Absorption and Dispersion of Forced Spherical and Cylindrical Sound Waves According to the Navier-Stokes Equations

Abstract

In his classical paper on infinitesimal sound waves in perfect gases endowed with shear viscosity, bulk viscosity, and heat conduction, Kirchhoff [Ann. Phys. 136, 177–193 (1868)] considered not only plane waves but also a certain class of curved waves. He showed that the absorption and dispersion of these waves is determined by the solution of a biquadratic equation for the complex frequencies, but in discussing the behavior of particular waves he employed only one of the two pairs of roots, and this he linearized with respect to the driving frequency. The biquadratic itself, as extended by Langevin so as to hold for fluids obeying an arbitrary equation of state, has been solved exactly and generally by C. A. Truesdell [J. Rational Mech. Anal. 2, 693–762 (1953)]. In his comprehensive analysis and interpretation of the solutions, however, Truesdell considered only forced plane waves. In the present article I give a new version of Kirchhoff's approach to curved waves of expansion (an arbitrary motion may be regarded as the vector sum of an isochoric irrotational motion, a set of vorticity waves, and a set of expansion waves. Vorticity waves are not considered in this note. As has been remarked in varying degrees of generality by M. Lessen [J. Aero. Sci. 21, 849–850 (1954)]; Lagerstrom, Cole, and Trilling, [“Problems in the theory of viscous compressible fluids,” GALCIT (1949)]; S. Jarvis, Jr., J. Acoust. Soc. Am. 27, 70–73 (1954)]; and U. Yao-Tsu Wu [J. Math. Phys. 35, 13–27 (1956)] the propagation of linearized vorticity waves is independent of bunk viscosity and heat conduction) and show how Truesdell's results on plane waves may be adjusted so as to apply to the class of curved waves considered. Spherical and cylindrical waves are analyzed in some detail. Proper treatment of this problem requires dimensionless variables and hence some reference standard of length. Two alternatives are considered: (1) the unit of length is a linear dimension of a given oscillator, (2) the unit of length is the wavelength corresponding to the driving frequency according to the theory of nondissipative fluids. In the brief remarks of Kirchhoff, apparently only the first possibility was noticed. For the second class of spherical waves, the absorption coefficient per cm is shown always to be much less than that for the first class. There is a frequency-dependent factor which tends to increase the amplitude with the frequency. Similar analysis is carried out for cylindrical waves, which me shown to behave in a fashion intermediate between plane and spherical waves. In principle, the results are contained in Kirchhoff's paper. However, the analysis there, expecially as given in the expository work of Rayleigh [Theory of Sound (Dover Publications, New York, 1945), Vol. II, pp. 319–323] does not make clear how much depends on linearization with respect to frequency and how much does not. The purpose of Sec. 1 of this note is to show that Kirchhoff's method of reducing the analysis of certain curved waves to that of plane waves is valid exactly, at all frequencies.