## Abstract

Unlike the trapping of time-harmonic water waves by fixed obstacles, the oscillation of freely floating structures gives rise to a complex nonlinear spectral problem. Still, through a convenient elimination scheme the system simplifies to a linear spectral problem for a self-adjoint operator in a Hilbert space. Under symmetry assumptions on the geometry of the fluid domain, we present conditions guaranteeing the existence of trapped modes in a two-layer fluid channel. Numerous examples of floating bodies supporting trapped modes are given.

## 1. Introduction

The motion and stability of a floating body in the context of hydrodynamics and naval engineering is an age-old issue (e.g. [1,2]). Yet, the linearized equations describing the small amplitude motion of the coupled system consisting of an inviscid, incompressible, heavy fluid and a rigid body floating freely in it were derived only in the late 1940s by John [3]. The classical water-wave problem where an immersed obstacle is assumed to be fixed was thoroughly investigated over the past century (see [4–9] and references therein) but the coupled freely floating problem was largely ignored until recently.

In one of his pioneering works, John [10] showed uniqueness for the linearized water-wave problem. He concluded that the problem is unique if none of the vertical lines drawn downwards from the free surface intersects the obstacle immersed in a homogeneous fluid. Later generalized to other geometric configurations, a similar condition has been shown to hold also for fixed or freely floating interface-piercing obstacles in a two-layer fluid (cf. [11,12]).

Trapped modes in the water-wave problem are coupled free oscillations of the floating body with the surrounding fluid. If these non-trivial solutions exist, then the corresponding water-wave problem is non-unique. One should bear in mind, however, that John's uniqueness theorem and its generalizations are valid only in fluid domains which are unbounded in all horizontal directions. These results have thus no direct bearing on our work, since we search for trapped modes around floating obstacles in a three-dimensional fluid channel.

Trapped modes are spatially localized oscillations of finite energy. These modes exist at discrete frequencies, mathematically corresponding to eigenvalues in the discrete spectrum or embedded in the continuous spectrum. For a freely floating structure the modes of oscillation are characteristic of the combination of the structure and the fluid.

The study of trapped modes in the water-wave problem is relevant to offshore activities and structures, think e.g. of oil and gas drilling, piers and other floating structures subject to tides and/or wave motions. Harbour buoys and vessels in channels and fjords are also prone to this kind of oscillations.

McIver & McIver [13] constructed two-dimensional trapping structures by examining the streamlines of suitably chosen potentials. Their method was similar to the one used by Kyozuka & Yoshida [14] to obtain wave-free oscillating structures. Later, McIver & McIver [15] extended their previous work to three-dimensional structures having vertical axis of symmetry. Evans & Porter [16] showed by numerical computations that an immersed circular cylinder, making forced time-harmonic two-dimensional heave or sway motions, can create a local flow field in which no waves radiate to infinity at particular frequencies and depths of submergence of the cylinder. In [17], the same authors proved that pairs of semi-immersed circular cylinders, free to move in both heave and sway under natural hydrostatic restoring, can support trapped waves. Fitzgerald & McIver [18] obtained numerical evidence of the existence of passive trapped modes, that is, modes for which the net force on the structure exerted by the fluid oscillation vanishes. With the help of an inverse procedure, Kuznetsov [19] built trapped-mode solutions for two-dimensional freely floating structures named motionless floating structures. Kuznetsov & Motygin [20] extended these results to three dimensions considering motionless toroid-like surface-piercing structures.

Kuznetsov [19] rewrote the equations of motion for the coupled time-harmonic problem as a spectral boundary-value problem consisting of a differential equation and an algebraic system, coupled through boundary conditions, and Nazarov & Videman [21] wrote the problem in a suitable variational and operator form. Since the interaction of time-harmonic waves with freely floating objects gives rise to a quadratic operator pencil, a scheme that reduces the quadratic pencil to a linear spectral problem was also proposed in [21]. As a first step towards stratified fluids, in [12] we derived the linear system of equations governing the interaction of water waves with partially or totally immersed freely floating structures in a two-layer fluid and suggested a suitable variational formulation. Moreover, examples of configurations where the problem admits only the trivial solution were given.

