Pulse propagation and time reversal in random waveguides
33 Pages
English
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Pulse propagation and time reversal in random waveguides

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33 Pages
English

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Niveau: Secondaire, Lycée
Pulse propagation and time reversal in random waveguides Context: time-reversal experiments in underwater acoustics. Experimental observations: - robust spatial refocusing - diffraction-limited focal spot [1] W. A. Kuperman, W. S. Hodgkiss, H. C. Song, T. Akal, C. Ferla, and D. R. Jackson, Phase conjugation in the ocean, experimental demonstration of an acoustic time-reversal mirror, J. Acoust. Soc. Am. 103 (1998), 25-40. [2] H. C. Song, W. A. Kuperman, and W. S. Hodgkiss, Iterative time reversal in the ocean, J. Acoust. Soc. Am. 105 (1999), 3176-3184. Analysis of the mechanisms responsible for statistically stable time reversal.

  • iterative time

  • time harmonic

  • waveguide cross-section

  • perturbed wave

  • refocusing - diffraction-limited focal

  • source source


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Language English

Exrait

Pulsepropagationandtimereversalinrandomwaveguides

Context:time-reversalexperimentsinunderwateracoustics.

Experimentalobservations:
-robustspatialrefocusing
-diffraction-limitedfocalspot

[1]W.A.Kuperman,W.S.Hodgkiss,H.C.Song,T.Akal,C.Ferla,andD.R.

Jackson,Phaseconjugationintheocean,experimentaldemonstrationofanacoustic

time-reversalmirror,J.Acoust.Soc.Am.
103
(1998),25-40.

[2]H.C.Song,W.A.Kuperman,andW.S.Hodgkiss,Iterativetimereversalinthe

ocean,J.Acoust.Soc.Am.
105
(1999),3176-3184.

Analysisofthemechanismsresponsibleforstatisticallystabletimereversal.

Perfectacousticwaveguide

waveguidecross-section
2R⊂D

D

x

z

up∂1∂ρ
¯+

p
=
F
,
¯+

u
=0
,
for
x
∈D
and
z

R
.
t∂t∂Kp
istheacousticpressure,
u
istheacousticvelocity.
ρ
¯isthedensityofthemedium,
K
¯isthebulkmodulus.
Thesourceismodeledbytheforcingterm
F
(
t,
r
).

Waveequationwiththesoundspeed
c
¯=
K
¯

¯:
p2p∂1Δ
p

c
¯
2
∂t
2
=

.
F
for
x
∈D
and
z

R
.

Dirichletboundaryconditions

p
(
t,
x
,z
)=0for
x


D
and
z

R
.

Timeharmonicwaveequation
k
=
ω/c
¯


z
2
p
ˆ(
ω,
x
,z
)+Δ

p
ˆ(
ω,
x
,z
)+
k
2
(
ω
)
p
ˆ(
ω,
x
,z
)=0

SpectrumofΔ

withDirichletBC=infinitenumberofdiscreteeigenvalues


Δ

φ
j
(
x
)=
λ
j
φ
j
(
x
)
,
x
∈D

j
(
x
)=0
,
x


D
,
for
j
=1
,
2
,...

Numberofpropagatingmodes
N
(
ω
):

λ
N
(
ω
)

k
(
ω
)

N
(
ω
)+1
,

Propagatingmodes1

j

N
(
ω
):
p
ˆ
j
(
ω,
x
,z
)=
φ
j
(
x
)
e
±

j
(
ω
)
z

j
(
ω
)=
k
2
p

Evanescentmodes
j>N
(
ω
):

(ω)−λj.q
ˆ
j
(
ω,
x
,z
)=
φ
j
(
x
)
e
±
β
j
(
ω
)
z

j
(
ω
)=
λ
j

k
2
(
ω
)
.
p

=)ω(jbˆ−=)ω(jaˆ,)z()0,∞−(1)z()∞,0(1)x(jφzjβ−e)ω(jβ)ω(jcˆ∞+)x(jφzjβie)ω(jβ)ω(jaˆN=)z,x,ω(pExcitationConditionsforaSource

