2D and 3D Reconstruction Algorithms in the Fourier Domain for PlaneWave Imaging in Nondestructive Testing. Merabet, L., S. Robert, and C. Prada. IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control 66, no. 4 (2019): 772–788.
Résumé: © 19862012 IEEE. Timedomain planewave imaging (PWI) has recently emerged in medical imaging and is now taking to nondestructive testing (NDT) due to its ability to provide images of good resolution and contrast with only a few steered plane waves. Insonifying a medium with plane waves is a particularly interesting approach in 3D imaging with matrix arrays because it allows to tremendously reduce the volume of data to be stored and processed as well as the acquisition time. However, even if the data volume is reduced with plane wave emissions, the image reconstruction in the time domain with a delayandsum algorithm is not sufficient to achieve low computation times in 3D due to the number of voxels. Other reconstruction algorithms take place in the wavenumberfrequency (fk) domain and have been shown to accelerate computation times in seismic imaging and in synthetic aperture radar. In this paper, we start from timedomain PWI in 2D and compare it to two algorithms in the fk domain, coming from the Stolt migration in seismic imaging and the Lu theory of limited diffraction beams in medical imaging. We then extend them to immersion testing configurations where a linear array is facing a plane watersteel interface. Finally, the reconstruction algorithms are generalized to 3D imaging with matrix arrays. A comparison dwelling on image quality and algorithmic complexities is provided, as well as a theoretical analysis of the image amplitudes and the limits of each method. We show that the reconstruction schemes in the fk domain improve the lateral resolution and offer a theoretical and numerical computation gain of up to 36 in 3D imaging in a realistic NDT configuration.
MotsClés: 2D and 3D imaging; nondestructive testing (NDT); transducer arrays; ultrasounds; wavenumberfrequency (fk) migration


Negative reflection of elastic guided waves in chaotic and random scattering media. Gérardin, B., J. Laurent, F. Legrand, C. Prada, and A. Aubry. Scientific reports 9, no. 1 (2019): 2135.
Résumé: The propagation of waves in complex media can be harnessed either by taming the incident wavefield impinging on the medium or by forcing waves along desired paths through its careful design. These two alternative strategies have given rise to fascinating concepts such as time reversal or negative refraction. Here, we show how these two processes are intimately linked through the negative reflection phenomenon. A negative reflecting mirror converts a wave of positive phase velocity into its negative counterpart and vice versa. In this article, we experimentally demonstrate this phenomenon with elastic waves in a 2D billiard and in a disordered plate by means of laser interferometry. Despite the complexity of such configurations, the negatively reflected wave field focuses back towards the initial source location, thereby mimicking a phase conjugation operation while being a fully passive process. The superfocusing capability of negative reflection is also highlighted in a monochromatic regime. The negative reflection phenomenon is not restricted to guided elastic waves since it can occur in zerogap systems such as photonic crystals, chiral metamaterials or graphene. Negative reflection can thus become a tool of choice for the control of waves in all fields of wave physics.


Negative refraction of Lamb modes: A theoretical study. Legrand, F., B. Gérardin, J. Laurent, C. Prada, and A. Aubry. Physical Review B 98, no. 21 (2018).
Résumé: © 2018 American Physical Society. This paper provides a theoretical investigation of negative refraction and focusing of elastic guided waves in a freestanding plate with a steplike thickness change. Under certain conditions, a positive phase velocity (forward) Lamb mode can be converted into a negative phase velocity (backward) mode at such interface, giving rise to negative refraction. A semianalytical model is developed to study the influence of various parameters such as the material Poisson's coefficient, the steplike thickness, the frequency, and the incidence angle. To this end, all the Lamb and shear horizontal propagating modes and also a large number of their inhomogeneous and evanescent counterparts are taken into account. The boundary conditions applied to the stressdisplacement fields at the thickness step yields an equation system. Its inversion provides the transmission and reflection coefficients between each mode at the interface. The steplike thickness and Poisson's ratio are shown to be key parameters to optimize the negative refraction process. In terms of material, Duralumin is found to be optimal as it leads to a nearly perfect conversion between forward and backward modes over broad frequency and angular ranges. An excellent focusing ability is thus predicted for a flat lens made of two symmetric thickness steps. A laser ultrasonic experiment quantitatively confirms those theoretical predictions. This study paves the way toward the optimization of elastic devices based on negative refraction, in particular for cloaking or superfocusing purposes.


Towards realtime assessment of anisotropic plate properties using elastic guided waves. Bochud, N., J. Laurent, F. Bruno, D. Royer, and C. Prada. Journal of the Acoustical Society of America 143, no. 2 (2018): 1138–1147.
Résumé: © 2018 Acoustical Society of America. A method to recover the elastic properties, thickness, or orientation of the principal symmetry axes of anisotropic plates is presented. This method relies on the measurements of multimode guided waves, which are launched and detected in arbitrary directions along the plate using a multielement linear transducer array driven by a programmable electronic device. A modelbased inverse problem solution is proposed to optimally recover the properties of interest. The main contribution consists in defining an objective function built from the dispersion equation, which allows accounting for higherorder modes without the need to pair each experimental data point to a specific guided mode. This avoids the numerical calculation of the dispersion curves and errors in the mode identification. Compared to standard rootfinding algorithms, the computational gain of the procedure is estimated to be on the order of 200. The objective function is optimized using genetic algorithms, which allow identifying from a single outofsymmetry axis measurement the full set of anisotropic elastic coefficients and either the plate thickness or the propagation direction. The efficiency of the method is demonstrated using data measured on materials with different symmetry classes. Excellent agreement is found between the reported estimates and reference values from the literature.


Effect of microstructural elongation on backscattered field: Intensity measurement and multiple scattering estimation with a linear transducer array. Baelde, A., J. Laurent, P. Millien, R. Coulette, W. B. Khalifa, F. Jenson, F. Sun, M. Fink, and C. Prada. Ultrasonics 82 (2018): 379–389.
Résumé: © 2017 Elsevier B.V. The effect of microstructural elongation on ultrasonic backscattered fields was studied. Two methods for determining the elongation direction of macrozones in titanium alloys, using the anisotropic spatial coherence of the backscattered field, are presented. Both methods use a phased array attached on a rotative holder that records the array response matrix at several angles. Two titanium alloys were investigated: TA6V and Ti17. TA6V exhibited a strong macrozone elongation, whereas Ti17 macrozones were found equiaxial. The first method is based on the measurement of backscattered intensity in function of the probe angle relative to the macrozones elongation direction. An angular dependence of backscattered intensity is observed in presence of elongated scatterers, and their elongation direction is collinear with the probe direction corresponding to a minimal intensity. This variability is linked to both piezoelectric shape and the backscattered field spatial properties. The second method is based on the measurement of the relative proportion of single to multiple scattering in a diffusive media, using a simplified version of the single scattering filter developed in Aubry and Derode (2009). It allows the measurement of the level of multiple scattering: both titanium alloys exhibited strong multiple scattering. The elongation direction was determined as the direction of minimal multiple scattering. Furthermore, these results were confirmed by the measurement of the coherent backscattering cone on both samples.


Ultrasonic Imaging in Highly Attenuating Materials with Hadamard Codes and the Decomposition of the Time Reversal Operator. Lopez Villaverde, E., S. Robert, and C. Prada. IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control 64, no. 9 (2017): 1336–1344.
Résumé: © 19862012 IEEE. In this paper, defects in a high density polyethylene pipe are imaged with the total focusing method. The viscoelastic attenuation of this material greatly reduces the signal level and leads to a poor signaltonoise ratio (SNR) due to electronic noise. To improve the image quality, the decomposition of the time reversal operator method is combined with the spatial Hadamard coded transmissions before calculating images in the time domain. Because the Hadamard coding is not compatible with conventional imaging systems, this paper proposes two modified coding methods based on sparse Hadamard matrices with +1/0 coefficients. The SNRs expected with the different spatial codes are demonstrated, and then validated on both simulated and experimental data. Experiments are performed with a transducer array in contact with the base material of a polyethylene pipe. In order to improve the noise filtering procedure, the singular values associated with electronic noise are expressed on the basis of the random matrix theory. This model of noise singular values allows a better identification of the defect response in noisy experimental data. Finally, the imaging method is evaluated in a more industrial inspection configuration, where an immersion array probe is used to image defects in a butt fusion weld with a complex geometry.
MotsClés: Attenuating material; defect characterization; nondestructive evaluation (NDE); time reversal; ultrasonic array imaging


Causality, Nonlocality, and Negative Refraction. Forcella, D., C. Prada, and R. Carminati. Physical Review Letters 118, no. 13 (2017).
Résumé: © 2017 American Physical Society. American Physical Society.The importance of spatial nonlocality in the description of negative refraction in electromagnetic materials has been put forward recently. We develop a theory of negative refraction in homogeneous and isotropic media, based on first principles, and that includes nonlocality in its full generality. The theory shows that both dissipation and spatial nonlocality are necessary conditions for the existence of negative refraction. It also provides a sufficient condition in materials with weak spatial nonlocality. These fundamental results should have broad implications in the theoretical and practical analyses of negative refraction of electromagnetic and other kinds of waves.


Spatial Laplace transform for complex wavenumber recovery and its application to the analysis of attenuation in acoustic systems. Geslain, A., S. Raetz, M. Hiraiwa, M. A. Ghanem, S. P. Wallen, A. Khanolkar, N. Boechler, J. Laurent, C. Prada, A. Duclos et al. Journal Of Applied Physics 120, no. 13 (2016).


