Loss of Resolution for the Time Reversal of Waves in Random Underwater Acoustic Channels

0.87 MB·English

by  Christophe Gomez

Loss of Resolution for the Time Reversal of Waves in Random
Underwater Acoustic Channels
Christophe Gomez∗
January 26, 2012
Abstract
Inthispaperweanalyzeatime-reversalexperimentinarandomunderwateracousticchannel.
2
In this kind of waveguide with semi-infinite cross section a propagating field can be decomposed
1
0 overthreekindsofmodes: thepropagatingmodes,theradiatingmodesandtheevanescentmodes.
2 Usinganasymptoticanalysisbasedonaseparationofscalestechniquewederivetheasymptotic
form of the the coupled mode power equation for the propagating modes. This approximation is
n
usedtocomputethetransverseprofileoftherefocusedfieldandshowthatrandominhomogeneities
a
J inside the waveguide deteriorate the spatial refocusing. This result, in an underwater acoustic
channel context, is in contradiction with the classical results about time-reversal experiment in
4
other configurations, for which randomness in the propagation medium enhances the refocusing.
2

] Key words. acoustic waveguides, random media, asymptotic analysis
P
A AMS subject classification. 76B15, 35Q99, 60F05
.
h
t Introduction
a
m
[ Thetime-reversalexperimentsofM.FinkandhisgroupinParishaveattractedconsiderableattention
because of the surprising effect of enhanced spatial focusing and time compression in random media.
1
The refocusing properties have numerous interesting applications, in detection, destruction of kidney
v
stones, and wireless communication for instance. This experiment is in two steps. In the first step
7
5 (see Figure 1 (a)), a source sends a pulse into a medium. The wave propagates and is recorded by
1 a device called a time-reversal mirror. A time-reversal mirror is a device that can receive a signal,
5 recordit,andresendittime-reversedintothemedium. Inthesecondstep(seeFigure1(b)),thewave
.
1 emitted by the time-reversal mirror has the property of refocusing near the original source location.
0 However, surprisingly, it has been observed that random inhomogeneities enhance refocusing [7, 8].
2
Time-reversal refocusing in one-dimensional propagation media is carried out in [6, 10]. In three-
1
dimensional randomly layered media [9], in the paraxial approximation [3, 5, 21], and in random
:
v waveguides[10,12],ithasbeenshownthatthefocalspotcanbesmallerthantheRayleighresolution
i
X formula λL/D (where λ is the carrier wavelength, L is the propagation distance, and D is the mirror
diameter),butthefocalspotisstilllargerthanthediffractionlimitλ/2. Moreover,in[14]theauthor
r
a propose a setup in a waveguide with random perturbations in the vicinity of the source in order to
obtain a superresolution effect, that is to refocus beyond the diffraction limit with a far-field time-
reversalmirror. Thesetupwasinspiredfrom[20]describingthissuperresolutioneffectexperimentally.
Time reversal of a broadband pulse, in the case of a waveguide with a bounded cross-section and
Dirichlet boundary conditions, is carried out in [12] and [10, Chapter 20]. In underwater acoustics
the waveguide model (see Figure 1) has a semi-infinite cross section, and therefore a wave field can
be decomposed into three kinds of modes: the propagating modes, the evanescent modes, and the
radiating modes. Consequently, waveguides with bounded cross-sections does not take into account
radiationlosseswhichcanbeencounteredinunderwateracoustics. Moreover,therefocusingenhance-
ment of the time-reversal experiment in random waveguides is closely related to the propagation of
the propagating modes power in the random medium. For waveguides with bounded cross-sections
∗Department of Mathematics, Stanford University, Building 380, Sloan Hall Stanford, California 94305 USA
([email protected]). Tel: +(1)650-723-1968. Fax: +(1)650-725-4066.
1
Source Ocean Source Ocean
Time−reversal Time−reversal
mirror mirror
Bottom Bottom
(a) (b)
Figure 1: Representation of the time-reversal experiment. In (a) we represent the first step of the
experiment, and in (b) we represent the second step of the experiment.
the total propagating mode power is conserved and even uniformly distributed after long-range prop-
agations[10,12], thatiswhyrefocusingenhancementcanbeobserved. Inunderwateracoustics, with
a far-field time-reversal mirror only the propagating modes contribute to the focal spot, and it has
been shown [15, 17] that the presence of radiating modes produces an effective dissipation on the
propagatingmodes,whichcannotbefullycompensatedbytimereversal. Asaresult,incontrastwith
all the results listed above regarding improvement of the time-reversal focusing thanks to the ran-
dom perturbations in the propagation medium, we show in this paper that random inhomogeneities
in a context of underwater acoustics deteriorate the refocusing property. The deterioration of the
refocusing property for the time-reversal experiment has already been observed in [1]. However, in
there context the authors considered a thermo-viscous wave model to incorporate viscosity effects in
wave propagation. In our context, the deterioration of the refocusing is only due to the random in-
homogeneities and the geometry of the propagation media. Therefore, it is interesting to understand
these effects and to characterize them on the focal spot resulting from the time-reversal experiment.
