Loudspeakers are often modelled as a rigid piston in an infinite baffle. This model is for real loudspeakers somewhat limited in two ways. One issue is that a loudspeaker 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 oforthogonal 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 quantitiesviz. the baffle-step response, sound power and directivity, and theacoustic center of the radiator. These quantities are compared withthose from a real loudspeaker.