## Abstract

The interaction of linear water waves with totally or partially submerged obstacles is considered in a two-layer fluid consisting of two immiscible liquid layers of different densities. A sufficient condition for the existence of trapped modes is established by introducing a trace operator that restricts the solutions to the free surface and the interface. The modes correspond to localized solutions of a spectral problem, decaying at large distances from the obstacles and belonging to the discrete spectrum below a positive cut-off value of the continuous spectrum. The sufficient condition is a simple relation between the cut-off value and some geometrical constants, namely the surface integrals taken over the cross sections of the submerged parts of the obstacles and the line integrals along the parts of the free surface and the interface pierced by the obstacles.

## 1. Introduction

The mathematical problem of interaction of linear water waves with submerged obstacles has a long history. The problem has numerous variants depending on the geometry of the fluid domain, direction and conditions on the incident and/or scattered waves, boundary conditions on the surfaces of the obstacles and so on, see the book by Kuznetsov *et al.* (2002) for more details. The question we address here is the existence of linear water waves trapped in the neighbourhood of fixed, submerged or surface piercing, horizontal cylinders in a two-layer fluid.

The existence of trapped waves (modes) above a submerged horizontal cylinder with a sufficiently small radius in a channel containing a homogeneous inviscid, incompressible liquid was first established by Ursell (1951) and later generalized to cylinders of any cross section and to fluid domains with both infinite and finite depth by Jones (1953) and Ursell (1987). In Evans *et al.* (1994), using a standard variational approach, the authors proved the existence of trapped modes in water-wave channels of constant depth containing a vertical, surface-piercing cylinder of uniform cross section which extends throughout the depth and is symmetrically placed with respect to the centreline of the channel.

Trapped modes are non-trivial solutions (eigenfunctions) of a spectral problem. The solutions established by Ursell (1951, 1987), Jones (1953) and Evans *et al.* (1994) all correspond to eigenvalues in the discrete spectrum of the problem. For a long time it was thought that trapped modes would not exist if the discrete spectrum is empty as in the two-dimensional case. This idea was based on the uniqueness condition (non-existence of non-trivial solutions) proved by John (1950) stating that the solution is unique if any vertical line drawn downward from the free surface does not intersect the surface-piercing or any other submerged body. The class of bodies for which uniqueness can be established was later widened by Simon & Ursell (1984), see also Kuznetsov *et al.* (1998). However, McIver (1996) disproved this conjecture showing the existence of trapped modes for a pair of surface-piercing bodies. The trapped modes she found are waves travelling between the two cylinders and cancelling each other at infinity and correspond to modes embedded in the continuous spectrum of the spectral problem. For extensions of this two-dimensional result, see Kuznetsov *et al.* (1998) and Motygin (1999) among others. The problem of existence of trapped modes has also been addressed numerically, cf. McIver & Evans (1985) and Porter & Evans (1998). Most of these results, along with many generalizations and numerous other references, can be found in Kuznetsov *et al.* (2002), see also Linton & McIver (2007).

Despite their obvious interest as a step towards more realistic, stratified fluids, layered fluid models have rarely been considered for the problem of existence or non-existence of trapped modes around submerged obstacles. The layered models bring about interfaces between the fluid layers that act like the free surface in guiding (internal) waves but if the constant-density fluids are assumed to be immiscible and gravitationally stable (a lighter fluid above a heavier one), the linear water-wave theory can be applied layer-wise as in the homogeneous case. Even the simple two-layer model is often used in geophysics in modelling large-scale atmospheric and oceanic flows with shallow-water dynamics and is a significant model for estuarine dynamics.

The first results about trapped modes in a two-layer fluid were obtained by Kuznetsov (1993) when he considered the existence of trapped modes above a submerged cylinder in the lower layer. Using the density difference as a small parameter in a formal perturbation analysis and reducing the equations to a problem in the lower layer, he studied the existence of trapped modes both on the free surface and at the interface between the two liquid layers. Later, Linton & Cadby (2003) computed the trapped mode frequencies for a circular, horizontal cylinder submerged either in the upper layer or in the lower layer. They also considered the case of a pair of identical, submerged, circular cylinders in the lower layer and predicted the existence of trapped modes embedded in the continuous spectrum. In Kuznetsov *et al.* (2003), the authors addressed the question of uniqueness and gave examples of two-dimensional structures supporting trapped modes. The problem of wave scattering by submerged obstacles in a two-layer fluid has been studied more frequently (Linton & McIver 1995; Sturova 1999; Cadby & Linton 2000; Linton & Cadby 2002).

