| <1/2 decreasing continuously as the disorder increases (i.e. mean free path decreases). The dependence of the exponent as a function of disorder has been explained by analytical calculations using weak localization techniques. This sub-diffusion can be understood in two ways : 1) from weak localization in 2D which essentially recognizes the existence of coherent interferences between arbitrarily complex wave paths when propagated forward and backward; 2) from the geometrical fractal structure of localized eigenmodes in 2D. 1.3 Non-destructive evaluations using ultrasonic waves in multiple-scattering heterogeneous media. We address the question of the identification of a new defect (a damage crack for instance) in a composite medium or a polycrystalline system. Standard ultrasonic non-destructive testing techniques rely on the measurement of the wave which is singly reflected by the scatterer. However, suppose that the medium is highly heterogeneous. In order to minimize the background reflections from the surrounding heterogeneities, low frequencies are used, which lead however to a severe loss of spatial resolution. Here, we report on a technique relying on state-of-the-art analytical calculations, which allows one to identify a new scatterer of the same scattering strength as the typical heterogeneities of the medium, in the large frequency regime, with a resolution of the order of the mean free path of the ultrasonic wave in the scattering medium. The technique consists in measuring the transmission or reflection speckle patterns at different frequencies before and after the introduction of the defect. Then, the average of the square of the difference of the speckle patterns before and after the introduction of the new defect over several tens of frequencies can be shown to be related to the space derivative of the Green function of the diffusion equation with a point source located at the position of the new defect [14,15]. A fit between the theoretical prediction obtained by solving the diffusion equation and the experimentally obtained averaged speckle patterns allows one in principle to retrieve the position of the defect. We have tested this theory by performing extensive numerical simulations with the "wave automaton" in a 2D system of size 256 by 512, a mean free path of 10 and a new scatterer added on one node at different positions inside the system. The positions retrieved from the proposed scheme are in remarkable agreement with their actual values [15]. Potential applications of our approach can be found in medical and industrial imaging in highly scattering systems. We can thus conclude that it is possible to "see" new defects through apparently completely opaque systems using the intrinsic coherent nature of the wave field in random systems! 1.4 Random distributed feedback tunable laser A laser needs two ingredients : 1) a gain so as to amplify pre-existing wave background and 2) a cavity to provide a feedback of the amplified photons which are returned to the amplified medium. Here, we report on the numerical test of the idea proposed a few years ago [16] that Anderson wave localization in a random media can lead to the existence of effective cavities (the localized eigenmodes) which are suitable for the laser feedback. Using the "wave automaton", we have constructed a weakly lossy disordered medium with disorder, presenting in addition a weak saturable gain on a single node of the lattice. Numerical simulations have shown that, starting from an initial white spectrum for the weak background wave noise, coherent amplification of a single frequency occurs, ultimately leading to the existence of a single frequency in the spectrum at large time, hence the demonstration of the laser effect in a disordered system. The "cavity" comes from the Anderson localization effect, as can be verified from the fact that the spatial distribution of the wave energy corresponding to the selected frequency turns out to be precisely one of the linear modes of the disordered medium (in absence of gain) which has the strongest overlap with the node on which the gain is applied. By sweeping the position of the node on which the gain is applied, various different frequencies are selected (in fact as many as the number of nodes) with varying spatial structures, thus leading to the concept of a random distributed feedback tunable laser [17]. In a sense, this result illustrates that a random medium acts similarly to the superposition of many different periodic systems. Similar ideas should be relevant to create coherent phonon sources from heterogeneous media. 