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Proc. R. Soc. A (2009) 465, 3799–3816
doi:10.1098/rspa.2009.0281
Published online 25 September 2009
A sufficient condition for the existence
of trapped modes for oblique waves in
a two-layer fluid
BY SERGEY A. NAZAROV1
AND
JUHA H. VIDEMAN2, *
1 Institute
of Mechanical Engineering Problems, Russian Academy
of Sciences, V.O., Bol’shoi pr., 61, 199178 St. Petersburg, Russia
2 CEMAT/Departamento de Matemática, Instituto Superior Técnico,
Av. Rovisco Pais 1, 1049-001 Lisboa, Portugal
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.
Keywords: water waves; trapped modes; sufficient condition; submerged obstacles;
spectral problem
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
*Author for correspondence ([email protected]).
Received 25 May 2009
Accepted 21 August 2009
3799
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S. A. Nazarov and J. H. Videman
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).
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Trapped modes in a two-layer fluid
3801
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 waterwave 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. ρ2 > ρ1 > 0. Suppose,
furthermore, that the motion is represented in a non-rotating frame, is generated
Proc. R. Soc. A (2009)
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3802
S. A. Nazarov and J. H. Videman
(a)
(b)
Figure 1. The cross section of the fluid domain with (a) and without (b) the obstacles.
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 ω1 , within the lower layer by ω2 , and the union of their boundaries by
∂ω1 and ∂ω2 , 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 1a).
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:
Ψ j (x, y, z, t) = Re{φ j (x, z)ei(ly−ωt) },
j = 1, 2,
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,
a two-dimensional spectral problem
we obtain
for the eigenpair (φ, Λ) = (φ 1 , φ 2 ), Λ consisting of the modified Helmholtz
equations,
ρ(∇ 2 − l 2 ) φ 1 = 0
in Ω 1
and
(∇ 2 − l 2 )φ 2 = 0
in Ω 2 ,
(2.1)
the linearized (spectral) boundary condition at the free surface,
ρφz1 = ρΛφ 1
Proc. R. Soc. A (2009)
on Γ 1 ,
(2.2)
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3803
Trapped modes in a two-layer fluid
the linearized transmission conditions at the interface,
ρ(φz1 − Λφ 1 ) = φz2 − Λφ 2
and
φz1 = φz2
on Γ 2
(2.3)
and of the Neumann boundary conditions (no normal flow) on the surface of the
rigid bodies
(2.4)
ρφn1 = 0 on Σ 1 and φn2 = 0 on Σ 2 .
Here, Λ = ω2 /g is a spectral parameter, ρ = ρ1 /ρ2 , ∇ = (∂x , ∂z ), φz = ∂z φ and
φn = ∂n φ with n denoting the outward normal vector. Moreover,
Ω 1 = Π 1 \ ω1 ,
Ω 2 = Π 2 \ ω2 ,
Σ 1 = ∂ω1 \ (γ 1 ∪ γ 2 )
and
Γ 1 = Υ 1 \ γ 1,
Γ 2 = Υ 2 \ γ 2,
Σ 2 = ∂ω2 \ γ 2 ,
where (figure 1b)
Π 1 = {(x, z) : −∞ < x < ∞, 0 < z < d} and
Π 2 = {(x, z) : −∞<x<∞, −∞<z<0}
and
Υ 1 = {(x, z) : −∞ < x < ∞, z = d},
Υ 2 = {(x, z) : −∞<x<∞, z = 0}.
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.
φ 1 , |∇φ 1 | → 0 when |x| → ∞
and
φ 2 , |∇φ 2 | → 0 when |x| → ∞ or z → −∞,
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.
ω1 = ω2 = ∅. A solution φ = (φ 1 , φ 2 ) of the form:
√
√
√
1
±i k 2 −l 2 x kz
±i k 2 −l 2 x −kz
2
±i k 2 −l 2 x kz
φ (x, z) = Ae
e + Be
e
and φ (x, z) = C e
e ,
where k > 0 is the wave number in the z-direction and A, B, C ∈ C solves the
Helmholtz equations (2.1) in Π 1 ∪ Π 2 . 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)
(k − Λ) Λ coth(dk) + Λρ − (1 − ρ)k = 0.
