## Abstract

A large class of problems in the field of fluid–structure interaction involves higher-order boundary conditions for the governing partial differential equation and the eigenfunctions associated with these problems are not orthogonal in the usual sense. In the present study, mode-coupling relations are derived by utilizing the Fourier integral theorem for the solutions of the Laplace equation with higher-order derivatives in the boundary conditions in both the cases of a semi-infinite strip and a semi-infinite domain in two dimensions. The expansion for the velocity potential is derived in terms of the corresponding eigenfunctions of the boundary-value problem. Utilizing such an expansion of the velocity potential, the symmetric wave source potentials or the so-called Green's function for the boundary-value problem of the flexural gravity wave maker is derived. Alternatively, utilizing the integral form of the wave source potential, the expansion formulae for the velocity potentials are recovered, which justifies the completeness of the eigenfunctions involved. As an application of the wave maker problem, oblique water wave scattering caused by cracks in a floating ice-sheet is analysed in the case of infinite depth.

## 1. Introduction

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 eigenfunction expansion method. The solutions are obtained by matching technique across the interface at the discontinuities, provided the eigenfunctions in each region form a complete set of orthogonal basis functions. Hence, using the orthogonal relation, the BVP with a partial differential equation as the governing equation can be reduced to a problem requiring the solution of a well-behaved infinite system of linear equations. However, if the eigenfunctions do not form an orthogonal basis or the algebraic system it produces does not contain diagonally dominant matrix, then the method becomes inefficient. In such cases, if we can establish an orthogonal relation, by which the Fourier coefficients of the eigenfunction expansion can be isolated and expressed in terms of boundary data, then the method becomes effectively simpler. 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). Sahoo *et al*. (2001) analysed the wave scattering by semi-infinite floating elastic plate in finite depth with the linearized theory of water waves and eigenfunction expansion method. Evans & 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. They also used the Green's function approach to derive the same explicit solution. Linton & Chung (2003) applied the technique of residue calculus in complex variables to obtain the solution for the problem of wave scattering caused by a semi-infinite floating ice sheet. 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. Hermans (2003) derived an integral equation method based on Green's theorem to describe the wave interaction with rigid or flexible dock of zero draught. Takagi (2004) applied the boundary integral equation method based on Green's function to investigate the hydroelastic behaviour of a floating elastic plate.

In the literature, the BVP involving higher-order boundary conditions have not been extensively studied with a view to establish the relevant orthogonal relations. Havelock (1929) solved the classical problem of wave maker, to find the two-dimensional water wave motions with free surface generated by a vertical plane wave-maker, with the eigenfunction expansion method. However, Havelock's expansion fails when the BVP involves a higher-order condition. Rhodes-Robinson (1971) investigated the wave-maker problem, taking into account the effect of surface tension which gives a third-order boundary condition. Green's function approach was adopted to derive the expansion theorem for the potential functions in the cases of both finite and infinite depths. Rhodes-Robinson (1979*a*,*b*) proposed orthogonal relations to study wave interaction, with rigid structures in the presence of surface tension in which the governing equation is the Laplace equation and the free surface condition is of third-order. This was later exploited by Chakrabarti & Sahoo (1998) to analyse the wave past porous structures in the presence of surface tension in cases of finite and infinite depths. Sahoo *et al*. (2001) generalized the orthogonal inner product of Rhodes-Robinson (1979*a*,*b*) to study the wave scattering caused by semi-infinite floating elastic plate in finite depths with the linearized theory, where the free surface condition is of fifth-order along with various types of edge conditions of higher-order to that of the governing equation for the consistency of the physical problem. In the recent past, Lawrie & Abrahams (1999) established an orthogonal relation for a general class of BVP of Helmholtz-type, partial differential equation with higher-order boundary conditions in a semi-infinite strip type domain and utilized these relations to solve problems of acoustics. The relation developed by Lawrie & Abrahams (1999) is in integral form and can be applied directly to obtain the solution of similar types of problems that arise while analysing problems in the fields of electromagnetism and elasticity. Lawrie & Abrahams (2002) applied the orthogonal relation developed by them to the three-dimensional duct problem bounded by thin elastic plates. The limitation of Lawrie & Abrahams (1999) result is that it is suitable for problems of finite strip that may be of finite or semi-infinite length and need not be of the same height. On the other hand, for problems in a half plane or quarter plane, the results of Lawrie & Abrahams (1999) cannot be applied directly and, to the authors' knowledge, there is no such existing relation available for the quarter plane problems in the literature.