2- Acoustic waves and quantum chaos. In 1989, we proposed to use the analogy between the problems of the field loosely called "quantum chaos" and those encountered in high frequency vibrations [18,19] to develop new methods of analysis and new techniques of calculations in this last field. The high frequency (HF) regime is defined as the regime where the wavelength of the wave of interest is small compared to the characteristic size of the structure. As a consequence, many modes participate in the response function of a system in the HF regime. Apart from rough approximations as in the Statistical Energy Analysis approach, there are essentially no theory to tackle this domain. In engineering applications, the HF regime is often encountered and pose formidable problems that are rarely satisfactorily addressed. The analogy and the ensuing techniques described briefly below have been developed in an attempt to improve on this state of affairs. 2.1 Spectral properties The basic question is to quantify the amount of information contained in complex spectra. In contradistinction with common wisdom, we have shown that a lot of information can be extracted beyond the smooth density of state approximation, in particular in the fluctuations of the density of state around its smooth average. In order to illustrate the method, we have examined two different systems : 1) a 3D elastodynamic experiment on aluminium blocks [20] and 2) numerical computations of the vibrations of 2D thin plates [21]. The measured spectra are analyzed with techniques borrowed from the theory of random matrices. The main conclusion is that fluctuations of the spectrum on small scales (involving a few mean eigenfrequency spacings) are well described by the model of Gaussian Orthogonal Ensemble (GOE) of random matrices. In addition, we show that the large scale oscillations of the spectrum (large frequency differences) are due to short periodic orbits (i.e. rays following trajectories that close on themselves) in the corresponding "classical" system (obtained by taking the "eikonal" limit of infinite frequency) and thus yield informations on the size and shape of the aluminium blocks! The influence of the classical ray trajectories is also felt on the eigenmodes of vibrations. It can be shown that some of them present a partial localization of the spatial vibration amplitude pattern ("scar") in the neighborhood of periodic orbits followed by geometrical rays. This result is important for the multipolar nature and the acoustic radiation efficiency of the structure [22]. An efficient and reliable numerical scheme has been developed to compute the acoustic radiation directivity and the total acoustic power radiated by isolated eigenmodes and by finite bandwidth excitations of a membrane of arbitray shape over the whole frequency domain [22]. Results have been obtained for the case of a membrane having the shape of a stadium. This stadium shape, while being simple enough, is in fact representative of the generic properties of complex structures presenting chaotic ray trajectories. The radiation directivity is given by the Fourier transform of the vibration amplitude distribution on the membrane and localization in emission directivty is thus simply controlled by the "scars" of eigenmodes made by resonance on periodic ray orbits. The dependence of the radiation efficiency as a function of the ratio cM/c of the membrane wave velocity cM over the air sound speed c and its important fluctuations from mode to mode has been explained by the theory of random matrices [22]. In the presence of absorption, always present in an experimental situation, eigenfrequencies overlap and any measured spectrum usually takes the form of a complicated "herb"-like function, generally believed to contain no information beyond the average density of state. It turns out that we have been able to show that two-point correlation functions of the spectrum can allow one to identify the nature of the underlying system (integrable or chaotic ray trajectories described respectively by Poissonian or GOE random matrix statistics) and in the same token to get access