Research project (4 mounth)

Laboratoire Ondes et Acoustique
Ecole Supérieure de Physique et de Chimie Industrielles, Paris, France

Acoustical propagation into multi-scale environments.

Acoustical propagation
in a waveguide can be considerated as unidimensional if the longitudinal dimensions, and the wavelengths are much greater than the transversal ones. If the waveguide respects this approximation, the propagation can be approached by the study of acoustical quadripoles placed in serie or in parallel, which is the case for wind instruments. Sound propagation is approximated by the plane mode. The upper modes, propagative or not, that could exist in the tube, are not taken into account even though their influence are not negligible which is the case for important section discontinuities for exemple. Therefore, the unidimensional approach is a drastic approximation but the experimental measurements are still possible.

Considering the unidimentional propagation, it is possible to consider an acoustic tube as an acoustical filter in reference to the electric quadripoles chains. Moreover, with some different acoustic cavities configurations, like wind instuments horns, on the one hand, and the holes systems, on the other hand, it is possible to reach a similar acoustical behaviour : some frequency bands are favorised and better transmitted than others, and inversely. Another example using acoustical filters is the exhaust valve, and more generally, tubulure sets into heat engines: in one case, some frequencies can be favorised ; in another case, it is possible to avoid the emission of specific frequencies in the vehicule environment.

Guided propagation, in particular in acoustics, was subjected to many studies. Generally, those studies have concerned either some simple particular stuctures as an element of complexe device like a wind instument, or complicated structures as an approximation of complexe structure for some exhaust valves, or very particular structures like periodical or disorderly structures. In those cases, some results close to the most modern physics of solids has been outlined . Nevertheless, a lot of structures with interesting physical characteristics have never been (or rarely) approached.

The hierarchic structures are an example of non-periodic structures but not really disorderly. A structure is calling hierarchic when, according to the observation degree, it presents a "self-similar" characteristic on several scales ; in other words, when the same structure watched with an appropriate magnification will present the same geometry. A typical structure example is the Cantor set: the unity segment is divided in three parts ; the central part is deleted, and the process is repeated in the remaining segments. The object watched with a magnification three times as much important (just watching the third on the right or on the left) has the same structure as the precedent. By iteration, if the process is executed infinitely, we obtain what we call a "Cantor dust" with an non-entire topologic dimension. Such objects, called fractals, does not exist in the literal meaning in the nature, but some fractals structures exist like silica aerogels.

Cantor structures have been already studied for ultrasounds. Allowed and forbidden frequency bands have been found similarly to the wind instruments case. So, an acoustic Cantor set was studied on the waves lengths domain corresponding to the wind instruments, or in the automobile industry.

The aim of this project was to determine the hierarchical structure's effects on the characteristics of an acoustical filter. Moreover, the aim of this study was to find a model that could represent acoustic propagation on an extreme porous middle, with a complex structure, "self-similar" on several scales, and with a very high tortuosity : silica aerogels.

Firstly, a modelisation of the acoustical propagation in a Cantor set hierarchical structure was developed in the time domain. Then, with the entry impedance mesurement, a simulation optimisation was performed and comparisons with experimental datas was carried out. The hierarchical and random structures was compared in order to evaluate the audible acoustical propagation characteristics that could have an interest on the envisaged applications.

Gibiat V., Barjau A., Castor K., Bertaud du Chazaud E.,
Acoustical propagation in a prefractal wave-guide.

Physical Review E, vol. 67, 066609 (2003).

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