The trapped modes we are interested in correspond to positive eigenvalues below the cut-off value of the continuous spectrum, i.e. they belong to the discrete spectrum of the spectral problem modelling the interaction of linear water waves with fixed obstacles totally or partially submerged in a two-layer fluid. Above the cut-off value one can consider the scattering problem but below this value all waves are non-radiating. Our main result is a sufficient condition for the existence of a trapped mode and is based on a standard, although seemingly unpublished, variational formulation of the spectral problem. By restricting the solutions of the variational problem to the free surface and the interface, we introduce a surface–interface trace mapping and prove that under certain conditions its discrete spectrum is non-empty and thus a trapped mode exists. This simply formulated geometrical condition relates the surface integrals of the cross sections of the submerged parts of the bodies and the line integrals taken over the parts of the free surface and the interface pierced by the bodies to the cut-off value.

The main ideas of this work can be traced back to Kamotskii & Nazarov (1999, 2003), who proposed a similar approach for studying localized elastic and electromagnetic waves (see also Kamotskii 2008), based on the theory of Birman–Krein–Vishik (Alonso & Simon 1980; Birman & Solomjak 1987), describing the spectral characteristics of semi-bounded, self-adjoint operators by closed, quadratic forms. The general theory does not, however, apply when the spectral parameter appears in the boundary conditions as in the linear water-wave problem. A pivotal modification of the approach was recently suggested by Nazarov (in press), see also Nazarov (2009), in the homogeneous single-layer case for two- and three-dimensional problems. This modification approach greatly simplifies the approach so that only the basic properties of continuous self-adjoint operators and their spectra are necessary for deriving a sufficient condition for the existence of trapped modes.

The paper is organized as follows. In §2, we formulate the problem, introduce our notation and determine the cut-off value and the corresponding eigensolution/eigenvalue pair by considering the (model) problem without obstacles. The variational formulation and the trace operator are introduced in a suitable Hilbert space in §3. The trace operator is positive, continuous and self-adjoint, and its spectrum is known to lie on the segment [0,* τ*] of the real axis with

*τ*denoting its norm. The condition that guarantees the existence of a trapped mode arises from a lower bound for

*τ*for which the discrete spectrum of the trace operator is non-empty. In §4 we list some general conclusions that can be drawn from the condition and consider a few particular cases. We also present a new and simple proof for the comparison principle, a method often used in proving existence of trapped modes (Jones 1953; Ursell 1987; Motygin 2008).

## 2. Formulation of the problem

Consider two homogeneous, incompressible, inviscid liquids lying on top of one another. Assume, for gravitational stability, that the constant density in the lower layer is greater than the one in the upper layer, i.e. *ρ*_{2}>*ρ*_{1}>0. Suppose, furthermore, that the motion is represented in a non-rotating frame, is generated from an irrotational initial state and that the fluids are immiscible. Hence, according to Kelvin’s circulation theorem, the motion remains irrotational in each fluid layer at all times.

We are interested in the interaction of the water waves with fixed, rigid bodies. The bodies can be submerged within only one of the layers or pierce the free surface and/or the interface. We will assume that the amplitude of the oscillation is small enough and the water depth large enough for the linear water-wave theory to apply (cf. Kuznetsov *et al.* 2002; Kundu & Cohen 2004), and that the wave frequency is much larger than the Coriolis frequency.

We fix the Cartesian coordinates with origin in the mean level of the interface between the infinite fluid layers and assume that the fluid domain has infinite depth and that the obstacles are fixed, rigid, cylindrical bodies, infinite in the *y*-direction and with bounded, constant cross sections in the (*x*,*z*)-plane. The union of the cross sections (or parts of them) lying within the upper layer is denoted by *ω*^{1}, within the lower layer by *ω*^{2}, and the union of their boundaries by *∂**ω*^{1} and *∂**ω*^{2}, respectively. The union of the segments of the free surface pierced by the bodies is denoted by *γ*^{1} and the union of the pierced interface segments by *γ*^{2} (figure 1*a*).