The main result of this paper is the analysis of the radiation losses on the refocused wave in the
time-reversal experiment in a context of underwater acoustics. In Theorem 3.7 and Theorem 3.8,
we show that the radiative losses affect the quality of the time-reversal refocusing in two different
ways. First, as expected according to the results obtained in [15], the amplitude of the refocused
wave decays exponentially with the propagation distance. Second, the width of the main focal spot
increases and converges to an asymptotic value, which is larger than the diffraction limit λ /(2θ)
oc
obtained in Proposition 3.1 (where λ is the carrier wavelength in the ocean section with index of
oc
p
refraction n , and θ = 1−1/n2).
1 1
The organization of this paper is as follows. In Section 1 we introduce the underwater waveguide
modelstudiedindetailin[24],andinSection2wepresentthemodedecompositionassociatedtothis
modelandwederivethecoupledmodeequations. InSection3westudythetime-reversalexperiment.
Wedescribeinasimplewaytherefocusedtransverseprofileintermsofthesolutionofthecontinuous
diffusive model obtained in [15], and describing the mode-power coupling between the propagating
and radiating modes.Thanks to this representation we show that the quality of the time-reversal
refocusing is deteriorated by the radiative losses in the ocean bottom.
1 Waveguide Model
We consider a two-dimensional linear acoustic waveguide model. The conservation equations of mass
and linear momentum are given by
∂u
ρ(x,z) +∇p=F(cid:15),
∂t q
(1)
1 ∂p
+∇.u=0,
K(x,z) ∂t
where p is the acoustic pressure, u is the acoustic velocity, ρ is the density of the medium, K is the
bulk modulus, and the source is modeled by the forcing term F(cid:15)(t,x,z) given by
q
F(cid:15)(t,x,z)=Ψ(cid:15)(t,x)δ(z−L )e .
S z
2
Figure 2: Illustration of the shallow-water waveguide model.
The third coordinate z represents the propagation axis along the waveguide. The transverse section
of the waveguide is the semi-infinite interval [0,+∞), and x ∈ [0,+∞) represents the transverse
coordinate. Here, F(cid:15) represents a point source localized at z = L , pointing in the z-direction,
S
with temporal and transverse profile given by Ψ(cid:15)(t,x). Let d > 0 be the bottom of the underwater
waveguide, the medium parameters are given by
√
 1 (cid:0)n2(x)+ (cid:15)V(x,z)(cid:1) if x∈[0,d], z ∈[0,L/(cid:15)]
1  K¯  x∈[0,+∞), z ∈(−∞,0)∪(L/(cid:15),+∞)
= 
K(x,z) 1n2(x) if or
 K¯  x∈(d,+∞), z ∈(−∞,+∞).
ρ(x,z)=ρ¯ if x∈[0,+∞), z ∈R,
and where V is a stochastic process describing the random perturbation of the propagation medium
(seeFigure2foranillustrationoftherandomunderwaterwaveguidemodel). Inthispaperweconsider
the Pekeris waveguide model. This kind of model has been studied for half a century [22] and in this
model the index of refraction n(x) is given by
(cid:26)
n >1 if x∈[0,d)
n(x)= 1
1 if x∈[d,+∞).
The Pekeris profile models an ocean with a constant sound speed, and where d represents the ocean
depth. Conditions corresponding to the Pekeris model can be found during the winter in Earth’s mid
latitudes and in water shallower than about 30 meters.
The perturbation V is assumed to be a continuous real-valued zero-mean stationary stochastic
process with φ-mixing properties [19]. More precisely, let
F =F =σ(V(x,z), x∈[0,d], 0≤z ≤u) and F =σ(V(x,z), x∈[0,d], ,u≤z),
u 0,u u,+∞
we assume that
sup |P(A|B)−P(A)|≤φ(u).
v≥0
A∈Fu+v,+∞
B∈F0,v
This φ-mixing property describes the decorrelating behavior of the random perturbation V through
the nonnegative function φ ∈ L1(R)∩L1/2(R) characterizing the decorrelation speed. Throughout
this paper, for the sake of simplicity and for explicit computations in Section 3.9.2 and Section 3.9.3,
we assume that the autocorrelation function of the random perturbation V is given by
E[V(x ,z )V(x ,z )]=γ (x ,x )e−a|z1−z2| ∀(z ,z ,x ,x )∈[0,+∞)2×[0,d]2.
