Advances in Differential Equations

Periodic solutions for a bidimensional hybrid system arising in the control of noise

Sorin Micu

Full-text: Access denied (no subscription detected) We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


We consider a simple model arising in the control of noise. We assume that the two-dimensional cavity $\Omega=(0,1)\times (0,1)$ is occupied by an elastic, inviscid, compressible fluid. The potential $\phi$ of the velocity field satisfies the linear wave equation. The boundary of $\Omega$ is divided in two parts, $\Gamma_{0}$ and $\Gamma_{1}$. The first one, $\Gamma_{0}$, is flexible and occupied by a dissipative vibrating string. The transversal displacement of the string, $W$, satisfies a non homogeneous one-dimensional wave equation. On $\Gamma_{0}$ the continuity of the normal velocities of the fluid and the string is imposed. The subset $\Gamma_{1}$ of the boundary is assumed to be rigid and therefore, the normal velocity of the fluid vanishes. This constitutes a non homogeneous dissipative system of two coupled wave equations in dimensions two and one respectively. The non homogeneous term acting on the flexible part of the boundary (an elastic force or an exterior source of noise) is assumed to be periodic. We are interested in the existence of periodic solutions of this system. Due to the localization of the damping term in a relatively small part of the boundary and to the effect of the hybrid structure of the system, the existence of periodic solutions holds for a restricted class of non homogeneous terms. Some resonance-type phenomena are also exhibited.

Article information

Adv. Differential Equations Volume 4, Number 4 (1999), 529-560.

First available in Project Euclid: 15 April 2013

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35L05: Wave equation
Secondary: 35B10: Periodic solutions 74F10: Fluid-solid interactions (including aero- and hydro-elasticity, porosity, etc.) 74H99: None of the above, but in this section 76Q05: Hydro- and aero-acoustics


Micu, Sorin. Periodic solutions for a bidimensional hybrid system arising in the control of noise. Adv. Differential Equations 4 (1999), no. 4, 529--560.

Export citation