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

Pierre-Etienne Druet. A theory of generalised solutions for ideal gas mixtures with Maxwell-Stefan diffusion. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020458


Yi-Ming Tai, Zhengyang Zhang. Relaxation oscillations in a spruce-budworm interaction model with Holling's type II functional response. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021027


Hua Zhong, Xiaolin Fan, Shuyu Sun. The effect of surface pattern property on the advancing motion of three-dimensional droplets. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020366


Soonki Hong, Seonhee Lim. Martin boundary of brownian motion on gromov hyperbolic metric graphs. Discrete & Continuous Dynamical Systems - A, 2021  doi: 10.3934/dcds.2021014


Shao-Xia Qiao, Li-Jun Du. Propagation dynamics of nonlocal dispersal equations with inhomogeneous bistable nonlinearity. Electronic Research Archive, , () : -. doi: 10.3934/era.2020116


Claudia Lederman, Noemi Wolanski. An optimization problem with volume constraint for an inhomogeneous operator with nonstandard growth. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020391


Shin-Ichiro Ei, Hiroshi Ishii. The motion of weakly interacting localized patterns for reaction-diffusion systems with nonlocal effect. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 173-190. doi: 10.3934/dcdsb.2020329


Peter Frolkovič, Viera Kleinová. A new numerical method for level set motion in normal direction used in optical flow estimation. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 851-863. doi: 10.3934/dcdss.2020347


Alex H. Ardila, Mykael Cardoso. Blow-up solutions and strong instability of ground states for the inhomogeneous nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2021, 20 (1) : 101-119. doi: 10.3934/cpaa.2020259


Yunfeng Jia, Yi Li, Jianhua Wu, Hong-Kun Xu. Cauchy problem of semilinear inhomogeneous elliptic equations of Matukuma-type with multiple growth terms. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3485-3507. doi: 10.3934/dcds.2019227


Dongfen Bian, Yao Xiao. Global well-posedness of non-isothermal inhomogeneous nematic liquid crystal flows. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1243-1272. doi: 10.3934/dcdsb.2020161

2019 Impact Factor: 1.27


  • PDF downloads (30)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]