In this paper, we study the trapping of water waves by freely floating obstacles in a two-layer fluid channel. We assume that the objects are symmetric with respect to the centreplane of the channel and, for expediency, place them in the middle of the channel. We provide examples of objects, totally immersed, surface-piercing or interface-piercing, supporting trapped modes. Most of our results are new also for homogeneous fluids.

In §2, we introduce the governing equations for the coupled system and state the stability conditions assumed to be fulfilled throughout the paper. In §3, we write the problem in a variational and operator form following the procedure proposed in [21] for the homogeneous fluid. In §4, we reason as in Evans *et al.* [22] and construct an artificial discrete spectrum by imposing symmetry conditions on the solution and the fluid domain, and considering an auxiliary problem in the half-channel. Consequently, we may rewrite the coupled operator equations as a linear spectral problem for a positive definite self-adjoint operator and derive a sufficient condition for the existence of trapped modes. This condition translates into an inequality for the sum of surface and volume integrals over the body plus some algebraic terms associated with the rigid body motions. Bearing in mind the stability conditions, we present in §5 several examples of freely floating bodies that satisfy the sufficient condition and, therefore, support trapped modes. We examine also the dependence of the trapping and stability conditions on the free parameters (obstacle dimensions, layer depths and density ratio). In the conclusion given in §6, we briefly discuss our results.

## 2. Equations of motion

Consider the mechanical system consisting of two homogeneous, incompressible, inviscid liquid layers lying on top of one another and of a partly immersed rigid body freely floating in it, i.e. floating under the effect of gravity without other external forces acting on the body. Assume that the constant density in the lower layer is greater than the one in the upper layer (*ρ*_{2}>*ρ*_{1}>0). The fluid layers are taken to be of finite depth and the entire fluid domain is in an open channel, being bounded laterally by rigid walls, from above by a free surface, and from below by a horizontal rigid bottom. The origin of Cartesian coordinates is fixed at the interface between the fluid layers in such a way that the (*x*,*y*)-plane coincides with its rest position and the *z*-axis points upwards.

We shall non-dimensionalize the equations in the frequency domain. Towards this end, we divide the components *x*, *y* and *z* by the channel's half width *l* and introduce a three-dimensional parallelepiped-shaped channel composed of the upper layer *Π*^{1} and the lower layer *Π*^{2} defined by
*H*_{j}>0 denote the dimensionless layer depths, i.e. *H*_{j}=*h*_{j}/*l*, with *h*_{j} standing for the (dimensional) depth of layer *j*. We represent the partly immersed rigid body by a bounded connected domain *Θ*^{1}=*B*∩*Π*^{1} and *Θ*^{2}=*B*∩*Π*^{2} denote the immersed parts (at their rest position) in the upper and lower layer, respectively. We assume that the wetted surfaces of *B*, defined by *Σ*^{1}=∂*Θ*^{1}∩*Π*^{1} and *Σ*^{2}=∂*Θ*^{2}∩*Π*^{1}, are Lipschitz (uniformly continuous) surfaces, in order to be able to define a (uniformly) continuous normal vector field on them. By *θ*^{1} and *θ*^{2}, we denote the cross-sectional areas (at their rest position) of the parts of the body piercing the free surface and the interface, respectively. The fluid domain is composed, within each layer, of *Γ*^{1} and *Γ*^{2} we denote the free surface and the interface (at their rest position) not pierced by the obstacles, i.e.
*Γ*^{b}, is a strip belonging to the plane *z*=−*H*_{2} (cf. figure 1).