to the value of the dissipation [23]. In particular, there is a strong interplay between GOE statistics and dissipation that leads to severe erroneous errors when neglected as done in the past. Since most structures for engineering applications are in the GOE or multiple-GOE universality class, our results bear important applications. During this work, new efficient algorithms which allow the computation of a large number (hundreds to thousands) of eigenfrequencies for clamped and freely supported plates and for 3D elastodynamical problems have been developed. An equivalence between the vibration problem of a thin plate and that of a membrane with a complex boundary condition has also been shown [21]. This last result allows to circumvent the problems of stability and precision associated to the calculation of thin plate vibration eigenfrequencies. Also, we have extended the algorithm to calculate a large number of eigenfrequencies and eigenmodes of coupled membranes of arbitrary shapes and study the statistical properties of the spectrum and eigenmode fluctuations [24]. Our results emphasize the large sensitivity of the detailled structure of the spectrum of a classically chaotic system with respect to perturbation such as couplings. However, the global statistical properties are very robust as they pertain to only a few different universality class. 2.2 Time-dependent properties : geometric theory of wave dynamics in chaotic enclosures. Spectral properties are only one facet of waves. As for transport of waves in random media described in =A71, the time-dependent properties of waves in cavities can provide new insights. Furthermore, it turns out to be the relevant view point for the problem of room acoustics, in which one is interested in describing the transient behavior of an acoustic wave launched from a source in an enclosure. Interesting links between the theoretical foundation of room acoustics, whose full wave theory has until recently been lacking, and chaotic ray trajectories in billiards have been recently studied, enabling a better quantification of the various regimes [25,26]. Recently semi-classical time-dependent Green function for the hyperbolic wave equation has been constructed using a summation over quasi-recurrent classical ray trajectories [27]. The finite resolution of the wave problem associated to the smallest wavelength allowed us to introduce a natural coarse-graining which permits to partition the classical rays into bundles forming a Cantor set [28] We have shown the existence of contributions in the sum which correspond to precursors to the classical ray arrival times., which embody the physics of multiple interferences and more precisely a diffraction correction associated to the presence of odd numbers of focal points along the classical ray trajectories. Our global formulation and Green function construction over the classical ray trajectories suitably enriched by their relevant phases and amplitudes provide a very good agreement with the direct numerical integration of the wave equation for integrable as well as for various billiard shapes, such as the Sina=EF and stadium billiards. In conclusion, this short review has shown a few examples where new interesting physics can spring out of analogies between different fields. I would like to thank warmly my collaborators C. Vanneste, O.Legrand, P. Sebbah, P. Mortessagne, D. Delande, R. Weaver, C. Schmit, O. Bohigas and S. de Toro Arias. REFERENCES [1] D.Sornette, "Acoustic waves in random media: I Weak disorder regime", Acustica 67, 199 (1989); "II Coherent effects and strong disorder regime", Acustica 67, 251 (1989); "III Experimental situations", Acustica 68, 15 (1989) [2] D.Sornette et B.Souillard, "A mean field approach to Anderson localization", Europhys.Lett. 13, 7-12 (1990) [3] J.P. Desidri and D. Sornette, "Band-edge localization and spatial textures of surface acoustic waves in weakly disordered 1D-superlattices, Europhys. Lett. 23, 165-170 (1993) [4] L.Macon, J.P. Desidri and D.Sornette, "Surface acoustic waves in a simple quasi-periodic system", Phys.Rev.B 40, 3605 (1989) [5] J.P. Desidri, L.Macon and D.Sornette, "First experimental observat ion of critical modes in quasi-periodic systems", Phys.Rev.Lett. 63, 390 (1989) [6] J.P. Desidri, O.Legrand, L.Macon and D.Sornette, "Cantor set spectra and self-similar critical modes in a 