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By considering the joint probability distribution of 4 fields we have obtained an analytical expression for the correlation of the phase first derivative as a function of the field correlation function:
And hence, the cumulative phase correlation function:
GeneratingTransducer
Hydrophone Detector
Fluidized Bed
Liquid Flow
Glass Beads
Distributor
p1p
p+1
p+2
p
p+1kp
kp 1
kp + 1
rp (0)rp + 1 (0)
∆rp+1 (τ)∆rp (τ)
• Setup:
0 5 10 15 20 25 30 35
1
0
1
NO
RM
AL
IZE
D
FIE
LD
T R A N S IT T IM E (µs)
0 .04
0 .00
0.04T R A N S M IT T E D
tS
T R A N S M IT T E DIN P U T
τ = 0 s τ = 0 s
FIE L D PH A SEA M P L IT U D E
τ = .75 s
TR A N SIT TIM E (µs)
τ = .75 s
π
π
π
π
π
π
0
0
0
τ = 0 s
τ = .75 s
17.5 18.0 18.5 17.5 18.0 18.5
τ = 1 .5 s
17.5 18.0 18.5
τ = 1 .5 s
τ = 1 .5 s
• Experiment:By repeating this procedure for each pulse i, we measure the
scattered field Ψ(t,τ),
amplitude A(t,τ),
wrapped phase Φ(t,τ)
as a function of “Field time” t – the propagation time of the pulse in the medium (µs) and
“Fluctuation time” τ (= i×trep ) – the time scale of the dynamics (ms → s).
Fit independent test for circular gaussian statistics
Phase StatisticsPhase Statistics in disordered mediain disordered media Applications to acoustics and seismology
D. AnacheMénier, B. A. van Tiggelen (LPMMC, Grenoble), J. Page (Univ Manitoba),
L. Margerin, P. Roux (LGIT, Grenoble)
Application to ultrasound acoustics Application to seismology
Web: http://lpm2c.grenoble.cnrs.fr/People/Anache/
Complex scalar field:
Sum of partial waves:
References:[1] A.Z. Genack, A.A. Chabanov, P.Sebbah and B.A. van Tiggelen,
Waves in random media, Encyclopedia of Condensed Matter Physics (2005, Elsevier), p 307317.
[2] M.L. Cowan, I.P. Jones, J.H. Page, and D.A. Weitz. Diffusing acoustic wave spectroscopy. Physical Review E, 65, June 2002.
[3] B. A. van Tiggelen, D. Anache et A. Ghysels, Role of Mean Free Path in Spatial Phase Correlation and Nodal Screening, Europhys. Lett. 74 999, 2006 .
[4] P. Sebbah, O.Legrand, B.A van Tiggelen, A.Z. Genack, Statistics of the cumulative phase of microwaves in random media, Phys. Rev. Lett. E. 56, 1996.
Conclusion:We studied theoretically the CGS statistics at higher order.
Experiments extremely well modelled by circular gaussian statistics.
Phase allows to investigate the dynamics of such strongly scattering materials at short and long time scales; provides more accurate information than the more traditional field correlation measurements.
We currently study seismic data recorded at the Piñon Flat Observatory in California.
Hope to measure the mean free path in the crust.
Asymptotic power law decayPhase statistics
• 1
• 1
• 3
By considering the joint probability distribution of 4 fields we have obtained an analytical expression for the joint probability distribution of the first 3 derivatives:
3 fiting parameters Q R and S that depend in turn to derivatives of the field correlation function:
Phase Correlations
2D: exponential decay toward the saymptotic value: 3D: The asymptotic value varies logarithmically with [3]
• Unwrapped phase correlation:
Circular gaussian statistics: after a few mean free path phase becomes random and partial waves become independent so aplying central limit theorem:
Gaussian hypothesis
source
where the index denotes different times, position or frequencies.
Direct consequences:
No more oscillations mean free path= caracteristic length scale, new oportunity to measure the scattering mean free path.
Exponential decay due to multiple scattering Oscillation on the scale of the wavelength originate from a superposition of plane waves incident with arbitrary directions but with equal amplitude.
• Phase correlation and measure of the mean free path:
Field correlation function:
Correlation function of the phase derivative with respect to position:Probe of the early time behaviour of the particule motion:
µm² DAWStx τ/=with
• Statistics of phase derivatives
up to the 6th power in time
with evolution time: DAWS field correlation function[2]:
• Phase correlation:
0 .1 1 101E 4
1E 3
0 .01
0 .1
1
E X P E R IM E N T T H E O R Y (no cro ssover) T H E O R Y (t
c = 1 0τ
D AW S)
T H E O R Y (tc = 7τ
D AW S)
T H E O R Y (tc = 5τ
D AW S)
τ / τD A W S
CΦ
' τ 2
DA
WS
0.1 1 101E 4
1E 3
0 .01
0 .1
1
E X P E R IM E N T T H E O R Y (no crossover) T H E O R Y ( t
c = 10τ
D A W S)
T H E O R Y ( tc = 5τ
D AW S)
T H E O R Y ( tc = 4τ
D AW S)
τ / τD A W S
CΦ
' τ 2
DA
WS
m=1/2
Ballisticstop crossover model: m=1
2
2 22
1( ) m
c
relrel
Vr
0 2 4 6 8 1 00 .5
0 .4
0 .3
0 .2
0 .1
0 .0 E X P E R IM E N T (e rro r b a rs a re
s ta n d a rd e rro r fo r 9 tr ia ls ) T H E O R Y (τ
c = 1 0 τ
D A W S)
T H E O R Y (τc = 7 τ
D A W S)
T H E O R Y (τc = 5 τ
D A W S)
<Φ
C(
τ /2
) Φ
C(τ
/2
)>
τ /τD A W S
Good overall agreement, need the cumulative correlation function to discriminate between the two crossover models
A superposition of waves scattered by a disordered medium gives rise to a speckle pattern. It has already been shown that field correlations and intensity could provide information on the scatterers. Our motivation to study phase are the following: it is a genuine property of wave, it provides additional
information, there is no need to normalize the field with respect to the sensivity of sensors or to the source magnitude and finally, contrary to optics, in seismology, ultrasound acoustics and microwaves it is possible to measure the field directly; the wave length allows to compute the phase quite easily.
Congrès général de la Société Française de Physique – 913 Juillet 2007
• Phase statistics:Lamb waves scattered by cylindrical holes in a 2m2 plexiglass plate:
Setup:
Preliminary results:
k=111,8m1
l=1,4m
From the fitting parameters Q and R we can calculate:
Good agreement betwenn theory and experiment.