May  2007, 7(3): 619-628. doi: 10.3934/dcdsb.2007.7.619

Resonant oscillations of an inhomogeneous gas in a closed cylindrical tube


Department of Applied Mathematics, University College, Cork, Ireland


Department of Mathematics, University of British Columbia, Vancouver, V6T 1Z2, Canada

Received  September 2006 Revised  January 2007 Published  February 2007

Experimental work on the basic problem of resonant acoustic oscillations in a closed straight cylindrical tube goes back at least to Lettau [14]. He showed that, even for "small" piston velocities, shock waves traverse the tube. Shocks are a nonlinear phenomenon and a means of converting mechanical energy to heat. Betchov [1], followed by Chu and Ying [4]}, Gorkov [7] and Chester [2], gave the first satisfactory theoretical explanation of the phenomena. The interest at this time was in an understanding of noise excitation in jets and reciprocating engines. A completely new phenomenon emerged with the experiments of Lawrenson et al [13]. They showed that very high shockless pressures can be generated by resonant acoustic oscillations in specially shaped containers. They called this Resonant Macrosonic Synthesis (RMS) and indicated important technological applications. The first analytical results explaining RMS were given by Mortell & Seymour [18], showing good qualitative agreement with both experimental and numerical results. The challenge was to understand the interaction of the geometry with the nonlinearity. It was shown that when the geometry yields incommensurate eigenvalues, i.e. the higher modes are not integer multiples of the fundamental, the resulting motion is shockless. With no shocks, higher pressures resulted for the same energy input. Here we review the 'classical' resonance in a straight tube, and then show that shockless motions can be produced even in a straight tube by introducing a variable ambient density distribution.
Citation: Michael P. Mortell, Brian R. Seymour. Resonant oscillations of an inhomogeneous gas in a closed cylindrical tube. Discrete & Continuous Dynamical Systems - B, 2007, 7 (3) : 619-628. doi: 10.3934/dcdsb.2007.7.619

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