Order, Disorder, Velocity Dispersion and Time- Frequency Analysis

International Journal of Science and Research (IJSR) ISSN (Online): 2319-7064, Impact Factor (2013): 4.438 www.ijsr.net Order, Disorder, Velocity Dispersion and TimeFrequency Analysis Manell E. Zakharia Professor, International Technical Expert, French Embassy in India, Project Officer. Indo-French cooperation Indian Institute of Technology, IIT Jodhpur, ROOM 3108, Old Residency Road, Ratanada, Jodhpur - 342 011, India Abstract: The propagation of wideband pulses in multilayered periodic media shows a variable signal structure depending on the pulse bandwidth B (time resolution). Depending on the ratio between travel time in a layer and the time resolution, the propagated signal can follow different regimes: pulse trains, null (extinction) or dispersive waveform (chirp). The phenomena have been observed both on experimental and simulated data. The simulations have allowed a better understanding the phenomena of velocity dispersion: the presence of a large number of thin layers that could not be separated individually (thickness << time resolution i.e. each individual layer is acoustically “transparent”) can affect considerably the propagation and lead to a dispersive "equivalent medium". For such a case, increasing the bandwidth will lead to pulse train structure of propagated signal. Nevertheless, if layer thickness is reduced accordingly, velocity dispersion phenomenon occurs again. Several simulations on random thickness layers have shown that a thickness variation of about 1/8 wavelength (at the central frequency) lead to the destruction of velocity dispersion phenomena. The dispersion is definitely related to a periodicity of the layers. Time-frequency was use to characterize and quantify velocity dispersion. Group velocity was estimated accurately using group delay extracted from Wigner-Ville distribution. An example will be shown for Lamb waves in plates before investigating an industrial application: non destructive evaluation of electrical cables using ultrasounds. The difference between a "good cable" and bad cable" lies in the insulating material (polyethylene) whose structure can be either amorphous or polymerised (periodical structure at microscopic level). Here again the velocity dispersion is an indicator of "order". 1. Introduction In 1992, Marion and Coudin [1] have published a very interesting paper on experimental investigation of the propagation through a periodic medium. Although the paper was oriented towards geophysical applications, it can lead to several general observations on the propagation of sound in a layered media. The Relevant results are summarized in figure 1 extracted from reference [1]. The samples investigated consist in a sequence of layers of steel and plastic. The stack size is always the same (52 mm). The number of layer in a sample vary from 2 (A) to 64 (F). The same pattern is reproduced periodically for all stacks: 1/3 plastic, 2/3 steel. So, for each sample, the same amount of steel and plastic is present (same total thickness of each material) but it is ordered in a different layered geometry. The experimental results cover a frequency range from about 50 to about 500 kHz. When a short pulse is transmitted through a periodic system, the received signal is constituted of several components: direct propagation, single reflexiontransmission on interfaces and multiple reflexions and transmissions. For thick layers (A), several pulses can be observed and separated that correspond to the cited components. When layer thickness decreases, these echo components are getting closer and closer (B-C) until they cannot be separated any more. When pulses separation is no more possible (D to F), the stack behaves like an “equivalent medium” with two particular properties: • the minimum propagation time through stacks D-F is about twice the one for A-C • velocity dispersion occurs that is characterised by the arrival of low frequencies before high frequencies: output is not a succession of pulses but a frequency modulated signal (chirp). Figure 1: Propagation through multilayered media (time scale: 0-50 µs) such a thickness is acoustically transparent, the Velocity dispersion occurs when travel time through a layer multiplication of layers leads to quite different properties. It is about 1/8th of the time resolution. Although one layer of International Symposium on Ultrasonics-2015, 22-24 January 2015 Department of Physics, Rashtrasant Tukdoji Maharaj Nagpur University, Nagpur, Maharashtra, India 353 Licensed Under Creative Commons Attribution CC BY International Journal of Science and Research (IJSR) ISSN (Online): 2319-7064, Impact Factor (2013): 4.438 www.ijsr.net is important to note that, thanks to layering, one can build up composite media possessing a sound velocity much lower than the velocity in each constituent. 