Nonlinear secondary noise sources for passive defect detection using ultrasound sensors. Chehami, L., E. Moulin, J. De Rosny, C. Prada, E. Chatelet, G. Lacerra, K. Gryllias, and F. Massi. Journal Of Sound And Vibration 386 (2017): 283–294.
MotsClés: Frictioninduced vibrations; Noise correlation; Defect localisation


Laser beam shaping for enhanced ZeroGroup Velocity Lamb modes generation. Bruno, F., J. Laurent, P. Jehanno, D. Royer, and C. Prada. Journal Of The Acoustical Society Of America 140, no. 4 (2016): 2829–2838.


Ultrasonic imaging of defects in coarsegrained steels with the decomposition of the time reversal operator. Villaverde, E. L., S. Robert, and C. Prada. Journal Of The Acoustical Society Of America 140, no. 1 (2016): 541–550.


Particlelike wave packets in complex scattering systems. Gerardin, B., J. Laurent, P. Ambichl, C. Prada, S. Rotter, and A. Aubry. Physical Review B 94, no. 1 (2016).


Negative reflection of Lamb waves at a free edge: Tunable focusing and mimicking phase conjugation. Gerardin, B., J. Laurent, C. Prada, and A. Aubry. The Journal of the Acoustical Society of America 140, no. 1 (2016): 591.


Measurement of dispersion curves of circumferential guided waves radiating from curved shells: Theory and numerical validation. Chekroun, M., J. G. Minonzio, C. Prada, P. Laugier, and Q. Grimal. Journal of the Acoustical Society of America 139, no. 2 (2016): 790–799.
Résumé: © 2016 Acoustical Society of America. A method is proposed to evaluate in a noncontact way the phase velocity dispersion curves of circumferential waves around a shell of arbitrary shape immersed in a fluid. No assumptions are made about the thickness or the material of the shell. A geometrical model is derived to describe the shape of the radiated wavefronts in the surrounding fluid, and predict the positions of its centers of curvature. Then the timereversal principle is applied to recover these positions and to calculate the phase velocity of the circumferential waves. Numerical finitedifference simulations are performed to evaluate the method on a circular and on an elliptic thin shell. Different dispersion curves can be recovered with an error of less than 10%.


Effect of refracted light distribution on the photoelastic generation of zerogroup velocity Lamb modes in optically lowabsorbing plates. Raetz, S., J. Laurent, T. Dehoux, D. Royer, B. Audoin, and C. Prada. Journal of the Acoustical Society of America 138, no. 6 (2015): 3522–3530.
Résumé: © 2015 Acoustical Society of America. Zerogroup velocity (ZGV) Lamb modes are associated with sharp local acoustic resonances and allow, among other features, local measurement of Poisson's ratio. While the thermoelastic generation of Lamb waves in metal plates has been widely studied, the case of materials of lowoptical absorption remains unexplored. In materials such as glasses, the generation of bulk elastic waves has been demonstrated to be sensitive to the refracted light distribution. In this paper, a detailed analysis of the effect of light refraction on the laserbased generation of ZGV Lamb modes is presented. Experiments are performed on a bare glass plate without the need for an additional layer for light absorption or reflection. Using an appropriate tilted volume source, it is shown that the laserultrasonic technique allows noncontact measurement of the Poisson's ratio.


Investigation of interfacial stiffnesses of a trilayer using ZeroGroup Velocity Lamb modes. Mezil, S., F. Bruno, S. Raetz, J. Laurent, D. Royer, and C. Prada. Journal of the Acoustical Society of America 138, no. 5 (2015): 3202–3209.
Résumé: © 2015 Acoustical Society of America. ZeroGroup Velocity (ZGV) Lamb waves are studied in a structure composed of two plates bonded by an adhesive layer. The dispersion curves are calculated for a Duralumin/epoxy/Duralumin sample, where the adhesion is modeled by a normal and a tangential spring at both interfaces. Several ZGV modes are identified and their frequency dependence on interfacial stiffnesses and on the bonding layer thickness is numerically studied. Then, experiments achieved with laser ultrasonic techniques are presented. Local resonances are measured using a superimposed source and probe. Knowing the thicknesses and elastic constants of the Duralumin and epoxy layers, the comparison between theoretical and experimental ZGV resonances leads to an evaluation of the interfacial stiffnesses. A good agreement with theoretical dispersion curves confirms the identification of the resonances and the parameter estimations. This noncontact technique is promising for the local evaluation of bonded structures.


Experimental study of passive defect localization in plates using ambient noise. Chehami, L., J. D. Rosny, C. Prada, E. Moulin, and J. Assaad. IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control 62, no. 8 (2015): 1544–1553.
Résumé: © 2014 IEEE. Passive listening methodology has been shown to be a practical and effective method for passive structural health monitoring. In this work, this approach is applied experimentally to monitor the occurrence of defects in thin aluminum plates. A correlation matrix is estimated from noise vibrations recorded on a transducer array. A defect is localized by applying a beamforming algorithm to the difference between the correlation matrices obtained with and without the defect. We successfully detect defects for different kinds of noise sources. Moreover, we show that this technique is robust to detect massive inclusions, holes, and cracks. With a vibrometer, we observe that the fidelity of the estimated transient responses strongly depends on the number of uncorrelated noise sources. Finally, we show that the defect is successfully localized even if the noise source distribution is not uniform, provided that it remains spatially stationary between the states with and without defect. A simple theoretical framework is proposed to interpret these results.


Laser induced zerogroup velocity resonances in transversely isotropic cylinder. Laurent, J., D. Royer, T. Hussain, F. Ahmad, and C. Prada. Journal Of The Acoustical Society Of America 137, no. 6 (2015): 3325–3334.


Focusing on plates: Controlling guided waves using negative refraction. Philippe, F. D., T. W. Murray, and C. Prada. Scientific Reports 5 (2015).
Résumé: Elastic waves are guided along finite structures such as cylinders, plates, or rods through reflection, refraction, and mode conversion at the interfaces. Such wave propagation is ubiquitous in the world around us, and studies of elastic waveguides first emerged in the later part of the 19 th century. Early work on elastic waveguides revealed the presence of backward propagating waves, in which the phase velocity and group velocity are antiparallel. While backward wave propagation exists naturally in very simple finite elastic media, there has been remarkably little attention paid to this phenomenon. Here we report the development of a tunable acoustic lens in an isotropic elastic plate showing negative refraction over a finite acoustic frequency bandwidth. As compared to engineered acoustic materials such as phononic crystals and metamaterials, the design of the acoustic lens is very simple, with negative refraction obtained through thickness changes rather than internal periodicity or subwavelength resonant structures. A new class of acoustic devices, including resonators, filters, lenses, and cloaks, may be possible through topography optimization of elastic waveguide structures to exploit the unique properties of backward waves.


Tomographic diffractive microscopy with agile illuminations for imaging targets in a noisy background. Zhang, T., C. Godavarthi, P. C. Chaumet, G. Maire, H. Giovannini, A. Talneau, C. Prada, A. Sentenac, and K. Belkebir. Optics Letters 40, no. 4 (2015): 573–576.


Circumferential guided wave measurements of a cylindrical fluidfilled bonemimicking phantom. Nauleau, P., Q. Grimal, J. G. Minonzio, P. Laugier, and C. Prada. Journal of the Acoustical Society of America 135, no. 2 (2014): 994–1001.
Résumé: In the context of hip fracture risk prediction, measurement of guided waves could improve the assessment of cortical femoral neck properties. The decomposition of the time reversal operator (DORT) method was previously shown to be efficient to measure circumferential guided modes in an empty cortical bonemimicking tube of circular cross section. In this study, an adaptation of the DORT method is proposed to probe the same bonemimicking tube but filled with a marrowmimicking fluid. The contributions to the backscattered field of waves multiply reflected in the cavity of the tube interfere with those of circumferential guided waves. The former contributions are eliminated in the backpropagation image using ad hoc criterion determined with simulation. Eight portions of different guided modes were observed from experimental and simulated data. They were identified by comparison with theoretical predictions. This work confirms the feasibility of measuring guided waves in a fluidfilled tube of bonemimicking material with the DORT method. © 2014 Acoustical Society of America.


Full Transmission and Reflection of Waves Propagating through a Maze of Disorder. Gerardin, B., J. Laurent, A. Derode, C. Prada, and A. Aubry. Physical Review Letters 113, no. 17 (2014).


Non contact probing of interfacial stiffnesses between two plates by zerogroup velocity Lamb modes. Mezil, S., J. Laurent, D. Royer, and C. Prada. Applied Physics Letters 105, no. 2 (2014).


Temporal behavior of laser induced elastic plate resonances. Laurent, J., D. Royer, and C. Prada. Wave Motion 51, no. 6 (2014): 1011–1020.
MotsClés: Lamb modes; Zerogroup velocity resonance; Laserultrasound; Power law decay


Detection and localization of a defect in a reverberant plate using acoustic field correlation. Chehami, L., E. Moulin, J. De Rosny, C. Prada, O. B. Matar, F. Benmeddour, and J. Assaad. Journal Of Applied Physics 115, no. 10 (2014).


Active detection of a moving target in a waveguide with strong masking echoes. Benoit, Y., and C. Prada. The Journal of the Acoustical Society of America 133, no. 5 (2013): 3526.


Invariants of the time reversal operator and characterization of solid media: An overview. Prada, C. The Journal of the Acoustical Society of America 134, no. 5 (2013): 4033.


