The influence of solid boundaries upon aerodynamic sound

TLDR

Solid boundaries affect aerodynamic sound through reflection, diffraction, and the creation of a dipole field at the boundary limits of Lighthill’s quadrupole distribution. The study extends Lighthill’s general theory of aerodynamic sound to incorporate the influence of solid boundaries on the sound field. Using dimensional analysis, the authors show that dipole-generated sound intensity at large distances follows I ∝ P₀ U₀⁶ a₀⁻³ L² x⁻², where U₀ is the flow velocity, L the body length, a₀ the sound speed, and P₀ the fluid density. The dipole distribution is equivalent to the force exerted by unit boundary area on the fluid, is a more efficient sound generator than Lighthill’s quadrupoles at low Mach numbers, and its fundamental frequency is half that of the quadrupole sound.

Abstract

An extension is made to Lighthill’s general theory of aerodynamic sound, so as to incorporate the influence of solid boundaries upon the sound field. This influence is twofold, namely (i) reflexion and diffraction of the sound waves at the solid boundaries, and (ii) a resultant dipole field at the solid boundaries which are the limits of Lighthill’s quadrupole distribution. It is shown that these effects are exactly equivalent to a distribution of dipoles, each representing the force with which unit area of solid boundary acts upon the fluid. A dimensional analysis shows that the intensity of the sound generated by the dipoles should at large distances x be of the general form Ioc P0 U 6 0 a 0 -3 L 2 x -2 , where U 0 is a typical velocity of the flow, L is a typical length of the body, a 0 is the velocity of sound in fluid at rest and P 0 is the density of the fluid at rest. Accordingly, these dipoles should be more efficient generators of sound than the quadrupoles of Lighthill’s theory if the Mach number is small enough. It is shown that the fundamental frequency of the dipole sound is one half of the frequency of the quadrupole sound.