Title: Notes on the Exact Equations Governing the Propagation of Sound in Fluids
Abstract: The assumptions underlying the exact equations of motion for a thermoviscous fluid are reviewed and the complete equations are given, for reference convenience, in both tensor and vector form. The first- and second-order acoustic equations are then exhibited and used to obtain the source terms that account for the generation of vorticity and streaming. In order to preserve a broad base from which to make the approximations appropriate under various circumstances, all terms are retained explicitly including those arising from any functional dependence of the viscosity and thermal coefficients on the state variables. The distinction between spatial and material coordinate systems is carefully drawn and conversion transforms are derived rigorously and their use illustrated. The general properties of finite-amplitude waves are demonstrated by including the second-order terms in a plane-wave solution of the exact wave equation in material coordinates, with special concern for the effects of large amplitude on speed of propagation and on wave-form distortion. Sound absorption and dispersion measures for a viscous conducting fluid are analyzed in terms of Truesdell's recent exact solution of the first-order secular equation. These differ characteristically from the corresponding measures predicted for pure relaxation in a two-fluid mixture. It is concluded that a complete and adequate theory of sound absorption and dispersion will need to take into account both relaxation and viscothermal phenomena as well as their interaction, and that until such a general theory is available, the exact theory of viscothermal effects—rather than the crude linear approximation commonly, but inappropriately, called “classical”—should be used in computing the “excess” absorption and dispersion to be accounted for by relaxation processes. The exact solutions of the secular equation permit a new evaluation, in series form, of the characteristic acoustic impedance for a thermoviscous medium. The notes conclude with a revised account of the spectral character of thermal noise in the acoustic medium based on the quantum hypothesis and a merger of the concepts of architectural acoustics and specific-heat theory.
Publication Year: 1955
Publication Date: 1955-11-01
Language: en
Type: article
Indexed In: ['crossref']
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Cited By Count: 48
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