Since the motion is irrotational we may define three-dimensional velocity potentials, *Ψ*^{1}=*Ψ*^{1}(*x*,*y*,*z*,*t*) and *Ψ*^{2}=*Ψ*^{2}(*x*,*y*,*z*,*t*), in the upper and lower layers both satisfying the Laplace equation in their respective domains. The wave motion is assumed to be time harmonic, with non-zero wave frequency *ω*, and periodic in the *y*-direction with non-zero wave number *l*. Writing the velocity potentials as:
where both *ω* and *l* are considered real, and assuming constant ambient pressure at the free surface, continuity of the vertical velocity and pressure at the interface and small amplitude motion, we obtain a two-dimensional spectral problem for the eigenpair (*ϕ*,* Λ*)=((

*ϕ*

^{1},

*ϕ*

^{2}),

*) consisting of the modified Helmholtz equations, 2.1 the linearized (spectral) boundary condition at the free surface, 2.2 the linearized transmission conditions at the interface, 2.3 and of the Neumann boundary conditions (no normal flow) on the surface of the rigid bodies 2.4 Here,*

*Λ**Λ*=

*ω*

^{2}/

*g*is a spectral parameter,

*ρ*=

*ρ*

_{1}/

*ρ*

_{2}, ∇=(

*∂*

_{x},

*∂*

_{z}),

*ϕ*

_{z}=

*∂*

_{z}

*ϕ*and

*ϕ*

_{n}=

*∂*

_{n}

*ϕ*with

*n*denoting the outward normal vector. Moreover, where (figure 1

*b*) We assume that

*Ω*

^{1}and

*Ω*

^{2}are Lipschitz domains so that the normal vector is defined almost everywhere on

*∂*

*Ω*

^{1}and

*∂*

*Ω*

^{2}, in particular on the wetted part of the surfaces

*Σ*

^{1}and

*Σ*

^{2}. Equations (2.1)–(2.4) have been scaled by the constant densities in a way that leads to the most convenient form for our variational formulation. Note that, according to the linear theory and small amplitude motion, the interface at

*z*=0 as well as the free surface at

*z*=

*d*can be taken flat.

The trapped modes are non-trivial solutions to problems (2.1)–(2.4) such that the motion decays at infinity, i.e.
and correspond to guided waves trapped near the bodies and travelling along them in the *y*-direction.

### (a) The model problem

Consider the model problem of the wave motion in the absence of bodies, i.e. *ω*^{1}=*ω*^{2}=∅. A solution *ϕ*=(*ϕ*^{1},*ϕ*^{2}) of the form:
where *k*>0 is the wave number in the *z*-direction and solves the Helmholtz equations (2.1) in . The free-surface boundary condition (2.2) on *Υ*^{1} and the interface boundary conditions (2.3) on *Υ*^{2} yield the dispersion relation (Lamb 1932)
This quadratic equation for *Λ* has two real roots
where
It is easy to see that *Λ*_{1}(*k*)<*Λ*_{2}(*k*) for all positive *k*. Moreover, there is a cut-off value for *Λ*_{1}(*k*) at *k*=*l* below which no waves can propagate to infinity. In other words, it is the cut-off value for the continuous spectrum of the spectral problem (2.1)–(2.4) consisting of values of *Λ* for which there exist eigensolutions having infinite energy per unit length in the *y*-direction (Jones 1953; Kuznetsov *et al.* 2002). The cut-off value will be denoted by
The non-trivial solution corresponding to *Λ*_{†} may be written, up to a multiplication by an arbitrary non-zero constant, as
2.5

Note that (*ϕ*^{1},*ϕ*^{2}) solves the equations
with the boundary conditions
Therefore, integrating by parts, one readily obtains
2.6
where
2.7

## 3. Operator formulation and the sufficient condition

### (a) Variational formulation

Let us multiply the first equation in (2.1) by a test function and the second equation in (2.1) by , where denotes, as usual, the space of infinitely differentiable functions with compact support in *Ω*. Integrating by parts over *Ω*^{1} and *Ω*^{2}, taking into account the boundary conditions (2.2)–(2.4) and summing the resulting equations, we obtain the following variational formulation for problems (2.1)–(2.4):

Find a non-trivial *ϕ*=(*ϕ*^{1},*ϕ*^{2})∈*H*^{1}(*Ω*^{1})×*H*^{1}(*Ω*^{2}) and such that
3.1
for all *Ψ*=(*Ψ*^{1},*Ψ*^{2})∈*H*^{1}(*Ω*^{1})×*H*^{1}(*Ω*^{2}). On the other hand, if *ϕ*^{1} and *ϕ*^{2} are smooth enough, it is a standard matter to show, cf. Evans (1998), that any solution to problem (3.1) is also a classical solution solving problems (2.1)–(2.4).

