Numerical Analysis on Isotropic Elastic Waveguides by Mode-Matching Method - I. Analytical Foundations and General Algorithms

Abstract

A new method for computer-aided calculation, the mode- motching merhod, based on the Rayleigh expansion theorem, is pro- posed in order to analyze the boundary condition problem of elastic fields, and its analytical foundations and general algorithms are dis- cussed. The alastic fields are represented by scalar potential functions which are solutions of Helmholtz equations and expanded by the well- known separated solutions. Introducing the formal Green functions, the integral representations of the potentials are derived. The Rayleigh expansion theorem assures the existence of the infinite sequence of the truncated modal expansions which uniformly converges to the true field in arbitrarily shaped cross section. Through this theorem and the integral representations, the procedure of the mode-matching method is described simply so that the truncated modal expansions are made to fit for the boundary conditions in the least squares sense. Conse- quently this new method may ensure more precise analyses not only on the dispersion characteristics, but also on the field distributions of the particle velocity or the stress by small amount of computational efforts, than the other numerical methods reported so far. The general algo- rithm for the mode-matching method is described in the cases of forced and free vibrations of the homogeneous isotropic elastic waveguide with arbitrarily shaped cross section.