Application of the spectral method for computation of the spectrum of anisotropic waveguides

Abstract

Spectral method is formulated in cylindrical coordinates for the general case of waveguide with arbitrary anisotropy with the spatial dependence. According to the idea of this approach, matrix representation of operator in the right-hand side of governing equations is considered. As a result, the latter are cast into exact infinite set of integro-differential equations. Explicit expressions for their kernels expose coupling between axial and azimuthal harmonics. Coupling of axial harmonics vanishes in important case of waveguide with translational invariance in axial direction. It results in the set of differential equations, which is used to introduce practical approximation procedure. The latter yields generalized eigenvalue problem, which can be solved numerically for the spectrum of the operator. The spectrum is sorted according to eigenmodes’ properties. Thus dispersion curves of eigenmodes are constructed. Presented consideration can be adapted for waveguides of different physical nature (elastic, electromagnetic, etc.) and different geometry (rectangular, elliptical, etc.). Developed technique is verified by comparison with results of controlled laboratory measurements on anisotropic sample. Monopole, dipole and quadrupole normal modes for scaled borehole in anisotropic rock sample with TTI symmetry are considered. The comparison of spectral method results with the dispersion analysis of synthetic data is provided as well.