Modal surface impedances for two spheres in a thermoviscous acoustic medium

Abstract

Two spherical bodies are submerged in an infinite thermoviscous fluid; one body is motionless and the other is vibrating at high frequency with a surface pattern that is axisymmetric with respect to the line joining the centers of the two spheres. Both bodies are sufficiently good conductors that their surface temperatures deviate insignificantly from the ambient temperature. The acoustic stress and velocity fields on the vibrating surface may be conveniently related by a modal surface impedance matrix based on field expansions in Legendre functions. This impedance matrix, which of course accounts for the presence of the motionless sphere, is here obtained from the field equations of Epstein and Carhart [J. Acoust. Soc. Am. 25, 553–565 (1953)] through the application of translational addition theorems for bispherical coordinates [Y. A. Ivanov, DiffractionofElectromagneticWavesonTwoBodies, NASA Tech. Trans. F-597 (1970)]. Impedance matrices for fluids that exhibit thermoviscous boundary layers of various thicknesses are compared with counterparts produced by the thin-boundary-layer model [A. D. Pierce, Acoustics (McGraw-Hill, New York, 1981)]. Special attention is devoted to the case when the motionless sphere is sufficiently large relative to the vibrating sphere that it approximates a rigid wall.