1 1 2 2 0 1 2 1 2 1 2
From the conservation equations (1), we derive the wave equation for the pressure field,
1 ∂2p
∆p− =∇.F(cid:15),
c(x,z)2 ∂t2
3
q
where c(x,z) = pK(x,z)/ρ(x,z) is the sound speed profile, ∆ = ∂2 + ∂2, and c = K¯/ρ¯. In
x z
underwater acoustics it is natural to use a pressure-release condition since the density of air is very
small compared to the density of water. As a result, the pressure is very weak outside the waveguide,
and by continuity, the pressure at the free surface x = 0 is zero. This consideration leads us to
consider the Dirichlet boundary conditions
p(t,0,z)=0 ∀(t,z)∈[0,+∞)×R.
To study the focusing property of the time-reversal experiment we need to understand the wave
propagationintherandommedium. Todothat,weuseaseparationofscaletechniqueintroducedby
G.Papanicolaouandhiscoauthorsin[2]forinstance. Theimportantscaleparametersinourproblem
are: thewavelength,thecorrelationlengthandthestandarddeviationofthemediuminhomogeneities,
the propagation distance, and bandwidth of the pulse. This last scale parameter plays a key role in
the statistical stability of the time-reversal experiment.
First of all, the model of wave propagation considered in this paper is a linear models, so that the
pressurep(t,x,z)canbeexpressedasthesuperpositionofmonochromaticwavesbytakingitsFourier
transform. Here, the Fourier transform and the inverse Fourier transform, with respect to time, are
defined by
Z 1 Z
fb(ω)= f(t)eiωtdt, f(t)= fb(ω)e−iωtdω.
2π
Asaresult,inthehalf-spacez >L (resp.,z <L ),wegetthatp(ω,x,z)satisfiesthetime-harmonic
S S b
wave equation without source term
√
∂2p(ω,x,z)+∂2p(ω,x,z)+k2(ω)n2(x)p(ω,x,z)+ (cid:15)V(x,z)p(ω,x,z)1 (x)1 (z)=0, (2)
zb xb b b [0,d] [0,L/(cid:15)]
where k(ω)=ω/c is the wavenumber, and with Dirichlet boundary conditions p(ω,0,z)=0 ∀z. The
b
source term does not appear in (2) but induces the following jump conditions for the pressure field
across the plane z =L
S
pb(ω,x,L+S)−pb(ω,x,L−S) = Ψb(ω,x), (3)
∂ p(ω,x,L+)−∂ p(ω,x,L−) = 0.
zb S zb S
To study (2), we consider this equation as an operational differential equation
d2 (cid:0) (cid:1) √
p(ω,.,z)+R(ω) p(ω,.,z) + (cid:15)V(·,z)p(ω,.,z)1 (·)1 (z)=0
dz2b b b [0,d] [0,L/(cid:15)]
in H =L2(0,+∞), where R(ω) is an unbounded operator on H with domain
D(R(ω))=H1(0,+∞)∩H2(0,+∞),
0
and defined by
d2
R(ω)(y)= y+k2(ω)n2(x)y ∀y ∈D(R(ω)). (4)
dx2
In the next section we introduce the spectral decomposition of the operator R(ω) [24]. This decom-
position we will be used in what follows to decompose the field p(ω,x,z) and then understand the
b
stochastic effects undergoes during the propagation.
1.1 Spectral Decomposition in Unperturbed Waveguides
Thespectralanalysisoftheself-adjointPekerisoperator(4)iscarriedoutin[24]. Tousethisspectral
decomposition, we are interested in solutions of (2) such that
(cid:16) (cid:17) (cid:16) (cid:17)
p(ω,.,.)1 (z)∈C0 (L ,+∞),H1(0,+∞)∩H2(0,+∞) ∩C2 (L ,+∞),H ,
b (LS,+∞) S 0 S
(cid:16) (cid:17) (cid:16) (cid:17)
p(ω,.,.)1 (z)∈C0 (−∞,L ),H1(0,+∞)∩H2(0,+∞) ∩C2 (−∞,L ),H .
b (−∞,LS) S 0 S
According to [24], the spectrum of the unbounded operator (4) is given by
Sp(cid:0)R(ω)(cid:1)=(cid:0)−∞,k2(ω)(cid:3)∪(cid:8)β2 (ω),...,β2(ω)(cid:9).