In the present paper, expansion formulae are derived based on orthogonal mode-coupling relations in the case of semi-infinite strips as well as quarter-plane problems in order to tackle a general class of problems of the type as discussed above. In the case of finite depth, an equivalent form of the orthogonal relation of Lawrie & Abrahams (1999) is obtained for the semi-infinite strip. Then, in the case of a quarter-plane problem, the velocity potential is expanded, utilizing the basic definition of Fourier sine transform and its inversion formula, in to an integral form with terms containing the higher-order derivatives of the modal functions on the boundary. The form of the expansion formula provides the motivation behind the theoretical development of a new orthogonal mode-coupling relation. The mode-coupling relation corresponding to the quarter-plane problem takes the discrete form in the case of the semi-infinite strip and was not explained otherwise in the earlier literature where negligible progress is being made in this context. An equivalent integral form of the mode-coupling relation is derived in the case of the quarter plane with boundary conditions relevant for the problems of wave–structure interaction, which is similar to the form of Lawrie & Abrahams (1999). In addition, the expansion formulae in both the cases are alternatively derived with a suitable application of Green's second identity and the appropriate Green's function, which confirms the uniqueness of the two formulae and the completeness of the eigenfunctions involved. The general form of the Green's functions satisfying the higher-order boundary conditions are obtained by two alternative methods, namely (i) applying the expansion theorem along with the symmetric property of the Green's function and (ii) constructing the source potentials satisfying appropriate boundary conditions. As an application of the expansion formulae and the mode-coupling relation, the explicit form of the solution for the flexural gravity wave-maker problem is obtained for the case of both infinite and finite depths. Finally, as a direct application of the flexural gravity wave-maker problem, the half-plane problem of oblique wave scattering caused by an infinite crack in a floating ice-sheet is analysed and the computed numerical results are compared with known existing results in the literature.

## 2. Generalized Fourier expansions

Fourier transform and Fourier series act as important tools in finding the solution to a large class of BVPs in semi-infinite domains and semi-infinite strips. Depending on the behaviour of the function and its derivative at one end of the boundary and knowing the functional behaviour at the far field, that is, at infinity, the appropriate integral transform, such as the Fourier integral transform, Fourier sine or cosine transform, is applied to reduce the dimension of the partial differential equation in the half plane or quarter plane of a BVP. It is in this context that the Fourier sine transform is applied in a suitable manner to derive the expansion formulae for the velocity potentials in the quarter plane of a class of BVPs with a higher-order boundary condition in one of the boundaries that arise in the broad area of fluid–structure interaction. The corresponding BVP in a semi-infinite strip follows by considering the discrete form of the appropriate integral transform used in the expansion formulae. These expansion formulae can be easily extended to a half plane or infinite strip by using the geometrical symmetry of the problem under study and are deferred here to avoid repetition.

### (a) The general boundary-value problem

For the sake of simplicity, the general BVP considered here is two-dimensional in nature. In the present context, the BVP considered arises in the broad area of fluid–structure interaction, where the fluid is assumed to be inviscid and incompressible and the fluid motion is irrotational and simple harmonic in time with angular frequency *ω*. Thus, there exists a velocity potential *Φ*(*x*,*y*, *t*) of the form . In the present paper, we follow the convention that the *x*-axis is horizontal and the *y*-axis is vertical. The fluid region in the case of fluid of finite depth is the semi-infinite strip 0<*x*<∞, 0<*y*<*h* and, in the case of infinite depth, is the quarter plane 0<*x*<∞, 0<*y*<∞. The spatial velocity potential *ϕ* satisfies the partial differential equation(2.1)in the fluid domain. On the structural boundary located at *y*=0, the velocity potential *ϕ* satisfies the boundary condition of the form as given by(2.2)where *L* is the linear differential operator of the form(2.3)with *c*_{k} the known constants. Keeping in mind various physical problems of concern, only the even derivatives in *x* are included. The rigid bottom boundary condition is given by(2.4)

Finally, the far field radiation condition is of the form(2.5)where the *p*_{0} satisfies the relation(2.6)

### (b) Expansion of the potential function

The BVP satisfying equation (2.1) along with the boundary conditions (2.2) and (2.4) and the radiation condition (2.5) are not of standard Sturm–Liouville type and the eigenfunctions involved are not orthogonal in the usual sense. It is in this context that more general expansion formulae based on the eigenfunctions are developed along with appropriate mode-coupling relations in both the cases of finite and infinite depths.

