A limiting form for the nearfield of the baffled piston

The circular piston on the planar, infinite, rigid baffle and the pulsating sphere are fundamental (if not classic) problems in acoustic radiation theory. Those who work in the area of radiation and scattering are very familiar with the solutions to these two problems. The solution to a somewhat more difficult problem—the radially vibrating polar cap on an otherwise rigid sphere—is also available and is well documented. Although mathematically difficult to prove, the polar cap problem (or more precisely its general solution) reduces to that of the pulsating sphere in one limiting form, and to that of the piston on the baffle in another. This notion—that the solution to the polar cap problem subsumes the solutions to the two classic radiation problems of linear acoustics—is instructive and useful, if not profound. Its significance is in the use of the general solution of the polar cap problem to generate a convergent infinite series solution for the nearfield of the piston on the baffle. Extension computations have been performed in limiting cases to demonstrate this convergence.