### (b) Trace operator and its spectrum

Let *H* be the function space consisting of elements *ϕ*=(*ϕ*^{1},*ϕ*^{2})∈*H*^{1}(*Ω*^{1})×*H*^{1}(*Ω*^{2}) and equipped with the scalar product
where (⋅,⋅)_{Ω} denote the integrals on the first line in equation (3.1). In the Hilbert space *H*, we introduce the operator *T* by the formula
Thanks to the trace inequality in a Lipschitz domain, cf. Evans & Gariepy (1992); note that *∂**Ω*^{1} and *∂**Ω*^{2} are straight lines at large distances;
the operator is continuous. Besides, it is obviously positive and symmetric and, thus, self-adjoint; note that 0<*ρ*<1.

The spectral problems (2.1)–(2.4) can now be written with the help of operator *T*, as
where *μ*=*Λ*^{−1} is a new spectral parameter. The only exception is the point *μ*=0, which corresponds to and does not influence the spectrum of problems (2.1)–(2.4). The eigenvalue *μ*=0 of *T* has infinite multiplicity and, observing that
the associated eigenspace becomes
and has infinite dimension.

The continuous spectrum of *T* is inherited from the continuous spectrum of the original problem which lies on . The continuous spectrum of *T* is thus and the essential spectrum results from adding the eigenvalue *μ*=0 of infinite multiplicity. Since the operator *T* is positive, continuous and symmetric, therefore, self-adjoint, its spectrum belongs to the segment [0,*τ*] of the real axis in the complex plane with *τ* denoting the operator norm of *T* and *Σ*_{d}(*T*) its discrete spectrum (Birman & Solomjak 1987). As for the discrete spectrum, there are two possibilities (figure 2*a*,*b*).

(i) The norm

*τ*of the operator*T*coincides with*μ*_{†}and, thus, its discrete spectrum is empty;(ii) The norm

*τ*is strictly greater than*μ*_{†}and since*τ*∈*Σ*(*T*), cf. Birman & Solomjak (1987), the discrete spectrum is definitely non-empty;*τ*∈*Σ*_{d}(*T*).

### (c) A sufficient condition for the existence of a trapped mode

Consider a trial function defined by
where (*ϕ*^{1},*ϕ*^{2}) are taken from equation (2.5) and ϵ≪1 is a small positive parameter. It follows that
where
and wherein the compact sets (*i*=1,2) we have used the approximation *e*^{−ϵ|x|}=1+*O*(ϵ).

Similarly, we obtain
where
Note that owing to equation (2.6), we have *B*=*Λ*_{†}*A*. We thus conclude that, with certain positive constants *c*_{p} it holds
where *P*=*P*_{1}+*P*_{2} and *V* =*V*_{1}+*V*_{2}. This implies that if the condition
3.2
is satisfied, then there exists a small ϵ>0 such that . Hence, equation (3.2) guarantees that the discrete spectrum of *T* is non-empty and is a sufficient condition for the existence of a trapped mode.

Substituting *ϕ*_{1} and *ϕ*_{2} in the formulae for *A*,*P* and *V* we obtain the expressions, cf. equation (2.7),
and the condition (3.2) becomes

### (d) Non-necessity of the sufficient condition

Condition (3.2) is not necessary. To see this, assume that the cross section of a body occupies the region depicted in figure 3. Then *P*_{1}=*ρ**L*,*P*_{2}=0 and it is obvious that for any *L*>0 there exists *h*>0, small enough, such that *V*_{1}+*V*_{2}−*Λ*_{†}(*P*_{1}+*P*_{2})<0, that is, condition (3.2) is violated. On the other hand, in Nazarov (2008) it was shown that for any δ>0 and , there exists *h*_{0}=*h*_{0}(δ,*N*)>0 such that for *h*∈(0,*h*_{0}(δ,*N*)) the spectral problem admits at least *N* linearly independent trapped modes corresponding to *N* eigenvalues *Λ*∈(0,δ). Even though the fluid domain in Nazarov (2008) was not exactly the same as here, the result being based only on a local analysis is clearly valid in our case.