This quadratic equation for Λ has two real roots
Λ1 = k
where
σ=
Proc. R. Soc. A (2009)
1 − e−2dk
,
σ + e−2dk
Λ2 = k,
ρ2 + ρ1 1 + ρ
.
=
ρ2 − ρ 1 1 − ρ
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3804
S. A. Nazarov and J. H. Videman
It is easy to see that Λ1 (k) < Λ2 (k) for all positive k. Moreover, there is a cut-off
value for Λ1 (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
Λ† := Λ1 (l) = l
1 − e−2dl
.
σ + e−2dl
The non-trivial solution corresponding to Λ† may be written, up to a
multiplication by an arbitrary non-zero constant, as
Φ 1 (z) = (elz + (1 − ρ + ρe2dl )e−lz ),
Φ 2 (z) = ρ(1 − e2dl )elz ,
0 < z < d,
(2.5)
−∞ < z < 0.
Note that (Φ 1 , Φ 2 ) solves the equations
ρ∂z2 Φ 1 (z) − ρl 2 Φ 1 (z) = 0,
0 < z < d;
and
∂z2 Φ 2 (z) − l 2 Φ 2 (z) = 0,
−∞ < z < 0,
with the boundary conditions
ρΦz1 (d) = ρΛ† Φ 1 (d),
ρΦz1 (0) − ρΛ† Φ 1 (0) = Φz2 (0) − Λ† Φ 2 (0) and
Φz1 (0) = Φz2 (0).
Therefore, integrating by parts, one readily obtains
0
d
1
2
2
1
2
(|∂z Φ 2 (z)|2 + l 2 |Φ 2 (z)|2 ) dz
ρ (|∂z Φ (z)| + l |Φ (z)| ) dz +
0
−∞
= Λ†
1
2
1
2
(Φ (0) − ρΦ (0)) ,
ρΦ (d) +
1−ρ
1
2
(2.6)
where
Φ 1 (d)2 = (1 − ρ)2 e2dl (σ + e−2dl )2 ,
1
(Φ 2 (0) − ρΦ 1 (0))2 = ρ 2 (1 − ρ)e4dl (σ + e−2dl )2 .
1−ρ
(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 ψ 1 ∈ Cc∞ (Ω 1 ) and the
second equation in (2.1) by ψ 2 ∈ Cc∞ (Ω 2 ), where Cc∞ (Ω) 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):
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3805
Trapped modes in a two-layer fluid
Find a non-trivial φ = (φ 1 , φ 2 ) ∈ H 1 (Ω 1 ) × H 1 (Ω 2 ) and Λ ∈ C such that
ρ
∇φ 1 · ∇ψ 1 dΩ + ρl 2
φ 1 ψ 1 dΩ +
∇φ 1 · ∇ψ 1 dΩ + l 2
φ 1 ψ 1 dΩ
1
1
2
2
Ω
Ω
Ω
Ω
1
φ 1 ψ 1 dΓ +
(φ 2 − ρφ 1 )(ψ 2 − ρψ 1 ) dΓ ,
(3.1)
=Λ ρ
1
2
1
−
ρ
Γ
Γ
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 (Ω 2 ) and equipped with the scalar product
1
φ, ψ
= ρ(∇φ 1 , ∇ψ 1 )Ω 1 + ρl 2 (φ 1 , ψ 1 )Ω 1 + (∇φ 2 , ∇ψ 2 )Ω 2
+ l 2 (φ 2 , ψ 2 )Ω 2 ,
φ, ψ ∈ H ,
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
T φ, ψ
= ρ(φ 1 , ψ 1 )Γ 1 +
1
(φ 2 − ρφ 1 , ψ 2 − ρψ 1 )Γ 2 ,
1−ρ
φ, ψ ∈ H .
