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An Asymptotic Method for the Solution of Nonlinear Wave Equations with Application to Oscillations in Gas-filled Tubes

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In this paper we consider second order hyperbolic differential equations with weak nonlinearities and generalized boundary conditions. The boundary conditions can contain second order time derivatives and small perturbations describing for example nonlinear and dissipative effects or external forces like oscillating boundaries.

To determine asymptotic solutions, we extend the Krylov-Bogoliubov-Mitropolskii-method by combining it with the theory of Sturm-Liouville problems involving the eigenvalue parameter in the boundary conditions.

This method is applied to nonlinear oscillations in gas-filled tubes with vibrating spring-mass-systems coupled at the ends.

Zusammenfassung

Wir untersuchen schwach nichtlineare partielle Differentialgleichungen zweiter Ordnung vom hyperbolischen Typ mit verallgemeinerten Randbedingungen. Die Randbedingungen dÜrfen sowohl zweite Zeitableitungen als auch kleine Störungen wie z. B. nichtlineare Terme, Reibungsglieder oder äu ßere Kräfte (schwingender Kolben etc.) enthalten. Es gelingt uns, monofrequente Näherungslösungen zu bestimmen, indem wir die asymptotische Methode von Krylov, Bogoliubov und Mitropolskii mit Hilfe der Theorie der Eigenwertprobleme, die den Eigenwertparameter linear in den Randbedingungen enthalten, erweitern.

Die Methode wird angewandt, um die nichtlinearen Schwingungen in einem luftgefüllten Rohr mit angekoppelten Feder-Masse-Systemen zu berechnen.

Sommaire

On examine des équations différentielles hyperboliques du second ordre avec des non-linéarités faibles et des conditions aux limites généralisées, ces dernières pouvant contenir des dérivées du second ordre par rapport au temps et de petites perturbations dues, par exemple, à des effets non-linéaires dissipatifs, ou à des forces extérieures comme celles qu'induisent des frontières oscillantes.

Pour déterminer des solutions asymptotiques, on a procédé à une extension de la méthode de Krylov-Bogoliubov-Mitropolskii en la combinant à la théorie des problèmes de Sturm-Liouville qui incorporent aux conditions aux limites le paramètre des valeurs propres.

Pour terminer on applique cette méthode aux oscillations non-linéaires qui se produisent dans des tubes remplis de gaz couplés à leurs extrémités avec des systèmes vibrants masse-ressort.
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Document Type: Research Article

Publication date: September 1, 1990

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  • Acta Acustica united with Acustica, published together with the European Acoustics Association (EAA), is an international, peer-reviewed journal on acoustics. It publishes original articles on all subjects in the field of acoustics, such as general linear acoustics, nonlinear acoustics, macrosonics, flow acoustics, atmospheric sound, underwater sound, ultrasonics, physical acoustics, structural acoustics, noise control, active control, environmental noise, building acoustics, room acoustics, acoustic materials, acoustic signal processing, computational and numerical acoustics, hearing, audiology and psychoacoustics, speech, musical acoustics, electroacoustics, auditory quality of systems. It reports on original scientific research in acoustics and on engineering applications. The journal considers scientific papers, technical and applied papers, book reviews, short communications, doctoral thesis abstracts, etc. In irregular intervals also special issues and review articles are published.
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