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Aerodynamic sound in a relaxing medium

Published online by Cambridge University Press:  12 April 2006

J. T. C. Liu
J. T. C. Liu
Affiliation:
Division of Engineering, Brown University, Providence, Rhode Island 02912
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Abstract

In this paper we formulate the aerodynamic sound problem for a relaxing medium in a rather general way, independent of the details of the relaxation process. The medium is characterized by an appropriate relaxation time τ0 and by a frozen (af0) and an equilibrium (ae0) sound speed. The equation describing aerodynamic sound in such a medium is the familiar one describing acoustic waves in a non-equilibrium medium but subjected to aerodynamic sound sources expressed in terms of a frozen and an equilibrium form of the Lighthill stress tensor. The far-field result for both compact and non-compact sources in the frequency range ω [Gt ] τ0−1 can be expressed as the ratio of far-field densities for the relaxing and non-relaxing propagation medium: \[ \frac{\rho}{\rho_L} = \exp\left[-\left(1-\frac{a^2_{e_0}}{a^2_{f_0}}\right)\frac{x}{2a_{f0}\tau_0}\right], \] where x is the observation distance and the subscript L stands for ‘Lighthill’. The result for the main radiated aerodynamic sound, which comes from sources in the range ω [Lt ] τ0−1, essentially propagates in a manner described by the lower-order equilibrium waves, the diffusive effects from the higher-order waves being small, and the result for compact sources is a restatement of Lighthill's result in terms of the equilibrium propagation speed with the source region identically in equilibrium. For non-compact sources the propagation is still given by ae0 but the source region is now understood to encompass relaxation effects, the details of which are left unspecified.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 83 , Issue 4 , 21 December 1977 , pp. 775 - 783
Copyright
© 1977 Cambridge University Press

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