Schlumberger Moscow Research, Pudovkina Street 13, Moscow 119285, Russia.
J Acoust Soc Am. 2013 Sep;134(3):1739-53. doi: 10.1121/1.4817839.
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.
由于轴向和角向谐波之间的耦合，具有空间依赖性的任意各向异性波导的光谱计算是一项具有挑战性的任务。该问题在圆柱坐标系中通过扩展一般情况下的谱方法来解决。通过考虑控制方程右侧算子的矩阵表示，可以将其精确地重新表述为一组无限的积分微分方程。该研究的重要部分是考虑不同谐波的耦合，这从这些方程的核中显而易见。如果波导在轴向方向上具有平移不变性，则轴向谐波的耦合消失。实用的逼近和截断过程产生了广义特征值问题，可以通过数值求解该问题来获得算子的整个谱，并构建本征模的色散曲线。该谱方法通过与倾斜横向各向同性各向异性岩石样本中 scaled boreholes 的单极子、偶极子和四极子正常模式的色散曲线的测量结果进行了对比验证。此外，还讨论了通过谱方法计算的色散曲线与通过合成数据计算的色散曲线之间的比较。
