VII. The extinction of sound in a viscous atmosphere by small obstacles of cylindrical and spherical form

Abstract

The theory of the incidence of waves of sound in a non-viscous air upon small obstacles of cylindrical or spherical form is well known to students of mathematical physics; it has been treated in Lord Rayleigh’s ‘Theory of Sound,’ and in Prof. Lamb’s ‘Treatise on Hydrodynamics.’ The corresponding problems for a viscous air have not, however, been worked out, and this paper is devoted to an investigation of these problems. The solutions of the equations of vibration of a viscous gas with reference to cylindrical and spherical surfaces were given by Prof. L a m b in a paper entitled “On the Motion of a Viscous Fluid Contained in a Spherical Vessel” and published in the ‘Proceedings of the London Mathematical Society’ in 1884. It is easy to obtain solutions suitable to the case of divergent waves; the functions involved are Bessel functions with a complex argument. An analytical expression for the secondary waves diverging from the obstacle is obtained without difficulty. It then remains to find an expression for the loss of energy to the primary waves. In calculating this loss of energy it is necessary to consider the dissipation of energy by friction in the immediate neighbourhood of the obstacle in addition to the energy which is carried away to a distance by the secondary waves. This was pointed out to me by Prof. Lamb, at whose suggestion this paper was written. In obtaining an expression for the energy dissipated by friction I at first made use of the dissipation function. This method led to exactly the same results as that finally adopted, but the mathematics involved were cumbrous, and the physical ideas, on which they were based, were somewhat obscure. Another disadvantage of this method was that it was necessary to calculate separately the scattered and the dissipated energy. I have to thank Prof. Lamb for his kindness in pointing out to me the method of calculating the lost energy adopted in this paper. The result has been to make the paper more clear and readable. I have succeeded in obtaining expressions for the energy lost to the primary waves in the case of spherical and cylindrical obstacles. As might he expected, the problem of the cylindrical obstacle presents greater analytical difficulty than that of the spherical obstacle, and in the former case it is necessary to obtain different approximate expressions according to the diameter of the obstacle. The results for wires of 10 -1 cm. radius and for wires of 10 -3 cm. radius can be obtained without much difficulty, but when the radius of the wire is of order 10 -2 cm. it is necessary to perform very laborious calculations in order to arrive at intelligible results. The energy lost to the primary waves is, in all cases, very great compared with what would he lost in a non-viscous air, but the ratio of the lost energy to that incident upon the obstacle is at most of order 10 -2 .