Data set reduction for ultrasonic TFM imaging using the effective aperture approach and virtual sources. Bannouf, S., S. Robert, O. Casula, and C. Prada. Journal of Physics: Conference Series 457 (2013).
Résumé: The Total Focusing Method (TFM) is sometimes described in the literature as the “gold standard” compared to conventional imaging techniques. TFM is an algorithm that usually postprocesses the full matrix of data, also called Full Matrix Capture (FMC). Realtime piloting of such an algorithm is heavy due to the large number of firings (N for a Nelement array) and the large number of signals (N×N) to process that tend to decrease the frame rate and, consequently, the inspection speed. This problem can be overcome to some extent if only a few elements are activated which is equivalent to using a sparse array in transmit. The PSF (Point Spread Function) provides information about important images parameters: lateral resolution and contrast. An algorithm based on PSF optimization is proposed to obtain both the number of transmit pulses and the location of the active elements. However, reducing the number of emissions induces a loss in transmitted energy. To compensate it, each transmit pulse is carried out by multiple transmit elements that emulate a single “virtual” element. The method is evaluated on experimental data in a realistic NDT configuration by comparison of images obtained with FMC and SMC (Sparse Matrix Capture) acquisitions. © Published under licence by IOP Publishing Ltd.
MotsClés: Active elements; Conventional imaging; Effective aperture; Full matrix captures; Lateral resolution; Multiple transmit; Sparse matrices; Virtual sources; Imaging techniques; Matrix algebra; Nondestructive examination; Optical transfer function; Ultrasonic applications; Algorithms


Analysis of the time reversal operator for a scatterer undergoing small displacements. Philippe, F. D., C. Prada, M. Fink, J. Garnier, and J. De Rosny. Journal of the Acoustical Society of America 133, no. 1 (2013): 94–107.
Résumé: The method of the time reversal operator decomposition is usually employed to detect and characterize static targets using the invariants of the time reversal operator. This paper presents a theoretical and experimental investigation into the impact of small displacements of the target on these invariants. To find these invariants, the time reversal operator is built from the multistatic response matrix and then diagonalized. Two methods of recording the multistatic response matrix while the target is moving are studied: Acquisition either element by element or column by column. It is demonstrated that the target displacement generates new significant eigenvalues. Using a perturbation theory, the analytical expressions of the eigenvalues of the timereversal operator for both acquisition methods are derived. We show that the distribution of the new eigenvalues strongly depends on these two methods. It is also found that for the column by column acquisition, the second eigenvector is simply linked to the scatterer displacements. At last, the implications on the Maximum Likelihood and Multiple Signal Classification detection are also discussed. The theoretical results are in good agreement with numerical and 3.4 MHz ultrasonic experiments. © 2013 Acoustical Society of America.
MotsClés: Analytical expressions; Eigenvalues; Element by elements; Experimental investigations; Multistatic; Multiple signal classification; Perturbation theory; Response matrices; Small displacement; Target displacement; Theoretical result; Time reversal operator decompositions; Timereversal operator; Ultrasonic experiments; Mergers and acquisitions; Ultrasonic applications; Wavelet analysis; Eigenvalues and eigenfunctions; water; algorithm; article; computer simulation; equipment; equipment design; mathematical computing; methodology; motion; nonlinear system; pressure; radiation scattering; signal processing; sound; sound detection; statistical model; theoretical model; time; transducer; ultrasound; vibration; Algorithms; Computer Simulation; Equipment Design; Likelihood Functions; Linear Models; Models, Theoretical; Motion; Nonlinear Dynamics; Numerical Analysis, ComputerAssisted; Pressure; Scattering, Radiation; Signal Processing, ComputerAssisted; Sound; Sound Spectrography; Time Factors; Transducers; Ultrasonics; Vibration; Water


Characterization of mechanical properties of a hollow cylinder with zero group velocity Lamb modes. Cès, M., D. Royer, and C. Prada. Journal of the Acoustical Society of America 132, no. 1 (2012): 180–185.
Résumé: Hollow cylinders used in the industry must be regularly inspected. Elastic guided waves, similar to Lamb modes in a plate, can propagate in the axial direction or around the circumference. They are sensitive to geometrical and mechanical parameters of the cylindrical shell. The objective of this paper is to show that zero group velocity (ZGV) Lamb modes can be used to bring out anisotropy and to measure elastic constants of the material. This study provides experimental and numerical investigations on a Zirconium alloy tube extensively used by the nuclear industry in reactor core components. A noncontact method, based on laser ultrasound techniques and ZGV Lamb modes, demonstrates that the difference observed between axial and circumferential guided waves cannot be explained by an isotropic model. Then, a transverse isotropic model is used for the Zircaloy tube. Four of the five elastic constants are directly extracted from ZGV resonance frequencies. The last one is deduced from the measured dispersion spectra. With this complete set of constants, a good agreement is obtained between theoretical and experimental dispersion curves for both axially and circumferentially propagating guided waves. © 2012 Acoustical Society of America.
MotsClés: Axial direction; Circumferential guided wave; Core components; Cylindrical shell; Dispersion curves; Dispersion spectra; Elastic guided waves; Hollow cylinders; Isotropic models; Lamb modes; Laser ultrasound technique; Mechanical parameters; Noncontact methods; Numerical investigations; Resonance frequencies; Transverse isotropic; Zerogroup velocity; Zircaloy; Cylinders (shapes); Elastic constants; Light velocity; Nuclear industry; Tubes (components); Zirconium alloys; Guided electromagnetic wave propagation


Characterization of circumferential guided waves in a cylindrical cortical bonemimicking phantom. Nauleau, P., E. Cochard, J.  G. Minonzio, Q. Grimal, P. Laugier, and C. Prada. Journal of the Acoustical Society of America 131, no. 4 (2012): EL289–EL294.
Résumé: The femoral neck cortical shell was recently demonstrated to act like a waveguide for circumferential waves. Femoral neck assessment with ultrasound could be enhanced by guided waves measurement. In this study, the decomposition of the time reversal operator (DORT) method is used to measure the phase velocities of circumferential guided modes in a circular tube with dimensions characteristic of femoral neck. The tube is made of a bonemimicking material. Five guided modes are obtained and compared to theoretical predictions. The work substantiates the feasibility of measuring guided waves in a relatively thick tube of attenuating material with the DORT method. © 2012 Acoustical Society of America.
MotsClés: Circular tubes; Circumferential guided wave; Circumferential waves; Cortical shell; Decomposition of the time reversal operator; Femoral necks; Guided modes; Theoretical prediction; Thick tubes; Bone; Tubes (components); Guided electromagnetic wave propagation; article; audiovisual equipment; bone density; echography; femur neck; human; physiology; ultrasound; Bone Density; Femur Neck; Humans; Models, Anatomic; Ultrasonics


Imaging changes in scattering media from Time Reversal of the Coda wave Difference (TRECOD). Bonneau, L., C. Prada, M. Fink, and A. Tourin. Waves in Random and Complex Media 22, no. 1 (2012): 109–120.
Résumé: We propose a new method for monitoring temporal changes in a complex scattering environment. It is referred to as Time Reversal of the Coda wave Difference (TRECOD). The impulse responses of the probed medium are recorded between all the emitter/receiver pairs of a transceiver array. A Fourier transform of these responses then produces the socalled backscattering transfer matrix at each frequency of the transceiver bandwidth. The matrix acquisition is repeated and the image of the temporal changes occurring between two matrix acquisitions is formed by propagating numerically the significant singular vectors of the matrix difference. Smallscale experiments with ultrasound are used to validate the method and demonstrate its interest for the monitoring of a fluid injection in a porous medium behind a strongly scattering obstacle. © 2012 Copyright Taylor and Francis Group, LLC.
MotsClés: Coda waves; Fluid injections; matrix; Porous medium; Scattering environment; Scattering media; Singular vectors; Smallscale experiment; Temporal change; Time reversal; Transceiver array; Transfer matrixes; Porous materials; Transceivers; Transfer matrix method; Scattering


Thin layer thickness measurements by zero group velocity Lamb mode resonances. Cès, M., D. Clorennec, D. Royer, and C. Prada. Review of Scientific Instruments 82, no. 11 (2011).
Résumé: Local and noncontact measurements of the thickness of thin layers deposited on a thick plate have been performed by using zero group velocity (ZGV) Lamb modes. It was shown that the shift of the resonance frequency is proportional to the mass loading through a factor which depends on the mechanical properties of the layer and of the substrate. In the experiments, ZGV Lamb modes were generated by a Nd:YAG pulsed laser and the displacement normal to the plate surface was measured by an optical interferometer. Measurements performed at the same point that the generation on the noncoated face of the plate demonstrated that thin gold layers of a few hundred nanometers were detected through a 1.5mm thick Duralumin plate. The shift of the resonance frequency (1.9 MHz) of the fundamental ZGV mode is proportional to the layer thickness: typically 10 kHz per m. Taking into account the influence of the temperature, a 240nm gold layer was measured with a 4 uncertainty. This thickness has been verified on the coated face with an optical profiling system. © 2011 American Institute of Physics.
MotsClés: Gold layer; Lamb modes; Layer thickness; Mass loadings; Nd:YAG pulsed laser; Noncontact measurements; Optical interferometer; Optical profiling; Plate surfaces; Resonance frequencies; Thick plate; Thin layer thickness measurement; Thin layers; Zerogroup velocity; Gold; Light velocity; Mechanical properties; Natural frequencies; Neodymium lasers; Thickness measurement; Gold coatings


