The completeness of the set of modes for various waveguides and its significance for the near-field interaction with vibrating structures

The completeness of the set of modes for various waveguides and its significance for the near-field interaction with vibrating structures

Tsouvalas, A.
Hendrikse, H.
Metrikine, A.

Civil Engineering and Geosciences

Structural Engineering

2014-06-30

Modal decomposition is often used in geophysics and acoustics for the solution of problems related to wave propagation in elastic or acousto-elastic waveguides. One of the key elements of this method is the solution of an eigenvalue problem for obtaining the roots of the characteristic equation, which may signify either frequencies or wavenumbers. The nature of the roots for the majority of the elastic systems allows for a line search along the real or the imaginary axis in the complex plane. Nonetheless, there exist cases in which eigenvalues become complex-valued requiring thus the use of more advanced algorithms. Most up-to-date algorithms for the solution of complex eigenvalue problems are based on the principle of the argument method for a first gross estimation of the location and the number of the roots within a predefined region, followed by a bisection or steepest descent method for a refinement of the final root position. These techniques are shown to be efficient when the roots are located distinctly apart. In the case of elastic or acousto-elastic waveguides, complex roots do exist and often lie close to each other, thereby not allowing for an efficient application of such algorithms. In this study, an approach is presented in which the real and the imaginary parts of the characteristic equation of several types of waveguides are treated separately in order to define the locus of points in the complex plane where roots may lie. It is shown that next to the real-valued roots, which correspond to propagating modes in the medium, infinitely many imaginary and complex-valued ones do exist which complete the eigenvalue spectrum. The contribution of the latter is rather significant in the vicinity of a load and is very essential for the source-waveguide interaction.

characteristic equation
complex eigenvalues
acousto-elastic waveguides
principle of the argument method

http://resolver.tudelft.nl/uuid:78c5857b-b183-4f41-9621-1aa396809702

University of Porto, Faculty of Engineering, Department of Civil Engineering

978-972-752-165-4

EURODYN 2014: Proceedings of the 9th International Conference on Structural Dynamics, Porto, Portugal, 30 June - 2 July 2014

Institutional Repository

conference paper

(c) 2014 The Author(s)