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Modeling a Loudspeaker as a Flexible Spherical Cap on a Rigid Sphere

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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 moving rigid spherical cap. Series expansions, valid in the whole space on and outside the sphere, for the pressure due to a harmonically excited, flexible 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. This has the advantage that 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 arametrization is required. This is achieved by using recent results on argument scaling in orthogonal Zernike polynomials. The efficacy of the approach is exemplified by calculating various acoustical quantities with particular attention to certain velocity profiles that vanish at the rim of the cap to a desired degree. These quantities are: the sound pressure, polar response, baffle-step response, sound power, directivity, and acoustic center of the radiator. The associated inverse problem, in which the velocity profile is estimated from pressure measurements around the sphere, is feasible as well since the number of expansion coefficients to be estimated is limited. This is demonstrated with a simulation.

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