N(ω) 1
4
The continuous part of the spectrum comes from the fact that our waveguide model is semi-infinite.
For the discrete part, the modal wavenumber β (ω) are positive and
j
k2(ω)<β2 (ω)<···<β2(ω)<n2k2(ω).
N(ω) 1 1
Regarding the spectral decomposition, there exists a resolution of the identity Π of R(ω) such that
ω
∀y ∈H, ∀r ∈R,
N(ω)
Π (r,+∞)(y)(x)= X (cid:10)y,φ (ω,.)(cid:11) φ (ω,x)1 (cid:0)β (ω)2(cid:1)
ω j H j (r,+∞) j
j=1
Z k2(ω)
(cid:10) (cid:11)
+ y,φ (ω,.) φ (ω,x)dγ1 (r),
γ H γ (−∞,k2(ω))
r
and ∀y ∈D(R(ω)), ∀r ∈R,
N(ω)
Π (r,+∞)(R(ω)(y))(x)= X β (ω)2(cid:10)y,φ (ω,.)(cid:11) φ (ω,x)1 (cid:0)β (ω)2(cid:1)
ω j j H j (r,+∞) j
j=1
Z k2(ω)
(cid:10) (cid:11)
+ γ y,φ (ω,x) φ (ω,x)dγ1 (r).
γ H γ (−∞,k2(ω))
r
Let us describe more closely the discrete and the continuous part of the decompositions.
(cid:8) (cid:9)
Discrete part of the decomposition ∀j ∈ 1,...,N(ω) , the jth eigenvector is given in [24] by
(cid:26) A (ω)sin(σ (ω)x/d) if 0≤x≤d
j j
φ (ω,x)=
j Aj(ω)sin(σj(ω))e−ζj(ω)x−dd if d≤x,
where
q q
σ (ω)=d n2k2(ω)−β2(ω), ζ (ω)=d β2(ω)−k2(ω),
j 1 j j j
and
v
u 2/d
A (ω)=u . (5)
j t1+ sin2(σj(ω)) − sin(2σj(ω))
ζj(ω) 2σj(ω)
Here, σ (ω),...,σ (ω) are the solutions on (0,n k(ω)dθ) of the following equation,
1 N(ω) 1
y
tan(y)=− , (6)
p
(n kdθ)2−y2
1
p
and such that 0<σ (ω)<···<σ (ω)<n k(ω)dθ, with θ = 1−1/n2. This last equation ad-
1 N(ω) 1 1
(cid:0) (cid:1)
mitsexactlyonesolutionovereachintervaloftheform π/2+(j−1)π,π/2+jπ forj ∈{1,...,N(ω)},
where
(cid:20) (cid:21)
n k(ω)d
N(ω)= 1 θ ,
π
and [·] stands for the integer part. From (6), we have the following results [16] which are used to
obtain the main result of this paper in Section 3.9.3.
Lemma 1.1 Let α∈(1/3,1), we have as N(ω)→+∞
(cid:16) (cid:17)
sup |σj+1(ω)−σj(ω)−π|=O N(ω)21−23α .
j∈{1,...,N(ω)−[N(ω)α]−1}
sup (cid:12)(cid:12)σj+2(ω)−2σj+1(ω)+σj(ω)(cid:1)(cid:12)(cid:12)=O(cid:0)N(ω)1−3α(cid:1).
j∈{1,...,N(ω)−[N(ω)α]−2}
5
Let us note that, ∀η ∈[0,1[, we have
sup |σ (ω)−jπ|=O(N(ω)α−1), (7)
j
j∈{1,...,N(ω)α}
and
lim sup kφ (ω,·)−φ (·)k =0
j j H
N(ω)→+∞j∈{1,...,N(ω)α}
with
( q
2sin(jπx) if x∈[0,d]
φ (x)= d d
j
0 if x≥d.
This result means that in the limit of large number of propagating modes the low order propagat-
ing modes are very similar in shape to those of a perfect bounded waveguide with pressure-release
boundary conditions at x = 0 and x = d. This approximation does not hold anymore for high or-
der propagating modes, but the results of Lemma 1.1 mean that the distribution of solutions of (6)
is closed to the distribution of the eigenvalues of the transverse Laplacian associated to a perfect
bounded waveguide with pressure-release boundary conditions at x=0 and x=d.
