Pulse propagation and time reversal in random waveguides
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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
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
¯
/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:
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
jε
→
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<