In the case of fluid of finite depth using the eigenfunction expansion method, the velocity potential *ϕ*(*x*,*y*) satisfying the governing equation (2.1), the boundary conditions (2.2) and (2.4) and the radiation condition (2.5) can be expanded in the generalized form(2.7)where the eigenfunctions *I*_{n}(*y*) are of the form as given by(2.8)with *p*_{n} satisfying the relations(2.9)and the unknown functions *A*_{n}(*x*) are of the form(2.10)

The eigenfunctions, *I*_{n}(*y*), are not orthogonal in the usual sense. It is in this context that we will mention the relevant form of the orthogonal relation satisfied by *I*_{n}(*y*) which was given by Lawrie & Abrahams (1999) for the two-dimensional problem with a rigid bottom boundary and give an equivalent form of the orthogonal relation, which is referred to as mode-coupling relation in the present paper.

(Lawrie & Abrahams 1999) *In the case of finite depth, the eigenfunctions I*_{n}(*y*) *given by equation* *(2.8)* *satisfy the orthogonal relation*(2.11)*where* . *The function P*(*p*_{n}) *is the characteristic polynomial and corresponds to the action of the operator L*(∂/∂*x*) *given by the relations* *(2.3)* *on the function A*_{n}(*x*)*, that is, P*(*p*_{n})≡*L*(i*p*_{n}).

An equivalent form of the orthogonal relation is presented in the following theorem.

*The equivalent form of the orthogonal relation* *(2.11)* *is given by the mode-coupling relation*(2.12)*which satisfies*

*On integrating by parts the relation* *(2.11)* *and using the dispersion relation as in relations* *(2.9)* *for finite depth, the above form of the mode-coupling relation is derived in a direct manner*.

Note that the orthogonal relation defined in Lawrie & Abrahams (1999) is for boundary conditions of the type (2.2) at both ends of a Helmholtz equation as the governing equation, which reduces to relation (2.11) in the context of the present study. Sahoo *et al*. (2001) have used a similar mode-coupling relation as in relation (2.12) to deal with the water wave scattering caused by a semi-infinite floating elastic plate in the case of finite depth.

*The unknown functions A*_{n}(*x*)*, as in relation* *(2.10)**, are given by**which is equivalent to*

The proof follows by utilizing the mode-coupling relation as given in theorem 2.2 to the expression (2.7) and using relation (2.10).

It may be noted that the mathematical motivation behind the construction of the orthogonal relation is not apparent from the work of Lawrie & Abrahams (1999) in the case of fluid of finite depth. On the other hand, in the case of fluid of infinite depth, the expansion of the velocity potential is derived through the direct application of the Fourier sine transform, from which the mode-coupling relation is easily obtained.

For the sake of simplicity, in order to find general eigenfunction expansion type formulae, we will discuss in detail the case for *k*_{0}=2 in equation (2.2) for the case of infinite depth, which has direct application in wave interaction caused by elastic plate/ice sheet. However, the general expansion formulae will follow in a similar manner, which will be mentioned for the sake of completeness, deferring the proof. Hence, in the present case, the boundary condition at *y*=0 will be of the form(2.13)

*The velocity potential ϕ*(*x*,*y*) *satisfying the governing equation* *(2.1)* *and the boundary conditions* *(2.4), (2.13)* *and the radiation condition* *(2.5)* *in the case of infinite water depth is given by*(2.14)*with p*_{n} *satisfying*(2.15)*where* *is the Fourier sine transform of, say, ψ*(*x*,*y*)*, where**and is given by*(2.16)*and*(2.17)

*The unknown functions, A*_{n}(*x*)*, are given by*(2.18)*where the functions I*_{n}(*y*) *are of the form*(2.19)

The proof is based on the direct application of the Fourier sine transform for *ψ*(*x*,*y*) in the quarter plane. The condition *ψ*(*x*, 0)=0 enables us to apply the Fourier sine transform in the present case and is indeed the basis of the present theorem. We define the Fourier sine transform of *ψ*(*x*,*y*) as and it is given by(2.20)

Using equation (2.1) and then integrating by parts, is obtained as given by equation (2.16).

Further, from equation (2.20), using Fourier sine inversion formula, we have(2.21)

Equation (2.21) is an ordinary differential equation (ODE) in *y* of fifth-order with the solution as given by equation (2.14). Equations (2.14) and (2.20) are the mixed transform pair where the unknown functions *A*_{n}(*x*) are to be obtained for the full solution of the ODE, which in turn proves the theorem.

