## Abstract

This article is concerned with the existence of rigid freely floating structures capable of supporting trapped modes (time-harmonic water waves of finite energy in an unbounded domain). Under the usual assumptions of linear water-wave theory, a condition guaranteeing the existence of trapped modes is derived, and structures satisfying this geometric condition are shown to exist in a three-dimensional water channel. The sufficient condition arises from the application of variational principles to a conveniently formulated linear spectral problem, the main effort being the construction of a reduction scheme that turns the quadnic operator pencil associated with the original coupled system into a linear self-adjoint spectral problem. An example of floating bodies supporting at least four trapped modes is given.

## 1. Introduction

Linearized equations governing the small amplitude motion of a coupled system consisting of a rigid object floating freely in an incompressible inviscid liquid were first derived by John (1949). Yet, practically all the analyses by John (1949) and in other early work (see John 1950; Ursell 1951; Jones 1953; Garipov 1967) were focused on the problem of interaction of water waves with *fixed* rigid obstacles. The latter problem is considerably more accessible and has been largely investigated over the past 60 years (see the monograph by Kuznetsov *et al.* (2002); the review article by Linton & McIver (2007) and all the references cited therein).

The linearized water-wave problem has a great number of variants, many of them related with classical problems of hydrodynamics such as ship waves or edge waves or scattering and radiation problems (e.g. Kuznetsov *et al.* 2002). Here, we are concerned with time-harmonic motions called (motion) trapped modes associated with free oscillations of a heavy liquid around a freely floating object and achieved by assuming that at large times, all motions are simple harmonic (see John 1950).

The trapped modes are non-trivial solutions to a homogeneous water-wave problem at a particular oscillation frequency. If a structure supports trapped modes, then the solution to the corresponding water-wave problem with suitable radiation conditions at infinity is non-unique. In 1950, John proved that the solution to the radiation problem is unique if the frequency of the harmonic motion is large enough and the freely floating object satisfies a certain geometrical condition (no point of the immersed surface lies below a point of the free surface). Since then, the problem of trapping of water waves by freely floating structures has rarely been considered in full generality, notable exceptions being the articles of Beale (1977), who studied the corresponding initial-value problem, and Kuznetsov (2010), who considered the two-dimensional problem and observed that motionless freely floating structures support trapped mode solutions. In most recent work, attention has been either restricted to the two-dimensional case (McIver & McIver 2006; Evans & Porter 2007; Porter & Evans 2009) or the motion of the floating structure constrained to heaving motion (McIver & McIver 2007; Newman 2008; Porter & Evans 2008).

In contrast to water-wave trapping by fixed obstacles, which corresponds to solving a linear eigenvalue problem in the frequency domain, a freely floating object turns the trapping problem into a quadratic operator pencil owing to the additional equation governing the motion of the body itself. At the same time, one is faced with a coupled system composed of a scalar equation for the velocity potential and an algebraic system for the complex amplitude vector of the rigid body displacements and rotations.

The main purpose of this work is to create a scheme that reduces the investigation of the quadratic eigenvalue problem to the study of the spectrum of a linear self-adjoint operator in Hilbert space. To analyse the latter problem, one can use the powerful techniques of operator theory in Hilbert spaces (e.g. Birman & Solomjak 1987). Indeed, a reduction scheme in the form of a suitable trace operator (cf. Nazarov 2008) has led to a series of simple sufficient conditions guaranteeing the existence of trapped modes around fixed structures in different geometrical configurations (see Nazarov 2009*a*–*c*; Nazarov & Videman 2009, 2010).

The operator formulation of the water-wave problem requires a number of preparatory measures, as opposed to similar spectral boundary-value problems for the Helmholtz equation in acoustic and quantum waveguides, wherein the self-adjoint operator arises immediately and naturally, not least because the spectral parameter appears in the Steklov boundary condition. Our reduction scheme will be realized in several steps:

— first, we set forth a variational formulation for the spectral water-wave problem,

— second, we introduce a trace operator and write the variational problem as a quadratic operator pencil ,

— third, we express the quadratic pencil as a linear pencil

*B*−*ωD*, and— finally, we turn the linear pencil into a self-adjoint operator , where is the identity operator and

*μ*=*ω*^{−1}.