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;
φ i ; L2 (∂Ω i ) ≤ cφ i ; H 1 (Ω i ),
i = 1, 2,
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
T φ = μφ,
where μ = Λ−1 is a new spectral parameter. The only exception is the point μ = 0,
which corresponds to Λ = ∞ and does not influence the spectrum of problems
(2.1)–(2.4). The eigenvalue μ = 0 of T has infinite multiplicity and, observing that
Tφ = 0
⇐⇒
ρφ 1 ; L2 (Γ 1 ) +
1
ρφ 1 − φ 2 ; L2 (Γ 2 ) = 0,
1−ρ
the associated eigenspace becomes
H 0 = {φ ∈ H : φ 1 = 0 on Γ 1 , ρ φ 1 = φ 2 on Γ 2 }
and has infinite dimension.
The continuous spectrum of T is inherited from the continuous spectrum
of the original problem which lies on [Λ† , ∞). The continuous spectrum of
T is thus σc (T ) = {μ : μ ∈ (0, Λ−1
† ]} and the essential spectrum σe (T ) = {μ : μ ∈
−1
[0, Λ† ]} results from adding the eigenvalue μ = 0 of infinite multiplicity. Since
the operator T is positive, continuous and symmetric, therefore, self-adjoint, its
Proc. R. Soc. A (2009)
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3806
S. A. Nazarov and J. H. Videman
(a)
(b)
τ = μ†
µ
0
τ
μ†
0
µ
Figure 2. The essential spectrum of the operator T .
spectrum σ (T ) = σe (T ) ∪ σd (T ) belongs to the segment [0, τ ] of the real axis in
the complex plane with τ denoting the operator norm of T and σd (T ) its discrete
spectrum (Birman & Solomjak 1987). As for the discrete spectrum, there are two
possibilities (figure 2a,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 ∈ σd (T ).
(c) A sufficient condition for the existence of a trapped mode
Consider a trial function φ = (φ1 , φ2 ) ∈ H defined by
φ1 (x, z) = e−|x| Φ 1 (z)
and
φ2 (x, z) = e−|x| Φ 2 (z),
where (Φ 1 , Φ 2 ) are taken from equation (2.5) and 1 is a small positive
parameter. It follows that
T φ , φ = ρφ1 ; L2 (Γ 1 )2 + (1 − ρ)−1 φ2 − ρφ1 ; L2 (Γ 2 )2
= ρφ1 ; L2 (Υ 1 )2 + (1 − ρ)−1 φ2 − ρφ1 ; L2 (Υ 2 )2
− ρφ1 ; L2 (γ 1 )2 − (1 − ρ)−1 φ2 − ρφ1 ; L2 (γ 2 )2
1
= A − P1 − P2 + O(),
where
A = ρΦ 1 (d)2 + (1 − ρ)−1 (Φ 2 (0) − ρΦ 1 (0))2 ,
P1 = ρΦ 1 (d)2 meas(γ 1 ),
P2 = (1 − ρ)−1 (Φ 2 (0) − ρΦ 1 (0))2 meas(γ 2 ),
and wherein the compact sets ωi (i = 1, 2) we have used the approximation
e−|x| = 1 + O().
Similarly, we obtain
φ , φ = ρ∇φ1 ; L2 (Ω 1 )2 + ρl 2 φ1 ; L2 (Ω 1 )2 + ∇φ2 ; L2 (Ω 2 )2 + l 2 φ2 ; L2 (Ω 2 )2
= ρ∇φ1 ; L2 (Π 1 )2 + ρl 2 φ1 ; L2 (Π 1 )2 + ∇φ2 ; L2 (Π 2 )2 + l 2 φ2 ; L2 (Π 2 )2
− ρ∇φ1 ; L2 (ω1 )2 − ρl 2 φ1 ; L2 (ω1 )2 − ∇φ2 ; L2 (ω2 )2 − l 2 φ2 ; L2 (ω2 )2
1
= B − V1 − V2 + O(),
Proc. R. Soc. A (2009)
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Trapped modes in a two-layer fluid
3807
where
d
B =ρ
(|Φz1 (z)|2
V1 = ρ
V2 =
ω2
+ l |Φ (z)| ) dz +
2
1
0
2
0
ω1
−∞
(|Φz2 (z)|2 + l 2 |Φ 2 (z)|2 ) dz,
(|Φz1 (z)|2 + l 2 |Φ 1 (z)|2 ) dx dz
(|Φz2 (z)|2 + l 2 |Φ 2 (z)|2 ) dx dz.
