Wave Motion 37 (2003) 313–332 Wave propagators for the Timoshenko beam Dag V.J. Billger a , Peter D. Folkow b,∗ a The Imego Institute, Aschebergsgatan 46, S-411 33 Göteborg, Sweden b Department of Mechanics, Chalmers University of Technology, S-412 96 Göteborg, Sweden Received 9 January 2001; received in revised form 9 August 2002; accepted 15 August 2002 Abstract The propagation and scattering of waves on the Timoshenko beam are investigated by using the method of wave propagators. This method is more general than the scattering operators connected to the imbedding and Green function approaches; the wave propagators map the incoming field at an internal position onto the scattering fields at any other internal position of the scattering region. This formalism contains the imbedding method and Green function approach as special cases. Equations for the propagator kernels are derived, as are the conditions for their discontinuities. Symmetry requirements on certain coupling matrices originating from the wave splitting are considered. They are illustrated by two specific examples. The first being an unrestrained beam with a varying cross-section and the other a homogeneous, viscoelastically restrained beam. A numerical algorithm for solving the equations for the propagator kernels is described. The algorithm is tested for the case of a viscoelastically restrained, homogeneous beam. In a limit these results agree with the ones obtained for the reflection kernel by a previously developed algorithm for the imbedding reflection equation. © 2002 Elsevier Science B.V. All rights reserved. 1. Introduction In the recent past, several direct and inverse wave propagation problems for the Timoshenko beam have been addressed using time domain techniques. The start of such analysis in beam theory being marked by the discovery of the wave splitting of the Timoshenko beam equation by Olsson and Kristensson [1]. The wave splitting was used together with the Green function technique in [2] for the direct propagation of waves on a free, homogeneous beam. In [3] the imbedding method was applied to derive equations for the reflection and transmission operator kernels. The imbedding equations have subsequently been used in the investigation of direct problems [4,5] and have also found application in inverse beam problems such as the reconstruction of the varying cross-section of a free inhomogeneous beam, see [4]. Another inverse problem is treated in [6] and deals with the reconstruction of the layer properties of a homogeneous beam on a semi-infinite viscoelastic foundation. The present paper concerns the propagation of flexural waves on an inhomogeneous and viscoelastically restrained Timoshenko beam, utilizing the propagator technique and the wave splitting of a free homogeneous beam [1]. The wave splitting results in coupled equations for the split fields in the inhomogeneous and restrained region [3]. The various fields in this region are related through propagators, which are operators mapping the incoming field at a ∗ Corresponding author. Tel.: +46-31-772-1521; fax: +46-31-772-3827. E-mail address: peter.folkow@me.chalmers.se (P.D. Folkow). 0165-2125/02/$ – see front matter © 2002 Elsevier Science B.V. All rights reserved. PII:S0165-2125(02)00094-X