The calculation of resonance separatrices for the near-integrable case of the standard mapping

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.     

The evolution of nonlinear standing waves in bounded media

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.

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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.

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Presented at EUROMECH 73, Aix-en-Provence, April 1976.     

The evolution of nonlinear resonant oscillations in closed tubes: Solution of the standard mapping

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.     

Evolution of Nonlinear Sloshing in a Tank Near Half the Fundamental Resonance

This article describes the evolution of shallow water waves in a tank that is closed at one end and is periodically forced at half the fundamental frequency at the other end. The nonlinear response occurs at the same order as the linear response and is governed by a forced Korteweg–de Vries ( K dV ) equation. Unlike the corresponding problem for a gas (or the hydraulic limit), there may be nonperiodic (beating) solutions and multiple steady solutions for the same frequency. The addition of a component at the fundamental frequency to the piston input can be used to cancel the nonlinear effects and leave only the linear response in the steady state.     

The evolution of a finite rate periodic oscillation

Finite rate oscillations of a gas in a closed tube arise when the amplitude of the applied periodic piston velocity is small while its acceleration is unrestricted. The asymptotic form of the periodic motion for large acceleration is given. The evolution to the final periodic motion from the initial state of rest is constructed for a finite rate oscillation. Exact results for a piecewise linear piston velocity are used to illustrate the solutions.     

Nonlinear sloshing and passage through resonance in a shallow water tank

This paper is concerned with the effect of slowly changing the length of a tank on the nonlinear standing waves (free vibrations) and resonant forced oscillations of shallow water in the tank. The analysis begins with the Boussinesq equations. These are reduced to a nonlinear differential-difference equation for the slow variation of a Riemann invariant on one end. Then a multiple scale expansion yields a KdV equation with slowly changing coefficients for the standing wave problem, which is reduced to a KdV equation with a variable dispersion coefficient. The effect of changing the tank length on the number of solitons in the tank is investigated through numerical solutions of the variable coefficient KdV equation. A KdV equation which is “periodically” forced and slowly detuned results for the passage through resonance problem. Then the amplitude-frequency curves for the fundamental resonance and the first overtone are given numerically, as well as solutions corresponding to multiple equilibria. The evolution between multiple equilibria is also examined.Received: December 16, 2003     

Resonant hydraulic sloshing in a tank of variable depth

The forced resonant oscillations of a fluid in a tank of variable depth are considered within the hydraulic approximation. It is shown that for certain bottom topographies a continuous periodic output dominated by the first normal mode is possible. This contrasts with the case of a tank of constant depth, where hydraulic jumps are a feature of the motion. The amplitude and frequency of the output are connected by a cubic equation. The fluid response can act like that of a hard or soft spring, depending on the bottom topography. There is also a critical bottom topography that yields a higher order response amplitude.     

Standing waves in an open pipe: A nonlinear initial-boundary value problem

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.

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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.