Eigensolutions of the Helmholtz equation for a multiply connected domain with circular boundaries using the multipole Trefftz method

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

“…In this paper Eqs. (17) and (18) are calculated without using the addition theorem of Mathieu functions. In order to achieve this, two difficulties must be surmounted.…”

“…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.…”

The collocation multipole method for solving multiple scattering problems with circular boundaries

“…The addition theorem is often employed to transform the multipole expansion into one of the coordinate systems to satisfy the specified boundary conditions. In the case of the circular boundary, some applications can be seen in the interaction of waves with arrays of circular cylinders [9], the free vibration of circular membranes [10] and circular plates [11] and flexural wave scattering [12]. From a mathematical perspective, the procedure is elegant.…”

Dynamics of a circular membrane with an eccentric circular areal constraint: Analysis and accurate simulations

“…Investigations of membranes with complex boundary conditions and shapes include the work by Laura and Gutierrez [19], Wu and Wang [20], Kang and Lee [21], and Chen et al [22]. Investigations of non-homogeneous membranes include the work by Cortinez and Laura [23] and Cap [24], and that of accuracy of solutions include the work by Zhao and Stevens [25].…”