Motivated by the form of the Fourier transform as in equation (2.16), we define a mode-coupling relation as(2.22)

The functions *I*_{n}(*y*), given by equation (2.19), satisfy the relation (see appendix A for details)(2.23)

The eigenvalues *p*_{n} satisfy the relation (2.15). Further, it can be easily verified that

Applying the mode-coupling relation as defined in equation (2.22) on *ϕ*(*x*,*y*), given by equation (2.14), we have, after using relation (2.23),and the unknown coefficients are obtained as in equation (2.18)

Thus, the theorem is proved. ▪

*The velocity potential ϕ*(*x*,*y*) *satisfying the governing equation* *(2.1)* *and the boundary conditions* *(2.2), (2.4) and (2.5)* *in the case of infinite depth is given by*(2.24)*with p*_{n} *satisfying the relation*(2.25)*and*(2.26)

*In this case, the mode-coupling relation takes the form*(2.27)*where* , *n*=0, *I*, *II*,…,2*k*_{0}. *In addition,*(2.28)

*The unknown functions, A*_{n}(*x*)*, are obtained in the general form like the Fourier coefficients:*

The proof follows by taking the Fourier sine transform of the boundary condition (2.2) and proceeding in a similar manner as in theorem 2.4, the details of which are omitted here to avoid repetition.

*An equivalent form of the mode-coupling relation* *(2.22)* *is given by*(2.29)*where Q*_{mn}, *P*(*p*_{n}) *are the same as defined in* *theorem 2.1*.

We consider the case where *k*_{0}=2. The defined mode-coupling relation (2.22) can be rewritten as

Or, equivalently,which is equivalent toby using and the expression for *Q*_{mn} as in theorem 2.1.

The general case follows in a similar way by using for *n*=0, *I*, *II*,…,2*k*_{0}. ▪

## 3. Flexural gravity wave-maker problem based on Fourier expansion

In this section, as an application of the expansion theorems described in the previous section, wave motion generated by a vertical wave-maker for the flexural gravity waves is analysed in the case of both infinite and finite depths. We consider a semi-infinite domain 0<*x*<∞, 0<*y*<∞ in the case of infinite depth and a semi-infinite strip 0<*x*<∞, 0<*y*<*h* in the case of finite depth, which is filled with inviscid and incompressible fluid with irrotional flow. The *y*-axis is vertically downward positive and the *x*-axis is horizontal. The motion considered here is two-dimensional and time-harmonic in nature with angular frequency *ω* due to the harmonically oscillating vertical plane wave-maker placed at *x*=0 oscillating with velocity with outgoing waves produced at the far field. The fluid surface is covered entirely by a floating elastic plate which is under uniform compression/tension *T*. Under the assumption of the linearized theory of water waves, the spatial velocity potential *ϕ*(*x*,*y*) satisfies the Laplace equation (2.1) along with the bottom boundary conditions given in equation (2.4).

The boundary condition on the wave maker is given by(3.1)

The linearized free surface condition satisfied on the plate-covered free surface is given by (Schulkes *et al*. 1987):(3.2)where , , , =flexural rigidity of the plate, *E*=Young's modulus, *ν*=Poisson ratio, *ρ*_{w}=density of water, *ρ*_{i}=mass density of the elastic plate, *T*=uniform compressive/tensile force, *d*=thickness of the plate.

The radiation conditions are of the form given in equation (2.5) with *p*_{0} satisfying the dispersion relation in *p*, as given by(3.3)

For the uniqueness of the boundary-value problem, edge conditions are to be prescribed depending on the physical problem. Under the assumption that the elastic plate has a free edge, the shear force and the bending moment is prescribed at the edge which gives rise to(3.4)

As a direct application of theorem 2.4, the velocity potential in the case of infinite depth is given by(3.5)where *M*(*ξ*, *y*), *Δ*(*ξ*) are given by(3.6)and for *n*=0, *I*, *II*. The dispersion relation has a positive real root *p*_{0} that describes the progressive wave and two complex conjugate pairs, (*p*_{I}, *p*_{II}) and (*p*_{III}, *p*_{IV}), with and (see appendix B). Here, we have considered the two roots *p*_{I} and *p*_{II} with positive real parts lying in the first and fourth quadrants, respectively, and the other two roots are neglected because of the boundedness of the solution (see Chakrabarti 2000; Evans & Porter 2003). Applying the boundary condition (3.1) on the wave maker, and utilizing the mode-coupling relation (2.22), the unknown coefficients are obtained as(3.7a)(3.7b)where(3.7c)with *ϕ*_{xy}(0^{+},0)=*α*, *ϕ*_{xyyy}(0^{+},0)=*β* and *α*, *β* determined from the plate edge conditions (3.4). Applying the zero-shear force condition from equation (3.4) to *ϕ*(*x*,*y*) (described in equation (3.5)) and then substituting , *A*_{0}, *A*_{I}, *A*_{II}, from equations (3.7a) and (3.7b), we have(3.8a)with

Similarly, from the zero bending moment edge condition as in equation (3.4), we obtain(3.8b)whereand *C*_{n}, *I*_{n} and *ϵ*_{n} are as defined in equations (3.7b) and (3.7c). Solving equations (3.8a) and (3.8b), we findand that completes the solution.

