On the sound field of a resilient disk in an infinite baffle

The Rayleigh integral describing the near-field pressure of an axisymmetric planar monopole source with an arbitrary velocity distribution is solved with a method similar to that used by Mast and Yu [J. Acoust. Soc. Am. 118(6), 3457–3464 (2005)] for a rigid disk in an infinite baffle. The closed-form solution is in the form of a double expansion, which is valid for distances from the observation point to the center of the source that are greater than its radius. However, for the remaining immediate near field, the King integral is solved using a combination of Gegenbauer’s summation theorem and the Lommel expansion, resulting in a solution which is in the form of a triple expansion, reducing to a double expansion along the source’s axis of symmetry. These relatively compact solutions in analytic form do not require numerical integration and therefore present no numerical difficulties except for a singularity at the rim. As an example of a monopole source with an arbitrary velocity distribution, equations describing the radiation characteristics of a resilient disk in an infinite baffle are derived. Using Babinet’s principle, the pressure field of a plane wave passing through the complementary hole in an infinite rigid screen is calculated.