Note that owing to equation (2.6), we have B = Λ† A. We thus conclude that, with
certain positive constants cp it holds
T φ, φ
T φ , φ ≥
φ , φ φ∈H \{0} φ, φ
τ = sup
≥
1 1 − A−1 P − c2 2
−1 A − P − c1 ≥
−1
2
−1 Λ† A − V + c1 Λ† 1 − Λ−1
† A V + c2 ≥
1
−1
2
(1 − A−1 P − c2 2 )(1 + Λ−1
† A V − c2 )
Λ†
≥
1 2
1 + A−1 (Λ−1
† V − P) − c3 ,
Λ†
where P = P1 + P2 and V = V1 + V2 . This implies that if the condition
V1 + V2 − Λ† (P1 + P2 ) > 0
(3.2)
is satisfied, then there exists a small > 0 such that τ > Λ−1
† . 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 Φ1 and Φ2 in the formulae for A, P and V we obtain the
expressions, cf. equation (2.7),
A = ρ(1 − ρ)e2ld (σ + e−2ld )2 (1 − ρ + ρ e2ld ),
P1 = ρ(1 − ρ)2 e2ld (σ + e−2ld )2 meas(γ 1 ),
P2 = ρ 2 (1 − ρ)e4ld (σ + e−2ld )2 meas(γ 2 ),
2
V1 = 2ρl
e−2lz e4lz + (1 − ρ)2 + 2ρ(1 − ρ)e2ld + ρ 2 e4ld dx dz,
ω1
V2 = 2ρ l (e
2 2
Proc. R. Soc. A (2009)
2ld
− 1)
2
e2lz dx dz,
ω2
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3808
S. A. Nazarov and J. H. Videman
z
0
L/2
L
d–h
d–2h
Figure 3. A body producing trapped modes and violating the sufficient condition.
and the condition (3.2) becomes
2ρl
e−2lz e4lz + (1 − ρ)2 + 2ρ(1 − ρ)e2ld + ρ 2 e4ld dx dz
ω1
2
2ld
2
+ 2ρ l(e − 1)
e2lz dx dz
ω2
− ρ(1 − ρ)(e
2ld
− 1)(σ + e−2ld ) (1 − ρ)meas(γ 1 ) + ρe2ld meas(γ 2 ) > 0.
(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 P1 = ρL, P2 = 0 and
it is obvious that for any L > 0 there exists h > 0, small enough, such that
V1 + V2 − Λ† (P1 + P2 ) < 0, that is, condition (3.2) is violated. On the other
hand, in Nazarov (2008) it was shown that for any δ > 0 and N ∈ N, there exists
h0 = h0 (δ, N ) > 0 such that for h ∈ (0, h0 (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 V1 + V2 = Λ† (P1 + P2 )
Assume that
V1 + V2 = Λ† (P1 + P2 ).
(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 crosssectional area is a line segment in the upper layer as depicted in figure 4. We
have V1 = V2 = P1 = P2 = 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 φ1 and φ2 .
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Trapped modes in a two-layer fluid
3809
z
Σ1–
Σ1+
Figure 4. An infinitely long thin blade.
Assume, for the sake of contradiction, that the discrete spectrum of T is empty,
that is τ = Λ−1
† . Hence
Λ† T φ, φ
≤ φ, φ
,
φ ∈ H.
