Experimental and numerical investigations on parametric antenna operation in granular

Experimental and numerical investigations on parametric antenna operation in granular

materials

V. Tournat ^∗ , V. Aleshin ^∗ , V.E. Gusev ^† , B. Castagnède ^∗

∗ Laboratoire d’Acoustique de l’Université du Maine, UMR-CNRS 66-13, Université du Maine, Av.

Olivier Messiaen, 72000 Le Mans cedex 9, France ; E-mail : vincent.tournat@univ-lemans.fr ;

† Laboratoire de Physique de l’État Condensé, UMR-CNRS , Université du Maine, Av. Olivier Messiaen, 72000 Le Mans cedex 9, France ;

Abstract. Experimental and numerical results obtained in an unconsolidated granular ma- terial (glass beads of 150µm diameter) are presented. An important feature of such media, compared to homogeneous media, is the scattering of the waves due to the non-ideal pa- cking of the beads and the transformation of the propagative modes into the localized, when the wavelength becomes of the order of the characteristic size of the micro-structure (bead diameter). The process of self-demodulation in the case of pump waves diffusion is similar to thermal expansion caused by phonon conductivity in solids. A numerical scheme based on a recent model of parametric antenna for granular media is developed. Wide band demo- dulated secondary signals are compared to a numerical fit of a three dimensional model to obtain characteristic parameters of the transport of primary high frequency acoustic waves that can’t be detected due to their strong attenuation.

INTRODUCTION

An experiment and the associated numerical computation based on a recent theo- retical model of parametric transmitting antenna in granular materials [1] are presented.

In the theoretical model, frequency dependent absorption and scattering of the primary

waves are taken into account, and velocity dispersion through the difference of primary

and demodulated waves velocities. This model is analyzed in the case of 1-D geometry to

get qualitative tendencies. However, due to the influence of secondary signal diffraction,

changes in the emitted primary waves directivity pattern, it is necessary, when studying

quantitatively demodulated profiles, to take into account the three-dimensional geome-

try. Numerical computation is then necessary to fit experimental results.

NUMERICAL COMPUTATION Experimental situation

The experimental apparatus is presented on Fig.1. The ultrasonic emitter is a wide bandwidth piezo-electric transducer centered on 100kHz. The receiver is the same trans- ducer but used in the low frequency part of it’s bandwidth, where no change in it’s direc- tivity occurs. We measured that output electrical signal was proportional to the particle acceleration in the medium. Glass beads are 0.15mm diameter. The static force applied on the assembly of glass beads can be varied from 10N to 10kN (from 300Pa to 3MPa), which corresponds to initial static deformations of the medium from 5.10 ⁻⁶ to 2, 3.10 ⁻³ . The optimal distance between the emitter and the receiver in order to avoid reflections on the wall container influencing the direct broadband demodulated signal is around 6.5cm.

In this case, reflections arises 2.6 times later the first received signal.

Theoretical model

20 cm

screw to apply the static force

ultrasonic receiver container rigid armature

Ultrasonic emitter

force sensor glass beads

container plexiglas wall

F IG . 1. Experimental setup

The numerical computation is based on a theoretical model sub- mitted last year [1]. This model of parametric antenna in granular me- dia takes into account frequency de- pendent absorption, scattering and dispersion of longitudinal acoustic waves. A classical quadratic non- linearity is derived from a Taylor ex- pansion of the Hertz non-linearity of contacts between beads, neglecting clapping contacts [3, 4].

Following a traditional ap- proach [5], the 3-D propagation equation for the demodulated wave is written in this form :

∂ ²

∂t ² − c ² ₀ ∆ _⊥ − c ² ₀ ∂ ²

∂x ²

U _Ω = − ε ρ 0

∂

∂x hW _ω i, (1)

where c ₀ is the phase sound velocity in the granular medium, ρ ₀ its density at rest, ∆ _⊥ the transverse Laplacian, U _Ω the demodulated wave displacement along the x direction and hW _ω i the total energy density of the primary waves. This last is divided into two parts hW _ωb i for ballistically propagating primary waves (described with the propagation equation 2) and hW _ωd i for diffusive ones (described with the diffusion equation 3).

∂

∂t hW _ωb i + c _g (ω) ∂

∂x hW _ωb i + 1

τ(ω) hW _ωb i = 0 (2)

∂

∂t hW _ωd i = D(ω) ∂ ²

∂x ² + ∆ _⊥

hW _ωd i − 1

τ a (ω) hW _ωd i + 1

τ s (ω) hW _ωb i (3) with c _g (ω) the group velocity of primary waves packets, τ(ω) the attenuation time, τ _a (ω) the absorption time, τ s (ω) the scattering time ( ¹ _τ = _τ ¹ _a + _τ ¹ _s ), and D(ω) the diffusivity assumed to be equal to τ _s c ² _g /3.