Again as a direct application of theorem 2.2, the velocity potential in case of finite depth is given by(3.9)where *I*_{n} are given byand *p*_{n} satisfy the dispersion relationswith and *p*_{I} lies in the first quadrant. The complex roots *p*_{I}, *p*_{II} are chosen to have positive real parts in order to satisfy the behaviour of the solution at infinity. The unknown constants are given by(3.10)(3.11)with

It is observed that all the unknowns are expressed in terms of the two unknowns, *U*_{y}(0) and *U*_{yyy}(0), which can be determined from the appropriate edge conditions in a similar manner as in the case of infinite depth. The general form of the expansion of the potential is given by equation (3.9) along with the set of equations (3.10) and (3.11).

## 4. Flexural gravity wave maker problem based on Green's function

In this section, we briefly outline an alternative derivation of the velocity potential of the flexural gravity wave-maker problem based on the Green's theorem. For that purpose, the Green's functions are derived below by two different methods, namely, (i) as a direct application of the generalized expansion theorems and (ii) by the utilization of the source functions in the velocity potential.

### (a) Derivation of Green's functions for flexural gravity waves based on Fourier expansions

The Green's function *G*(*x*,*y*; *x*_{0}, *y*_{0}) satisfies the governing Laplace equation in the respective fluid domain, except at (*x*_{0}, *y*_{0}), along with the free-surface boundary condition (3.2) and the respective bottom boundary condition (2.4) as appropriate for fluid of finite and infinite depths. The radiation condition in this case is of the form*p*_{0} satisfies the respective dispersion relations in equation (3.3) in the case of infinite and finite depth.

In addition, *G*(*x*,*y*; *x*_{0}, *y*_{0}) satisfies the condition(4.1)or equivalently,(4.2)where 0≤*y*, *y*_{0}<∞ in the case of infinite depth and 0≤*y*, *y*_{0}<*h* in the case of finite depth.

The Green's function *G*(*x*,*y*; *x*_{0}, *y*_{0}) in the case of infinite depth can be expressed as(4.3)with *M*(*ξ*, *y*) and *Δ*(*ξ*) given by equation (3.6). Using the condition (4.2) and the mode-coupling relation (2.22) for the functions *I*_{n} as given in equation (3.7*c*), the unknown constants *A*_{n} and the unknown function *A*(*ξ*) in the relation (4.3) are derived as(4.4)with *ϵ*_{n} as defined earlier in equation (3.7*b*).

For finite depth, the Green's function *G*(*x*,*y*; *x*_{0}, *y*_{0}) can be derived similarly and is given by(4.5)where

### (b) Alternative derivation of Green's function through the utilization of the source function

Here the source function is used directly to derive the Green's function for flexural gravity waves for both infinite and finite depth. Keeping the condition (4.1) in mind, we expand the Green's function for the case of infinite depth as(4.6)where and 0<*y*_{0}<∞ so that the singularity is submerged but not on the bottom. Using the plate-covered surface boundary condition (3.2) and results(4.7)and(4.8)we obtain

Hence, the Green's function is expressed as(4.9)

Now, the integrand has poles at *ξ*=*p*_{n}, *n*=0, *I*, …, *IV*, where (complex numbers) and the contour is indented below the pole at *ξ*=*p*_{0} (real number). It is a standard procedure to modify the above Green's function (4.9) into an alternative form by rotating the contours of integration, as given by(4.10)with *ϵ*_{n} as defined in equation (3.7*b*). Combining the expressions (4.8) and (4.10) gives rise to the Green's function obtained in the relation (4.3).

Proceeding in a similar way as in the case of infinite depth, the Green's function in the case of finite depth is expanded in the form(4.11)with *A*(*ξ*) and *B*(*ξ*) given bywhere the contour in the first integral is indented below the pole *ξ*=*p*_{0} (there are also poles at , where ). By rotating the contour of integration in equation (4.11) as in the case of infinite depth and utilizing the relationthe Green's function as in (4.5) is derived.