Adaptive projection method applied to threedimensional ultrasonic focusing and steering through the ribs. Cochard, E., J. F. Aubry, M. Tanter, and C. Prada. Journal of the Acoustical Society of America 130, no. 2 (2011): 716–723.
Résumé: An adaptive projection method for ultrasonic focusing through the rib cage, with minimal energy deposition on the ribs, was evaluated experimentally in 3D geometry. Adaptive projection is based on decomposition of the timereversal operator (DORT method) and projection on the noise subspace. It is shown that 3D implementation of this method is straightforward, and not more timeconsuming than 2D. Comparisons are made between adaptive projection, spherical focusing, and a previously proposed timereversal focusing method, by measuring pressure fields in the focal plane and rib region using the three methods. The ratio of the specific absorption rate at the focus over the one at the ribs was found to be increased by a factor of up to eight, versus spherical emission. Beam steering out of geometric focus was also investigated. For all configurations projecting steered emissions were found to deposit less energy on the ribs than steering timereversed emissions: thus the noninvasive method presented here is more efficient than stateoftheart invasive techniques. In fact, this method could be used for realtime treatment, because a single acquisition of backscattered echoes from the ribs is enough to treat a large volume around the focus, thanks to real time projection of the steered beams. © 2011 Acoustical Society of America.
MotsClés: 3D geometry; Backscattered; Beamsteering; Focal Plane; Invasive techniques; Minimal energy; Noise subspace; Noninvasive methods; Pressure field; Projection method; Real time; Specific absorption rate; Timereversal focusing; Timereversal operator; Timereversed; Ultrasonic focusing; Noninvasive medical procedures; Three dimensional; Ultrasonics; Focusing; absorption; article; comparative study; equipment design; Fourier analysis; human; image quality; instrumentation; methodology; pressure; pressure transducer; radiation scattering; rib; signal processing; time; ultrasound therapy; Absorption; Equipment Design; Fourier Analysis; Humans; Phantoms, Imaging; Pressure; Ribs; Scattering, Radiation; Signal Processing, ComputerAssisted; Time Factors; Transducers, Pressure; Ultrasonic Therapy


Edge resonance and zero group velocity Lamb modes in a free elastic plate. Cès, M., D. Clorennec, D. Royer, and C. Prada. Journal of the Acoustical Society of America 130, no. 2 (2011): 689–694.
Résumé: The local resonances of a free isotropic elastic plate are investigated using laser ultrasonic techniques. Experimental results are interpreted in terms of zero group velocity Lamb modes and edge mode. At a distance from the edge larger than the plate thickness a sharp resonance is observed at the frequency where the group velocity of the first symmetrical Lamb mode vanishes. Close to the edge of the plate, the resonance due to the edge mode dominates. Both zero group velocity and edge resonances appear at the theoretically predicted frequencies. These frequencies do not vary with the distance from the edge of the plate and the transition between the two modes of vibration, at about the plate thickness, is abrupt. Using a laser excitation on the edge, the amplitude profile of the normal displacement at the edge resonance frequency was determined. © 2011 Acoustical Society of America.
MotsClés: Edge resonance; Elastic plate; Group velocities; Lamb modes; Laser ultrasonic techniques; Local resonance; Normal displacement; Plate thickness; Zerogroup velocity; Laser excitation; Light velocity; Ultrasonic testing; Resonance; article; elasticity; equipment design; instrumentation; interferometry; motion; solid state laser; sound detection; theoretical model; time; ultrasound; vibration; Elasticity; Equipment Design; Interferometry; Lasers, SolidState; Models, Theoretical; Motion; Sound Spectrography; Time Factors; Ultrasonics; Vibration


Thin layer Characterization by ZGV Lamb modes. Cès, M., D. Clorennec, D. Royer, and C. Prada. Journal of Physics: Conference Series 269, no. 1 (2011).
Résumé: Ultrasonic nondestructive testing of plates can be performed with Lamb modes guided by the structure. Non contact generation and detection of the elastic waves can be achieved with optical means such as a pulsed laser source and an interferometer. With this setup, we propose a method using zero group velocity (ZGV) Lamb modes rather than propagating modes. These ZGV modes have noteworthy properties, in particular their group velocity vanishes, whereas their phase velocity remains finite. Thus, a significant part of the energy deposited by the pulsed laser can be trapped in the source area. For example, in a homogeneous isotropic plate and at the minimum frequency of the S 1Lamb mode a very sharp resonance can be observed, the frequency of which only depends on the plate thickness, for a given material. In fact, other ZGV modes exist and the set of ZGV resonance frequencies provide a local and absolute measurement of Poisson's ratio. These nonpropagating modes can also be used to characterize multilayered structures. Experimentally, we observed that a thin (500 nm) gold layer deposited on a thick (1.5 mm) Duralumin plate induces a sensitive downshift of the set of ZGV resonance frequencies. This shift, which is typically 5 kHz for the S 1Lamb mode at 1.924 MHz, can be approximated by a formula providing the layer thickness. Thickness down to 100 nm can be estimated by this method. Such a sensitivity with conventional ultrasound inspection by acoustic microscopy would require an operating frequency in the GHz range. © Published under licence by IOP Publishing Ltd.
MotsClés: Absolute measurements; Acoustic microscopy; Gold layer; Group velocities; Isotropic plates; Lamb modes; Layer thickness; Multilayered structure; Noncontact; Operating frequency; Plate thickness; Poisson's ratio; Propagating mode; Pulsed laser sources; Resonance frequencies; Source area; Thin layers; Ultrasonic nondestructive testing; Zerogroup velocity; Gold coatings; Light velocity; Natural frequencies; Nondestructive examination; Plates (structural components); Poisson ratio; Ultrasonic applications; Ultrasonics; Pulsed lasers


Negative refraction and focusing of elastic Lamb waves at an interface. Bramhavar, S., C. Prada, A. A. Maznev, A. G. Every, T. B. Norris, and T. W. Murray. Physical Review B – Condensed Matter and Materials Physics 83, no. 1 (2011).
Résumé: We study negative refraction and focusing of elastic waves in a simple mechanical system comprised of a freestanding plate with a step change in thickness. A pointfocused and intensitymodulated laser source is used to excite backwardpropagating Lamb waves on one side of the step, and the displacement field is probed using an optical interferometer. Conversion between forward and backwardpropagating modes at the interface leads to negative refraction, and we demonstrate for the first time the operation of a flat lens, similar to that predicted by Veselago in negativeindex media, for guided elastic waves in isotropic media. We propose that guided elastic waves provide a convenient and powerful experimental test bed for the study of negativeindex physics. © 2011 American Physical Society.


Detection and imaging of human beings behind a wall using the dort method. Davy, M., T. Lepetit, J. de Rosny, C. Prada, and M. Fink. Progress in Electromagnetics Research 110 (2010): 353–369.
Résumé: In recent years, through the wall imaging has become a topic of intense research due to its promising applications in police, fire and rescue or emergency relief operations. In this paper, we propose to use the DORT method (French acronym for Decomposition of the Time Reversal Operator) to detect and localize a moving target behind a wall. One of the DORT method major strengths is that detection remains possible through a distorting medium. In this paper, the DORT method is successfully applied to detect and track moving human beings behind a thick concrete wall. The smallest detectable displacement is also investigated.
MotsClés: Concrete wall; Decomposition of the time reversal operator; Emergency relief; Human being; Moving targets; Through the wall imaging; Law enforcement; Walls (structural partitions)


Quantitative Assessment of Arterial Wall Biomechanical Properties Using Shear Wave Imaging. Couade, M., M. Pernot, C. Prada, E. Messas, J. Emmerich, P. Bruneval, A. Criton, M. Fink, and M. Tanter. Ultrasound in Medicine and Biology 36, no. 10 (2010): 1662–1676.
Résumé: A new ultrasoundbased technique is proposed to assess the arterial stiffness: the radiation force of an ultrasonic beam focused on the arterial wall induces a transient shear wave (~10 ms) whose propagation is tracked by ultrafast imaging. The large and highfrequency content (100 to 1500 Hz) of the induced wave enables studying the wave dispersion, which is shown experimentally in vitro and numerically to be linked to arterial wall stiffness and geometry. The proposed method is applied in vivo. By repeating the acquisition up to 10 times per second (theoretical maximal frame rate is ~100 Hz), it is possible to assess in vivo the arterial wall elasticity dynamics: shear modulus of a healthy volunteer carotid wall is shown to vary strongly during the cardiac cycle and measured to be 130 ± 15 kPa in systole and 80 ± 10 kPa in diastole. (Email: ). © 2010 World Federation for Ultrasound in Medicine & Biology.
MotsClés: Acoustic radiation force; Arterial stiffness; Elastography; Ultrasound; Vascular imaging; Acoustic radiation force; Arterial stiffness; Elastography; Ultrasound; Vascular imaging; Acoustic emissions; Acoustic radiators; Acoustic wave propagation; Acoustic wave transmission; Biomechanics; Shear waves; Stiffness; Ultrasonic testing; Ultrasonics; Shear flow; adult; animal experiment; arterial stiffness; artery compliance; artery wall; article; biomechanics; carotid artery; diastole; echography; heart cycle; human; human experiment; in vitro study; in vivo study; nonhuman; normal human; priority journal; quantitative analysis; systole; Young modulus; Biomechanics; Carotid Arteries; Elastic Modulus; Elasticity; Elasticity Imaging Techniques; Feasibility Studies; Humans; Image Processing, ComputerAssisted; Phantoms, Imaging; Reference Values; Shear Strength