Continuous part of the decomposition For γ ∈(−∞,k2(ω)), we have [24]
φ (ω,x)=
γ
(
A (ω)sin(η(ω)x/d) if 0≤x≤d
γ
A (ω)(cid:16)sin(η(ω))cos(cid:0)ξ(ω)x−d(cid:1)+ η(ω)cos(η(ω))sin(cid:0)ξ(ω)x−d(cid:1)(cid:17) if d≤x,
γ d ξ(ω) d
where
q
p
η(ω)=d n2k2(ω)−γ, ξ(ω)=d k2(ω)−γ,
1
and
s
dξ(ω)
A (ω)= .
γ π(cid:0)ξ2(ω)sin2(η(ω))+η2(ω)cos2(η(ω))(cid:1)
(cid:10) (cid:11)
Let us note that φ (ω,.) does not belong to H so that y,φ (ω,.) is not defined in the classical
γ γ H
way. In fact, we have
Z M
(cid:10) (cid:11)
y,φ (ω,.) = lim y(x)φ (ω,x)dx
γ H M→+∞ 0 γ
where the limit holds on L2(cid:0)−∞,k2(ω)(cid:1). Moreover, according to the following Plancherel equality
N(ω) Z k2(ω)
kyk2H =kΠω(−∞,+∞)(y)k2H = X (cid:12)(cid:12)(cid:10)y,φj(ω,.)(cid:11)H(cid:12)(cid:12)2+ (cid:12)(cid:12)(cid:10)y,φγ(ω,.)(cid:11)H(cid:12)(cid:12)2dγ,
j=1 −∞
the map which assigns to every element of H the coefficients of its spectral decomposition
Θ :H −→ Hω
ω
(cid:16)(cid:0)(cid:10) (cid:11) (cid:1) (cid:0)(cid:10) (cid:11) (cid:1) (cid:17)
y −→ y,φ (ω,.) , y,φ (ω,.)
j H j=1,...,N(ω) γ H γ∈(−∞,k2(ω))
is an isometry, from H onto Hω =CN(ω)×L2(cid:0)−∞,k2(ω)(cid:1).
2 Mode Coupling in Random Waveguides
Before describing the time-reversal experiment in our randomly perturbed waveguide model, we need
tounderstandhowthewaveisperturbedduringthepropagationthroughthemedium. Inthissection,
we study the random effects produced on the modal decomposition of p(ω,x,z) propagating in the
b
perturbed section [0,L/(cid:15)].
Using the resolution of the identity Π associated to Pekeris operator R(ω), we have
ω
N(ω) Z k2(ω)
X
p(ω,x,z)= p (ω,z)φ (ω,x)+ p (ω,z)φ (ω,x)dγ,
b bj j bγ γ
j=1 −∞
6
where p(ω,z)=Θ (p(ω,.,z)) and Θ is defined in Section 1.1.
b ω b ω
For the sake of simplicity in the presentation of the forthcoming asymptotic analysis, we will
restrict ourself to solutions of the form
N(ω) Z k2(ω)
X
p(ω,x,z)= p (ω,z)φ (ω,x)+ p (ω,z)φ (ω,x)dγ. (8)
b bj j bγ γ
j=1 ξ
Thisassumptionistantamounttoneglectingtheroleplayedbytheevanescentmodesduringtheprop-
agation in the random medium. Nevertheless, as it has been observed in [15, 10] that these modes
play no role in the refocusing process. The reason is that these modes only imply a mode-dependent
and a frequency-dependent phase modulations without remove any energy from the propagating and
radiating modes, However, the dispersion phenomena are compensated by the time-reversal mecha-
nism. Moreover, we assume that (cid:15)(cid:28)ξ and therefore we have two distinct scales. We will consider in
a first time the asymptotic (cid:15) goes to 0 and in a second time the asymptotic ξ goes to 0.
2.1 Coupled Mode Equations
Accordingtothepressurefielddecomposition(8),wegiveinthissectionthecoupledmodeequations,
which describes the coupling mechanism between the amplitudes of the two kinds of modes, prop-
agating and radiating modes. In the random section [0,L/(cid:15)], p(ω,z) satisfies the following coupled
b
equation in Hω =CN(ω)×L2(ξ,k2(ω)).