ˆSourcelocalizedintheplane
z
=0:

.F
(
t,
x
,z
)=
f
(
t
)
δ
(
x

z#"XXj
=1
j
=
N
+1
pp∞N#"+
Xp
b
ˆ
j
(
ω
)
e


j
z
φ
j
(
x
)+
Xp
d
ˆ
j
(
ω
)
e
β
j
z
φ
j
(
x
)
j
=1
β
j
(
ω
)
j
=
N
+1
β
j
(
ω
)

ewhti

)p
β
j
(
ω
)
f
ˆ(
ω
)
φ
j
(
x
0
)
,
2c
ˆ
j
(
ω
)=

d
ˆ
j
(
ω
)=

β
2
j
(
ω
)
f
ˆ(
ω
)
φ
j
(
x
0
)
.
p

zFor
k
(
ω
)
z

1:

()ω(NXj
=1
β
j
(
ω
)
p
ˆ(
ω,
x
,z
)=
p
a
ˆ
j
(
ω
)
φ
j
(
x
)
e

j
(
ω
)
z

δ)0x
Perturbedwaveguide:Timeharmonicapproach

xpind/2
02/d-

prtL /
e
2
z

ρ
(
r
)
∂∂
u
t
+

p
=
F
,
K
1(
r
)
∂∂tp
+

u
=0
,

8K=1
<
1
(1+
ε
ν
(
x
,z
))for
x
∈D
,z

[0
,
L/ε
2
]
KK
(
x
,z
)
:
1
for
x
∈D
,z

(
−∞
,
0)

(
L/ε
2
,

)
ρ
(
x
,z
)=
ρ
¯
∈D∈−∞∞
PerturbedwaveequationwithDirichletboundaryconditions:

forx,z(,)2Δ
p
ˆ(
ω,
x
,z
)+
k
(1+
ε
ν
(
x
,z
))
p
ˆ(
ω,
x
,z
)=0
.

Wavemodeexpansions:

φj(x)qˆj(z)∞Np
ˆ(
x
,z
)=
φ
j
(
x
)
p
ˆ
j
(
z
)+
XXj
=1
j
=
N
+
Right-goingandleft-goingmodeamplitudes
a
ˆ
j
(
z
)and
b
ˆ
j
(
z
):

j1”“”“zdpβp
ˆ
j
=
p
1
a
ˆ
j
e

j
z
+
b
ˆ
j
e


j
z
,dp
ˆ=

j
a
ˆ
j
e

j
z

b
ˆ
j
e


j
z
,
j

Nj≤

Coupledmodeequations

Neglectevanescentmodes.

ˆlie(βl−βj)z+ˆbl−e(iβl+βj)zCoupledmodeequationsfor
j

N
:
dz
2
1

l

N
β
j
β
l
da
ˆ
j
=
iεk
2
X
C
p
jl
(
z
)

a

lldz
2
1

l

N
β
j
β
l
db
ˆ
j
=

iεk
2
X
C
p
jl
(
z
)

a
ˆ
e
i
(
β
l
+
β
j
)
z
+
b
ˆ
e
i
(
β
j

β
l
)
z

C
jl
(
z
)=
φ
j
(
x
)
φ
l
(
x
)
ν
(
x
,z
)
d
x
ZDBoundaryconditions:

hwere

Rescaling:

La
ˆ
j
(0)=
a
ˆ
j,
0
,b
ˆ
j
(
2
)=0
ε

zzεεa
ˆ
j
(
z
)=
a
ˆ
j
(
2
)
,b
ˆ
j
(
z
)=
b
ˆ
j
(
2
)
εε֒

Diffusionapproximationtheorem.

