Sound radiation from a resilient spherical cap on a rigid sphere

It has been argued that the sound radiation of a loudspeaker is modeled realistically by assuming the loudspeaker cabinet to be a rigid sphere with a resilient spherical cap. Series expansions, valid in the whole space outside the sphere, for the pressure due to a harmonically excited cap with an axially symmetric velocity distribution are presented. The velocity profile is expanded in functions orthogonal on the cap, rather than on the whole sphere. As a result, only a few expansion coefficients are sufficient to accurately describe the velocity profile. An adaptation of the standard solution of the Helmholtz equation to this particular parametrization is required. This is achieved by using recent results on argument scaling of orthogonal (Zernike) polynomials. The approach is illustrated by calculating the pressure due to certain velocity profiles that vanish at the rim of the cap to a desired degree. The associated inverse problem, in which the velocity profile is estimated from pressure measurements around the sphere, is also feasible as the number of expansion coefficients to be estimated is limited. This is demonstrated with a simulation.