2. Numerical modelling The simulated signal, for the reference case, is a pulse covering the frequency range up to 500 kHz. The reference case (a) corresponds to a stack geometry leading to velocity dispersion: it is clear from observing zero crossings that low frequencies travel faster than high frequencies, in this case. It was shown [2] that the propagation through a layered medium can be modelled using propagator matrix method. We will not go into the details of such model but try to use it in order to understand the velocity dispersion observed experimentally. Several simulations have been run using various signal parameters and stack geometries. In the figure 2b, the stack geometry has been kept unchanged while the bandwidth has been increase by a factor of 4 (0-2 MHz), i.e. the time resolution has been improved by a factor of 4. In this case, the dispersive nature of the waves has disappeared and the output signal is a sequence of individual pulses (as in figure 1a). Figures 2 show some representative examples in the case where the travel time is the same in both layer (layer thickness ratio adjusted to velocity ratio in order to have a travel time of 10 ฀s in each layer). With the same time resolution (B = 2 MHz), the layer thickness has been divided by 4 for figure 2c. The signal looks exactly the same with a scale factor of 4 in time. Similarly, for 500 kHz bandwidth, when the thickness is multiplied by 4, the output is the same as in figure 2b. The reference case consists of a stack of 32 layers of steel and plastic (same properties as [1] and [2]: steel/plastic, velocity: 5535 / 2487 m/s, density: 7.90 / 1.21 T/m3). Figure 2: Illustration of scale factor in multilayered propagation. These figures clearly illustrate the existence of relationship between the observation bandwidth and the layer thickness that can lead to velocity dispersion in the stack. Like other dispersion equations this relationship is more complex than just a linear one. A close look at output spectra has shown the resonant aspect of the power spectrum of transmitted signals as well as its band-pass band-stop structure. Nevertheless, it is clear that physical phenomena involved possess multiple-scale aspects. As for other dispersive propagation geometries (waveguides, surface waves…) the dispersion relation is connected to a time repetition of a signal component. It is essential to know how regular this repetition must be and which jitter can lead to the destruction of the dispersion phenomena. For this purpose, a series of simulations have been conducted with random layer thickness. In fact, we started with previous periodic structures (with velocity dispersion) and added random noise on the layer thickness. Hundreds of simulations showed that as long as the standard deviation of the error on thickness is less than λ 0 /8 (λ 0 being the wavelength corresponding to the central frequency), the dispersion phenomena is present. It starts fading out when the error exceeds this value and disappear totally for large errors (output signal become a series of spikes). This experiment and the associated simulations have illustrated the complexity of the propagation in a periodic medium. In particular, they have pointed out that, when the travel time in an elementary layer is less than the time resolution, the stack behaves as an “equivalent medium” possessing a slower velocity and velocity dispersion. This dispersion disappears progressively when the “order” in the material is reduced (random thickness). 3. Time-Frequency propagation analysis of dispersive Introduction The velocity dispersion expresses a strong relationship between the time and the frequency domain: every frequency is travelling at a different speed and no time measurement can be achieved without prior frequency International Symposium on Ultrasonics-2015, 22-24 January 2015 Department of Physics, Rashtrasant Tukdoji Maharaj Nagpur University, Nagpur, Maharashtra, India 354 Licensed Under Creative Commons Attribution CC BY International Journal of Science and Research (IJSR) ISSN (Online): 2319-7064, Impact Factor (2013): 4.438 www.ijsr.net information. Many methods have tried to overcome this • direct access to power spectrum and envelope by strong relationship that operates either the time or the computing function marginals frequency domain. They are commonly using strong • conservation of bandwidth and duration approximations (narrow band, small dispersion…) that • compatibility with linear filtering and probably the most reduce their potential use in a general case. As the velocity important property: dispersion is a time-frequency