1D-quasicrystal", Physica D 38, 56 (1989) [7] L.Macon J.P. Desidri and D.Sornette, "Localisation of surface acoustic waves in quasi-periodic systems", Phys.Rev.B 44, 6755-6772 (1991) [8] C. Vanneste, P. Sebbah and D.Sornette, "A wave automaton for time-dependent wave propagation in random media", Europhys.Lett. 17, 715-720 (1992); 18, 567 (1992) (Erratum) [9] D.Sornette, O.Legrand, F.Mortessagne, P.Sebbah and C. Vanneste, "The wave automaton for the time-dependent Schrdinger, classical wave and Klein-Gordon equations", Phys.Lett.A 178, 292-300 (1993); C. Vanneste, P. Sebbah and D. Sornette, "General wave equations modelled by the Wave Automaton", Europhys.Letters 21 (7), 794 (1993); [10] S. De Toro Arias, D.Sornette and C.Vanneste, to be published [11] P. Sebbah, D. Sornette and C. Vanneste, "A Wave Automaton for wave propagation in the time domain : I. periodic systems", J.Phys. I France 3, 1259-1280 (1993) [12] P. Sebbah, D. Sornette and C. Vanneste, "A Wave Automaton for wave propagation in the time domain : II. random systems", J.Phys. I France 3, 1281-1302 (1993) [13] P.Sebbah, C.Vanneste and D.Sornette, "Anomalous diffusion in two-dimensional Anderson localization dynamics", Phys.Rev. B 48, 12506, 1 Nov (1993)-I [14] S.Feng and D.Sornette, "Acoustic nondestructive evaluation of heterogeneous materials in the multiple scattering regime", J.Acoust.Soc.Am. 90, 1742-1747 (1991) [15] C. Vanneste, S. Feng and D.Sornette, "Non-destructive Evaluations in multiple scattering media", Europhys. Lett. 24, 339-344 (1993) [16] D.Sornette, Distributed cavity laser by Anderson localiztaion in two dimensions, DRET grant N=B089/34271 (1989) [17] P. Sebbah, D. Sornette and C. Vanneste, "Wave automaton for wave propagation in random media", in Proceedings of "Photon migration in random media", Orlando, USA, march 1994; P. Sebbah, C. Vanneste and D. Sornette, "Numerical study of wave propagation in nonlinear disordered media : random distributed feedback tunable laser", In proceedings of the Workshop on Optical Telecommunications Fibres and Components for Systems Applications (COST), University of Nice, France, April 18-19 (1994) [18] D.Sornette, "High frequency vibrations and quantum chaos", DRET grant 89/543 (1989) [19] O. Legrand and D. Sornette, "Quantum chaos and classical waves", Lecture Notes in Physics 392, 267-274 (1991); D. Sornette, "Vibrations de plaques et chaos quantique", Revue Franaise de Mcanique, numero special 1991, p.364-381 (1991); "Vibrations hautes frequences des structures", Aux Frontires du Domaine, Acoustique et Vibrations, Science et Defence 92 (Dunod, Paris, 1992), p.185-205. [20] O. Bohigas, O.Legrand, C. Schmit and D.Sornette, "Comment on Spectral Statistics in Elastodynamics", J.Acoust.Soc.Am.89, 1456-1458 (1991); D. Delande, D. Sornette and R. Weaver, "A reanalysis of experimental high frequency spectra using periodic orbit theory", J.Acoust.Soc.Am. 96, nb03, 1873-1880 (1994). [21] O.Legrand, C.Schmit and D.Sornette, "High frequency plate vibrations and quantum chaos", Europhys.Lett. 18, nb02, 101-106 (1992) [22] D. Delande and D. Sornette, "Acoustic radiation from membranes at high frequencies : the quantum chaos regime", J.Acoust.Soc.Am. submitted [23] O.Legrand, F. Mortessagne and D. Sornette, "Spectral rigidity in the large modal overlap regime : beyond the Ericson-Schroeder hypothesis", J.Phys.I France 5 (to appear, August 1995) [24] C.Schmit and D.Sornette, "Properties of connected membranes using quantum chaos methods", Acta Acustica submitted [25] O.Legrand and D.Sornette, "Test of Sabine's reverberation time in ergodic auditorium within geometrical acoustics", J.Acoust.Soc.Am.88(2), 865 (1990); F. Mortessagne, O.Legrand and D.Sornette, "Role of the absorption distribution and generalization of exponential reverberation law in chaotic rooms", J.Acoust.Soc.Am.94, 154-161 (1993); "Renormalization of exponential decay rates by fluctuations of barrier encounters", Europhys.Lett. 20 (4), 287-293 (1992). [26] F. Mortessagne, O.Legrand and D.Sornette, "Transient chaos in room acoustics", Chaos 3, 529-541 (1993) [27] F. Mortessagne, O. Legrand and D. Sornette, "Geometric theory of wave dynamics in chaotic billiards", Europhys.Lett. submitted [28] O.Legrand and D.Sornette, "Fractal set of recurrent orbits in billiards", Europhys.Lett.11,583 (1990) | |
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