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  • 1
    Electronic Resource
    Electronic Resource
    The evolution of nonlinear resonant oscillations in closed tubes: Solution of the standard mapping (1992)
    Woodbury, NY : American Institute of Physics (AIP)
    Chaos 2 (1992), S. 201-214 
    Details
    ISSN: 1089-7682
    Source: AIP Digital Archive
    Topics: Physics
    Notes: The concern of this paper is the evolution of small-amplitude resonant oscillations of an inviscid gas in a closed tube. The evolution of the oscillations of the gas generated at and near the fundamental frequency and half the fundamental frequency, where experiment shows that shocks are a feature of the final periodic motion, is examined. The basis for the analysis of this nonlinear initial value, boundary value problem on a semi-infinite strip is the Dissipative Standard Map. Since the purpose is to elucidate how the initial rest state of the gas evolves to the final periodic state, the focus of the analysis is on how a prescribed initial curve is mapped, under the Dissipative Standard Map, onto an invariant curve. The method used is to approximate the Dissipative Standard Map, in the small rate limit, by the partial differential equation appropriate to the resonance in question.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.165906
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    Paper (German National Licenses)
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  • 2
    Electronic Resource
    Electronic Resource
    Standing waves in an open pipe: A nonlinear initial-boundary value problem (1973)
    Springer
    Zeitschrift für angewandte Mathematik und Physik 24 (1973), S. 473-487 
    Details
    ISSN: 1420-9039
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Description / Table of Contents: Zusammenfassung Zeitabhängige Störungen in einem mit Gas gefüllten offenen Rohr werden beschrieben mit Hilfe einer nicht-linearen Theorie, welche sowohl Amplituden-Dispersion wie auch gegenseitige Beeinflussung von Wellenzügen berücksichtigt. Für Reflektionen am offenen Ende muss die Theorie die nicht-linearen Korrekturen zweiter Ordnung zu den Charakteristiken berücksichtigen, um lineare und nicht-lineare Laufzeiten unterscheiden zu können. Die Anfangswertaufgabe wird auf die Lösung einer Funktional-Differenzen-Gleichung reduziert, welche das Signal am Rohrende bestimmt. Wenn das Signal gedämpft wird durch Zulassung von Abstrahlung am Rohrende, kann die Stossbildung vermieden werden.
    Notes: Summary Transient disturbances in a column of gas in an open tube are described using a nonlinear theory which includes both amplitude dispersion and wave interactions. For reflection from an open end, the theory must include the second nonlinear correction to the characteristic to distinguish the linear and nonlinear travel times. The initial value problem is reduced to solving a functional difference equation which determines the signal on a boundary. When the signal is damped by allowing radiation of energy at the boundaries, it may be possible to prevent shock formation.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01588151
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    Paper (German National Licenses)
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  • 3
    Electronic Resource
    Electronic Resource
    The evolution of resonant oscillations in closed tubes (1983)
    Springer
    Zeitschrift für angewandte Mathematik und Physik 34 (1983), S. 845-866 
    Details
    ISSN: 1420-9039
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Description / Table of Contents: Zusammenhang Ein Rohr mit einem geschlossenen Ende ist mit einem reibungsfreien Gas gefüllt und wird durch einen schwingenden Kolben am anderen Ende des Rohres angeregt. Angenommen wird eine eindimensionale isentrope Bewegung des Gases. Es wird die Entwicklung der Bewegung im Gas untersucht, ausgehend vom ursprünglichen Zustand der Ruhe bis zum Erreichen des periodischen Bewegungszustandes. Die Bewegung wird durch einen Ähnlichkeitsparameter gekennzeichnet. Es wird angezeigt wie Stöße sich formen und innerhalb des Resonanzbandes erhalten bleiben, während außerhalb des Resonanzgebietes Stöße zunächst auch entstehen, aber später abklingen, so daß im wesentlichen die lineare Lösung übrigbleibt.
    Notes: Summary An inviscid gas is contained in a tube which is closed at one end and is excited by an oscillating piston at the other end. The motion of the gas is assumed to be one-dimensional and isentropic. The evolution of the gas motion, from the initial rest state to the final periodic state, is examined. The motion is characterised by a similarity parameter. It is shown how shocks form and are sustained within the resonant band, while outside the resonant band shocks form but decay to leave essentially the linear solution.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00949060
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    Paper (German National Licenses)
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  • 4
    Electronic Resource
    Electronic Resource
    The calculation of resonance separatrices for the near-integrable case of the standard mapping (1988)
    Springer
    Zeitschrift für angewandte Mathematik und Physik 39 (1988), S. 861-873 
    Details
    ISSN: 1420-9039
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: Abstract The standard mapping arises in many physical applications, including the analysis of nonlinear resonant acoustic oscillations in a closed tube. A perturbation expansion, in powers of the amplitude parameter, is given for the calculation of the fixed points of various orders and the associated separatrices. It is shown how exact homoclinic orbits can be calculated numerically. Explicit analytic expressions are given for the separatrices associated with the first four resonances when the perturbation parameter is small.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00945123
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    Paper (German National Licenses)
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  • 5
    Electronic Resource
    Electronic Resource
    The evolution of nonlinear standing waves in bounded media (1977)
    Springer
    Zeitschrift für angewandte Mathematik und Physik 28 (1977), S. 33-46 
    Details
    ISSN: 1420-9039
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Description / Table of Contents: Zusammenfassung Die Entwicklung von Störungen kleiner Amplitude in einem begrenzten Medium wird untersucht, mit festen und nahezu festen Endbedingungen. Die berücksichtigten physikalischen Effekte sind Amplituden-Dispersion, Frequenz-Dispersion und Dissipation sowohl durch Abstrahlung von Energie aus dem Medium wie auch durch die Deformationsgeschwindigkeit im Medium. In der nicht-linearen geometrischen Akustik ist die Transportgleichung, welche das von einer Wellenkomponente übertragene Signal bestimmt, die einfache Wellengleichung, bezw. die Korteweg-de Vries-Gleichung, die gedämpfte einfache Wellengleichung und die Burgers-Gleichung.
    Notes: Summary The evolution of small amplitude disturbances in a bounded medium, under fixed and nearly fixed end conditions, is considered. The various physical effects accounted for are amplitude dispersion, frequency dispersion and dissipation due to both radiation of energy out of the medium and rate-dependence of the medium. In a nonlinear geometrical acoustics theory the transport equations which determine the signal carried by a component wave have the form of a simple wave equation, Korteweg-de Vries equation, damped simple wave equation and Burgers' equation.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01590706
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    Paper (German National Licenses)
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