### (e) Existence of trapped modes when *V*_{1}+*V*_{2}=*Λ*_{†}(*P*_{1}+*P*_{2})

Assume that
3.3
The sufficient condition (3.2) is not satisfied but we can still prove the existence of a trapped mode in most situations. To fix ideas and to simplify the presentation, consider a two-dimensional body (an infinitely long thin blade) whose cross-sectional area is a line segment in the upper layer as depicted in figure 4. We have *V*_{1}=*V*_{2}=*P*_{1}=*P*_{2}=0 so that equation (3.3) is valid. Following Kamotski & Nazarov (2003), who examined the existence of trapped modes in acoustic waveguides, we will examine the existence of trapped modes by redefining the trial functions and .

Assume, for the sake of contradiction, that the discrete spectrum of *T* is empty, that is . Hence
3.4
Now consider the trial function defined by
where *Ψ*^{1} is a smooth function with compact support in the neighbourhood of the obstacle. Repeating the calculations of the previous section, we obtain
where *ϕ*=(*ϕ*^{1},*ϕ*^{2}) and *Ψ*=(*Ψ*^{1},0). Therefore, owing to equation (3.3) and since *Λ*_{†}*A*=*B*, it follows from equation (3.4) that
Since (*ϕ*^{1},*ϕ*^{2}) solves the model problem in , integrating by parts yields
3.5
where the sum over ± indicates that the integrals are calculated on both faces of the blade. If *∂*_{n}*ϕ*^{1} does not vanish almost everywhere, that is, if the blade is not vertical, then we may fix *Ψ*^{1} in such a way that . This means that, for ϵ>0 small enough, equation (3.5) is violated which is a contradiction. Thus, equation (3.4) is not true and, consequently, the discrete spectrum of *T* is non-empty so that a trapped mode exists.

The same argument is obviously valid in a much more general situation, with bodies intersecting with the free surface and the interface or not, as long as equation (3.4) holds true and the bodies have a piecewise smooth boundary. As for the vertical blade, we refer to the uniqueness example, non-existence of trapped modes, presented in Kuznetsov *et al.* (2003).

## 4. Particular cases

### (a) A submerged body (or bodies) touching neither the free surface nor the interface

In this case *P*_{1}=*P*_{2}=0 and *V* >0; thus the condition (3.2) is trivially satisfied for any non-empty union of bodies of arbitrary shape. Recall that this result is known to be true for a homogeneous (one-layer) fluid, cf. Ursell (1951, 1958), Jones (1953), but in a two-layer fluid it has been established only if the density difference between the fluid layers is sufficiently small, cf. Kuznetsov (1993, 1995).

### (b) A body in the upper layer piercing the surface

Consider first the model case where the body has constant width, say *L*, and occupies the entire depth. We have
Note that, in view of equation (2.6), it holds
4.1
On the other hand, one readily sees that
and hence owing to equation (4.1)
4.2

Now, in the case depicted in figure 5, we have so that for sufficiently small *h*>0 it still holds
and a trapped mode exists. This also means that John’s uniqueness result, cf. John (1950), Simon & Ursell (1984), cannot be valid in the two-fluid case if the interface is not pierced.

### (c) A submerged body piercing the interface

Assuming again for simplicity that the body has constant width *L*, inequality (4.2) implies that if condition (3.2) is to be satisfied, a sufficiently large part of the body has be to located in the upper layer. Hence, if
where , and meas(*γ*^{2})=*L* (figure 6), then it is easy to see that the sufficient condition (3.2) is satisfied for sufficiently small *h*>0 and sufficiently large *H*>0.

