Loudspeakers are often modelled as a rigid piston in an infinite baffle. As a model for real loudspeakers, this approach is limited in two ways. One issue is that a loudspeaker cone is not rigid, and a second issue is that a loudspeaker is mostly used in a cabinet. Both issues are addressed here by developing the velocity of the radiator in terms of orthogonal polynomials known from optical diffraction theory as Zernike circle polynomials. Using these polynomials we develop semi-analytic expressions for the sound pressure from the radiator in two different cases: as a flexible flat radiator mounted in an infinite baffle, and as the cap of a rigid sphere. In the latter case the comparison is done not only for the pressure but also for other quantities viz. the baffle-step response, sound power and directivity, and the acoustic centre of the radiator. These quantities are compared with those from a real loudspeaker. Finally, in the case of the baffled-piston radiation the spatial impulse response is presented.
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