Laser ultrasonic inspection of plates using zerogroup velocity lamb modes. Clorennec, D., C. Prada, and D. Royer. IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control 57, no. 5 (2010): 1125–1132.
Résumé: A noncontact laserbased ultrasonic technique is proposed for detecting small plate thickness variations caused by corrosion and adhesive disbond between two plates. The method exploits the resonance at the minimum frequency of the S1 Lamb mode dispersion curve. At this minimum frequency, the group velocity vanishes, whereas the phase velocity remains finite. The energy deposited by the laser pulse generates a local resonance of the plate. This vibration is detected at the same point by an optical interferometer. First experiments show the ability to image a 1.5m deep corroded area on the back side of a 0.5mmthick duralumin plate. Because of the finite wavelength of the S1 zero group velocity (ZGV) mode, the spatial resolution is limited to approximately twice the plate thickness. With the same technique we investigate the state of adhesive bonds between duralumin and glass plates. The S 1Lamb mode resonance is strongly attenuated when plates are rigidly bonded. In the case of thin adhesive layers, we observed other resonances, associated with ZGV modes of the multilayer structure, whose frequencies and amplitudes vary with adhesive thickness. Experiments were carried out on real automotive adhesively bonded structures and the results were compared with images obtained by Xray radiography. © 2010 IEEE.
MotsClés: Adhesive bond; Adhesive layers; Adhesive thickness; Adhesively bonded structures; Disbond; Finite wavelength; Glass plate; Group velocities; Lamb modes; Laser ultrasonics; Laserbased ultrasonics; Local resonance; Multilayer structures; Noncontact; Optical interferometer; Plate thickness; Spatial resolution; Two plates; Zerogroup velocity; Light velocity; Pulsed laser applications; Resonance; Ultrasonic testing; Ultrasonics; X ray radiography; Plates (structural components)


Experimental study of the invariants of the timereversal operator for a dielectric cylinder using separate transmit and receive arrays. Davy, M., J.  G. Minonzio, J. De Rosny, C. Prada, and M. Fink. IEEE Transactions on Antennas and Propagation 58, no. 4 (2010): 1349–1356.
Résumé: The decomposition of the time reversal operator (DORT method) is applied to electromagnetic waves in order to characterize a dielectric cylinder. It consists in determining the Time Reversal Invariants of the Time Reversal Operator. Here, this matrix is built from the interelement responses between distinct transmit and receive arrays. In this paper experimental results obtained between 2 and 4 GHz are compared to a theoretical model, developed in an other paper (Minonzio , Theory of the timereversal operator for a dielectric cylinder using separate transmit and receive arrays, IEEE Trans. Antennas Propag., vol. 57, pp. 23312340, 2009). The DORT method is then applied to an inverse problem to determine the diameter and the permittivity of the cylinder. It is shown experimentally that different experimental parameters can be estimated from the singular values of the time reversal operator. © 2006 IEEE.
MotsClés: Antenna array processing; Electromagnetic inverse problem; Electromagnetic scattering; Decomposition of the time reversal operator; Dielectric cylinder; Electromagnetic inverse problems; Electromagnetic scattering; Experimental parameters; Experimental studies; matrix; Singular values; Theoretical models; Time reversal; Timereversal operator; Antenna arrays; Array processing; Cylinders (shapes); Dielectric devices; Electromagnetic wave scattering; Electromagnetic waves; Inverse problems; Microwave frequencies; Quantum theory; Electromagnetism


Influence of noise on subwavelength imaging of two close scatterers using time reversal method: Theory and experiments. Davy, M., J.  G. Minonzio, J. de Rosny, C. Prada, and M. Fink. Progress in Electromagnetics Research 98 (2009): 333–358.
Résumé: Although classical imaging is limited by the Rayleigh criterion, it has been demonstrated that subwavelength imaging of two pointlike scatterers can be achieved with probing sensors arrays, even if the scatterers are located in the far field of the sensors. However, the role of noise is crucial to determine the resolution limit. This paper proposes a quantitative study of the influence of noise on the subwavelength resolution obtained with the DORTMUSIC method. The DORT method, French acronym for decomposition of the time reversal operator, consists in studying the invariants of the time reversal operator. The method is combined here with the estimator MUSIC (MUltiple SIgnal Classification) to detect and image two close metallic wires. The microwaves measurements are performed between 2.6GHz and 4GHz. Two wires of λ/100 diameters separated by λ/6 are imaged and separated experimentally. To interpret this result in terms of noise level, the analytical expression of the eigenvectors of the time reversal operator perturbed by the noise is established. We then deduce the noise level above which the subwavelength resolution fails. Numerical simulations and experimental results validate the theoretical developments.
MotsClés: Analytical expressions; Classical imaging; Decomposition of the time reversal operator; Eigenvectors; Far field; Metallic wire; Multiple signal classification; Music method; Noise levels; Numerical simulation; Quantitative study; Rayleigh criterion; Resolution limits; Sensors array; Subwavelength imaging; Subwavelength resolution; Theoretical development; Timereversal methods; Timereversal operator; Twopoint; Computer simulation; Scattering; Sensor arrays; Wavelet analysis; Wire; Quantum theory


Theory of the timereversal operator for a dielectric cylinder using separate transmit and receive arrays. Minonzio, J.  G., M. Davy, J. de Rosny, C. Prada, and M. Fink. IEEE Transactions on Antennas and Propagation 57, no. 8 (2009): 2331–2340.
Résumé: The DORT method applies to scattering analysis with arrays of transceivers. It consists in the study of the timereversal invariants. In this paper, a large dielectric cylinder is observed by separate transmit and receive arrays with linear polarizations, E or H, parallel to its axis. The decomposition of the scattered field into normal modes and projected harmonics is used to determine the theoretical timereversal invariants. It is shown that the number of invariants is about 2k1a, where a is the cylinder radius and k1 the wave number in the surrounding medium. Furthermore, this approach provides approximated expressions of the two first invariants for a subresolved cylinder, i.e., when the cylinder diameter is smaller than the resolution width of the arrays. The two first invariants are also expressed in the small object limit for k1 a < 0.5. AMS subject classifications. 35B40, 35P25, 45A05, 74J20,78M35 © 2009 IEEE.
MotsClés: Antenna array processing; Electromagnetic inverse problem; Electromagnetic scattering; Cylinder diameters; Dielectric cylinder; Electromagnetic inverse problem; Electromagnetic scattering; Linear polarization; Normal modes; Scattered field; Scattering analysis; Small objects; Subject classification; Timereversal; Timereversal operator; Wave numbers; Antenna arrays; Array processing; Dielectric devices; Dielectric materials; Electromagnetic wave scattering; Electromagnetism; Inverse problems; Microwave frequencies; Cylinders (shapes)


Ultrasonic focusing through the ribs using the DORT method. Cochard, E., C. Prada, J. F. Aubry, and M. Fink. Medical Physics 36, no. 8 (2009): 3495–3503.
Résumé: Thermal ablation induced by high intensity focused ultrasound has produced promising clinical results to treat hepatocarcinoma and other liver tumors. However skin burns have been reported due to the high absorption of ultrasonic energy by the ribs. This study proposes a method to produce an acoustic field focusing on a chosen target while sparing the ribs using the decomposition of the timereversal operator (DORT method). The idea is to apply an excitation weight vector to the transducers array which is orthogonal to the subspace of emissions focusing on the ribs. A linear array of transducers has been used to measure the set of singular vectors associated with a chest phantom, made of three human ribs immersed in water, and to produce the desired acoustic fields. The resulting propagating fields have been measured both in the focal plane and in the plane of the ribs using a needle hydrophone. The ratio of the energies absorbed at the focal point and on the ribs has been enhanced up to 100fold, as demonstrated by the measured specific absorption rates. © 2009 American Association of Physicists in Medicine.
MotsClés: Adaptive focusing; DORT; High intensity focused ultrasound; Noninvasive surgery; Phased array; acoustics; article; energy absorption; phantom; priority journal; radiation energy; rib; ultrasound; ultrasound transducer; Absorption; Humans; Male; Phantoms, Imaging; Ribs; Thorax; Time Factors; Ultrasonic Therapy


Lamb mode spectra versus the Poisson ratio in a free isotropic elastic plate. Royer, D., D. Clorennec, and C. Prada. Journal of the Acoustical Society of America 125, no. 6 (2009): 3683–3687.
Résumé: The variation, with material parameters, of Lamb modes is investigated. Vibration spectra of tractionfree elastic plates are generally presented, for a given isotropic material, as a set of dispersion curves corresponding to the various Lamb mode branches. Here, the spectrum variations, with the Poisson ratio , are plotted in a dimensionless coordinate system in the form of a bundle of curves for each Lamb mode. Except for the fundamental antisymmetric mode A0, this representation highlights the same behavior for all Lamb modes. VT denoting the shear wave velocity, the (ω,k) plane can be divided into two angular sectors separated by the line of slope VT 2. In the upper one, corresponding to a phase velocity V=ωk larger than VT 2, dispersion curves are very sensitive to the plate material parameters. In the lower sector (V< VT 2) all the branches, whatever the value of the Poisson ratio (0<0.5), are gathered into a thin pencil. Moreover, curves of a given bundle cross the boundary line at coincidence points equally spaced. These properties and a specific behavior observed for =0 are explained in terms of Laḿ wave solutions of the characteristic equations of Lamb modes. © 2009 Acoustical Society of America.
MotsClés: Antisymmetric; Boundary lines; Characteristic equation; Coordinate system; Coincidence points; Dispersion curves; Elastic plate; Isotropic materials; Lamb modes; Material parameter; Shear wave velocity; Vibration spectra; Wave solution; Graph theory; Plates (structural components); Poisson distribution; Poisson equation; Wave equations; Poisson ratio; article; elasticity; mathematical analysis; Poisson distribution; priority journal; shear strength; vibration; waveform