(3.4)
Now consider the trial function φ = (φ1 , φ2 ) ∈ H defined by
φ1 (x, z) = e−|x| Φ 1 (z) + 1/2 Ψ 1 (x, z),
φ2 (x, z) = e−|x| Φ 2 (z)
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
1
T φ , φ = A − P1 − P2 + 2 1/2 T Φ, Ψ + O(),
1
φ , φ = B − V1 − V2 + 2 1/2 Φ, Ψ + O(),
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
2 1/2 Φ, Ψ − Λ† T Φ, Ψ + O() ≥ 0.
Since (Φ 1 , Φ 2 ) solves the model problem in Π 1 ∪ Π 2 , integrating by parts yields
1/2
Ψ 1 (x, z)∂n Φ 1 (z) ds + O() ≥ 0,
(3.5)
2ρ
±
1
Σ±
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 ± Σ 1 Ψ 1 (x, z)∂n Φ 1 (z) ds < 0.
±
This means that, for > 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).
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3810
S. A. Nazarov and J. H. Videman
h
Figure 5. A surface-piercing body satisfying the sufficient condition.
4. Particular cases
(a) A submerged body (or bodies) touching neither the free surface
nor the interface
In this case P1 = P2 = 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
V10 = ρl(1 − e−2ld ) (1 − ρ)2 + ρ 2 e4ld + (1 + 2ρ − 2ρ 2 )e2ld L,
Λ† P10 = ρ(1 − ρ)2 l(e2ld − 1)(σ + e−2ld )L,
V20 = ρ 2 l(e2ld − 1)2 L
and
Λ† P20 = ρ 2 (1 − ρ)le2ld (e2ld − 1)(σ + e−2ld )L.
Note that, in view of equation (2.6), it holds
V10 + V20 = Λ† (P10 + P20 ).
(4.1)
On the other hand, one readily sees that
V10 − Λ† P10 = ρ 2 l(e2ld − 1)(2 + ρ e2ld − ρ)L > 0,
and hence owing to equation (4.1)
V20 − Λ† P20 < 0.
(4.2)
Now, in the case depicted in figure 5, we have V10 − V1 = O(h) so that for
sufficiently small h > 0 it still holds
V1 − Λ† P1 > 0,
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Trapped modes in a two-layer fluid
z
0
3811
L
d–h
–H
Figure 6. An interface-piercing body satisfying the sufficient condition.
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
ω1 = {(x, z) : 0 < x < L, 0 < z < d − h} and
ω2 = {(x, z) : 0 < x < L, − H < z < 0},
where h, H ∈ R, 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
Π−1 = {(x, z) : −∞ < x < −H (z), 0 < z < d},
Υ−1 = {(x, z) : −∞ < x < −H (z), z = d},
Π−2 = {(x, z) : −∞ < x < −H (z), −∞ < z < 0},
Υ−2 = {(x, z) : −∞ < x < −H (z), z = 0}.
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):
∂n φ 1 (x, z) = 0 at {(x, z) : x = −H (z), 0 < z < d},
∂n φ 2 (x, z) = 0 at {(x, z) : x = −H (z), ∞ < z < 0}.
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
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3812
S. A. Nazarov and J. H. Videman
z
x
x = –H(z)
Figure 7. A fluid domain bounded by a steep cliff.
Φ 1 (z) and Φ 2 (z) can be defined as previously. However, the sufficient condition
is slightly altered. Indeed, using the trial function φ = (φ1 , φ2 ) ∈ H with
φ1 (x, z) = ex Φ 1 (z)
and
φ2 (x, z) = ex Φ 2 (z),
where 0 < 1, we get
1
T φ , φ = A − P1 − P2 − P3 + O(),
where
A = ρΦ 1 (d)2 + (1 − ρ)−1 (Φ 2 (0) − ρΦ 1 (0))2 ,
P1 = ρΦ 1 (d)2 meas(γ 1 ),
P2 = (1 − ρ)−1 (Φ 2 (0) − ρΦ 1 (0))2 meas(γ 2 ),
P3 = H (0)(1 − ρ)−1 (Φ 2 (0) − ρΦ 1 (0))2 .
