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 nonperiodic structures but not really disorderly.
A structure is calling hierarchic when, according
to the observation degree, it presents a "selfsimilar"
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 nonentire 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, "selfsimilar"
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.
