Natural modes of circular membranes and plates weakened by a finite number of circular holes

Natural mode analysis of an acoustic cavity with multiple elliptical boundaries by using the collocation multipole method

“…The addition theorem is often employed to transform the multipole expansion into one of coordinate systems to satisfy the specified boundary conditions. For the circular boundary, some applications can be seen in the water wave scattering problem [16], the free vibration of circular membranes [18,19], the free vibration of circular plates [19,20] and the flexural wave scattering [21]. Furthermore, Chatjigeorgiou and Mavrakos [22,23] proposed an analytical approach to the hydrodynamic diffraction by multiple elliptical cylinders using the addition theorem of the Mathieu functions derived from Graf's addition theorem for the Bessel functions.…”