Similarly, we obtain
1
φ , φ = Λ† A − V1 − V2 − V3 − V4 + O(),
where
V1 = ρ
(|Φz1 (z)|2 + l 2 |Φ 1 (z)|2 ) dx dz,
1
ω
V2 = (|Φz2 (z)|2 + l 2 |Φ 2 (z)|2 ) dx dz,
ω2
V3 = ρ
H (z)(|Φz1 (z)|2 + l 2 |Φ 1 (z)|2 ) dx dz
ω1
V4 =
H (z)(|Φz2 (z)|2 + l 2 |Φ 2 (z)|2 ) dx dz,
ω2
and the condition that guarantees the existence of a trapped mode reads as:
V1 + V2 + V3 + V4 − Λ† (P1 + P2 + P3 ) > 0.
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(4.3)
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Trapped modes in a two-layer fluid
3813
h
Figure 8. A cliff coastline producing a trapped mode.
Note that if H (0) < 0, then P3 is negative and this term can actually help in
establishing the existence of a trapped mode. The terms V3 and V4 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 H (z) = H∞ < 0 for
z < h (figure 8), condition (3.2) is met and a trapped mode exists.
(e) The limit case ρ → 0
Letting (formally) ρ → 0 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
∇ 2 − l 2 φ 1 = 0 in Ω 1
φz1 = Λφ 1 on Γ 1 and φn1 = 0 on Σ 1 ∪ Γ 2 .
The dispersion relation corresponding to the model problem in the absence of
bodies has the root
Λ1 = k tanh(kd)
and the cut-off value is
Λ† = l tanh(ld).
Moreover, Φ 1 (z) = elz + e−lz and repeating the calculations of §3c one easily
shows that the condition that secures the existence of a trapped mode is
−1
A−1 Λ−1
† V1 − A P1 > 0,
where
−1
A
Λ−1
† V1
= 2l(sinh(2ld))
−1
cosh(2lz) dx dz,
ω1
A−1 P1 = meas(γ 1 ).
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).
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S. A. Nazarov and J. H. Videman
(ii) In general, if the condition
cosh(2lz) dx dz − sinh(2ld)meas(γ 1 ) > 0
2l
ω1
is satisfied, there exists a trapped mode.
(iii) If we consider the flow in a semi-infinite channel of finite depth
Π−1 = {(x, z) : −∞ < x < −H (z), 0 < z < d},
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 A−1 Λ−1
† V1 > 0, 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 ω1 , ω2 ω•1 , ω•2 such that
ωp ⊂ ω•p ,
γ p = γ•p ,
p = 1, 2
and
ω1 ∪ ω2 = ω•1 ∪ ω•2 .
In other words, the unions of bodies ω•p are larger in cross-sectional area than ωp
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 Ω•p 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 cutoff value, τ > μ† , and denote by φ1 = (φ11 , φ12 ) ∈ H the eigenfunction of T
corresponding to the eigenvalue μ1 = τ , i.e.
T φ1 , φ1 = τ φ1 , φ1 .
Since H ⊂ H• and γ p = γ•p , we have φ1 ∈ H• and
T• φ1 , φ1 • = T φ1 , φ1 .
On the other hand, as Ω• Ω p it holds
p
φ1 , φ1 • < φ1 , φ1 ,
because the function φ1 cannot vanish on a set of positive area. As for the norm τ• ,
we can now derive the lower bound
T• φ• , φ• • T• φ1 , φ1 • T φ1 , φ1 = τ > μ† .
≥
>
τ• = sup
φ1 , φ1 •
φ1 , φ1 φ• ∈H• \{0} φ• , φ• •
From here it follows that the discrete spectrum of T• is also non-empty and
μ•1 > μ1 .
(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).
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Trapped modes in a two-layer fluid
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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.
The first author is grateful for the financial support of RFFI (grant no. 07-01-00476).
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