relationship, we will describe • possibility of computing group delay (and thus group it explicitly in the time-frequency domain. As physicians, velocity) from distribution moment: energy distribution is one of our major concerns. For a time∫Wz (t , f ).t .dt d frequency energy representation that is compatible with τ g ( f ) 21= = π dt arg Z ( f ) ∫Wz (t , f ).dt propagation (time invariance properties), the general expression for finite energy signals (s) is given by the socalled Cohen’s class [3], [4]: Thank to all these properties, the propagation can simply be 2 iπξ ( s −t ) 2 iπ fdescribed τ τ = ρ s (t , f ) g e .g (ξ ,τ ).s *( s − 2 ).s ( s + 2 ).e − τ d ξ .ds.dinτ the time-frequency plane: an impulsive signal is a vertical line while a pure tone is a horizontal one. Figures 3 illustrate the propagation from point a to point b in the time-frequency plane for both dispersive and non dispersive Where g is a smoothing (2D) window while WV z is the socases: while, for non-dispersive case, the propagation is a called Wigner-Ville distribution: simple horizontal translation of the elementary pattern (time−2 iπ f τ frequency representation of the transmitted signal), the τ τ WVs (t , f ) = s *(t − 2 ).s (t + 2 ).e dτ propagation in a dispersive (and absorbent) media lead to a translation and rotation of the pattern as well as to frequency Aside the time invariant properties, the Wigner-Ville dependant attenuation. distribution possesses several particular properties that are essential for the interpretation of the propagation of Thanks to the properties of the Wigner-Ville distribution, the acoustics waves in dispersive media: pattern amplitude can be directly related to the absorption and the first moment can be used to estimate group delay. [ ] ∫∫∫ ∫ Figures 3 [5]: Illustration of the propagation in the time frequency plane Figures 4 show experimental Lamb waves that have travelled respectively 6 (a) and 12 (b) cm on a plate. For Application to Lamb waves in a plate each signal the Wigner-Ville distribution has been computed Such techniques have been applied to the estimation of and displayed. The time dispersion is straightforward when group velocity of Lamb waves (measurement of group delay comparing time axes. For each Wigner-Ville distribution, in two points) and validated experimentally [6], [7]. group delay was computed and then group velocity was obtained and compared to theoretical values (figures 5). International Symposium on Ultrasonics-2015, 22-24 January 2015 Department of Physics, Rashtrasant Tukdoji Maharaj Nagpur University, Nagpur, Maharashtra, India 355 Licensed Under Creative Commons Attribution CC BY International Journal of Science and Research (IJSR) ISSN (Online): 2319-7064, Impact Factor (2013): 4.438 www.ijsr.net Figures 4 [6]: propagation of lamb (dispersive) waves top: received signal, bottom: Wigner-Ville representation, Vertical scale: 0-2 MHz, 4dB/grey level. Figure 5 show clearly the relevance of the method for both qualitative and quantitative description of the propagation in dispersive media: the comparison of theoretical and experimental velocity dispersion shows an error of less of 5% (comparable to experiment accuracy). Figure 5 [6]: velocity dispersion of Lamb waves, over an aluminium plate of 10 mm Solid curves correspond to theoretical values; dots correspond to experimental values; dot size gives an order of magnitude of experimental error, discrepancy < 5%. operation, the heating (that could lead to melting) is generated from inside to outside (while manufacturing heating is from outside to inside). For industrial control, an ultrasound evaluation of cables and cable materials has been conducted. The size of monomers is about a few hundred of microns. When not reticulated, they form a random arrangement in longer polymer molecules. When heated, they move freely and the electrostatic forces lead them to be arranged periodically. The propagation in reticulated polymer is very similar to the propagation in a periodic stack while the propagation in nonreticulated one is similar to random stack. The results on material samples showed that the propagation in non reticulated polymers is non-dispersive while it becomes dispersive in reticulated ones. After testing polymer samples, several experiments were conducted with real (bad and good) cables (immersed in water for coupling). Figure 6 shows a typical example of a cable echo: it is constituted mainly of two components: • outer echo component: this first component is due to the (rough) interface between the water and the cable • inner echo component: this second component is due to the reflexion on the metallic conductor and carries information on two-way travel of ultrasonic waves in the insulator. We have