and

3.2 Singularities in the upper layer

To develop multipoles singular at y = f >0 and polar co-ordinates are again defined via(9). The solutions of Helmholtz equation singular at y = f >0 are ${K_n}{\text{(}}\gamma r{\text{)cos}}n\theta$ and ${K_n}{\text{(}}\gamma r{\text{)sin}}n\theta $,n≥1 . Let ψ^s_n and ψ^a_ndenote the symmetric and antisymmetric multipoles satisfying(54) and (55)for upper and lower fluid except at(0,f)with boundary conditions(56) and (6)to(8) and represent an outgoing waves at infinity. Near the point(0,f), the behaviours of $\varphi _n^{{\text{I}}s, a}$ are given by

Also they represent outgoing waves as |x|→∞ .

The mutipoles are constructed as(Linton and Cadby, 2002)

where the contour is indented below the poles k=μ₁ and k=μ₂ in the complex k-plane.

The far-field forms of the multipoles, in the upper fluid, is given by

where as |x|→±∞. Here $A_n^{{\text{(}}j{\text{)}}}{\text{(}}{k_1}{\text{)}}$,$A_n^{{\text{(}}j{\text{)}}}{\text{(}}{k_2}{\text{)}}$, and $B_n^{{\text{(}}j{\text{)}}}{\text{(}}{k_1}{\text{)}}$,$B_n^{{\text{(}}j{\text{)}}}{\text{(}}{k_2}{\text{)}}$ are the residues of $A_n^{{\text{(}}j{\text{)}}}{\text{(}}k{\text{)}}$ and $B_n^{{\text{(}}j{\text{)}}}{\text{(}}k{\text{)}}$ respectively at k=μ₁ and k=μ₁, given in(42) and (43).

Using(88) and (89), we find

and

Now using the representations(90) and (91)it can be shown that

and The last contour integrals in(92) and (93)can be written in the form the integrals being in the sense of Cauchy principal value.

After substituting(94),(95)in(92),(93)respectively and letting f→h in(92) and (93)we obtain the symmetric and antisymmetric wave-free potentials with singularity in the upper surface of the two-layer and are given by

and the last integrals in(96) and (97)being in the sense of Cauchy principal value.

In particular, choose c₀=1,c_i=0,i=1,2,…,m₀, the BVP becomes the BVP for two-layer fluid with free surface(Linton and Cadby, 2002) and the multipoles exactly coincide with those for the case of two-layer fluid with free surface(Linton and Cadby, 2002) and the wave-free potentials become the wave-free potentials for two-layer fluid with free surface. Similarly, if choose c₀=1−εK,c₁=0,c₂=D,c_i=0,i=3,4,…,m₀, then the BVP becomes the BVP for two layer fluid with ice-cover boundary condition(3)(Das and M and al, 2007) and obtain the corresponding multipoles(Das and M and al, 2007) and wave-free potentials. If we let c₀=1,c_i=0,i=1,2,…,m₀ and ρ→0 in this problem then it can be shown that the multipoles and wave-free potential functions go over to the single layer multipoles evaluated by Thorne(1953) and wave-free potential evaluated by Das and M and al(2010a). Thus by letting ρ→0 in the above analysis we recover the results for the single layer fluid.

4 Three dimensional multipoles and wave-free potentials

Here the velocity potential

describing the fluid motion exists where $\varphi {\text{(}}x{\text{,}}y{\text{,}}z{\text{)}}$ is a complex valued function and ω is the angular frequency. Let the potential in the upper layer be $\varphi _{\text{I}}^m$ and that in the lower layer be $\varphi _{{\text{II}}}^m$(m=0,1, the potential functions for the heave and sway problems being denoted by ψ₀ and ψ₁, respectively). The potential functions satisfy the Laplace’s Eqs.(1),(2)with

and equations(6) and (7)represent the linearized boundary conditions at the interface y=0, while the free-surface condition with higher-order derivatives at y=h is where the differential operator where c_m(m=0,1,…,m₀)are known constants. Keeping in mind various physical problems involving fluid structure interaction, only the even order partial derivatives in x are considered in the differential operator $N$ and (8)is the bottom condition of the lower layer.

