Second-sound phenomena in inviscid, thermally relaxing gases
Pedro M. Jordan
Discrete & Continuous Dynamical Systems - B 2014, 19(7): 2189-2205 doi: 10.3934/dcdsb.2014.19.2189
We consider the propagation of acoustic and thermal waves in a class of inviscid, thermally relaxing gases wherein the flow of heat is described by the Maxwell--Cattaneo law, i.e., in Cattaneo--Christov gases. After first considering the start-up piston problem under the linear theory, we then investigate traveling wave phenomena under the weakly-nonlinear approximation. In particular, a shock analysis is carried out, comparisons with predictions from classical gases dynamics theory are performed, and critical values of the parameters are derived. Special case results are also presented and connections to other fields are noted.
keywords: Cattaneo--Christov gas hyperbolic Burgers equation traveling waves. Nonlinear acoustics shock waves
Introduction to the special volume ``Mathematics of nonlinear acoustics: New approaches in analysis and modeling''
Pedro M. Jordan Barbara Kaltenbacher
Evolution Equations & Control Theory 2016, 5(3): i-ii doi: 10.3934/eect.201603i
Over the last 12--15 years, there has been a resurgence of interest in the study of nonlinear acoustic phenomena. Using the tools of both classical mathematical analysis and computational physics, researchers have obtained a wide range of new results, some of which might be described as remarkable. As with almost all trends in science, the reasons for this revival are varied: they range from practical applications (e.g., the need to improve our understanding of high-intensity ultrasound); to the development of numerical schemes which are better at capturing the physics of nonlinear compressible flow; to new acoustic models which lend themselves to study by analytical methods.

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