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On Sound Generated Aerodynamically. I. General Theory

M. J. Lighthill
Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences
Vol. 211, No. 1107 (Mar. 20, 1952), pp. 564-587
Published by: Royal Society
Stable URL: http://www.jstor.org/stable/98943
Page Count: 24
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On Sound Generated Aerodynamically. I. General Theory
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Abstract

A theory is initiated, based on the equations of motion of a gas, for the purpose of estimating the sound radiated from a fluid flow, with rigid boundaries, which as a result of instability contains regular fluctuations or turbulence. The sound field is that which would be produced by a static distribution of acoustic quadrupoles whose instantaneous strength per unit volume is ρ vivj + pij - a0 2ρ δ ij, where ρ is the density, vi the velocity vector, pij the compressive stress tensor, and a0 the velocity of sound outside the flow. This quadrupole strength density may be approximated in many cases as ρ 0vivi. The radiation field is deduced by means of retarded potential solutions. In it, the intensity depends crucially on the frequency as well as on the strength of the quadrupoles, and as a result increases in proportion to a high power, near the eighth, of a typical velocity U in the flow. Physically, the mechanism of conversion of energy from kinetic to acoustic is based on fluctuations in the flow of momentum across fixed surfaces, and it is explained in section 2 how this accounts both for the relative inefficiency of the process and for the increase of efficiency with U. It is shown in section 7 how the efficiency is also increased, particularly for the sound emitted forwards, in the case of fluctuations convected at a not negligible Mach number.

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