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Vibration des films smectiques librement suspendus. Effets non-linéaires, points diaboliques et oscillateurs auto-ajustables
Vibration des films smectiques librement suspendus. Effets non-linéaires, points diaboliques et oscillateurs auto-ajustables
M. Brazovskaia, H. Dumoulin and P. Pieranski.
"Nonlinear effects in vibrating smectic films."
Physical Review Letters 76, no. 10 (1996): 1655.
Dumoulin, H., M. Brazovskaia and P. Pieranski.
"Motion of islands on vibrating smectic films."
EPL (Europhysics Letters) 35, no. 7 (1996): 505.
M. Brazovskaia and P. Pieranski
« Self-tuning behaviour of vibrating smectic films”
Phys. Rev. Lett. 80 (1998) 5595
M. Brazovskaia and P. Pieranski,
"Diabolical pionts in the resonance spectra of vibrating smectic films",
Phys. Rev. E. 58 (1998) 4077
M. Ben Amar, P. Patrıcio da Silva, N. Limodin, A. Langlois, M. Brazovskaia, C. Even, I. V. Chikina, and P. Pieranski.
"Stability and vibrations of catenoid-shaped smectic films."
The European Physical Journal B 3, no. 2 (1998): 197-202.
I. Kraus, Ch Bahr, I. V. Chikina, and P. Pieranski.
"Can one hear structures of smectic films?."
Physical Review E 58, no. 1 (1998): 610.
M. Brazovskaia, and Pawel Pieranski.
"Smectic films and quantum billiards."
In Liquid Crystals: Physics, Technology, and Applications, 3318 (1998) 382-385,
International Society for Optics and Photonics, 1998.
Diabolical points in smectic drums
Smectic Films can be easily pierced by a glass fiber as shown on the right. The fiber hinders vibrations z(x,y,t) of the film and introduces an additional boundary condition :
z(xo,yo) = 0
As a consequence, frequencies of eigenmodes depend now on coordinates (xo,yo) of the fiber :
fn = fn(xo,yo) = 0
The functions fn = fn(xo,yo) for different eigenmodes n can be represented as surfaces in 3D. It has been pointed out by Michael Berry that these surfaces can intersect each other only in isolated points called diabolical points.
Experiments with smectic drums have shown that such diabolical points exist in the spectrum of a rectangular drum pierced by a glass fiber.
Self tuning behaviour of vibrating smectic films
The wave equation that rules the drumhead oscillations of freestanding smectic films is similar to the two-dimensional Schrödinger equation. This makes vibrating smectic films the classical analog of appropriate quantum billiards. In particular, a rectangular smectic film carrying a very small inclusion, like a small metallic ball, is equivalent to a rectangular quantum billiard with a quasi-point-like scatterer.
We point out that the ball, free to move in the plane of the film, tends to adjust its position as a function of the excitation frequency.
Stability and vibrations of a catenoid
Catenoid is a surface of revolution obtained by rotation of a catenary curve. It is also a minimal surface spanned between two parallel coaxial circles of radius R and separated by the distance H. For a given H and R and H/R<1.3254, two surfaces of zero mean curvature can be spanned between the two supporting circles: one is stable and the other one unstable.
We made experiments with smectic membranes supported by two circular frames. In particular, we measured frequencies of eigenmodes of vibration of such smectic catenoids.
As expected the frequency of the fundamental mode tends to zero when the aspect ratio approaches the critical value of 1.3254.
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