Characterization of an elastic target in a shallow water waveguide by decomposition of the timereversal operator. Philippe, F. D., C. Prada, J. De Rosny, D. Clorennec, J.  G. Minonzio, and M. Fink. Journal of the Acoustical Society of America 124, no. 2 (2008): 779–787.
Résumé: This paper reports the results of an investigation into extracting of the backscattered frequency signature of a target in a waveguide. Retrieving the target signature is difficult because it is blurred by waveguide reflections and modal interference. It is shown that the decomposition of the timereversal operator method provides a solution to this problem. Using a modal theory, this paper shows that the first singular value associated with a target is proportional to the backscattering form function. It is linked to the waveguide geometry through a factor that weakly depends on frequency as long as the target is far from the boundaries. Using the same approach, the second singular value is shown to be proportional to the second derivative of the angular form function which is a relevant parameter for target identification. Within this framework the coupling between two targets is considered. Small scale experimental studies are performed in the 3.5 MHz frequency range for 3 mm spheres in a 28 mm deep and 570 mm long waveguide and confirm the theoretical results. © 2008 Acoustical Society of America.
MotsClés: Waveguides; Experimental studies; Form function; Frequency ranging; Modal interference; Second derivatives; Shallow water waveguides; Singular values; Small scale; Target identification; Target signatures; Timereversal operator; Waveguide geometry; Targets; water; article; decomposition; elasticity; frequency analysis; geometry; priority journal; theoretical study; waveform; Acoustics; Elasticity; Models, Theoretical; Motion; Sound; Sound Spectrography; Steel; Surface Properties; Time Factors; Water


Local vibration of an elastic plate and zerogroup velocity Lamb modes. Prada, C., D. Clorennec, and D. Royer. Journal of the Acoustical Society of America 124, no. 1 (2008): 203–212.
Résumé: Elastic plates or cylinders can support guided modes with zero group velocity (ZGV) at a nonzero value of the wave number. Using laserbased ultrasonic techniques, we experimentally investigate some fascinating properties of these ZGV modes: resonance and ringing effects, backward wave propagation, interference between backward and forward modes. Then, the conditions required for the existence of ZGV Lamb modes in isotropic plates are discussed. It is shown that these modes appear in a range of Poisson's ratio about the value for which the cutoff frequency curves of modes belonging to the same family intercept, i.e., for a bulk wave velocity ratio equal to a rational number. An interpretation of this phenomenon in terms of a strong repulsion between a pair of modes having a different parity in the vicinity of the cutoff frequencies is given. Experiments performed with materials of various Poisson's ratio demonstrate that the resonance spectrum of an unloaded elastic plate, locally excited by a laser pulse, is dominated by the ZGV Lamb modes. © 2008 Acoustical Society of America.
MotsClés: Alumina; Cutoff frequency; Lasers; Light velocity; Plates (structural components); Poisson distribution; Poisson equation; Pulsed laser applications; Pulsed laser deposition; Resonance; Ultrasonic testing; Acoustical Society of America (ASA); Backward wave propagation; Elastic plates; Guided modes; Isotropic plates; Lamb modes; Locally excited (LOC); Nonzero values; Rational numbers; Resonance spectrum of; Ultrasonic techniques; Wave numbers; Wave velocity (IGC:D7); Zerogroup velocity; Poisson ratio; article; dispersion; phenomenology; Poisson distribution; priority journal; spike wave; velocity; vibration


Power law decay of zero group velocity Lamb modes. Prada, C., D. Clorennec, and D. Royer. Wave Motion 45, no. 6 (2008): 723–728.
Résumé: Elastic plates or cylinders can support guided modes with zero group velocity (ZGV) for nonzero wave numbers. At these ZGVpoints of the dispersion curves the acoustic energy does not propagate in the waveguide, resulting in sharp resonance effects. In this paper, using laserbased ultrasonic techniques, we investigate the timedecay of the mechanical displacement for ZGV Lamb modes excited by a pulsed laser in various thin metallic plates. In the first microseconds of the local plate vibration, we observed a t 1 / 2 decay due to the second order term in the dispersion relation. This effect is dominant because the first order term, proportional to the group velocity, vanishes for ZGVmodes. After this power law decay, the mechanical displacement undergoes an exponential decay corresponding to the wave damping. Then, the local attenuation of the plate material can be estimated at the ZGVresonance frequency. © 2008.
MotsClés: Lamb modes; Time decay; Zero group velocity; Attenuation equalizers; Damping; Estimation; Natural frequencies; Wave equations; Elastic plates; Laserbased ultrasonic techniques; Mechanical displacement; Wave damping; Zero group velocity (ZGV); Elastic waves; Attenuation equalizers; Damping; Elastic waves; Estimation; Natural frequencies; Wave equations; damping; estimation method; laser method; power law; resonance; ultrasonics; wave attenuation


Characterization of an elastic cylinder and an elastic sphere with the timereversal operator: Application to the subresolution limit. Minonzio, J.  G., F. D. Philippe, C. Prada, and M. Fink. Inverse Problems 24, no. 2 (2008).
Résumé: The Décomposition de l'Opérateur de Retournement Temporel method applies to scattering analysis with arrays of transducers. It comprises the study of the timereversal invariants which correspond to the eigenvectors of the timereversal operator or to the singular vectors of the array response matrix K. In this paper, the decomposition of the scattered pressure into normal modes of vibrations is used to determine the theoretical timereversal invariants for a large elastic object such as a cylinder or sphere. For an N transducers onedimensional array, the timereversal operator is dimension N. It is shown that the dimension is reduced to 2k0a for a cylinder or a sphere, where a is the scatterer radius and k0 is the wave number in the surrounding fluid. Furthermore, this approach provides analytical expressions of symmetric and antisymmetric singular values and vectors in the subresolution limit, i.e. when the scatterer diameter is smaller than the array resolution cell. These results are verified experimentally and in good agreement with the original point of view: for a small scatterer, one singular value dominates and the associated singular vector focuses isotropically on the scatterer. © 2008 IOP Publishing Ltd.
MotsClés: Eigenvalues and eigenfunctions; Electromagnetic wave scattering; Matrix algebra; Transducers; Vectors; Elastic cylinders; Elastic sphere; Mathematical operators


Simulation and measurement of the optical excitation of the S1 zero group velocity Lamb wave resonance in plates. Balogun, O., T. W. Murray, and C. Prada. Journal of Applied Physics 102, no. 6 (2007).
Résumé: Recent reports on the thermoelastic generation of Lamb waves in isotropic elastic plates show that a laser source efficiently excites a resonance that occurs at the minimum frequency of the first order symmetric (S1) Lamb mode. The group velocity of the Lamb wave goes to zero at this frequency while the phase velocity remains finite, and the resonance is referred to as the S1 zero group velocity (S1 ZGV) resonance. The S1 ZGV resonance can be employed for the nondestructive evaluation of the elastic properties of plates or plate thickness. A model for the generation of elastic waves in plates using an intensitymodulated continuous wave laser source is developed and used to study the behavior of the S1 ZGV resonance. The effects of the laser source parameters on the generation of the S1 ZGV resonance are explored, and the spatial distribution of the displacement produced at the resonance frequency is determined. The predicted displacement spectrum of Lamb waves generated in micron scale plates is found to compare well with experimental measurements. In addition, experimental measurements demonstrate that the S1 ZGV resonance can be used to map subsurface features in thin (4 μm) membranes at high ultrasonic frequencies (700 MHz). © 2007 American Institute of Physics.
MotsClés: Elastic waves; Group velocity dispersion; Photoexcitation; Plates (structural components); Resonance; Ultrasonics; Isotropic elastic plates; Resonance frequency; Ultrasonic frequencies; Zero group velocity Lamb wave resonance; Surface waves


Experimental detection and focusing in shallow water by decomposition of the time reversal operator. Prada, C., J. De Rosny, D. Clorennec, J.  G. Minonzio, A. Aubry, M. Fink, L. Berniere, P. Billand, S. Hibral, and T. Folegot. Journal of the Acoustical Society of America 122, no. 2 (2007): 761–768.
Résumé: A rigid 24element sourcereceiver array in the 1015 kHz frequency band, connected to a programmable electronic system, was deployed in the Bay of Brest during spring 2005. In this 10 to 18m deep environment, backscattered data from submerged targets were recorded. Successful detection and focusing experiments in very shallow water using the decomposition of the time reversal operator (DORT method) are shown. The ability of the DORT method to separate the echo of a target from reverberation as well as the echo from two different targets at 250 m is shown. An example of active focusing within the waveguide using the first invariant of the time reversal operator is presented, showing the enhanced focusing capability. Furthermore, the localization of the scatterers in the water column is obtained using a rangedependent acoustic model. © 2007 Acoustical Society of America.
MotsClés: Backscattering; Data acquisition; Frequency bands; Signal receivers; Waveguide components; Acoustic model; Time reversal operator (DORT method); Water column; Water; article; decomposition; echolocation; electronics; frequency modulation; priority journal; receiver operating characteristic; sound detection; sound intensity; water content; Kinetics; Models, Theoretical; Sound; Sound Localization; Time; Water