ξ
d2 √ NX(ω)
p (ω,z)+β2(ω)p (ω,z)+ (cid:15)k2(ω) Cω(z)p (ω,z)
dz2bj j bj jl bl
l=1
√ Z k2(ω)
+ (cid:15)k2(ω) Cω (z)p (ω,z)dγ0 =0,
jγ0 bγ0
ξ
(9)
d2 √ NX(ω)
p (ω,z)+γ p (ω,z)+ (cid:15)k2(ω) Cω(z)p (ω,z)
dz2bγ bγ γl bl
l=1
√ Z k2(ω)
+ (cid:15)k2(ω) Cω (z)p (ω,z)dγ0 =0,
γγ0 bγ0
ξ
where the coupling coefficients Cω(z) are defined by:
Z d
Cω(z)=(cid:10)φ (ω,.),φ (ω,.)V(.,z)(cid:11) = φ (ω,x)φ (ω,x)V(x,z)dx,
jl j l H j l
0
Z d
Cω (z)=C (z)=(cid:10)φ (ω,.),φ (ω,.)V(.,z)(cid:11) = φ (ω,x)φ (ω,x)V(x,z)dx, (10)
jγ γj j γ H j γ
0
Z d
Cω (z)=(cid:10)φ (ω,.),φ (ω,.)V(.,z)(cid:11) = φ (ω,x)φ (ω,x)V(x,z)dx.
γγ0 γ γ0 H γ γ0
0
Next, we decompose the wave field p(ω,z) using the amplitudes of the generalized right- and
b
left-going modes ba(ω,z) andbb(ω,z), which are given by
1 (cid:16) (cid:17)
pbj(ω,z)= p baj(ω,z)eiβj(ω)z+bbj(ω,z)e−iβj(ω)z ,
β (ω)
j
d q (cid:16) (cid:17)
dzpbj(ω,z)=i βj(ω) baj(ω,z)eiβj(ω)z−bbj(ω,z)e−iβj(ω)z ,
1 (cid:16) √ √ (cid:17)
pbγ(ω,z)= γ1/4 baγ(ω,z)ei γz +bbγ(ω,z)e−i γz ,
d (cid:16) √ √ (cid:17)
dzpbγ(ω,z)=iγ1/4 baγ(ω,z)ei γz−bbγ(ω,z)e−i γz
∀j ∈(cid:8)1,...,N(ω)(cid:9)andalmosteveryγ ∈(ξ,k2(ω)). From(9),thisdecompositionallowsustoobtain
a first order differential system instead of a second order one, so that we obtain the coupled mode
7
equation in Hξω×Hξω for the amplitudes (ba,bb),
ddzba(ω,z)=√(cid:15)Haa(ω,z)(cid:0)ba(ω,z)(cid:1)+√(cid:15)Hab(ω,z)(cid:0)bb(ω,z)(cid:1) (11)
ddzbb(ω,z)=√(cid:15)Hba(ω,z)(cid:0)ba(ω,z)(cid:1)+√(cid:15)Hbb(ω,z)(cid:0)bb(ω,z)(cid:1). (12)
This system is complemented with the boundary conditions
(cid:18) (cid:19)
L
ba(ω,0)=ba(cid:15)0(ω) and bb ω, (cid:15) =0
where
p
β (ω)
ba(cid:15)j,0(ω)= 2j (cid:10)Ψb(cid:15)(ω,·),φj(ω)(cid:11)He−iβj(ω)LS, ∀j ∈(cid:8)1,...,N(ω)(cid:9),
(13)
ba(cid:15)γ,0(ω)= γ12/4(cid:10)Ψb(cid:15)(ω,·),φγ(ω)(cid:11)He−i√γLS, for almost every γ ∈(ξ,k2(ω)).
(cid:8) (cid:9)
For j ∈ 1,...,N(ω) , a (ω ) represents the initial amplitude of the jth propagating mode, and
bj,0 0
for γ ∈ (ξ,k2(ω)), a (ω) represents the initial amplitude of the γth radiating mode at z = 0. The
bγ,0
initial conditions for the right-going mode a (ω ) comes from (3), and the second condition for the
b0 0
left-goingmodesmeansthatnowaveiscomingfromtherighthomogeneouswaveguide. Thecoupling
operator Haa(ω,z), Hab(ω,z), Hba(ω,z), and Hbb(ω,z) in (11) and (12) are defined by:
Haja(ω,z)(y)=Hbjb(ω,z)(y)= ik22(ω)hNX(ω)pβC(jωωl()zβ)(ω)ylei(βl(ω)−βj(ω))z
l=1 j l
Z k2(ω) Cω (z) √ i
+ q jγ0 √ yγ0ei( γ0−βj(ω))zdγ0 ,
ξ β (ω) γ0
j
Haγa(ω,z)(y)=Hbγb(ω,z)(y)= ik22(ω)hNX(ω)pC√γωγlβ(z()ω)ylei(βl(ω)−√γ)z
l=1 l
+Z k2(ω) Cγωγ0(z) y ei(√γ0−√γ)zdγ0i,
γ1/4γ01/4 γ0
ξ
Hajb(ω,z)(y)=Hbja(ω,z)(y)= ik22(ω)hNX(ω)pβC(jωωl()zβ)(ω)yle−i(βl(ω)+βj(ω))z
l=1 j l
Z k2(ω) Cω (z) √ i
+ q jγ0 √ yγ0e−i( γ0+βj(ω))zdγ0 ,
ξ β (ω) γ0
j
Haγb(ω,z)(y)=Hbγa(ω,z)(y)= ik22(ω)hNX(ω)pC√γωγlβ(z()ω)yle−i(βl(ω)+√γ)z
l=1 l
+Z k2(ω) Cγγ0(z) y e−i(√γ0+√γ)zdγ0i.