Boundary conditions associated with this system of equations are zero low- frequency demodulated displacement in the plan of the primary waves emitter, frequency dependent directivity pattern on the surface of the primary waves emitter, continuity of energy flux at the emitter, and zero diffusive energy flux at the emitter. Directivity pat- terns have been measured using a vibrometer on the emitter surface for each emitting frequency. A numerical study revealed that, in our case, influence of this primary direc- tivity pattern on the demodulated wave profile is negligible. Calculation of the solution of this problem is done numerically using Fourier transforms (time and low frequency) and Hankel transforms (radius and radial wave numbers).

An optimization function is used to minimize differences between experimental and numerical profiles. Parameters c ₀ , c _g , τ a , τ s and ε are adjustable for the optimiza- tion process. Optimizations have been performed over a wide range of primary waves frequencies (40-1000kHz), for different static pressures (0,3-300kPa) and propagation distances (6-12cm).

FIRST RESULTS

Numerical fits of two experimental demodulated signals are presented on Fig.2.

Agreement is very good until the time 0.6ms (5 − 10%), where the experimental si- gnal begins to oscillate. These oscillations can be associated to the finite frequency response of the receiver especially in the very low frequency range (under 200Hz).

Another possible explanation is the fact we do not take into account in the model intermediate coherent effects due to scattering on the propagation of primary waves, considering scattered waves as diffusive. Moreover we neglect the process of mode conversion (between longitudinal and shear waves), that we have experimentally ob- served. So, it’s possible to estimate roughly the parameters of the granular medium using individual fits. For example, in the case illustrated in Fig.2 these parameters are c ₀ = 225m/s, c _g = 202, 5m/s, τ a = 0, 35ms for the 70kHz frequency of primary waves and c ₀ = 225m/s, c _g = 202, 0m/s, τ _a = 0, 19ms for 140kHz. Nevertheless, in the limits of the proposed approach, it’s difficult to extract frequency or pressure dependencies of these values.

CONCLUSIONS AND PERSPECTIVES

A numerical scheme associated with a model of parametric antenna in granular

materials is in development to fit experimentally obtained demodulated profiles. The

improvement of experimental (detailed frequency response of the receiver), theoretical

(scattering, mode conversion, non-linearity) and numerical parts is in progress. The ulti- mate goal of this analysis is to extract a frequency dependence of the model parameters, i.e. to have information on velocity dispersion, absorption and scattering of high fre- quency primary waves that can’t be transmitted through the sample thickness (and can’t be detected). Consequently, this work may find some applications in the non-destructive testing of materials.

0,2 0,4 0,6 0,8

-1,2 -1,0 -0,8 -0,6 -0,4 -0,2 0,0 0,2 0,4 0,6

0,2 0,4 0,6 0,8

-1,2 -1,0 -0,8 -0,6 -0,4 -0,2 0,0 0,2 0,4 0,6

Frequency of primary wave: 70kHz

Normalized demodulated profiles

Time (ms)

Frequency of primary wave: 140kHz

Normalized demodulated profiles

Time (ms)

F IG . 2. Numerical fits (in dashed line) of experimentally obtained demodulated profiles (in continuous line) for primary waves frequency of 70kHz and 140kHz respectively. Propagation distance is 9,3cm and static pressure is 300Pa.

ACKNOWLEDGEMENTS

This work is supported in the frame of DGA contract n ^o 00.34.026.

REFERENCES

1. Tournat, V., Gusev, V.E., Castagnède, B., “The influence of ballistics to diffusion transition in primary waves propagation on parametric antenna operation in granular media”, Phys. Rev. E, accepted for publication (2002).

2. Moussatov, A., Castagnède, B., Gusev, V.E., “Observation of nonlinear interaction of acoustic waves in granular materials : demodulation process”, Phys. Lett. A, 283, 216–223 (2001).

3. Zaitsev, V.Y., Kolpakov, A.B., Nazarov, V.E., “Detection of acoustic pulses in river sand : Experiment”, Acous. Phys., 45, 202-208 (1999).

4. Zaitsev, V.Y., Kolpakov, A.B., Nazarov, V.E., “Detection of acoustic pulses in river sand : Theory”, Acous. Phys., 45, 305-310 (1999).

5. Novikov, B.K., Rudenko, O.V., Timochenko, V.I., Nonlinear Underwater Acoustics (ASA, NewYork,

1987).