Theforwardscatteringapproximation

Diffusion-approximation=

multi-dimensionaldiffusionprocess.
Couplingcoefficientsbetweenleftandright-goingmodes:

EC[jl(0)Cjl(z)]cosβ(j(ω)+βl(ω))z)d∞Z0Couplingcoefficientsbetweenright-goingmodes:

EC[jl(0)Cjl(z)]cosβ(j(ω)−βl(ω))z)dzz,,,j,jll==,1,1
,N
(
ω
)
.

,N(ω).∞Z0Wecanneglecttheleft-going(backward)propagatingmodesifthefirsttypeof
coefficientsarenegligiblecomparedtothesecondones.

erudcedssytem:εda
ˆ=1
M
(
z
)
a
ˆ
ε
(
z
)
2εεzd2ββ2M
jl
(
z
)=
p
ikC
jl
(
z
)
e
i
(
β
l

β
j
)
z
lj

(ω,z))j=1,N,covnegreniidstirubtoinasεThemodeamplitudes(
a
ˆ


0toa
diffusionprocess
(
a
ˆ
j
(
ω,z
))
j
=1
,

,N
whoseinfinitesimalgeneratoris

L
=41Γ
j
(
lc
)
(
ω
)
A
jl
A
jl
+
A
jl
A
jl
+21Γ
j
(
l
1)
(
ω
)
A
jj
A
ll
XX´`j
6
=
lj,l
i+4Γ
j
(
ls
)
(
ω
)(
A
ll

A
jj
)
,
Xl=6j∂∂A
jl
=
a
ˆ
j
∂a
ˆ
l

a
ˆ
l
∂a
ˆ
j
=

A
lj
.

4∞Γ
(
c
)
(
ω
)=
ω
Zjl
2
c
¯
4
β
j
(
ω
)
β
l
(
ω
)
0
Γ
j
(
cj
)
(
ω
)=

Γ
j
(
cn
)
(
ω
)
,
Xj=6n

ocs(β(j(ω)−βl(ω))z)E[Cjl()0Cjl(z)]zdifj=6l,
Meanmodeamplitudes

jTheexpectedvaluesofthemodeamplitudes
E
[
a
ˆ
ε
(
ω,z
)]convergeas
ε

0to

E
[
a
ˆ
j
(
ω,z
)]givenby

֒→

epxoentnailadmipgnE
[
a
ˆ
j
(
ω,z
)]=exp(
q
j
(
ω
)
z
)
a
ˆ
j
0
(
ω
)

Re(
q
j
(
ω
))
<
0

oftehmeanamlptidues.
P

Meanmodepowers

Themodepowers(
|
a
ˆ

(
ω,z
)
|
2
)
j
=1
,

,N
convergeindistributionas
ε

to
(
P
j
(
ω,z
))
j
=1
,

,N
whoseinfinitesimalgeneratoris

«„»X֒

diffusionon
H
N
=(
N
P
j
=
R
02
onPNwhere
R
02
=
|
a
ˆ
j,
0
|
2
X1=jThemeanmodepowers
E
[
|
a
ˆ

(
ω,z
)
|
2
]convergeto
P
j
(1)
(
ω,z
)
j

j
(
cn
)
(
ω
)
P
n
(1)

P
j
(1)
dP
(1)
X“”
dz
n
6
=
j
startingfrom
P
j
(1)
(
ω,z
=0)=
|
a
ˆ
j,
0
|
2
,
j
=1
,

,N
.
Thisshowstheasymptotic
equipartition
ofmodeenergy:

(1)
1
2
z
«„j
=
s
1
,
u

p

,N
˛
P
j
(
ω,z
)

NR
0
˛

C
exp

L
equip
where1=secondeigenvalueofΓ
(
c
)
.
L
equip

1=j,0≥jP,N,,1=j)jPjP∂∂)jP−lP(+jP∂∂lP∂∂−jP∂∂jPlP)ω(lj)c(Γl=6j=PL