Modeling of wave dispersion along cylindrical structures using the spectral method

Abstract

Algorithm and code are presented that solve dispersion equations for cylindrically layered media consisting of an arbitrary number of elastic and fluid layers. The algorithm is based on the spectral method which discretizes the underlying wave equations with the help of spectral differentiation matrices and solves the corresponding equations as a generalized eigenvalue problem. For a given frequency the eigenvalues correspond to the wave numbers of different modes. The advantage of this technique is that it is easy to implement, especially for cases where traditional root-finding methods are strongly limited or hard to realize, i.e., for attenuative, anisotropic, and poroelastic media. The application of the new approach is illustrated using models of an elastic cylinder and a fluid-filled tube. The dispersion curves so produced are in good agreement with analytical results, which confirms the accuracy of the method. Particle displacement profiles of the fundamental mode in a free solid cylinder are computed for a range of frequencies.

Modelling the Generation and Propagation of Ultrasonic Signals in Cylindrical Waveguides

“…This numerical method, which is computationally simple and reportedly does not suffer from the problems associated with large diameter waveguides at high frequencies, has been recently applied to model multi-layered cylindrical waveguides (Karpfinger et al, 2008).…”

Modeling of axisymmetric wave modes in a poroelastic cylinder using spectral method

“…The problem is solved by numerical interpolation using spectral differentiation matrices (DM's). This approach is explained in detail for axisymmetric waves in cylindrical structures in Karpfinger et al (2008).…”

“…The differential operators L ± and L s are discretized using DMs (Karpfinger et al, 2008) and combined in a 3N ϫ 3N matrix. N is the number of Chebyshev points discretizing the cylinder along its radius.…”

“…An alternative approach to modeling wave propagation in circular structures was recently introduced by Adamou and Craster (2004) based on the spectral method. Karpfinger et al (2008) extended the spectral method to axisymmetric waves for arbitrary fluid and solid layers. In this letter we extend the spectral method to cylindrical poroelastic structures.…”

Guided waves' dispersion curves in anisotropic viscoelastic single- and multi-layered media

“…As pointed out in [22], spectral methods were introduced in the 1970s in the field of fluid dynamics by Kreiss & Oliger [23], Orszag [24] and Fornberg [25] and have remained a standard computational tool in the field ever since. Recently, the SCM has been successfully used in the fields of seismology and geophysics to study wave propagation, see for instance [26][27][28]. Guided wave problems in other contexts such as electromagnetic waveguides can also be tackled by means of the SCM, see for instance [29], where metal-insulator-metal electromagnetic lossless and lossy waveguides are studied.…”