Under the usual assumptions of the linearized water-wave theory, cf. [8], we introduce the single-valued velocity potentials *Φ*^{(1)}(*x*,*y*,*z*,*t*) and *Φ*^{(2)}(*x*,*y*,*z*,*t*) in the upper and lower fluid regions of the channel and the vector *t*, the displacement of the centre of mass of the body from its rest position *x*_{0} (components *a*_{j}, *j*=1,2,4, respectively, for surge, sway and heave) and the angular displacement of the body about the axis through the centre of mass *x*_{0} (components *a*_{j},*j*=3,5,6, respectively, for yaw, roll and pitch).

Assuming that the motion of the coupled system is time harmonic, with radian frequency *ω*>0, we can write the velocity potentials *Φ*^{(j)}, *j*=1,2, and the displacement vector ** a** as

*φ*

^{(1)},

*φ*

^{(2)},

**),**

*α**ω*) (see also [1])

*φ*

^{(1)},

*φ*

^{(2)},

**) consisting of two scalar functions**

*α**φ*

^{(1)}and

*φ*

^{(2)}, and a number vector

**. The variables**

*α**φ*and

**and the spectral parameter**

*α**ω*have been scaled as in [20], §4, except for the length scale for which we have used the channel's half width

*l*instead of the diameter of the floating body as in [20]. We have also scaled all densities by

*ρ*

_{2}, thus defining

*ρ*=

*ρ*

_{1}/

*ρ*

_{2}and

*ρ*

_{B}=

*ρ*

_{b}/

*ρ*

_{2}, with

*ρ*

_{b}=

*ρ*

_{b}(

**) denoting the density distribution within the obstacle.**

*x*Above, *D*(** x**) is a 3×6 matrix given by

*M*is a 6×6 (scaled) symmetric, positive definite matrix, defined by

*M*takes the form

*K*can be defined block-wise as

*K*′ a matrix of the form

For the equilibrium position, the linear momentum balance requires that
*x*_{F}=(*x*_{F},*y*_{F},*z*_{F}) is the centre of buoyancy defined by

To ensure stability of the equilibrium position, it is sufficient that the quadratic form represented by matrix *K*′ be positive definite. Defining the second-order moment of inertia with respect to a horizontal line rotated counterclockwise by an angle *ζ* relative to the *x*-axis and passing through the area centre of *θ*^{j} by
*K*′ can be shown to be positive definite, cf. [12], if

## 3. Variational and operator formulation

Let *ρ*_{j}Δ*φ*^{(j)}=0 by a test function *Ω*^{j}, using the boundary and transmission conditions and summing the resulting equations, we obtain (cf. [12]):
_{Ωj}, (⋅,⋅)_{Σj} and (⋅,⋅)_{Γj} denote the usual scalar products in [*L*^{2}(*Ω*^{j})]^{3}, *L*^{2}(*Σ*^{j}) and *L*^{2}(*Γ*^{j}), respectively. Taking the inner product between equation (2.1g) and the vector *φ*^{(j)}∈*H*^{1}(*Ω*^{j}) correspond to finite energy (trapped mode) solutions decaying, together with their gradients, to zero when

Let *H* be the Hilbert space composed of elements *φ*=(*φ*^{(1)},*φ*^{(2)})∈*H*^{1}(*Ω*^{1})×*H*^{1}(*Ω*^{2}) and equipped with the scalar product
^{1/2}. This norm guarantees the finiteness of the wave energy, as desired. In this case, no radiation condition is required. Next, we introduce the linear operators *A* and *T* through
*A* is associated with the kinetic energy and operator *T* is associated with the potential energy of the wave motion. We also define the operator *S* is a compact operator given that the boundary of the floating body is a compact set and, consequently, the set of traces of functions in *H*^{1}(*Ω*^{j}) is compactly embedded into *L*^{2}(*Σ*^{j}) (e.g. [23]). This guarantees that the trace and the Steklov–Poincaré inequalities hold in our infinite fluid domain and thus shows that the norm ∥⋅∥ is equivalent to the usual norm in *H*^{1}(*Ω*), see [21] for more details.