### (c) Utilization of Green's theorem to derive the velocity potentials

The expansion formulae for the velocity potentials of the wave-maker problem of flexural gravity waves are derived through the utilization of the Green's theorem for both infinite and finite depth. The boundary-value problems for infinite and finite depth are the same as described in §3 by the equations (2.1), (2.4), (2.5), (3.1) and (3.2). By defining(4.12)which is a combination of equal wave sources at (±*x*_{0},*y*_{0}) with zero-normal velocity on the wave maker, that is, , and applying Green's theorem to the potential function *ϕ*(*x*,*y*) and *G*^{mod}(*x*,*y*;*x*_{0},*y*_{0}) along with the relations (2.1) and (3.2) and the result , the potential can be expressed as(4.13)where is the semi-infinite axis (0,∞) in the case of infinite depth and the finite interval (0,*h*) in the case of finite depth. Then, substituting the explicit forms (4.3) and (4.5) of Green's function, the expansion formulae for the velocity potential in both the cases of infinite and finite depth are derived as in the relations (3.5) and (3.9), respectively.

As an application of the wave-maker problem, in the next section, the oblique wave scattering caused by a crack in an ice sheet in water of infinite depth is analysed in detail. Utilizing the geometrical symmetry of the physical problem, the associated BVP defined in the half plane is converted to a BVP to be solved in the quarter plane.

## 5. Oblique wave scattering by a crack in an ice sheet

Cracks in floating ice sheets are common in the cold regions of Arctic and Antarctic oceans and are formed due to the convergent or divergent stresses caused by changes in wind, the water flow beneath the sheet or aircrafts landing and taking off. Also, cracks may develop in the ice sheet between the islands because of local meteorological disturbances. Kouzov (1963) gave a solution to the problem of hydroelastic waves scattered by two thin elastic plates with identical properties separated by a crack. He used the integral representation to reduce the problem to a Riemann–Hilbert problem and explicitly solved the problem. Fox & Squire (1990, 1994) analysed the scattering of ocean waves by a shore fast sea ice that is modelled as a semi-infinite elastic plate in the case of finite depth by applying appropriate matching conditions across the interface and the conjugate gradient method. Barrett & Squire (1996) obtained a numerical solution to the problem of a single crack in an otherwise perfect ice sheet for the case of finite depth ocean. Squire & Dixon (2000) applied the Green's function approach to study the ice-coupled wave propagation across an open crack in water of infinite depth. Evans & Porter (2003) analysed the problem of scattering of obliquely incident waves caused by a narrow crack in an ice-sheet floating on water of finite depth with the eigenfunction expansion method as well as the application of Green's function. The eigenfunction expansion method used by Evans & Porter (2003) can be considered a direct application of the flexural gravity wave-maker problem in the case of water in finite depth. In the present section, as an application of the wave-maker problem in the case of water in infinite depth, the oblique wave scattering caused by a crack in an ice sheet is re-investigated and closed-form solutions are obtained in an elegant manner that was investigated by Williams & Squire (2002) through the application of Green's integral theorem. Various known results of physical importance available in the literature are reproduced from the analytical expressions.

### (a) Mathematical formulation

The problem under consideration is three-dimensional in nature and the physical problem is similar to that of Evans & Porter (2003), except that the fluid is of infinite depth in this case and occupies the region −∞<*x*, *z*<∞, 0≤*y*<∞. The ice sheet of thickness *d* is modelled as two semi-infinite elastic plates separated at *x*=0 due to an open crack and floating on an undisturbed water surface *y*=0, −∞<*x*, *z*<∞ (as in figure 1). With the assumption that the fluid is inviscid and incompressible and that the motion is time harmonic in nature with angular frequency *ω*, there exists a velocity potential *Φ*_{t}(*x*,*y*,*z*,*t*) of the form . The spatial velocity potential *ϕ*_{t}(*x*,*y*) satisfies the partial differential equation(5.1)with *l* being the component of the wavenumber in the *z* direction, the bottom condition (2.4), the linearized ice-covered free surface condition (3.2) with *Q*=0 and the usual far field radiation condition. In addition, assuming that near the crack in the ice sheet, free edge condition is satisfied (the condition of zero bending moment and shear stress), the velocity potential *ϕ*_{t} will satisfy the condition as given by(5.2)where *ν* is Poisson's ratio and *ν*_{1}=2−*ν*.