In the final and most involved step, we use an algebraic elimination trick and an idea (e.g. Evans *et al.* 1994) that the motion is antisymmetric with respect to the centreplane where artificial Dirichlet boundary conditions are imposed.

The paper is organized as follows. In §2, we set the problem in a water channel, write down the equations of motion and, by assuming that the coupled motion is time harmonic, express the problem in the frequency domain. In §3, we give a variational and operator formulation for the spectral problem as a quadratic pencil in a suitable Hilbert space. Next, we introduce our reduction scheme and write the problem as a linear pencil *B*−*ωD*. In §5, we consider two identical freely floating structures and make a symmetry assumption on their relative position, which guarantees that the self-adjoint operator *B* is positive definite. Consequently, we express the problem in the normal form and, by analysing its spectrum, obtain a sufficient condition for the existence of a trapped mode. In §6, we derive simplified forms of the sufficient conditions and in §7, consider particular freely floating structures supporting up to four distinct trapped mode solutions. Finally, in §8, we make comparisons with similar trapping conditions for fixed structures.

## 2. Equations of motion

Consider the mechanical system consisting of a three-dimensional rigid object *B* floating freely in an inviscid liquid under the effect of gravity. Assume that the coupled motion of the system takes place in an open channel bounded by rigid walls laterally and from below and by a free surface *Υ* from above. For simplicity, we derive the equations and conditions for one object only, but there is no difficulty in considering the more general case (see §5).

Let be the water channel, extending to infinity in the *x*-direction and with a bounded uniform cross section *ϖ* in the (*y*,*z*)-plane, and assume that the fluid occupies the domain , with denoting the submerged part of the body *B* (figure 1). Moreover, ignore surface tension and assume that the liquid is homogeneous and incompressible and the fluid motion is irrotational.

The small amplitude motion of the liquid is described by a velocity potential *ϕ*(*x*,*y*,*z*,*t*) satisfying the Laplace equation in *Ω*, the linearized kinematic/dynamic boundary condition
2.1
at the free surface , which is fixed at the equilibrium position at {*z*=0}, and the Neumann boundary condition (no normal flow)
2.2
at the bottom ∂*Π*\*Υ*. Here, is the cross section of the object at {*z*=0}, *g* is the acceleration due to gravity, and *ϕ*_{n}=**n**⋅∇*ϕ* is the normal derivative, with **n** denoting the normal vector outward to the fluid domain *Ω*. We assume that *Ω* is a Lipschitz domain so that **n** is defined almost everywhere at ∂*Ω* and that *B* is a connected set. Yet, both *Θ* and *θ* can contain several connected components. We also assume that all variables have been made dimensionless by rescaling, say in a way that the width of the channel at {*z*=0} is equal to *π* (cf. §7) and *g*=1.

The motion of the body *B* is coupled with the wave motion by the kinematic boundary condition on the immersed surface *Σ* of the body and by the equations of motion of the body itself. According to the linear theory, *Σ* can be assumed to be at the fixed equilibrium position of the body and the boundary condition takes the form (cf. John 1949)
2.3
where **x**=(*x*,*y*,*z*) and is a vector field describing, at time *t*, the position of the centre of mass (the components *a*_{j}, *j*=1,2,4) of the body and the angles of rotation (the components *a*_{j}, *j*=3,5,6) about the axes passing through the centre of mass. Hence, the components of **a**_{t} correspond to the translational and angular velocities of the object. Moreover, **x**_{0}=(*x*_{0},*y*_{0},*z*_{0}) are the coordinates of the equilibrium position of the centre of mass of the body *B* and the matrix
characterizes the rigid body motions, in conformity with the ordering of the components of the vector function **a**, with the first three columns corresponding to motions (surging, swaying and yawing) unaffected by the buoyancy forces. In other words, the displacements in the *x*-, *y*- and *z*-directions are described by the components *a*_{1}, *a*_{2} and *a*_{4}, respectively, and the angles of rotation about *z*-, *x*- and *y*-axes by the components *a*_{3}, *a*_{5} and *a*_{6}, respectively.