Problem (3.1) and (3.2) can now be written as
*S** is the adjoint operator of *S*, defined through

## 4. Artificial discrete spectrum

Omitting (formally) the terms including *S* and *S** in system (3.4) and (3.5) decouples the wave motion from the motion of the body and leads to the spectral problems
*ω*^{2}, i.e. the entire real line for *ω* [24,25]. The algebraic eigenvalue problem (4.2) is symmetric and finite-dimensional, hence its spectrum is real and fully discrete.

Since problem (4.1) and (4.2) differs from system (3.4) and (3.5) only by a compact perturbation (recall that the operator *S* is compact), the continuous spectra of the two problems coincide (cf. [26]). The continuous spectrum of system (3.4) and (3.5) covers thus the entire real line and, consequently, its discrete spectrum is empty and the eigenvalues, in case they exist, are necessarily embedded in the continuous spectrum.

The variational principles that we wish to employ here are only suited for studying discrete spectra. We will thus follow Evans *et al.* [22], see also [25,21], and create an artificial discrete spectrum for problem (3.4) and (3.5) by restricting ourselves to antisymmetric velocity potentials and to the rigid body movements which preserve antisymmetry with respect to the longitudinal centreplane of the channel, namely transverse horizontal translations (sway), rotations with respect to the vertical axis (yaw) and rotations with respect to the longitudinal axis (roll). Imposing artificial Dirichlet conditions on the velocity potentials at the centreplane {*y*=0} we can consider the problem in the half-channel wherein its continuous spectrum is the set *ω*_{†} denoting a positive cut-off value that leaves room for a discrete non-empty spectrum in the interval (−*ω*_{†},*ω*_{†}). Odd extensions of *φ*^{(j)} across {*y*=0} then show that the extended solution solves the problem in the entire domain (cf. [21]). The above argument is possible only if the fluid domain, i.e. the obstacle, is symmetric with respect to the centreplane {*y*=0}.

### (a) Auxiliary problem

Let the floating body *B* be symmetric with respect to the plane {*y*=0} and assume that its density *ρ*_{B} is even with respect to the *y*-variable so that the mass centre and the buoyancy centre lie on the centreplane.

The auxiliary problem consists of system (2.1a)–(2.1g), written in the half-channel, and complemented with the Dirichlet conditions
*b*, we may return to the original problem with symmetry assumptions imposed on the fluid domain and on the solution (*φ*^{(1)},*φ*^{(2)},** α**).

As for the assumptions on (*φ*^{(1)},*φ*^{(2)},** α**), note that the antisymmetry conditions

*y*=0}, to solve the original problem (2.1a)–(2.1g) in a symmetric fluid domain (cf. [22]). Moreover, the symmetry properties, with respect to {

*y*=0}, of the vector

### (b) Auxiliary problem without obstacles

Consider now the auxiliary problem with no obstacles and let λ=*ω*^{2}. Any function *φ*=(*φ*^{(1)},*φ*^{(2)}) of the form
*k*>0, solves equation (2.1a), the lateral and bottom boundary conditions (2.1d) and (2.1e) and the artificial Dirichlet conditions (4.3). The conditions at the free surface and the interface yield the dispersion relation (see also [27])
^{±}. There is a threshold value at *k*=*π*/2 below which no waves can propagate to infinity in the *x*-direction (cf. (4.6)). This cut-off value is denoted by λ_{†}=λ^{−}(*π*/2), where λ^{−} is the smaller solution of the previous quadratic equation when *k*=*π*/2. The continuous spectrum (in fact, the entire spectrum) of the auxiliary problem in the absence of obstacles is thus the set *ω*, with

The non-trivial solution corresponding to

### (c) Variational and operator formulation for the antisymmetric problem

The auxiliary problem can be seen as a compact perturbation of the decoupled problem (4.1) and (4.2) (considered in the half-channel with same symmetry conditions as the auxiliary problem); recall that the boundary of the object is compact so that *H*^{1}(*Ω*) is compactly embedded into *L*^{2}(*Σ*^{j}). Consequently, its continuous spectrum is inherited from the decoupled problem and, therefore, from the problem without obstacles.