### (b) Method of solution

The described physical problem is a problem in the half plane. As in the case of Evans & Porter (2003) who reduced the infinite strip problem to a semi-infinite strip problem, we will utilize the geometrical symmetry of the physical problem about the plane *x*=0 to reduce the half-plane BVP to a BVP in the quarter plane and apply the results of the wave-maker problem to determine the full solution, as described below.

The velocity potential *ϕ*_{t}(*x*,*y*) is written as the sum of symmetric and antisymmetric potential functions, *ϕ*_{t}(*ϕ*_{ts}+*ϕ*_{ta})/2, where *ϕ*_{ts} and *ϕ*_{ta} are even and odd functions, respectively, about *x*=0. Using the properties of even and odd functions, we have(5.3)

Thus, we need to consider only the region *x*>0. Since *ϕ*_{t} and ∂*ϕ*_{t}/∂*x* are continuous throughout the fluid region, it can be derived that(5.4)

Further, *ϕ*_{ts} and *ϕ*_{ta} are divided into incident and scattered potentials and written as for *x*≥0,(5.5)where are the symmetric and antisymmetric incident standing waves, respectively. The functions *ϕ*_{s} and *ϕ*_{a} represent the symmetric and antisymmetric scattered potentials and satisfy the governing equation (5.1) along with the bottom boundary condition (2.4) and the ice-covered boundary condition (3.2). However, the far field behaviour is now prescribed as(5.6)with . Thus, the complex amplitudes *R*_{t} and *T*_{t} of the reflected and transmitted waves, respectively, are given by . In addition, using the definitions of *ϕ*_{ts} and *ϕ*_{ta} from (5.5), the edge conditions (5.2) are modified to give(5.7)with

Now, we solve for the symmetric and antisymmetric potentials independently. First we consider the symmetric case. As an application of the wave-maker problem, symmetric potential *ϕ*_{s} can be expanded as(5.8)where *M*(ξ,*y*), *Δ*(*ξ*) are the same as defined in (3.6), with *Q*=0, , *n*=0,*I*,*II*, and *p*_{n} and the corresponding *γ*_{n} are governed by the dispersion relation . Using the continuity conditions (5.4) and the edge conditions (5.7) along with the mode-coupling relation (2.22), the unknown coefficients *A*_{n} and the unknown function *A*(*ξ*) are given by(5.9)with *ϵ*_{n} as defined in (3.7*b*).

By a similar approach, as in the case of the symmetric potential, the antisymmetric potential *ϕ*_{a} is given by(5.10)with the unknown constants *B*_{n} and the unknown function *B*(*ξ*) obtained as(5.11)with *C*_{n} defined for the symmetric case and *ϵ*_{n} as defined in equation (3.7*b*).

The integrals for *D*_{s} and *D*_{a} in equations (5.9) and (5.11), respectively, are integrable and can be easily evaluated. Once the unknown constants *A*_{0} and *B*_{0} are obtained (replacing *R*_{s} and *R*_{a} with *A*_{0} and *B*_{0}), *R*_{t} and *T*_{t} can be obtained from relations for complex amplitudes, as defined above.

### (c) Numerical results and discussion

Simple numerical computations are carried out to obtain the values of the reflection and transmission coefficients and the known results are validated. In the present context, the reflection and transmission coefficients are defined as *K*_{r}=|*R*_{t}|, *K*_{t}=|*T*_{t}|. Various values of the physical parameters used for the computational purpose are the same as provided by Squire & Dixon (2000) and are mentioned here for convenience: Young's modulus *E*=5 GPa, Poisson's ratio *ν*=0.3, water density *ρ*_{w}=1025.0 kg m^{−3} and ice density *ρ*_{i}=922.5 kg m^{−3}.

In figure 3, the magnitude of the reflection coefficient *K*_{r} and the transmission coefficient *K*_{t} are plotted against the time period T with *θ*=18° for different values of ice thickness; the result matches exactly with the results of Williams & Squire (2002). It appears from the curves that the low frequency waves pass through the crack where the crack reflects the high frequency waves more. There is also a finite wave period at which the reflection is zero and the transmission is perfect. This may be due to the change in phase of the reflected wave near the crack. After that point of zero reflection, the magnitude of reflection again increases, before asymptotically falling to zero. A similar observation was made by Evans & Porter (2003) in the case of oblique waves for water of finite depth.

In figure 4, the reflection and transmission coefficients *K*_{r} and *K*_{t} are plotted as a function of the non-dimensional wavelength with *θ*=30° for different values of the ice thicknesses and the nature of the curves are similar to the results obtained by Evans & Porter (2003) for the case of finite depth. The graph shows that transmission is almost perfect for long waves whereas the reflection is high for waves with short wavelength.