The equations of motion of the body can be expressed as a system of six second-order differential equations. In the absence of external forces, the linearized equations of motion for the conservation of linear and angular momentum read as (cf. John 1949; Mei *et al.* 2005)
2.4
where
is the mass matrix, with *ρ*(**x**) denoting the non-negative (non-trivial) distribution of density within the body, made dimensionless by assuming that the constant density of the liquid is equal to unity. Defining the various moments of *B* by
and noting that all first-order moments vanish by defining **x**_{0}, we obtain
The mass matrix is always symmetric and positive definite.

The first term on the right-hand side in equation (2.4) expresses the hydrodynamic forces (pressure force and torque) exerted on the surface of the immersed part of the body. The second term arises from the buoyancy forces. The 6×6 matrix *K*(**x**_{0}) can be expressed blockwise as
where all the blocks are 3×3 matrices; is the 3×3 null matrix. Recalling that is the submerged part of the body and the part of the free surface *Υ* pierced by the object and defining
we have (see Mei *et al.* 2005)
The matrix *K*(**x**_{0}) is symmetric and positive semi-definite and the submatrix *K*^{′}(**x**_{0}) positive definite provided
2.5
and the body is not totally submerged. Inequalities (2.5) are valid if the classical condition of stability of equilibrium holds, i.e. the moment of inertia of the water surface pierced by the object about any horizontal axis divided by the volume of the liquid displaced by the floating body must be larger than the distance between its centre of mass and centre of buoyancy (e.g. John 1949). In what follows, we assume that matrix *K*^{′}(**x**_{0}) is positive definite. The following conditions must also hold in the equilibrium position
2.6
namely that the mass of the displaced liquid equals the mass of the body (Archimedes law) and the centre of buoyancy lies on the same vertical line as the centre of gravity.

Assuming that the motion of the coupled system is time harmonic with the radian frequency *ω*, we express the velocity potential and the vector of displacements and rotations as
and write equations (2.1)–(2.4) and the Laplace equation as the following boundary-value problem for the unknowns ((*φ*,** α**),

*ω*): 2.7 2.8 2.9 2.10 and 2.11 with (

*φ*,

**) consisting of a scalar function**

*α**φ*and a number vector .

As we are primarily searching for trapped mode solutions that decay exponentially at infinity and, thus, have finite energy because of the cylindrical structure of the fluid domain at large distances (Nazarov 2009*c*), we do not need to impose radiation conditions at infinity. However, to deal with possible singular solutions, the fluid domain must be Lipschitz (Nazarov & Taskinen 2009) and the boundary-value problem (2.7)–(2.11) formulated as an integral identity, see the variational formulation in §3.

## 3. Variational and operator formulation

Multiplying equation (2.7) by a smooth scalar test function *ψ*, with compact support in , and integrating by parts over *Ω* using the boundary conditions (2.8)–(2.10), one obtains
3.1
where (⋅,⋅)_{Ω},(⋅,⋅)_{Σ} and (⋅,⋅)_{Γ} denote the standard scalar products in [*L*_{2}(*Ω*)]^{3},*L*_{2}(*Σ*) and *L*_{2}(*Γ*), respectively. Taking the inner product of equation (2.11) with the number vector , we get
3.2
where is the inner product in . Now, we introduce the following spectral variational formulation of equations (2.7)–(2.11): find a non-trivial and such that relations (3.1) and (3.2) are satisfied for any .

Let us equip the Hilbert space *H*^{1}(*Ω*) with the specific scalar product
and the norm . Recalling the following Steklov–Poincaré and trace inequalities in an infinite channel containing a compact obstacle with a Lipschitz boundary
3.3
and
3.4
we see that the norm ∥⋅∥_{Ω} is equivalent to the usual norm in *H*^{1}(*Ω*). Inequality (3.3) follows from Steklov–Poincaré inequalities in the cross section *ϖ* and in a truncated channel *Ω*_{R}={**x**∈*Ω*:|*x*|<*R*}, namely
3.5
and
3.6
where *γ* and *Γ*_{R} are parts of the free surface *Γ* laying on ∂*ϖ* and ∂*Ω*_{R}, respectively, *R*>0 is sufficiently large so that *Θ*⊂*Ω*_{R}, and *C*_{ϖ},*C*_{Ω,R} are constants. Integrating the two-dimensional inequality (3.5) in and adding the result to equation (3.6) yields (3.3). The trace inequality (3.4) emerges quite similarly. The trace inequality implies that the total energy is finite for any *φ*∈*H*^{1}(*Ω*).