Comment on “multiple scattering: The key to unravel the subwavelength world from the farfield pattern of a scattered wave”. De Rosny, J., and C. Prada. Physical Review E – Statistical, Nonlinear, and Soft Matter Physics 75, no. 4 (2007).
Résumé: Contrary to the main conclusion of Simonetti [Phys. Rev. E 73, 036619 (2006)], we maintain that multiple scattering (MS) is not the “key” for subwavelength detection. Indeed, even with no MS between subwavelength structures, subwavelength detection is still possible. Our statement is numerically confirmed. A simple mathematical argument explains this result. From our point of view, the incorrect conclusion of Simonetti comes from a misinterpretation of the Picard's theorem. © 2007 The American Physical Society.
MotsClés: Numerical methods; Theorem proving; Wavelength; Picard's theorem; Simonetti; Subwavelength structures; Multiple scattering


Local and noncontact measurements of bulk acoustic wave velocities in thin isotropic plates and shells using zero group velocity Lamb modes. Clorennec, D., C. Prada, and D. Royer. Journal of Applied Physics 101, no. 3 (2007).
Résumé: An original method for material characterization with acoustic waves is presented. The measurement of the longitudinal and shear wave velocities in thin isotropic plates or shells is performed locally on the same face without any mechanical contact. We exploit the resonance that occurs at the minimum frequency thickness product of the first order symmetric (S1) and of the second order antisymmetric (A2) Lamb modes. At these frequencies the group velocity vanishes, whereas the phase velocity remains finite. Then, the energy, which cannot propagate in the structure, is localized in a zone of diameter half the wavelength. The vibrations are excited in the thermoelastic regime by a laser pulse and detected at the same point by an optical interferometer. For these two Lamb modes we have computed the variations of the frequency thickness product versus Poisson's ratio. The resonance frequency ratio, which is independent of the plate or shell thickness, provides an absolute and local measurement of Poisson's ratio. Provided that the plate thickness is known, each resonance frequency allows us to determine in a single shot the bulk acoustic wave velocities VL and VT. Since it is based on frequency measurements, the method, tested on a large number of materials, is very accurate. © 2007 American Institute of Physics.
MotsClés: Interferometers; Poisson equation; Shear waves; Thin films; Velocity measurement; Antisymmetric Lamb modes; Frequency measurements; Mechanical contacts; Resonance frequency; Acoustic waves


Gaussian beams and Legendre polynomials as invariants of the time reversal operator for a large rigid cylinder. Aubry, A., J. De Rosny, J.  G. Minonzio, C. Prada, and M. Fink. Journal of the Acoustical Society of America 120, no. 5 (2006): 2746–2754.
Résumé: The DORT method (French acronym for decomposition of the time reversal operator) is an active remote sensing technique using an array of antennas for the detection and localization of scatterers. This method is based on the singular value decomposition of the interelement response matrix. In this paper an analytical expression of the singular vectors due to the reflection from a large rigid cylinder is provided. Depending on the array aperture, two asymptotic regimes are described. It is shown that the singular vectors correspond to HermiteGaussian modes for large apertures and Legendre polynomials for small ones. Using perturbation theory, the corresponding singular values are deduced. Theoretical predictions are in good agreement with experimental results. © 2006 Acoustical Society of America.
MotsClés: Acoustic fields; Antenna arrays; Polynomials; Remote sensing; Vectors; DORT method; Gaussian modes; Rigid cylinder; Singular value decomposition; Gaussian noise (electronic); acoustics; article; decomposition; mathematical analysis; mathematical computing; model; normal distribution; priority journal; remote sensing; technique; transducer


Multiple scattering between two elastic cylinders and invariants of the timereversal operator: Theory and experiment. Minonzio, J.  G., C. Prada, A. Aubry, and M. Fink. Journal of the Acoustical Society of America 120, no. 2 (2006): 875–883.
Résumé: The decompositionofthetimereversaloperator method is an ultrasonic method based on the analysis of the array response matrix used for detection and characterization. The eigenvalues and the eigenvectors of the timereversal operator (equivalent to the singular values and the singular vectors of the array response matrix) provide information on the localization and nature of scatterers in the insonified medium. Here, the eigenmodes of the timereversal operator are studied for two elastic cylinders: The effects of multiple scattering and anisotropic scattering are considered. Analytical expressions for the singular values are established within the isotropic scattering approximation. Then, the comparison with a complete model is presented, putting in evidence the importance of the anisotropy of the scattering. Experiments, carried out at central frequency 1.5 MHz on 0.25 mm diameter nylon and copper cylinders embedded in water, confirm the theory. In particular, the small cylinder limit and the effect of the dominant quadrupolar normal mode of nylon are discussed. © 2006 Acoustical Society of America.
MotsClés: Anisotropy; Approximation theory; Cylinders (shapes); Eigenvalues and eigenfunctions; Mathematical operators; Vectors; Elastic cylinders; Insonified medium; Isotropic scattering approximation; Quadrupolar normal mode; Acoustic wave scattering; acoustics; analytic method; anisotropy; article; decomposition; electromagnetic radiation; imaging system; model; priority journal; sound transmission; stimulus response; theory; ultrasound


Laser impulse generation and interferometer detection of zero group velocity Lamb mode resonance. Clorennec, D., C. Prada, D. Royer, and T. W. Murray. Applied Physics Letters 89, no. 2 (2006).
Résumé: In this letter, we describe experiments on the generation of the first order symmetric (S 1) Lamb mode by a pulsed yttrium aluminum garnet laser. The vibration of the plate is detected at the same point by a heterodyne interferometer. The acoustic signal is dominated by the resonance at the point of the dispersion curve where the group velocity vanishes. The time decay of the signal leads to the local attenuation coefficient of the material. The spectrum exhibits a very sharp peak, the frequency of which is sensitive to the plate thickness. For a 0.49mmthick Duralumin plate, thickness variations as small as 0.1 μm have been detected. Moving the detection point away from the source allows us to record the standing wave pattern resulting from the interference between the S 1 and S 2b Lamb waves having opposite wave vectors at the zero group velocity point. © 2006 American Institute of Physics.
MotsClés: Acoustic signal; Garnet laser; Heterodyne interferometer; Velocity point; Aluminum; Aluminum copper alloys; Interferometers; Laser pulses; Pattern recognition; Resonance; Spectrum analysis; Yttrium; Garnets


Laserbased ultrasonic generation and detection of zerogroup velocity Lamb waves in thin plates. Prada, C., O. Balogun, and T. W. Murray. Applied Physics Letters 87, no. 19 (2005): 1–3.
Résumé: A novel laserbased ultrasonic technique for the inspection of thin plates and membranes is presented, in which a modulated continuouswave laser source is used to excite narrow bandwidth Lamb waves. The dominant feature in the acoustic spectrum is a sharp resonance peak that occurs at the minimum frequency of the firstorder symmetric Lamb mode, where the group velocity of the Lamb wave goes to zero while the phase velocity remains finite. Experimental results with the laser source and receiver on epicenter demonstrate that the zerogroup velocity resonance generated with a lowpower modulated excitation source can be detected using a Michelson interferometer coupled to a lockin amplifier. This resonance peak is sensitive to the thickness and mechanical properties of plates and may be suitable, for example, for the measurement and mapping of nanoscale thickness variations. © 2005 American Institute of Physics.
MotsClés: Continuous wave laser; Ultrasonic technique; Zerogroup velocity; Bandwidth; Continuous wave lasers; Laser beam effects; Resonance; Thickness measurement; Velocity measurement; Ultrasonic waves


Characterization of subwavelength elastic cylinders with the decomposition of the timereversal operator: Theory and experiment. Minonzio, J.  G., C. Prada, D. Chambers, D. Clorennec, and M. Fink. Journal of the Acoustical Society of America 117, no. 2 (2005): 789–798.
Résumé: The decomposition of the timereversal operator provides information on the scattering medium. It has been shown [Chambers and Gautesen, J. Acoust. Soc. Am. 109, 26162624 (2001)] that a small spherical scatterer is in general associated with four eigenvalues and eigenvectors of the timereversal operator. In this paper, the 2D problem of scattering by an elastic cylinder, imbedded in water, measured by a linear array of transducers is considered. In this case, the array response matrix has three nonzero singular values. Experimental results are obtained with linear arrays of transducers and for wires of different diameters smaller that the wavelength. It is shown how the singular value distribution and the singular vectors depend on the elastic velocities c L, c T, the density p of each wire, and on the density ρ 0 and velocity c 0 of the surrounding fluid. These results offer a new perspective towards solution of the inverse problem by determining more than scattering contrast using conventional array processing like that used in medical ultrasonic imaging. © 2005 Acoustical Society of America.
MotsClés: Cylinders (shapes); Decomposition; Eigenvalues and eigenfunctions; Elasticity; Scattering; Transducers; Ultrasonic imaging; Medical ultrasonic imaging; Scattering medium; Subwavelegnth elastic cylinders; Timereversal operator; Acoustic waves; acoustics; article; decomposition; liquid; mathematical analysis; priority journal; signal processing; transducer; ultrasound; velocity; waveform