Let us now consider the original problem (2.1a) –(2.1g) with the symmetry assumptions on the fluid domain, the antisymmetry condition (4.4) and restrictions (4.5). Its continuous spectrum has been shown to be

We introduce the subspace
^{1/2}. Note that we can consider the scalar product 〈⋅,⋅〉 instead of 〈⋅,⋅〉_{H}, cf. (3.3), because *φ*∈*H*_{0} has zero mean due to the symmetry assumptions. This means that the operator *A* reduces to the identity operator and becomes positive definite.

Given that *α*_{1}=*α*_{4}=*α*_{6}=0, we will identify

### (d) Sufficient condition for the existence of a trapped mode

Unlike the trapping of water waves by fixed obstacles, the interaction of time-harmonic waves with freely floating objects gives rise to a quadratic eigenvalue problem. We will follow a scheme developed in [21] that reduces the quadratic problem to a linear spectral problem for a self-adjoint operator in a Hilbert space.

For any *ω*≠0, (4.9) and (4.10) can formally be reduced to a linear problem by defining *ξ*=*ωT*^{1/2}*φ*, where *T*^{1/2} is the operator square root of *T* (cf. [26]). Aiming at eliminating the singular part of the *K* matrix in the algebraic equation (4.10), we rewrite it without the first two components of ** α**, removing the rigid body movements not influenced by buoyancy. To this end, we define

**=**

*η**ωM*

**and write**

*α**S*into

*N*=

*M*

^{−1}block-wise as

*I*is the identity operator in

*H*

_{0}.

The antisymmetry of the functions belonging to the space *H*_{0} guarantees that the self-adjoint operator *μ*_{†}, *μ*_{†}], with *μ*_{†}=1/*ω*_{†}. For its discrete spectrum, there are two possibilities: either the norm of the operator coincides with *μ*_{†}, so that the discrete spectrum is empty, or the norm is greater than *μ*_{†}, and the discrete spectrum is non-empty since the norm belongs to its spectrum. Hence, if

## 5. Examples

There is no need to compute the supremum in (4.12). It suffices to show that the inequality is valid for a certain trial function. These trial functions have to solve neither the original problem nor the auxiliary model problem. They only have to belong to the function space over which the supremum in (4.12) is calculated. In particular, they need to decay at infinity. Keeping this in mind, we will now simplify the sufficient condition and examine its validity for a number of floating obstacles.

### (a) A totally immersed cube

Consider a cube totally immersed in the lower layer and floating symmetrically with respect to the centreplane {*y*=0} at its equilibrium position determined through Archimedes’ principle of flotation (2.3). Letting *d* be the side length and *δ*>0 the vertical distance from the top of the cube to the interface, we thus have
*ρ*_{B}(** x**) depends on

*z*in such a way that the centre of mass is below the centre of buoyancy, and on

*z*only so that the horizontal coordinates of the centres of mass and buoyancy satisfy conditions (2.4).

We will examine the existence of trapped modes by considering the velocity potential test function
*φ*_{ϵ} is defined through *φ*_{ϵ}(** x**)=e

^{−ϵ|x|}

*ϕ*

_{†}(

*y*,

*z*), where

*ϵ*≪1 is a small positive parameter. Since

*ϕ*

_{†}∉

*H*

_{0}, we have multiplied it by a function that provides the needed decay as

Since the cube is totally immersed in the lower layer, inequality (4.12) simplifies to (cf. [21] for similar computations)
_{Θ2} denotes the usual scalar product in [*L*^{2}(*Θ*^{2})]^{3}. Computing the first term, we obtain
*y*-axis, we have *I*^{B}=*d*^{3} in the light of Archimedes’ principle. Subtracting (5.3) from (5.2) and doing some algebra, inequality (5.1) becomes

If the cube is immersed in the upper layer, similar argument leads to the inequality