### (d) Steep cliff

Consider the case where the fluid domain is bounded by a vertical wall (a steep cliff) at *x*=0. The slope of the cliff is defined by the curve *x*=−*H*(*z*), where *H* is a piece-wise smooth function without a jump at *z*=0 and such that *H*(*d*)=0. The fluid layers and their surfaces in the (*x*,*z*)-plane are defined (without bodies) by
In the new fluid domain, the spectral problems (2.1)–(2.4) look the same except for the Neumann boundary conditions on the cliff (figure 7):
The variational and the operator formulations of the problem are not changed and neither is the dispersion relation nor the cut-off value *Λ*_{†}. The functions *ϕ*^{1}(*z*) and *ϕ*^{2}(*z*) can be defined as previously. However, the sufficient condition is slightly altered. Indeed, using the trial function with
where 0<ϵ≪1, we get
where
Similarly, we obtain
where
and the condition that guarantees the existence of a trapped mode reads as:
4.3
Note that if *H*(0)<0, then *P*_{3} is negative and this term can actually help in establishing the existence of a trapped mode. The terms *V*_{3} and *V*_{4} on the other hand can be negative and thus have an opposite effect. For example, recalling equation (4.2), we easily see that if *H*(*z*)=0 for *z*>*h* and for *z*<*h* (figure 8), condition (3.2) is met and a trapped mode exists.

### (e) The limit case

Letting (formally) in equations (2.1)–(2.4), the flow fluid reduces to a single homogeneous layer of finite depth with a flat bottom at *z*=0. Hence, *ϕ*^{2}≡0 and (*ϕ*^{1},*Λ*) satisfies the problem
The dispersion relation corresponding to the model problem in the absence of bodies has the root
and the cut-off value is
Moreover, *ϕ*^{1}(*z*)=e^{lz}+e^{−lz} and repeating the calculations of §3*c* one easily shows that the condition that secures the existence of a trapped mode is
where
Some immediate conclusions can be drawn.

(i) Any totally submerged body creates a trapped mode. In particular, an underwater ridge that can be understood as a submerged body that touches the ‘bottom’ at

*z*=0 creates a trapped mode, a well-known result first proved for a symmetric ridge by Jones (1953) and later generalized to any submerged ridge by Garipov (1967).(ii) In general, if the condition is satisfied, there exists a trapped mode.

(iii) If we consider the flow in a semi-infinite channel of finite depth where

*H*(*z*)≥0 for*z*∈[0,*d*], with*H*(*d*)=0, and repeat the calculations leading to condition (4.3), then meas(*γ*^{1})=0 and , thus trapped modes always exist. This situation corresponds to a sloping beach, for related results see Bonnet-Ben Dhia & Joly (1993).

### (f) The comparison principle

Let us compare two systems of bodies *ω*^{1}, *ω*^{2}, such that
In other words, the unions of bodies are larger in cross-sectional area than *ω*^{p} but they cover the same segments at the surface and at the interface. As before, we introduce a Hilbert space *H*_{•} equipped with the scalar product 〈⋅,⋅〉_{•} and define an operator *T*_{•} in *H*_{•} with the norm *τ*_{•}. Since the fluid domains *ω*^{p} and of the two problems differ only within a compact set, the essential spectrum of *T*_{•} is also [0,*μ*_{†}], i.e. the cut-off value *μ*_{†}>0 is the same for both problems.

Assume that the norm of the operator *T* is strictly greater than the cut-off value, *τ*>*μ*_{†}, and denote by the eigenfunction of *T* corresponding to the eigenvalue *μ*_{1}=*τ*, i.e.
Since and , we have *ϕ*_{1}∈*H*_{•} and
On the other hand, as it holds
because the function *ϕ*_{1} cannot vanish on a set of positive area. As for the norm *τ*_{•}, we can now derive the lower bound
From here it follows that the discrete spectrum of *T*_{•} is also non-empty and
4.4
In view of the max–min principle (e.g. Birman & Solomjak (1987), theorem 10.2.2), one readily shows that the total multiplicity of the discrete spectrum of operator *T*_{•} is not less than the one of *T* and that, moreover, the other eigenvalues in their discrete spectra, if enumerated in the increasing order, are also related as the first ones in equation (4.4).

Arguments of this kind are called comparison principles. For a homogeneous fluid they have been proved and exploited, e.g. Jones (1953), Ursell (1987) and Motygin (2008). However, the proof presented above is much simpler than any of the earlier ones.

## Acknowledgements

The first author is grateful for the financial support of RFFI (grant no. 07-01-00476).

## Footnotes

- Received May 25, 2009.
- Accepted August 21, 2009.

- © 2009 The Royal Society