Let us define the linear operators *A*,*T* and *S* by
Note that the operators *A*,*T* in *H*^{1}(*Ω*) and are all continuous. Moreover, *A* and *T* are symmetric, thus self-adjoint and positive.

We write problems (3.1) and (3.2) in an equivalent operator form as a quadratic pencil
3.7
and
3.8
where *S** stands for the adjoint of the operator *S*, i.e.

### Remark 3.1

If *ω*=0, problems (3.7) and (3.8) admit non-trivial solutions of the form *φ*=0, ** α**=(

*α*

_{1},

*α*

_{2},

*α*

_{3},0,0,0), with and |

*α*

_{1}|

^{2}+|

*α*

_{2}|

^{2}+|

*α*

_{3}|

^{3}≠0. These solutions are rigid body motions (horizontal translations and rotation about the

*z*-axis) that may exist independently of the time-harmonic motions. Indeed, before being able to show existence of trapped mode solutions, we need to eliminate these rigid body motions from systems (3.7) and (3.8).

## 4. Reduction scheme

For any *ω*≠0, the spectral problems (3.7) and (3.8) can be reduced to a linear pencil by defining *ξ*=*ωT*^{1/2}*φ*, ** η**=

*ωM*

**and**

*α***X**=(

*φ*,

*ξ*,

**,**

*α***). This leads to the system 4.1 where**

*η**T*

^{1/2}is a continuous self-adjoint operator in

*H*

^{1}(

*Ω*), defined as the positive square root of

*T*(e.g. Birman & Solomjak 1987, ch. 6),

*N*=

*M*

^{−1}is a symmetric and positive definite matrix, is the identity operator in

*H*

^{1}(

*Ω*), is the 6×6 identity matrix and the spectral parameter

*ω*appears only linearly. Problem (4.1) is clearly equivalent to systems (3.7) and (3.8); note in particular, the identities

*ξ*=

*ωT*

^{1/2}

*φ*and

**=**

*η**ωM*

**on the second and fourth lines in system (4.1).**

*α*

### Remark 4.1

It is straightforward to verify that if ((*φ*,*ξ*,** α**,

**),**

*η**ω*) solves problem (4.1), then ((

*φ*,−

*ξ*,−

**,**

*α***),−**

*η**ω*) is also an eigenpair for the same spectral problem. Hence, in view of remark 3.1, it is sufficient to consider positive values of

*ω*.

The positive, self-adjoint operator acting on **X** on the left-hand side of equation (4.1) cannot be positive definite as the matrix *K* is always singular. Therefore, we need to rewrite system (4.1) in an equivalent form where the first three components of ** α** are eliminated. With this in mind, we decompose the vectors

**and**

*α***as where**

*η*

*α*_{°}=(

*α*

_{1},

*α*

_{2},

*α*

_{3}),

*α*_{♮}=(

*α*

_{4},

*α*

_{5},

*α*

_{6}); similarly for

*η*_{°}and

*η*_{♮}. We also set where

*N*

_{°°},

*N*

_{°♮},

*N*

_{♮°},

*N*

_{♮♮},

*K*

_{♮♮}and are all 3×3 matrices,

*K*

_{♮♮}=

*K*

^{′}(

**x**

_{0}) and (

*N*

_{°♮})*=

*N*

_{♮°}. Moreover, we define the projection operators

*P*

_{°}and

*P*

_{♮}by Since , we can write ; similarly, . In other words, is one of the inverses of

*P*

_{°}.

Next, assume that *ω*>0 and consider the first three lines in the third block of equations in system (4.1). These can be written as **0**=*ω*(i*P*_{°}*S***φ*+*η*_{°}) from which it follows that any eigensolution **X**=(*φ*,*ξ*,** α**,

**) of system (4.1) must satisfy 4.2 The algebraic system relating**

*η***and**

*α***in system (4.1) reads as Using equation (4.2) to calculate**

*η*

*η*_{°}, we obtain Truncating the last two components in

**X**, we define an eigenvector and express system (4.1) as a linear pencil 4.3 where the operators and are given by and Note that and are both continuous, and self-adjoint because and

Next, we calculate
The second, third and the last two terms on the right-hand side combine to
with some constant *C*>0, where the inequality follows from the positive definiteness of matrix *N*. Therefore, recalling that *K*_{♮♮} is positive definite, we conclude that the operator is positive.

