The Influence of Solid Boundaries upon Aerodynamic Sound

TLDR: In this paper, an extension to Lighthill's general theory of aerodynamic sound was made to incorporate the influence of solid boundaries upon the sound field, and it was 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.

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 I$\propto $ $\rho \_{0}$ U$\_{0}^{6}$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 $\rho \_{0}$ 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.