# American Institute of Mathematical Sciences

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

## Resonant oscillations of an inhomogeneous gas in a closed cylindrical tube

 1 Department of Applied Mathematics, University College, Cork, Ireland 2 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
 [1] Alexander Krasnosel'skii. Resonant forced oscillations in systems with periodic nonlinearities. Discrete & Continuous Dynamical Systems, 2013, 33 (1) : 239-254. doi: 10.3934/dcds.2013.33.239 [2] Bernold Fiedler, Isabelle Schneider. Stabilized rapid oscillations in a delay equation: Feedback control by a small resonant delay. Discrete & Continuous Dynamical Systems - S, 2020, 13 (4) : 1145-1185. doi: 10.3934/dcdss.2020068 [3] Miroslav Bulíček, Eduard Feireisl, Josef Málek, Roman Shvydkoy. On the motion of incompressible inhomogeneous Euler-Korteweg fluids. Discrete & Continuous Dynamical Systems - S, 2010, 3 (3) : 497-515. doi: 10.3934/dcdss.2010.3.497 [4] Fabrizio Colombo, Irene Sabadini, Frank Sommen. The inverse Fueter mapping theorem. Communications on Pure & Applied Analysis, 2011, 10 (4) : 1165-1181. doi: 10.3934/cpaa.2011.10.1165 [5] John Banks. Topological mapping properties defined by digraphs. Discrete & Continuous Dynamical Systems, 1999, 5 (1) : 83-92. doi: 10.3934/dcds.1999.5.83 [6] Mads Kyed. On a mapping property of the Oseen operator with rotation. Discrete & Continuous Dynamical Systems - S, 2013, 6 (5) : 1315-1322. doi: 10.3934/dcdss.2013.6.1315 [7] Fabiana Maria Ferreira, Francisco Odair de Paiva. On a resonant and superlinear elliptic system. Discrete & Continuous Dynamical Systems, 2019, 39 (10) : 5775-5784. doi: 10.3934/dcds.2019253 [8] Yutong Chen, Jiabao Su. Resonant problems for fractional Laplacian. Communications on Pure & Applied Analysis, 2017, 16 (1) : 163-188. doi: 10.3934/cpaa.2017008 [9] Mike Boyle, Sompong Chuysurichay. The mapping class group of a shift of finite type. Journal of Modern Dynamics, 2018, 13: 115-145. doi: 10.3934/jmd.2018014 [10] Zhengxin Zhou. On the Poincaré mapping and periodic solutions of nonautonomous differential systems. Communications on Pure & Applied Analysis, 2007, 6 (2) : 541-547. doi: 10.3934/cpaa.2007.6.541 [11] Jacopo De Simoi. On cyclicity-one elliptic islands of the standard map. Journal of Modern Dynamics, 2013, 7 (2) : 153-208. doi: 10.3934/jmd.2013.7.153 [12] Jintai Ding, Sihem Mesnager, Lih-Chung Wang. Letters for post-quantum cryptography standard evaluation. Advances in Mathematics of Communications, 2020, 14 (1) : i-i. doi: 10.3934/amc.2020012 [13] Abbas Bahri. Attaching maps in the standard geodesics problem on $S^2$. Discrete & Continuous Dynamical Systems, 2011, 30 (2) : 379-426. doi: 10.3934/dcds.2011.30.379 [14] Maksym Berezhnyi, Evgen Khruslov. Non-standard dynamics of elastic composites. Networks & Heterogeneous Media, 2011, 6 (1) : 89-109. doi: 10.3934/nhm.2011.6.89 [15] Haifeng Chu. Surgery on Herman rings of the standard Blaschke family. Discrete & Continuous Dynamical Systems, 2018, 38 (1) : 63-74. doi: 10.3934/dcds.2018003 [16] Pavao Mardešić, David Marín, Jordi Villadelprat. Unfolding of resonant saddles and the Dulac time. Discrete & Continuous Dynamical Systems, 2008, 21 (4) : 1221-1244. doi: 10.3934/dcds.2008.21.1221 [17] Mapundi K. Banda, Michael Herty, Axel Klar. Gas flow in pipeline networks. Networks & Heterogeneous Media, 2006, 1 (1) : 41-56. doi: 10.3934/nhm.2006.1.41 [18] Martin Gugat, Falk M. Hante, Markus Hirsch-Dick, Günter Leugering. Stationary states in gas networks. Networks & Heterogeneous Media, 2015, 10 (2) : 295-320. doi: 10.3934/nhm.2015.10.295 [19] Robert Schippa. Generalized inhomogeneous Strichartz estimates. Discrete & Continuous Dynamical Systems, 2017, 37 (6) : 3387-3410. doi: 10.3934/dcds.2017143 [20] Jonathan E. Rubin, Justyna Signerska-Rynkowska, Jonathan D. Touboul, Alexandre Vidal. Wild oscillations in a nonlinear neuron model with resets: (Ⅱ) Mixed-mode oscillations. Discrete & Continuous Dynamical Systems - B, 2017, 22 (10) : 4003-4039. doi: 10.3934/dcdsb.2017205

2020 Impact Factor: 1.327