Using the Finite Product Method for solving eigenvalue problems formulated in cylindrical coordinates

Abstract

Analysis of free/forced wave propagation in multi-layered structures and structures under heavy fluid loading constitute classical problems of fluid-structure interaction. These waveguides and many similar ones support infinitely many waves and in some cases e.g. at high-frequency excitations or in near-field analysis it is necessary to account for a large number of them. Finding the dispersion curves from these transcendental dispersion relations is not a trivial task due to their ill-conditioned/unstable nature as well as numerical algorithms ability to solve transcendental equations. These issues can, however, be circumvented by using the Finite Product Method (FPM). The FPM is generic and has been used to solve dispersion equations of homogeneous waveguides derived in Cartesian coordinates with sine/cosine functions. However, it is yet to be formulated in cylindrical coordinates when Bessel functions are involved in dispersion equations. We focus in this paper on extending the method to the cylindrical problems and compound waveguides illustrated here by a fluid-filled cylindrical shell. The great advantage of the FPM is that it reduces the transcendental dispersion equation to a polynomial one easily solved by numerical algorithms but more importantly it delivers only authentic roots of the dispersion equation i.e. no spurious roots as often encountered when using Taylor approximations etc.