γ1/4γ01/4 γ0
ξ
Let us remark that we have the following global conservation relations
kba(ω,z)k2Hω −kbb(ω,z)k2Hω =kba(ω,0)k2Hω −kbb(ω,0)k2Hω ∀z ∈[0,L/(cid:15)],
ξ ξ ξ ξ
kba(ω,L/(cid:15))k2Hω +kbb(ω,0)k2Hω =kba(ω,0)k2Hω.
ξ ξ ξ
However, in our context, the coupling mechanism between the right- and the left-going mode is not
veryconvenienttostudythetime-reversalexperiment. Theasymptoticbehaviorofthewholecoupling
mechanismbetweentheright-andtheleft-goinginarandomwaveguidewithaboundedcrosssection
is carried out in [13], but this study leads to technical difficulties because of the waveguide geometry
in our context. Consequently, for the sake of simplicity we introduce in the following section the
forward scattering approximation, which allows us to neglect the coupling mechanism between the
right- and the left-going under certain conditions.
8
2.2 Propagator and Forward Scattering Approximation
Beforeintroducingthisapproximation,letusdefinetherescaledprocessesaccordingtothesizeofthe
random section [0,L/(cid:15)],
(cid:16) z(cid:17) (cid:16) z(cid:17)
ba(cid:15)(ω,z)=ba ω, (cid:15) and bb(cid:15)(ω,z)=bb ω, (cid:15)
which satisfy in Hω the rescaled coupled mode equation
ξ
ddzba(cid:15)(ω,z)= √1(cid:15)Haa(cid:16)ω,z(cid:15)(cid:17)(cid:0)ba(cid:15)(ω,z)(cid:1)+ √1(cid:15)Hab(cid:16)ω,z(cid:15)(cid:17)(cid:0)bb(cid:15)(ω,z)(cid:1)
ddzbb(cid:15)(ω,z)= √1(cid:15)Hba(cid:16)ω,z(cid:15)(cid:17)(cid:0)ba(cid:15)(ω,z)(cid:1)+ √1(cid:15)Hbb(cid:16)ω,z(cid:15)(cid:17)(cid:0)bb(cid:15)(ω,z)(cid:1),
with the two-point boundary conditions
ba(cid:15)(ω,0)=ba(cid:15)0(ω) and bb(cid:15)(ω,L)=0.
The propagator P(cid:15)(ω,z) is defined has being the unique solution of the following differential
equation
d 1 (cid:16) z(cid:17)
P(ω,z)= √ H ω, P(cid:15)(ω,z) with P(cid:15)(ω,0)=Id,
dz (cid:15) (cid:15)
so that
(cid:20)a(cid:15)(ω,z)(cid:21) (cid:20)a(cid:15)(ω,0)(cid:21)
b =P(cid:15)(ω,z) b .
bb(cid:15)(ω,z) bb(cid:15)(ω,0)
According to the symmetry of H(ω,z) the propagator has the following particular form
" #
Pa(ω,z) Pb(ω,z)
P(cid:15)(ω,z)= (cid:15) (cid:15) .
Pb(ω,z) Pa(ω,z)
(cid:15) (cid:15)
where, Pa(ω,z) and Pb(ω,z) are two operators representing respectively the coupling between the
(cid:15) (cid:15)
right-going modes and the coupling between the right-going and left-going modes.