Having a solution (**X**^{′},*ω*), with **X**^{′}=(*φ*,*ξ*,*α*_{♮},*η*_{♮}) and *ω*>0, to problem (4.3), we conclude that (**X**,*ω*), with **X**=(*φ*,*ξ*,** α**,

**) solves problem (4.1), where Moreover, ((**

*η**φ*,

**),**

*α**ω*) is a non-trivial solution to problems (3.7) and (3.8) and, equivalently, to problems (2.7)–(2.11). Thus, the positive parts (

*ω*>0) of the spectra of problems (4.3), (3.7) and (3.8) are the same.

### Remark 4.2

The reduction scheme works also for a totally submerged body. In that case, we just need to eliminate one more component from vector ** α** and work with a 2×2 submatrix

*K*

_{♮♮}, which is positive definite since by admitting the stability of equilibrium, we assume that the centre of buoyancy lies above the centre of gravity.

## 5. Derivation of a sufficient condition under symmetry assumptions

Let us first identify the continuous spectrum of problems (3.7) and (3.8). Setting (formally) *S*=0 in problems (3.7) and (3.8), leads to the decoupled problem
5.1
and
5.2
The spectral problem (5.1), with the spectral parameter *λ*=*ω*^{2}, corresponds to the operator formulation of the water-wave problem around fixed, rigid obstacles. It is well known (see Groves 1997 and also Nazarov 2009*c*) that the continuous spectrum of this problem is the semi-interval for *λ*, thus the whole real line for *ω*. On the other hand, the spectrum of the algebraic eigenvalue problem (5.2) is real and purely discrete because the problem is symmetric and finite dimensional. Finally, the boundary of the floating object is compact, thus the set of traces of functions belonging to *H*^{1}(*Ω*) is compactly embedded into *L*_{2}(*Σ*), which implies that the terms associated with operator *S* in problems (3.7) and (3.8) constitute a compact symmetric perturbation to problems (5.1) and (5.2). Therefore, the continuous spectrum of problems (3.7) and (3.8) still coincides with the whole real line (see Gohberg & Krein 1969).

We will now restrict ourselves to motions whose normal component with respect to the centreplane of the channel is antisymmetric. In this way, we can consider a problem whose continuous spectrum has a positive cut-off value. At the same time, the positive operator in equation (4.3) becomes positive definite.

Assume that the uniform channel cross section *ϖ* is symmetric with respect to the centreplane {*y*=0}. Moreover, assume that there are two identical bodies *B*_{±} symmetric with respect to reflection (mirror-image symmetry) across the centreplane, i.e. *B*_{∓}={(*x*,*y*,*z*):(*x*,−*y*,*z*)∈*B*_{±}}.

### Remark 5.1

It is straightforward to write the problem for two (or more) bodies. We have presented the calculations for only one body so as not to obscure the presentation of the variational and operator formulation and, in particular, of the reduction scheme. Moreover, after symmetry assumptions, the problem reduces to the single body case.

Reasoning as in Evans *et al.* (1994), we impose an artificial Dirichlet boundary condition at {*y*=0} and consider the following auxiliary problem in the half-channel *Ω*_{+}={(*x*,*y*,*z*)∈*Ω*:*y*>0} (for one body only)
5.3
5.4
5.5
5.6
5.7
and
5.8where *Γ*_{0}={(*x*,*y*,*z*)∈*Ω*:*y*=0} and where we have set

Once a solution ((*φ*^{+},*α*^{+}),*ω*^{+}) to problems (5.3)–(5.8) has been found, it can be extended smoothly to *Ω*, by an odd extension across {*y*=0} (see Evans *et al.* (1994) for similar considerations). In the other half-channel *Ω*_{−}={(*x*,*y*,*z*)∈*Ω*:*y*<0}, the solution ((*φ*^{−},*α*^{−}),*ω*^{−}) is determined from ((*φ*^{+},*α*^{+}),*ω*^{+}) mirror symmetry with respect to {*y*=0} through the formulae
The extended solution (*φ*,*α*^{+},*α*^{−},*ω*) clearly satisfies all necessary equations and boundary conditions for the system of two freely floating bodies.