Resolution enhancement and separation of reverberation from target echo with the time reversal operator decomposition. Folégot, T., C. Prada, and M. Fink. Journal of the Acoustical Society of America 113, no. 6 (2003): 3155–3160.
Résumé: Time reversal operator (TRO) decompositions are performed in a model of an ocean wave guide containing a target and having different kinds of bottom. The objective is to study the effects of bottom reverberation and absorption by means of ultrasonic experiments. It is shown experimentally that the echo from a target can be separated from the bottom reverberation. Reverberation eigenvectors are back propagated in the wave guide leading to focus on the bottom. An amplitude correction is applied to both reverberation and signal eigenvectors to compensate for bottom absorption and thus to improve target resolution. © 2003 Acoustical Society of America.
MotsClés: Acoustic wave absorption; Backpropagation; Eigenvalues and eigenfunctions; Mathematical operators; Ultrasonic applications; Water waves; Target resolution; Reverberation; poly(methyl methacrylate); water; acoustics; absorption; acoustics; article; Fourier transformation; priority journal; sea; signal noise ratio; time; ultrasound


Time reversal techniques in ultrasonic nondestructive testing of scattering media. Prada, C., E. Kerbrat, D. Cassereau, and M. Fink. Inverse Problems 18, no. 6 (2002): 1761–1773.
Résumé: Time reversal techniques are adaptive methods that can be used in nondestructive evaluation to improve flaw detection through inhomogeneous and scattering media. Two techniques are presented: the iterative time reversal process and the DORT (French acronym for decomposition of the time reversal operator) method. In pulse echo mode, iterative time reversal mirrors allow one to accurately control wave propagation and focus selectively on a defect reducing the speckle noise due to the microstructure contribution. The DORT method derives from the mathematical analysis of the iterative time reversal process. Unlike time reversal mirrors, it does not require programmable generators and allows the simultaneous detection and separation of several defects. These two procedures are presented and applied to detection in titanium billets where the grain structure renders detection difficult. Then, they are combined with the simulation code PASS (phased array simulation software) to form images of the samples.
MotsClés: Adaptive systems; Computer simulation; Computer software; Iterative methods; Microstructure; Mirrors; Nondestructive examination; Speckle; Ultrasonic testing; Time reversal techniques; Inverse problems


Ultrasonic nondestructive testing of scattering media using the decomposition of the timereversal operator. Kerbrat, E., C. Prada, D. Cassereau, and M. Fink. IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control 49, no. 8 (2002): 1103–1113.
Résumé: In ultrasonic nondestructive testing, the iterative timereversal process is an adaptive technique that can be used to detect flaws in complex samples with a large array of transducers. The decomposition of the timereversal operator (D.O.R.T.) method is a detection technique that is derived from the mathematical analysis of the iterative timereversal process. Contrary to timereversal techniques, the D.O.R.T. method does not require programmable generators, and it allows the simultaneous detection and separation of several defects. In this paper, the method is applied to a Ti64 titanium cylindrical sample to separate the echo of a defect from the speckle due to microstructure contribution. The grain structure of this titanium alloy makes detection very difficult and, for large depths, conventional techniques do not allow the detection of small flaws with a satisfactory signaltonoise ratio. The efficiency of the D.O.R.T. method to detect a flat bottom hole with a diameter of 0.4 mm located at a depth of 140 mm in a titanium alloy sample is shown.
MotsClés: Acoustic wave scattering; Mathematical operators; Microstructure; Signal to noise ratio; Titanium alloys; Ultrasonic transducers; Timereversal operators; Acoustic emission testing


Detection of cracks in a thin airfilled hollow cylinder by application of the DORT method to elastic components of the echo. Kerbrat, E., D. Clorennec, C. Prada, D. Royer, D. Cassereau, and M. Fink. Ultrasonics 40, no. 18 (2002): 715–720.
Résumé: The DORT method (Décomposition de l'Opérateur de Retournement Temporel in French) is a scattering analysis technique which uses arrays of transducers. This method is efficient for detection of selective focusing on pointlike scatterers. It has been also applied to analyze the scattering by an airfilled cylindrical steel shell immersed in water. It was shown that the diagonalization of the time reversal operator allows us to separate the different elastic components of the scattered field. Here, we apply the method to detect flaws in hollow cylinders. In this case, the dominant components are the three circumferential waves (A0, A1 and S0 Lamb modes). Each Lamb mode corresponds to an invariant of the time reversal operator. The dispersion curves of these waves are calculated from these invariants. Resonance frequencies of the shell are deduced from the frequency dependence of the eigenvalues of the time reversal operator. It is shown that the presence of a crack (0.2 mm in depth) affects significantly the eigenvalue distribution of the time reversal operator. Thus, the DORT method offers a new means for detecting defects in a shell. © 2002 Elsevier Science B.V. All rights reserved.
MotsClés: Crack; DORT method; Hollow cylinder; Lamb waves; Nondestructive evaluation; Acoustic wave scattering; Cracks; Cylinders (shapes); Eigenvalues and eigenfunctions; Natural frequencies; Nondestructive examination; Transducers; Lamb waves; Acoustic wave propagation


Ultrasound propagation through a rotational flow: Numerical methods compared to experiments. Manneville, S., C. Prada, M. Tanter, M. Fink, and J.  F. Pinton. Journal of Computational Acoustics 9, no. 3 (2001): 841–852.
Résumé: Sound propagation through a vortex is studied numerically using two different techniques: raytracing and finitedifferences. Geometrical acoustics and raytracing are shown to yield a good picture of the interaction between a sound wave and a vortex when the ratio of the vortex radius to the acoustic wavelength is larger than one. In particular, this technique allows to take into account finitesize effects such as edge waves and the results are compared to experimental data. The interest of the finitedifference approach is demonstrated for cases where sound scattering occurs. We show the ability of such a simulation to account for both sound scattering and finitesize effects. Those two numerical techniques are compared and their validity is investigated.


Multiple receiver wind profiling techniques for the boundary layer in the presence of hydrometeors. Van Baelen, J., L. Hirsch, and C. Prada. Geophysical Research Letters 24, no. 21 (1997): 2667–2670.
Résumé: A UHF boundary layer wind profiler has been operated in a multiple receiver mode. The data sets collected are used to estimate the horizontal wind using different multiple receiver analysis techniques performed in the time and frequency domains. Those results are also compared with simultaneous Doppler beam swinging measurements. The particularity of the data sets used for this intercomparison is that they were recorded while hydrometeors were present in the boundary layer. The large mean fall speed and the broad fall velocity distribution of rain drops causes a much more rapid decay of diffraction patterns than in the case of scattering by snow flakes or micro turbulence. Therefore, analysis methods which do not account for the effect of the decay of the diffraction patterns show an overestimation of the wind velocity, and do so in a different way whether snow or rain is present. To the contrary, algorithms that account for the effect of diffraction pattern decay appear to reasonably estimate the actual horizontal wind and compare well with the Doppler measurements. Copyright 1997 by the American Geophysical Union.
MotsClés: Atmospheric turbulence; Boundary layers; Electromagnetic wave diffraction; Frequency domain analysis; Light scattering; Rain; Snow; Time domain analysis; Hydrometeors; Multiple receiver wind profiling; Wind


The iterative time reversal process: Analysis of the convergence. Prada, C., J.  L. Thomas, and M. Fink. Journal of the Acoustical Society of America 97, no. 1 (1995): 62–71.
Résumé: The efficiency of a time reversal acoustic mirror to focus on a reflective target through an inhomogeneous media has been demonstrated. In a multitarget media, the ability of such a mirror to work in an iterative mode in order to focus selectively on the strongest target was shown [C. Prada, F. Wu, and M. Fink, J. Acoust. Soc. Am. 90, 1119 (1991)]. The theory of how the iterative time reversal process is built is based on a matrix formalism and treats the array of L transducers in a given medium as a linear system of L inputs/L outputs. The system is characterized at each frequency by its transfer matrix K and the time reversal iterative process is then described by a time reversal operator K*K. Because of the reciprocity principle, this operator is Hermitian. The following result is shown: If the scattering medium is a set of well resolved targets of different reflectivities then each eigenvector of the operator K*K with nonzero eigenvalue corresponds to one of the targets in the set and provides the optimum phase law to focus on it. Furthermore, the eigenvalue is proportional to the reflectivity of the target. In particular, the 'brightest' target is associated to the eigenvector of greatest eigenvalue so that the iterative time reversal process leads to a wave focusing on this target. This analysis is illustrated by numerical and experimental results.
MotsClés: acoustics; article; brightness; mathematical analysis; priority journal; signal transduction; sound; surface property; waveform; acoustic transducers; convergence; experimental study; iterative time reversal


The iterative time reversal mirror: A solution to selffocusing in the pulse echo mode. Prada, C., F. Wu, and M. Fink. Journal of the Acoustical Society of America 90, no. 2 I (1991): 1119–1129.
Résumé: Accurate focusing in an inhomogeneous medium is difficult to implement. In order to focus on a reflective target we have extended the concept of optical phaseconjugate mirrors, valid for monochromatic signals, to broadband pulses such as those used in ultrasound echography. The transducer's linear response to the acoustic pressure allows one to replace the phase conjugation by a time reversal operation on the pulse echo signals. The time reversal mirror is an array of transmitreceive transducers. A first incident wave is reflected by the target. The received signals are stored in shift registers, reversed in time, and then reemitted. The main advantage of this process is that waves distorted by the propagation through an aberrating medium are corrected by the mirror operation and the back propagation through the medium. When the medium contains several reflectors, this time reversal process can be iterated in order to focus on the most reflective one. Theoretical results on this principle, numerical simulations, and experimental results with a 1D array working at a central frequency of 3 MHz are presented.
MotsClés: acoustics; article

