Repulsion of dispersion curves of quasidipole modes of anisotropic waveguides studied by finite element method.

In this letter repulsion of phase-velocity dispersion curves of quasidipole eigenmodes of waveguides with non-circular cross section in non-axisymmetric anisotropic medium is studied by the semi-analytical finite element technique. Borehole waveguide is used as an example. The modeling helps in clarifying the nature of this phenomenon, which is accompanied by the rotation of the orientation of two quasidipole modes with frequency and by the exchange of their behavior at near-crossover point. The dispersion curves cross only in the presence of exact symmetry. Such a scenario is the alternative to the stress-induced anisotropy crossing of dispersion curves.

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.

Dispersion properties of helical waves in radially inhomogeneous elastic media.

In this paper, a method describing dispersion curve calculation for waves propagating in radially layered, inhomogeneous isotropic elastic waveguides is developed. Particular emphasis is placed on the helical waves with noninteger azimuthal wavenumbers, which can be potentially applied in such fields as nondestructive evaluation, acoustic tomography, etc., stipulating their practical importance. To solve the problem under consideration, the matrix Riccati equation is formulated for an impedance matrix. The use of the latter yields a simple form of the dispersion equation. Numerical computation of dispersion curves can encounter difficulties, which are due to potential singularities of the impedance matrix and the necessity to separate roots of the dispersion equation. These difficulties are overcome by employing the Cayley transform and invoking the parametric continuation method. The method developed by the authors is demonstrated by calculating dispersion diagrams in support of helical waves for several models of practical interest. Such computations for an inhomogeneous layer and its approximation by a set of homogeneous layers using a transfer matrix and Riccati equation methods revealed higher computational accuracy of the latter. Dispersion curves calculated for layers with different types of inhomogeneity demonstrated significant discrepancies at low frequencies.