### Remark 5.2

The assumption that the channel and the obstacle are mirror symmetric about the centreplane restricts the motion of the bodies with respect to each other (the motion of the other body is recovered by reflection from the first one), but does not cause any restrictions on the motion of the body in the half-channel.

If the object is rigid and fixed, the water-wave problem corresponding to equations (5.3)–(5.8) reads as
5.9
5.10
5.11
and
5.12
or in an equivalent variational form
The continuous spectrum of problems (5.9)–(5.12) does not fill the entire real line, but only the subset . The eigenfunction *ϕ*_{†}∈*H*^{1}(*ϖ*_{+}) associated with the cut-off value *ω*_{†}>0 is a non-trivial solution to the corresponding spectral problem in the cross section *ϖ*_{+}={(*y*,*z*)∈*ϖ*:*y*>0} of the half-channel (see Nazarov 2009*c*). Therefore, recalling that systems (5.3)–(5.8) can be understood as a compact perturbation of problems (5.9)–(5.12), the continuous spectrum of problems (5.3)–(5.8) is also given by .

Dropping the index + and defining the Hilbert space at *Γ*_{0}}, we write problems (5.3)–(5.8) in forms (3.7) and (3.8), with operators *A*,*S* and *T* defined in . Besides, problem (5.9)–(5.12) reads as *Aφ*=*ω*^{2}*Tφ*.

The operator *A* has now become positive definite, i.e.
with some constants *c*,*C*_{A}>0, and we obtain
5.13
where *C*_{K}, *C*>0, and
It follows that there exists a self-adjoint, positive definite operator , the positive square root of , such that (cf. Birman & Solomjak 1987, ch. 10). Hence, defining and *μ*=1/*ω*, we can rewrite the spectral problem (4.3) in the form
5.14
where is a self-adjoint operator. The operator inherits from problems (5.3)–(5.8) the continuous spectrum [−*μ*_{†},0)∪(0,*μ*_{†}], where *μ*_{†}=1/*ω*_{†}. The value *μ*=0, being an eigenvalue of infinite multiplicity, is also part of the essential spectrum [−*μ*_{†},*μ*_{†}] of , but as it corresponds to , it does not influence the spectrum of the original problem.

If the discrete part of the spectrum of is non-empty, a trapped mode exists. Now the operator is self-adjoint and bounded. Hence its norm, defined by
and also −*m* in view of remark 4.1, must be elements in its spectrum (cf. Birman & Solomjak 1987). It follows that (cf. Nazarov 2009*a*) if
5.15
then a trapped mode exists. Recalling that the operator is self-adjoint and positive definite, condition (5.15) becomes
5.16
This is our sufficient condition in its most general form. There is of course no need (or even possibility) to calculate the supremum in equation (5.16). It suffices to choose a test function and verify, for example numerically, that the inequality is valid. In §6, we will consider some particular test functions.

## 6. Analysis of the sufficient condition

Recall first that
and
Now, let us consider the test function
6.1
where *ϵ*≪1 is a small positive parameter and
Since , we are allowed to insert into equation (5.16). We also have , which guarantees for sufficiently small *ϵ*>0 that the expression
is positive. This implies that condition (5.16) is satisfied if
6.2

Next, let us define *η*^{ϵ}=(−i*P*_{°}*S***φ*^{ϵ},*η*_{♮}) and write
The identities (see Nazarov 2009*b*,*c*)
and
and the observation that in any compact set yield
where *η*_{†}=(−i*P*_{°}*S***ϕ*_{†},*η*_{♮}). Note that even if *ϕ*_{†}∉*H*^{1}(*Ω*), the term *S***ϕ*_{†} is well understood as the integration is over a compact set *Σ*. For small enough *ϵ*>0, condition (6.2) thus becomes
6.3

Condition (6.3) is still too general to be examined analytically. In what follows, we will single out some particular test functions of the form (6.1), which leads to simple geometric conditions guaranteeing the existence of trapped modes. We emphasize that by no means have we exhausted the possibilities to derive useful conditions but only considered a few cases which can be worked out easily without numerical computations or exhaustive calculations.