studied this particular component in details. Application to industrial control, ultrasonic NDE [7], [8] High voltage cable manufacturing consists in extruding an insulating polymer (around a metallic conductor). In its original form, the polymer is amorphous (monomer molecules are randomly distributed) and is fragile from both mechanical and thermal point of view. The manufacturing process consists then in heating the cable in order to reticulate it (thermo-chemical reaction). The reticulated polymer will possess the desired mechanical and thermal properties. It is essential that the inner part of the polymer (close to metallic conductor) is fully reticulated as, in Figure 6 [7]: Example of cable echo (timescale 0-2 µs) International Symposium on Ultrasonics-2015, 22-24 January 2015 Department of Physics, Rashtrasant Tukdoji Maharaj Nagpur University, Nagpur, Maharashtra, India 356 Licensed Under Creative Commons Attribution CC BY International Journal of Science and Research (IJSR) ISSN (Online): 2319-7064, Impact Factor (2013): 4.438 www.ijsr.net The time-frequency analysis (Wigner-Ville) of inner echo component was carried out for tens of cables in different points. • Time spreading doubled for a good cable • Additional absorption is associated to this velocity dispersion (frequency spreading reduction of about 40%). Figures 7 show typical time-frequency plots for both good”” and “bad cables”. Figure 7b shows that the dispersive nature of the insulator is visible on inner component: Figure 7 [7], [8]: Typical time-frequency pattern of inner cable echo 4. Conclusion The propagation in multilayered media can lead to velocity dispersion when the travel time through a single layer is comparable to time resolution and when the layering is periodic with regularity better than ฀/8: the presence of a large number of thin layers (thickness << time resolution, i.e. each individual layer is acoustically transparent) that could not be separated individually, can affect considerably the propagation and lead to a dispersive "equivalent medium". The equivalent composite material obtained by such a construction possesses an equivalent sound velocity much lower that the velocity in each component. The dispersion disappears when the difference in layer thickness exceeds ฀/8. The relationship between “geometrical order” and “velocity dispersion” has been observed in both macroscopic and microscopic (polymer) cases. Velocity dispersion can be fully characterized and evaluated quantitatively (group velocity) without any approximation by using Wigner-Ville time-frequency analysis of dispersed signal even in complex cases (several propagation modes present). An illustration has been given for Lamb waves. Finally an industrial control application has been shown: control of electrical cables. The order properties and the associated velocity dispersion observed on reticulated polymers sampled were also present in cable echo and were successfully used for characterizing good cables. [3] L. Cohen, Time frequency distribution. Proceeding of IEEE VOI77 No 7, 1989; pp. 941-981. [4] P. Flandrin, Time-Frequency/Time-Scale Analysis, 386 p., Academic Press, San Diego, 1999. [5] Application of time-frequency analysis to Non Destructive Evaluation M. E. Zakharia and J.-P. Sessarego, in Proceedings of The IEEE Ultrasonics Symposium, Lake Buena Vista, Florida (USA), December 1991, B.R. McAvoy Ed., vol. 2, pp. 809818. [6] J.P. Sessarego, J. Sageloli, P. Degoul, P. Flandrin and M Zakharia, Analyse temps-fréquence de signaux en milieux dispersifs. Application à I’étude des ondes de Lamb". J. Acoustique 3 (1990) pp. 273-280. [7] Pascal COMBE, Application des méthodes de déconvolution à la caractérisation ultrasonore de l’isolant de câbles électriques et analyse tempsfréquence des réponses acoustiques. Non destructive testing of electric cables by ultrasounds using deconvolution and time-frequency analysis. PhD Thesis INP, Grenoble. Sept. 1988. [8] Ultrasonic transducer response correction via inverse filtering. Characterization in the time-frequency plane. M. E. Zakharia, in Proceedings of The Ultrasonics International Conference, Madrid (Spain), July 1989, Butterworth Scientific Publishers, pp. 813-820. References [1] Marion, D., and Coudin, P., 1992, From ray to effective medium theories in stratified media: and experimental study, 62nd Ann. Intern. Mtg., Soc. Expl. Geophys., Expanded Abstract, 1341-1343 [2] Jens. M. Hovem, Acoustic waves in finely layered media, Geophysics, vol. 60, N°4, July-August 1995, pp. 1217-1221. International Symposium on Ultrasonics-2015, 22-24 January 2015 Department of Physics, Rashtrasant Tukdoji Maharaj Nagpur University, Nagpur, Maharashtra, India 357 Licensed Under Creative Commons Attribution CC BY