Keywords: waveguides, waves, wavemodes, finite element, parametric analysis, radiation conditions, eigenvalue problem, frequency domain.

Some important engineering problems deal with infinite waveguides which have constant cross section, boundary conditions and material properties along its transverse axis. However, there may be a section of the waveguide which is inhomogeneous, may have obstacles, irregular and complicated geometry as well as boundary conditions different for the infinite part of the waveguide. In such sections the waves may totally or partially reflect, but in the concatenated homogeneous part of the waveguide, which stretches to infinity, the waves propagate only away from the source of excitation, yet some of them may be decaying or even standing waves. An analysis of such wave motion, striving for reasonably exact results, may be very difficult task even when there is no irregular section in the waveguide. This paper shows the procedure for the analysis of wave motion, which in principle yields exact results, is simple to apply regardless of the diversities of the waveguide, and in addition yields evident parametric insight into the filtering effects of the waves in inhomogeneous waveguides.

Analytical solutions of wave motion in the waveguides are practically feasible only for simple cases, see for instance[1]. For other cases we use numerical or semi- numerical methods. Boundary element methods use the free-space fundamental solution, see for instance [2], which is not suitable for waveguides. Finite element methods are suitable for modelling the diversities of the wave guide, but we can always model only a finite section of it. Thus, on the limiting boundaries of the FE mesh, the radiation conditions have to be somehow fulfilled. Several methods and techniques, including the so called "infinite" finite elements or artificial material damping, are used for this purpose. A more extensive overview of these methods is given in reference [3]. The last group of methods, if we my say so, are the operator methods, see for instance[4]. To conclude briefly, we could say in general for all these methods that they are either approximate, or are difficult to employ, and yield no evident engineering insight into the wave phenomena.

By the method presented in this paper we have to model the discussed section of the waveguide by finite elements. On the limit of the FE mesh, where the wave propagates only in outward direction of the waveguide, we set an artificial boundary. There the proper radiation conditions are encountered. The essence of the procedure is the way in which we are computing these displacements and stresses, and the inherent feature of the wave phenomenon we are taking the advantage of. From the theory of wave motion in waveguides we know, see [1], that there exist certain complex wave modes of displacements and stresses, which propagate at constant velocity, and do not alter their forms. On the fictive boundary they are proportional to those on the adjacent neighbouring boundary. These two boundaries form a "cell", an important item employed by [3]. By FE modelling of the cell, and considering the wave motion in the frequency domain, the aforementioned proportionality yields one system of equations. The same relation can be written with the aid of the transfer matrix yielding the second system of equations. Merging both systems of equations yields the eigenvalue problem. Eigenvectors are the wave modes with constant velocities, and eigenvalues tell us in which direction they propagate. The displacements on the fictive boundary are a linear combination of only outward propagating wave modes. The weighting factors for this combination are determined by a given excitation and the transfer matrix from the fictive boundary to the excitation boundary. With their help we get a clear parametric insight of the wave motion. The method is simple and efficient. It is numerically easy to compute the wave motion and the propagating wave modes even for complicated cross sections of the inhomogeneous waveguides. The first numerical example presented in the paper is simple only because the exact theoretical solution, used for the comparison with numerical results, is feasible. In fact, although the second example is only modestly more complicated, and may represent an interesting engineering problem, the analytical solution is not given. However, this example illuminates the advantages of the presented procedure – its simplicity of the application and the quantitative and qualitative parametric study of the filtering effect of wave motion due to irregularity in waveguide.

1

Achenbach, J.D., "Wave Propagation in Elastic Solids", North Holland Publishing Company, Amsterdam, New York and Oxford, 1973.

2

Brebbia, C.A. & Telles, J.C.F & Wrobel, L.C., "Boundary Element Techniques", Springer-Verlag: Berlin, N.Y., Tokyo, 1984

3

Wolf, J.P. & Chonming Song, "Finite element modelling of unbounded media", John Wiley & Sons, Chichester, New York., Brisbane, Toronto, 1995.

4

Premrov, M., Umek A. & Špacapan I., "n iterative FEM for solving elasto-dynamics in infinite domains", Zeitschrift für angewandte Mathematik und Mechanik, 2000, 80, suppl. 3, p.749-750.

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