The forward scattering approximation is widely used in the literature. In this approximation the
couplingbetweenforward-andbackward-propagatingmodesisassumedtobenegligiblecomparedto
thecouplingbetweentheforward-propagatingmodes. Thephysicalexplanationofthisapproximation
isasfollows. Thecouplingbetweenaright-goingpropagatingmodeandaleft-goingpropagatingmode
involves coefficients of the form
Z +∞
E[Cω(0)Cω(z)]cos(cid:0)(β (ω)+β (ω))z(cid:1)dz,
jl jl l j
0
wherethecoefficientsCω(z)aredefinedby(10),andthecouplingbetweentworight-goingpropagating
modes or two left-going propagating modes involves coefficients of the form
Z +∞
E[Cω(0)Cω(z)]cos(cid:0)(β (ω)−β (ω))z(cid:1)dz
jl jl l j
0
(cid:8) (cid:9)2
∀(j,l)∈ 1,...,N(ω) . The forward scattering approximation consists in assuming that
Z +∞
E[Cω(0)Cω(z)]cos(cid:0)(β (ω)+β (ω))z(cid:1)dz =0 ∀(j,l)∈(cid:8)1,...,N(ω)(cid:9)2,
jl jl l j
0
so that Pb(ω,z) = 0, that is there is no coupling between right-going and left-going propagating
(cid:15)
modes. Therefore, this approximation holds if the power spectral density of the process V, i.e. the
Fourier transform of its z-autocorrelation function, possesses a cut-off wavenumber. We refer to
[12, 15] for justifications on the validity of this approximation. As a result, under this approximation
we can neglect the left-going propagating modes in the asymptotic (cid:15)→0, and then consider only the
simplified coupled amplitude equation on [0,L]
d 1 (cid:16) z(cid:17)
a(cid:15)(ω,z)= √ Haa ω, (a(cid:15)(ω,z)) with a(cid:15)(ω,0)=a(cid:15)(ω).
dzb (cid:15) (cid:15) b b b0
9
(a) (b)
Figure 3: Representation of the time-reversal experiment. In (a) we represent the first step of the
experiment, and in (b) we represent the second step of the experiment.
Finally, we introduce the transfer operator Tξ,(cid:15)(ω,z), which is the solution of
d 1 (cid:16) z(cid:17)
Tξ,(cid:15)(ω,z)= √ Haa ω, Tξ,(cid:15)(ω,z) with Tξ,(cid:15)(ω,0)=Id. (14)
dz (cid:15) (cid:15)
From this equation, one can easily check that the transfer operator Tξ,(cid:15)(ω,z) is unitary since Haa is
skew-Hermitian and
∀z ≥0, a(cid:15)(ω,z)=Tξ,(cid:15)(ω,z)(a(cid:15)(ω)).
b b0
3 Time Reversal in a Waveguide
Time-reversalexperimentswithsonarinshallowwater[18,23]werecarriedoutbyWilliamKuperman
and his group in San Diego. This experiment is carried out in two steps. In the first step (see Figure
3 (a)), a source sends a pulse into the medium. The wave propagates and is recorded by a device
called a time-reversal mirror. A time-reversal mirror is a device that can receive a signal, record it,
and resend it time-reversed into the medium. In other words, what is recorded first is send out last.
In the second step (see Figure 3 (b)), the wave emitted by the time-reversal mirror has the property
ofrefocusingneartheoriginalsourcelocation,andithasbeenobservedthatrandominhomogeneities
enhance refocusing [3, 6, 5, 7, 8, 9, 10, 21]. This experiment has already been analyzed in waveguides
with bounded cross-section in [10, Chapter 20] and [12, 14]. However, in contrast with all these
results where the random medium improve the refocusing, we show in this section in a context of
underwater acoustics that the random inhomogeneities deteriorate the refocusing property. In [1] the
authors observe such deterioration but in their context it is induced by viscosity effect in the wave
propagationmodel. Weshowinthissectionthatthiseffectissimplyinducedbytheinhomogeneitiesof
thepropagationmediumthroughthecouplingmechanismbetweenthepropagatingandtheradiating
modes.
3.1 First Step of the Experiment
In the first step of the experiment (see Figure 4), a source sends a pulse into the medium, the wave
propagates and is recorded by the time-reversal mirror located in the plane z = L/(cid:15). We assume
that the time-reversal mirror occupies the transverse subdomain D ⊂ [0,d] and in the first step of
M
the experiment the time-reversal mirror plays the role of a receiving array. The transmitted wave is
recorded for a time interval (cid:2)t0,t1(cid:3) and is re-emitted time-reversed into the waveguide toward the
(cid:15) (cid:15)
source. We have chosen such a time window because it is of the order of the total travel time of the
section [0,L/(cid:15)].
In this paper, the source profile Ψ(cid:15)(t,x) is given, in the frequency domain, by
(cid:18) (cid:19)
1 ω−ω
Ψb(cid:15)(ω,x)= fb 0
q (cid:15)q (cid:15)q
  (15)
N(ω) Z
X
× φj(ω,x0)φj(ω,x)+ φγ(ω,x0)φγ(ω,x)dγ,
j=1 (−S,−ξ)∪(ξ,k2(ω))
10