### (b) A rectangular cuboid floating at the interface

Consider a homogeneous rectangular cuboid of side length *d* and height *h* floating at its equilibrium position at the interface (figure 2). Letting *ρ*_{B} (*ρ*<*ρ*_{B}<1) be the constant density of the cuboid, the distance from its top to the interface is *th*, where, in view of Archimedes’ principle, *t*=(1−*ρ*_{B})/(1−*ρ*). We thus have

The centres of mass and buoyancy of the cuboid are given by
*z*_{0}−*z*_{F}>0 ∀ *t*∈]0,1[, which means that not all configurations satisfy the stability condition (2.5). Given the square bottom shape, *I*^{B}=*d*^{2}*hρ*_{B} and

To derive a trapping condition, we consider the following algebraic test function:
*x*=0} and {*y*=0} passing through the centre of mass, the sufficient condition (4.12) reduces to
*d*/*h*)^{2}=*ϵ*+6(1−*t*)*t* and choosing *ϵ*>0 small enough, both conditions (5.4) and (5.7) can be satisfied for all values of the problem parameters. In other words, given the mass of the floating cuboid there always exists a stable configuration (width/height ratio) that supports trapped modes.

In figure 3, we show regions in *ρ*–*d* parameter space where both the stability and the trapping condition are met for fixed values of parameters *H*_{1}, *H*_{2}, *ρ*_{B} and of the cuboid height *h*. Note that smaller values of *ρ* favour the existence of trapped modes, since *ρ* (cf. [27]). Moreover, increasing the height of the cuboid, and keeping other parameters fixed, enlarges the region where the trapping condition is satisfied.

### (c) Freely floating ellipsoidal obstacles

Given that the (vertical) floating position of the obstacle, the density ratio *ρ* and the obstacle mass are linked through Archimedes’ principle of flotation, only two of the three parameters are free. In the previous example, we examined the trapping conditions in terms of the density ratio *ρ* and the floating position or, equivalently, the obstacle mass. Now, we will fix the obstacle with its geometric centre at the interface, or free surface, so that only the density ratio *ρ* or, equivalently, the obstacle mass is a free parameter.

#### (i) Interface-piercing immersed ellipsoid

Consider an ellipsoid floating at the interface (figure 4). Its immersed parts are defined by
*b*<1 and *x*_{0},*y*_{0})=(*x*_{F},*y*_{F})=(0,0) and the equilibrium position of the ellipsoid is such that *z*_{0}=0. The buoyancy centre is
*ζ*∈[0,*π*[. Taking into account that *b*>*c*.

The sufficient condition (5.6) simplifies to
*H*_{2}≫*H*_{1} (the lower layer is much deeper than the upper layer), this condition becomes
*a*, we show the region in the *c*–*b* plane where both the stability and the sufficient condition hold for *ρ*=0.9, *H*_{1}=1 and *H*_{2}≫*H*_{1}. Note that *c* is bounded from above by *b* due to the stability condition (*b*>*c*) and from below through the sufficient condition (5.9). For *H*_{1} is less than unity smaller values of *ρ* (or, equivalently, lighter obstacles) give a somewhat larger region. It is seen that small obstacles have to be almost spherical (assuming that *a*=*b*) for the trapping conditions to be satisfied, but larger obstacles can be increasingly ellipsoidal.

When *H*_{2}≫*H*_{1}≫1, i.e. the fluid layers are deep compared with the channel width *l* (recall that all length variables are scaled to *l*), the sufficient condition (5.9) ceases to depend on *ρ*. Moreover, there is a lower bound *c*/*b*≈0.59, below which the condition is never satisfied.

#### (ii) Ellipsoid floating at the free surface

In this case, the immersed part is described by (figure 6)
*b*<1 and *c*<*H*_{1}. Assuming that the body is homogeneous, *z*_{0}=*H*_{1} and *x*_{F}=(0,0,−3*c*/8+*H*_{1}). The stability condition is the same as above, but the sufficient condition reduces to (5.8) except for the factor (1−*ρ*)/(1+*ρ*), which is now absent. If *H*_{2}≫*H*_{1}, the sufficient condition reads as

Smaller values of *ρ* favour wave trapping with the results being more visible when *b*, we have depicted the region in the *c*–*b* plane where both the stability and the sufficient condition hold for *ρ*=0.1, *H*_{1}=1 and *H*_{2}≫*H*_{1}. In an almost homogeneous fluid (*a*=*b*) obstacles.

