Synthesis of the radiated sound field from a vibrating plate-strip with free edges

A multipole expansion in elliptical coordinates was employed to calculate the directivity pattern of the radiation from free vibrations of a plate-strip with free edges. The Helmholtz equation can be separated in elliptical coordinates and an exact solution for a cylindrical geometry with elliptical cross section can be constructed by Mathieu functions. The radiation problem was solved by mapping the normal velocity of the rectangular cross section of the plate-strip onto an acoustically equivalent elliptical cross section and applying a multipole expansion. A comparison between this method and a standard wave-number transform showed good agreement in the case of a baffled plate, which proved the validity of this procedure. The directivity patterns of baffled and unbaffled plates are compared in order to estimate the influence of hydrodynamic short-circuit around free edges on the radiation. The compact representation of the sound field by the source-strengths is advantageous for the determination of the interference of superimposed eigenmodes for radiation. For applications in active noise control, a direct optimization criteria for secondary forces with respect to the radiated soundpower was obtained. It permits the study of efficiency and stability of the active measure with regard to number and location of the secondary forces.