In figure 5, the effect of incidence angle on the reflection and transmission coefficients is observed. In all the cases, as the angle of incidence approaches 90°, the reflection coefficient sharply rises to unity. For shorter wavelengths, the reflection coefficients are higher and decrease to almost zero at some point.

## 6. Conclusion

The expansion formulae for the velocity potentials are derived in a quarter plane by utilizing the orthogonal mode-coupling relations for a class of BVPs with higher-order boundary conditions through the direct application of Fourier sine transform. The equivalent form of the orthogonal mode-coupling relation derived is used in the expansion of the velocity potential in a semi-infinite strip domain. Using the expansion formulae, explicit analytical solutions of the flexural gravity wave-maker problems in both the cases of semi-infinite strip and quarter plane are derived, which are also alternatively obtained through an application of Green's integral theorem. Appropriate source potentials are derived by two different methods used in the expansion formulae based on Green's integral theorem. The developed source potential or Green's function can be easily utilized in the boundary integral equation formulation of the wave structure interaction problem. The use of the orthogonal mode-coupling relation reduces the BVP into a system of linear algebraic equations that can be solved by simple numerical computation. As an example, the oblique water wave scattering caused by cracks in floating ice sheets in a half plane is analysed, which shows the application of the quarter-plane wave-maker problem to a problem in the half plane. The non-integral terms of the mode-coupling relation clearly demonstrate the presence of the edge conditions in the solution and are automatically satisfied to provide a unique solution to the boundary-value problem, which are prescribed based on physical requirements. Various results derived in the present paper can easily be applied to handle a large class of problems arising in the broad area of fluid–structure interaction in a lucid manner.

## Acknowledgments

S.R.M. acknowledges the support received from the National Board for Higher Mathematics through a post-doctoral fellowship and J.B. acknowledges the support received from CSIR, New Delhi, India, through a Junior Research Fellowship. The research work is partially supported by the Naval Research Board, New Delhi, India.

## Appendix

To obtain the mode-coupling relation (2.22) along with equation (2.23), let us calculate the integral for *m*≠*n* where *I*_{n}(*y*) s are as given by equation (2.19) and satisfy the condition (2.13). So we havesince each *I*_{n}(*y*) satisfies the condition (2.13). Applying integration by parts and using the boundary condition (2.4) for infinite depth,which can be rewritten as(A1)after making use of the dispersion relation and the resultand

Furthermore, for *m*=*n*, by the use of the relation (2.15), we have(A2)

Combining (A 1) and (A 2), we have the relation as defined in equation (2.23).

## Appendix

*The polynomial* , *with D, K>0 and real Q has one positive real root and four complex roots distributed in each quadrant*.

We observe first that the polynomial has no purely imaginary roots as *F*(*p*) is an odd function with real coefficients.

*Case 1*. *Q*>0

*F*(*p*) has a positive root . Further, since *F*′(*p*) does not have any real roots, *F*(*p*) cannot have any other real root. The next job is to analyse the existence of complex roots in each quadrant.

Let us consider the closed contour OPQO (as in figure 2) with PQ being the circular arc of radius *R*. Then, on the positive imaginary axis, . Since the function *F*(*p*) is in the second quadrant, as *y* varies from 0 to ∞, the argument of the function varies from *π* to *π*/2 so that the change in argument is −*π*/2. Next, on the arc PQ, *p*=*R*e^{iθ}, *π*/2<*θ*<*π*, gives

As *θ* changes from *π*/2 to *π*, the argument of the function *F*(*p*) varies from 5*π*/2 to 5*π*. Hence, the change in argument on the arc is 5*π*/2. Now, on the negative real axis, the values of *F*(*p*) always remain negative and continuous, making the change in argument 0. Thus, the total change in the argument of the function over the whole contour is 2*π*. As a result, by the argument principle of complex function, the number of roots in the second quadrant becomes 1 (as the number of roots=(1/2*π*) times the total change in argument). Thus, the above equation has one complex root in the second quadrant and, consequently, the complex conjugate in the third quadrant. Since the equation has no other real or purely imaginary root, the other two roots are complex and situated in the first and fourth quadrant.

*Case 2*. *Q*<0

By using the transformation *p*=i*α*, *F*(*p*) is modified as . Since the function *G* has only one negative imaginary root , *F* has a unique positive real root. Proceeding in a similar manner to that in case 1, it can be proved that the other four roots are distributed in each quadrant of the complex plane.

- Received December 17, 2004.
- Accepted August 3, 2005.

- © 2005 The Royal Society