On the oscillations near and at resonance in open pipes

L. van Wijngaarden

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Abstract

This paper is concerned with resonance oscillations occurring when a piston executes small oscillations on one end of a pipe which is open to the atmosphere at the other end. According to linear theory very large amplitudes of pressure and velocity oscillations in the gas in the pipe result when the piston is oscillated with an angular frequency nearpgrao/2L, where ao is the sound velocity of the gas and L the length of the pipe. In the theory of resonators, due to Helmholtz and Rayleigh and discussed in section 1, radiation from the open end is taken into account. Then resonance occurs at a frequency slightly belowohgr o , and amplitudes are still very large, as is shown in section 1. Therefore a nonlinear theory is developed here, analogous to previous work on resonance oscillations in closed pipes. In section 2 the boundary conditions at the open end are formulated based on the fact that the reservoirconditions are constant at inflow but vary at outflow, since the gas issues as a jet. This difference results in a net efflux of energy to be balanced by the work done by the piston. In sections 3¿7 a perturbation theory is developed in terms of the characteristics of motion. The pertinent perturbation parameter is suggested by the energy balance. An ordinary differential equation for the first order perturbation in the quasi-steady state is obtained in section 7. In section 8 experimental results are presented together with results obtained from numerical integration of the above mentioned equation. The results, showing a satisfactory agreement, indicate that further experimental investigation on the conditions at the open end are needed.
Original languageEnglish
Pages (from-to)225-240
JournalJournal of engineering mathematics
Volume2
Issue number3
DOIs
Publication statusPublished - 1968

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Pipe
Pistons
Oscillation
Gases
Acoustic wave velocity
Energy balance
Ordinary differential equations
Resonators
Energy Balance
Parameter Perturbation
Hermann Von Helmholtz
Boundary conditions
Resonator
Experimental Investigation
Radiation
Rayleigh
Numerical integration
Perturbation Theory
Atmosphere
Ordinary differential equation

Keywords

  • IR-50400

Cite this

van Wijngaarden, L. / On the oscillations near and at resonance in open pipes. In: Journal of engineering mathematics. 1968 ; Vol. 2, No. 3. pp. 225-240.
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On the oscillations near and at resonance in open pipes. / van Wijngaarden, L.

In: Journal of engineering mathematics, Vol. 2, No. 3, 1968, p. 225-240.

Research output: Contribution to journalArticleAcademic

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T1 - On the oscillations near and at resonance in open pipes

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PY - 1968

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N2 - This paper is concerned with resonance oscillations occurring when a piston executes small oscillations on one end of a pipe which is open to the atmosphere at the other end. According to linear theory very large amplitudes of pressure and velocity oscillations in the gas in the pipe result when the piston is oscillated with an angular frequency nearpgrao/2L, where ao is the sound velocity of the gas and L the length of the pipe. In the theory of resonators, due to Helmholtz and Rayleigh and discussed in section 1, radiation from the open end is taken into account. Then resonance occurs at a frequency slightly belowohgr o , and amplitudes are still very large, as is shown in section 1. Therefore a nonlinear theory is developed here, analogous to previous work on resonance oscillations in closed pipes. In section 2 the boundary conditions at the open end are formulated based on the fact that the reservoirconditions are constant at inflow but vary at outflow, since the gas issues as a jet. This difference results in a net efflux of energy to be balanced by the work done by the piston. In sections 3¿7 a perturbation theory is developed in terms of the characteristics of motion. The pertinent perturbation parameter is suggested by the energy balance. An ordinary differential equation for the first order perturbation in the quasi-steady state is obtained in section 7. In section 8 experimental results are presented together with results obtained from numerical integration of the above mentioned equation. The results, showing a satisfactory agreement, indicate that further experimental investigation on the conditions at the open end are needed.

AB - This paper is concerned with resonance oscillations occurring when a piston executes small oscillations on one end of a pipe which is open to the atmosphere at the other end. According to linear theory very large amplitudes of pressure and velocity oscillations in the gas in the pipe result when the piston is oscillated with an angular frequency nearpgrao/2L, where ao is the sound velocity of the gas and L the length of the pipe. In the theory of resonators, due to Helmholtz and Rayleigh and discussed in section 1, radiation from the open end is taken into account. Then resonance occurs at a frequency slightly belowohgr o , and amplitudes are still very large, as is shown in section 1. Therefore a nonlinear theory is developed here, analogous to previous work on resonance oscillations in closed pipes. In section 2 the boundary conditions at the open end are formulated based on the fact that the reservoirconditions are constant at inflow but vary at outflow, since the gas issues as a jet. This difference results in a net efflux of energy to be balanced by the work done by the piston. In sections 3¿7 a perturbation theory is developed in terms of the characteristics of motion. The pertinent perturbation parameter is suggested by the energy balance. An ordinary differential equation for the first order perturbation in the quasi-steady state is obtained in section 7. In section 8 experimental results are presented together with results obtained from numerical integration of the above mentioned equation. The results, showing a satisfactory agreement, indicate that further experimental investigation on the conditions at the open end are needed.

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