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On linear and nonlinear acoustics in stratified variable-area ducts and atmospheres and Lighthill's proposition

Published online by Cambridge University Press:  11 March 2021

C. D. Matzner Open the ORCID record for C. D. Matzner [Opens in a new window]  and
S. Ro Open the ORCID record for S. Ro [Opens in a new window]
C. D. Matzner*
Affiliation:
David A. Dunlap Department of Astronomy and Astrophysics, University of Toronto, 50 St. George Street, TorontoM6J 2K2, Canada
S. Ro
Affiliation:
Astronomy Department and Theoretical Astrophysics Center, University of California, Berkeley, CA94720, USA
*
†Email address for correspondence: [email protected]
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Abstract

We consider linear and nonlinear waves in a stratified hydrostatic fluid within a channel of variable area, under the restriction of one-dimensional flow. We derive a modified version of Riemann's invariant that is related to the wave luminosity. This quantity obeys a simple dynamical equation in linear theory, from which the rules of wave reflection are easily discerned; and it is adiabatically conserved in the high-frequency limit. Following a suggestion by Lighthill, we apply the linear adiabatic invariant to predict mildly nonlinear waves. This incurs only moderate error. We find that Lighthill's criterion for shock formation is essentially exact for leading shocks, and for shocks within high-frequency waves. We conclude that approximate invariants can be used to accurately predict the self-distortion of low-amplitude acoustic pulses, as well as the dissipation patterns for weak shocks, in complicated environments such as stellar envelopes. We also identify fully nonlinear solutions for a restricted class of problems.

JFM classification

Compressible Flows: Gas dynamics Compressible Flows: Shock waves
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 915 , 25 May 2021 , A32
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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References

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