### (a) Velocity potential test function

Choosing the test function 6.4 the sufficient condition (6.3) simplifies to 6.5 In §7, we will provide an example of a floating body satisfying inequality (6.5).

### (b) Algebraic test function

Considering , we have
and obtain from equation (6.3) the sufficient condition
6.6
recall that *N*_{♮♮} and *K*_{♮♮} are positive definite. Condition (6.6) can be expressed as
6.7
where and (*N*_{♮♮})^{−1} has the form
If matrix *L* has at least one positive eigenvalue, then condition (6.7) is satisfied for some *α*_{♮}≠0 and a trapped mode exists.

If the body is symmetric with respect to the cross-sectional planes {*x*=*x*_{0}} and {*y*=*y*_{0}} passing through its centre of gravity, then , so that matrices and *K*_{♮♮} become diagonal. Consequently, a trapped mode exists if at least one of the following three inequalities is satisfied:
6.8
6.9
and
6.10

### Remark 6.1

Even though the test functions *α*_{♮}=(*α*_{4},0,0), *α*_{♮}=(0,*α*_{5},0) and *α*_{♮}=(0,0,*α*_{6}) leading to conditions (6.8)–(6.10), respectively, correspond to heaving, rolling and pitching motions, the actual trapped mode solution ** α** can oscillate in all modes of motion. The symmetry assumptions on the body about the cross-sectional planes do not impose restrictions on the motion itself.

## 7. A floating bottle and the multiplicity of trapped modes

Consider a rectangular channel *Π* of width *π* and depth *H* and two freely floating identical objects resembling square bottles floating with their neck upwards (figure 2*a*). Let 2*d* be the thickness and *h* the height of the body of the bottles and *ε* the height and width of the submerged part of the neck (figure 2*b*). Assume that *ε*≪*d* is small (*d*∼*h*) and that the density distribution of the object is such that . Since , this ensures the validity of the stability conditions (2.5).

### Remark 7.1

One can always make small, even zero, by redistributing the mass of the body along the *z*-axis without changing its equilibrium position (basically by making it bottom heavy). Note that a similar redistribution of mass may also be necessary to satisfy the stability conditions (2.5).

We will now show that the sufficient condition (6.5) is satisfied for the objects in figure 2. First, we write
and observe that the cut-off value of the continuous spectrum and the corresponding (conveniently scaled) eigenfunction for problems (5.3), (5.5) and (5.6) in the rectangular half-channel without obstacles *Π*_{+} are
It is easy to see that ; note that *ϕ*_{†} does not depend on *x*. The main contribution to the term arises from the surface integrals over the lateral sides at, say *y*=*d* and *y*=3*d*, as the submerged surface area of the neck is of the order . Therefore,
and it follows that, up to the multiplication by ,
where we have used the Archimedes law (cf. equation (2.6)) and observed that
Similarly,

If the channel is deep, i.e. if *H*≫1 and *h*≪*H*, we have
and
Hence, for large enough *h*,
and a trapped mode exists.

### Remark 7.2

The algebraic conditions (6.8)–(6.10) are also satisfied for the body in figure 2*b* because and the integrals over *θ* are negligible with respect to the integrals over the body *B*.

It is natural to wonder whether the trapped modes whose existence we have established are distinct. To answer this question, we follow Nazarov (2008) and introduce mutually orthogonal test functions. Hence, let , *j*=1,2,3, denote three linearly independent vectors in and consider the algebraic test functions
Consider also the test function (cf. equation (6.4)) and define
where . While satisfying conditions (6.5)–(6.10), we have guaranteed that *μ*^{(p)}>*μ*_{†}, *p*=1,2,3,4; note that condition (6.7) is satisfied with any as the matrix *L* is positive definite. The functions and can be made orthogonal among themselves because
and the matrix *ω*_{†}(*N*_{♮♮})^{−1}+*K*_{♮♮} is symmetric and positive definite, so that we can choose as the three linearly independent vectors the eigenvectors of *ω*_{†}(*N*_{♮♮})^{−1}+*K*_{♮♮} for which for *j*≠*k*. Moreover,
since *N*_{♮°}*P*_{°}*S***φ*^{ϵ} and , are vectors in .