4.1 Singularities in the lower layer

The velocity potential singular at(0,f,0)describing the motion in the lower fluid, then multipole singular potential functions are solutions of the Laplace’s equation which are singular at r=0, satisfy the boundary conditions(6)to(8) and (98). These can be constructed using the method given by Thorne(1953). A solution of Laplace’s equation in the spherical polar co-ordinate system(r,θ,α) and singular at r=0 is , where P_n^m are associated Legendre functions. This has the integral representation, valid for y>f(Thorne,1953)

where J_m are Bessel functions and $R = {{\text{(}}{x^2} + {z^2}{\text{)}}^{1/2}}$ . Let the multipole potentials $\varphi _{{\text{I}}n}^m{\text{cos}}m\alpha$ and $\varphi _{{\text{II}}n}^m{\text{cos}}m\alpha$(in the notation of Cadby and Linton(2000),m=0,1)be the singular solutions of the Laplace’s equation and satisfy the free surface boundary condition with higher order derivatives(98), the interface conditions(6) and (7) and behave as outgoing waves as R→∞. Then $\varphi _{{\text{I}}n}^m$ and $\varphi _{{\text{II}}n}^m$ are obtained as where The path of integration in the integrals in(101),(102)is chosen to be indented below the poles at k=k₁ and k=k₂. It can be shown that as R→∞, only the contributions to the integrals from these indentations, to the potential functions ψ, prevail and behave as waves having outgoing nature.

The far-field forms of the multipoles, in the lower layer, is given by

as R→∞, where ${C_{{k_1}}}$ and ${C_{{k_2}}}$ being the residues of C(k) at k=k₁ and k=k₂ respectively, which are given by Using(106), we find Now using the representations(108), it can be shown that

This is the wave-free potential having singularity at(0,f,0). Letting f→0 in(109)it is obtained the wave-free potentials having singularity near the interface between two-layer and is given by

4.2 Singularities in the upper layer

To develop multipoles singular at y = f >0. The solution of Laplace’s equation singular at y=f>0 is ${r^{ - n - 1}}P_n^m{\text{(cos}}\theta {\text{)cos}}m\alpha$ . Let $\varphi _n^{m, s}$ is denote the multipoles satisfying(1),(2)except at(0,f,0)with boundary conditions(6),(7) and (98). The mutipoles are constructed as(Cadby and Linton, 2000)

where

The path of the integration in the integrals in(111) and (112)is indented below the poles at k=k₁ and k=k₂ on the real k-axis.

The far-field forms of the multipoles, in the upper fluid, is given by

where As R→∞. Here A₁(k₁),A₁(k₂) and B₁(k₁),B₁(k₂)are the residues of A₁(k) and B₁(k)respectively at k=k₁ and k=k₂, given by

Using(116), it is found that

Now using the representation(119)it can be shown that

The last contour integrals in(120)can be written in the form:

the integral being in the sense of Cauchy principal value.

Thus(120)reduces to

This is the wave-free potential having singularity at(0,f,0). Letting f→h in(122)it is obtained the wave-free potentials with singularity in the upper surface of the two-layer and are given by

In particular, choose c₀=1,c_i=0,i=1,2,…,m₀, the BVP becomes the BVP for two-layer fluid with free surface(Cadby and Linton, 2000) and the multipoles exactly coincide with those for the case of two-layer fluid with free surface(Cadby and Linton, 2000) and the wave-free potentials become the wave-free potentials for two-layer fluid with free surface. Similarly, if choose c₀=1−εK,c₁=0,c₂=D,c_i=0,i=3,4,…,m₀, then the BVP becomes the BVP for two layer fluid with ice-cover boundary condition(3)(Das and M and al, 2010b) and obtain the corresponding multipoles(Das and M and al, 2010b) and wave-free potentials. If we let c₀=1,c_i=0,i=1,2,…,m₀ and ρ→0 in this problem then it can be shown that the multipoles and wave-free potential functions go over to the single layer multipoles evaluated by Thorne(1953) and wave-free potential evaluated by Dhillon and M and al(2013). Thus by letting ρ→0 in the above analysis we recover the results for the single layer fluid.

5 Conclusions

Wave-free potentials and multipoles in two-layer fluid with a free surface condition with higher order derivatives for non-oblique and oblique waves(two dimensions) and also three dimension are constructed in a symmetric manner. Appropriate modifications of the wave-free potentials can be made in the circumstances when the two-layer fluid are of uniformly finite depth for both the layers having a free surface conditions with higher order derivatives. In particular, these are obtained taking into account of the effect of the presence of surface tension at the free surface and also in the presence of an ice-cover modelled as a thin elastic plate. Also for limiting case, it can be shown that the multipoles and wave-free potential functions go over to the single layer multipoles and wave-free potential.