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# I. Nonlinear Gas Oscillations in Pipes. II. Wavetrains with Small Dissipation

## Citation

Jiménez Sendín, Javier (1973) I. Nonlinear Gas Oscillations in Pipes. II. Wavetrains with Small Dissipation. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/HRSK-FC56. https://resolver.caltech.edu/CaltechTHESIS:08172011-101324443

## Abstract

In part I of this thesis we study theoretically the problem of forced acoustic oscillations in a pipe. The oscillations are produced by a moving piston in one end of the pipe, while a variety of boundary conditions ranging from a completely closed end to a completely open mouth are considered at the other end. All these boundary conditions are modelled by two parameters: a length correction and a reflection coefficient equivalent to the acoustic impedance. The linear theory predicts large amplitudes near resonance and non-linear effects become crucially important. By expanding the equations of motion in a series of the Mach number, both the amplitude and waveform of the oscillations are predicted there. In both the open and closed-end cases the need for shock waves in some range of parameters is found. The amplitude of the oscillation is different for the two cases, however, being proportional to the square root of the piston amplitude in the closed end case, and to the cube root in the open end. This part of the thesis was first published in the Journal of Fluid Mechanics. In part II we modify the averaged Lagrangian method used by Whitham to analyze slowly varying non-linear wavetrains to include cases with a small dissipation. To do this, we use a pseudo-variational principle introduced by Prigogine in which the Lagrangian depends on the variable and the solution of the problem, and which can be used to describe irreversible processes. We prove the corresponding averaged equations to all orders and describe practical ways to use them to lowest order.

Item Type: Thesis (Dissertation (Ph.D.)) (Applied Mathematics) California Institute of Technology Engineering and Applied Science Applied Mathematics Public (worldwide access) Whitham, Gerald Beresford Unknown, Unknown 22 May 1973 CaltechTHESIS:08172011-101324443 https://resolver.caltech.edu/CaltechTHESIS:08172011-101324443 10.7907/HRSK-FC56 No commercial reproduction, distribution, display or performance rights in this work are provided. 6586 CaltechTHESIS INVALID USER 17 Aug 2011 17:32 17 Jul 2024 17:04

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