Continuity of the self-adjoint operator implies, in particular, that the operator is lower semi-bounded and its essential spectrum coincides with the closed interval [−*μ*_{†},*μ*_{†}]. From the max–min principle (see Birman & Solomjak 1987, theorem 10.2.2) applied to the operator , it then follows that if
where **E**_{p} is a subspace of with codim(**E**_{p})=*p*−1, then has at least *p* eigenvalues corresponding to *p* distinct trapped mode solutions. Now, every subspace of of codimension 3 must contain linear combinations of the form
because one cannot exclude all functions of this type from **E**_{4} owing to the orthogonality of , , and . We thus obtain
where , and conclude that the freely floating body in figure 2*b* generates at least four distinct trapped mode solutions.

Presented slightly more generally, we have proven the following result.

### Theorem 7.1

*Consider two identical freely floating objects mirror symmetric with respect to each other across the centreplane {y=0} and symmetric with respect to the cross-sectional planes passing through their centres of gravity and assume that their centres of buoyancy coincide with the centres of gravity. Then, if the matrix L in condition (6.7) is positive definite and condition (6.5) is satisfied, there exist at least four distinct trapped mode solutions to the corresponding water-wave problem in a symmetric water-wave channel.*

If the sufficient condition (5.16) is satisfied for three (two) mutually orthogonal test functions, then still at least three (two) distinct trapped modes are generated. For example, the matrix *L* in condition (6.7) can be shown to be positive definite for bodies for which condition (6.5) is not met.

## 8. Comparison with trapping conditions for fixed structures

Condition (6.5) was obtained by considering a test function with zero rigid body displacements. It is thus comparable with a similar sufficient condition guaranteeing the existence of a trapped mode around two *fixed* bodies, mirror symmetric with respect to the centreplane. In Nazarov (2009*c*), this condition was shown to be
8.1
where *ω*_{†}>0 and *ϕ*_{†}∈*H*^{1}(*ϖ*) stand for the same (cut-off) eigenvalue and eigenfunction in the cross section of the half-channel as above. Condition (8.1) guarantees that the first two terms in equation (6.5) are non-negative, but does not ensure the existence of a trapped mode for a freely floating body since .

Similarly, assuming that , (*ω*_{f}<*ω*_{†}), is a trapped mode solution generated by a fixed body, and inserting this solution in the form **X**^{′}=(*φ*_{f},*ω*_{f}*T*^{1/2}*φ*_{f},0,0) into condition (5.16), one obtains
8.2
where we have taken into account that . The two terms in equation (8.2) have again opposite signs and one cannot guarantee that the inequality is valid. Yet, if fortuitously *P*_{°}*S***φ*_{f}=0, then the trapped mode **X**^{′} is supported by a freely floating body (see also Kuznetsov (2010) in this connection).

Reciprocally, assuming that problems (3.1) and (3.2) admit a trapped mode solution ((*φ*_{1},*α*_{1}),*ω*_{1}), with 0<*ω*_{1}<*ω*_{†}, it follows that
and
where we have chosen a real-valued *φ*_{1} and, accordingly, written *α*_{1}=i**a**_{1}, with . Subtracting the equations yields
If the matrix is positive, that is, if
8.3
then it follows that
In other words, we have found a function satisfying the inequality , *φ*_{1}〉_{Ω}, which guarantees the existence of a trapped mode supported by a fixed body (cf. Nazarov 2009*c*).

### Theorem 8.1

*Consider two identical partially submerged objects in an infinite water channel and assume that both the objects and the channel are mirror symmetric with respect to the centreplane of the channel. If there exists a trapped mode solution ((φ*_{1}*,**α*_{1}*),ω*_{1}*), with ω*_{1}*∈(0,ω*_{†}*), for the problem when the objects are freely floating and if condition (8.3) is met, then the problem with the bodies fixed admits a trapped mode solution (φ*_{f}*,ω*_{f}*), with ω*_{f}*∈(0,ω*_{1}*).*

## Acknowledgements

S.A.N. was supported by RFFI (grant no. 09-01-00759) and J.H.V. by the project UTAustin/MAT/0035/2008.

- Received May 6, 2011.
- Accepted July 14, 2011.

- This journal is © 2011 The Royal Society