Calculating the spectrum of anisotropic waveguides using a spectral method

The computation of the spectrum of a waveguide with arbitrary anisotropy with spatial dependence is a challenging task due to the coupling between axial and azimuthal harmonics. This problem is tackled in cylindrical coordinates by extending a spectral method for the general case. By considering the matrix representation of the operator on the right-hand side of the governing equations, the latter are exactly reformulated as an infinite set of integro-differential equations. Essential part of this study is taking into account the coupling of different harmonics, which becomes evident from the kernels of these equations. Provided a waveguide is translationally invariant in the axial direction, the coupling of axial harmonics vanishes. A practical approximation and truncation procedure yields a generalized eigenvalue problem, which can be solved numerically to obtain the entire spectrum of the operator and to construct the dispersion curves for the eigenmodes. The spectral method is tested against the results from the measurements of dispersion curves for the monopole, dipole, and quadrupole normal modes of scaled boreholes in tilted transverse isotropy anisotropic rock sample. Besides, the comparison of dispersion curves calculated by the spectral method and those computed from the synthetic data is discussed.