Pulse propagation and time reversal in random waveguides

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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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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

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
¯
/ρ
¯:
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
,φ
j
(
x
)=0
,
x
∈
∂
D
,
for
j
=1
,
2
,...

Numberofpropagatingmodes
N
(
ω
):

λ
N
(
ω
)
≤
k
(
ω
)
<λ
N
(
ω
)+1
,

Propagatingmodes1
≤
j
≤
N
(
ω
):
p
ˆ
j
(
ω,
x
,z
)=
φ
j
(
x
)
e
±
iβ
j
(
ω
)
z
,β
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
,β
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
−
iβ
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
iβ
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
iβ
j
z
+
b
ˆ
j
e
−
iβ
j
z
,dp
ˆ=
iβ
j
a
ˆ
j
e
iβ
j
z
−
b
ˆ
j
e
−
iβ
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: