Acoustics: Sound Fields, Transducers and Vibration, 2nd Edition

12.5. Radiation from a point source on a sphere

This problem is essentially the same as the last one when d   =   R. However, this time we shall introduce the property of orthogonality to obtain a simple solution. By reciprocity, the resulting expression can be used to obtain the pressure at a point on the sphere due to a source at some point in space. This is a useful model for the diffraction effects of the human head on sound arriving at one ear, assuming a hard sphere model of the head. Unlike the pulsating sphere, only an infinitesimally small part of the surface is oscillating, so that the velocity distribution is described by

u ˜ ( R , θ ) = { u ˜ 0 , 0 ≤ θ ≤ δ 0 , δ ≤ θ ≤ π ,

(12.39) ...