## 6. Conclusion

We have recast the equations modelling the interaction of water waves with freely floating structures in a two-layer fluid in a convenient operator form as a spectral problem for a self-adjoint operator in Hilbert space. With the intention of studying the existence of trapped modes through variational techniques, we have considered the problem in an open channel (waveguide) and imposed symmetry assumptions on the fluid domain and on the solution itself. Assuming that the body is floating in the middle of the channel, the antisymmetry of the velocity potentials reduces the admissible rigid body motions to swaying, rolling and yawing, with only rolling being influenced by buoyancy. The symmetry restrictions allow us to seek trapped modes embedded in the continuous spectrum.

The variational principles give rise to an inequality which can be interpreted as a sufficient condition for the existence of trapped modes. Given the fluid domain and the floating structure, this inequality can be computed as a sum of volume and surface integrals but, in general, only by numerical means. To illustrate the value of the sufficient condition for the existence of trapped modes, we have chosen three examples of floating structures that meet the condition.

In the first example, we showed that a totally immersed cube not piercing the interface supports trapped modes. The result seems to be new even for a homogeneous fluid and holds for any stable, which means necessarily bottom-heavy, floating cube. Generalization to other immersed objects is likely to be straightforward.

In the other examples, the obstacles were assumed to be homogeneous, which emphasized the role of stability. When a homogeneous rectangular cuboid floats at the interface, see §5*b*, its position is stable only if the cuboid is sufficiently wide compared with its height. At the same time, the less stable is the position, the more likely it is for the sufficient condition to be satisfied. In general, given the mass of the cuboid one can always choose its dimensions in a way that it satisfies the sufficient condition.

In §5*c*, we examined interface or free-surface piercing ellipsoidal obstacles. We assumed that the width of the channel along the longitudinal axis, 2*a*, exceeds the other dimensions of the obstacle, so that the stability condition is only influenced by the cross-channel width 2*b* and the height 2*c* of the obstacle, in accordance with the sufficient condition related to the rolling reference motion. Almost spherical obstacles of almost any size were seen to satisfy the trapping conditions. For larger (compared to the channel width) obstacles, more ellipsoidal shapes *c*<*b* are also admissible.

We investigated also the effect of the density ratio *ρ*=*ρ*_{1}/*ρ*_{2} on the trapping conditions in §5*b*,*c*. At the interface, a decrease in *ρ* runs counter to the existence of trapped modes, but on the free surface smaller values of *ρ* favour the existence of trapped modes. In the virtual absence of stratification (*ρ*≈1), the stability and trapping conditions can be simultaneously satisfied on the free surface only for almost spherical obstacles. The results of the last two sections, although apparently similar, should be compared with caution, since in the first example, cf. §5*b*, we let the vertical position (equivalently, the body mass) be a free parameter, but in §5*c* we fixed the geometric centre of the ellipsoid at the free surface/interface, so that any change in *ρ* must be accompanied by a corresponding change in the body mass due to Archimedes’ principle of flotation.

The algebraic trial functions are easier to manipulate than the velocity potential ones and seem to be more helpful for surface or interface piercing obstacles. On the other hand, when the obstacle is totally immersed in one of the fluid layers it is the velocity potential trial function that gives the most general results. One should keep in mind though that the trial functions are not actual trapped mode solutions.

## Funding statement

GASD was supported by an FCT grant SFRH/BPD/70578/2010 and JHV was partially funded by the FCT Project PTDC/MAT-CAL/0749/2012.

- Received May 17, 2014.
- Accepted July 15, 2014.

- © 2014 The Author(s) Published by the Royal Society. All rights reserved.