Global periodic attractor for strongly damped wave equations with time-periodic driving force

2004 ◽  
Vol 45 (9) ◽  
pp. 3462-3467 ◽  
Author(s):  
Hongyan Li ◽  
Shengfan Zhou ◽  
Fuqi Yin
Keyword(s):  
Driving Force ◽  
Wave Equations ◽  
Periodic Attractor ◽  
Global Periodic Attractor ◽  
Time Periodic ◽  
Strongly Damped

On the Parametrical Excitation of Nonlinearly Coupled Wave Equations

1995 ◽  
Author(s):  
A. H. P. van der Burgh ◽  
P. Kuznetsov ◽  
S. A. Vavilov
Keyword(s):  
Wave Equations ◽  
Operator Method ◽  
Dynamical Behaviour ◽  
Existence Theory ◽  
Rigorous Analysis ◽  
Coupled Wave Equations ◽  
Time Periodic Solutions ◽  
Transversal Vibrations ◽  
Time Periodic ◽  
Coupled Wave

Abstract In this paper a mathematical model for the study of the interaction of longitudinal and transversal vibrations in a stretched string is presented. The study implies an existence theory for time periodic transversal vibrations generated by a horizontal excitation of one of the end-points of the string. The conditions for the existence of this parametrically excited time periodic vibrations are evaluated in a practical application. The innovative character of the results obtained concern the application of an operator method to a system of nonlinearly coupled wave equations modeling the dynamical behaviour of a strectched string where unite elasticity is taken into account. It may be known that in the literature little attention has been paid to a rigorous analysis of time periodic solutions for systems of partial differential equations.

scholarly journals On the existence of solutions of strongly damped nonlinear wave equations

2000 ◽  
Vol 23 (6) ◽  
pp. 369-382 ◽  
Author(s):  
Jong Yeoul Park ◽  
Jeong Ja Bae
Keyword(s):  
Nonlinear Wave ◽  
Existence And Uniqueness ◽  
Wave Equations ◽  
Existence Of Solutions ◽  
Hyperbolic Type ◽  
Nonlinear Wave Equations ◽  
Gamma Delta ◽  
Uniqueness Of Solutions ◽  
Alpha Beta ◽  
Strongly Damped

We investigate the existence and uniqueness of solutions of the following equation of hyperbolic type with a strong dissipation:utt(t,x)−(α+β(∫Ω|∇u(t,y)|2dy)γ)Δu(t,x)                                −λΔut(t,x)+μ|u(t,x)|q−1u(t,x)=0,     x∈Ω,t≥0            u(0,x)=u0(x),          ut(0,x)=u1(x),      x∈Ω,  u|∂Ω=0, whereq>1,λ>0,μ∈ℝ,α,β≥0,α+β>0, andΔis the Laplacian inℝN.

Nonlinear Effects in Resonantly Driven Surface Waves on Fluids

1972 ◽  
Vol 50 (19) ◽  
pp. 2235-2243
Author(s):  
F. L. Curzon ◽  
G. N. Ionides
Keyword(s):  
Surface Waves ◽  
Electric Fields ◽  
Driving Force ◽  
Wave Amplitude ◽  
Nonlinear Effects ◽  
Microwave Resonator ◽  
Electrode Geometry ◽  
Fluid Surface ◽  
Electric Stress ◽  
Time Periodic

The results presented in this paper show that fluid surface waves, resonantly driven by spatially nonuniform, time periodic electric fields, exhibit nonlinear effects when the wave amplitude ξ exceeds a significant fraction of the distance D between the driver electrode and the fluid surface. The phase difference between the surface wave and the driving force, as well as the dependence of wave amplitude on the electric stress are computed and compared with experimental results. For ξ/D exceeding ~0.7 (dependent on electrode geometry) the surface waves excited are unstable (also confirmed experimentally). The experiments are performed on surface waves on mercury contained in a cylindrical microwave resonator. Shifts in the microwave resonant frequency (caused by the surface waves) monitor the displacement of the fluid surface.

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