\r\ndomain usually requires significant computational effort. Hence, the

\r\nconsidered domain must be truncated to a smaller domain of interest.

\r\nIn addition, special boundary conditions, which absorb the outward

\r\ntravelling waves, need to be implemented in order to describe the

\r\nsystem domains correctly. In this work, the linear one dimensional

\r\nwave equation is approximated by utilizing the Fourier Galerkin

\r\napproach. Furthermore, the artificial boundaries are realized with

\r\nabsorbing boundary conditions. Within this work, a systematic work

\r\nflow for setting up the wave problem, including the absorbing

\r\nboundary conditions, is proposed. As a result, a convenient modal

\r\nsystem description with an effective absorbing boundary formulation

\r\nis established. Moreover, the truncated model shows high accuracy

\r\ncompared to the global domain.","references":"[1] R. Clayton and B. Engquist, \u201cAbsorbing boundary conditions for\r\nacoustic and elastic wave equations,\u201d Bulletin of the Seismological\r\nSociety of America, vol. 67, no. 6, pp. 1529\u20131540, 1977. [Online].\r\nAvailable: http:\/\/www.bssaonline.org\/content\/67\/6\/1529.abstract\r\n[2] G. Aschauer, A. Schirrer, and S.-M. Jakubek, \u201cRealtime-capable finite\r\nelement model of railway catenary dynamics in moving coordinates,\u201d\r\nIEEE Multi-Conference on Systems and Control, 2016.\r\n[3] A. Facchinetti, L. Gasparetto, and S. Bruni, \u201cReal-time catenary models\r\nfor the hardware-in-the-loop simulation of the pantograph\u2013catenary\r\ninteraction,\u201d Vehicle System Dynamics, vol. 51, no. 4, pp. 499\u2013516, 2013.\r\n[4] B. Engquist and A. Majda, \u201cAbsorbing boundary conditions for\r\nnumerical simulation of waves,\u201d Proceedings of the National Academy\r\nof Sciences, vol. 74, no. 5, pp. 1765\u20131766, 1977. [Online]. Available:\r\nhttp:\/\/www.pnas.org\/content\/74\/5\/1765.abstract\r\n[5] W. Guo and Z.-C. Shao, \u201cStrong stability of an unstable wave equation\r\nby boundary feedback with only displacement observation,\u201d IEEE\r\nTransactions on Automatic Control, vol. 57, no. 9, pp. 2367\u20132372, 2012.\r\n[6] C. Cerjan, D. Kosloff, R. Kosloff, and M. Reshef, \u201cA nonreflecting\r\nboundary condition for discrete acoustic and elastic wave equations,\u201d\r\nGeophysics, vol. 50, no. 4, pp. 705\u2013708, 1985.\r\n[7] J. Sochacki, R. Kubichek, J. George, W.-R. Fletcher, and S. Smithson,\r\n\u201cAbsorbing boundary conditions and surface waves,\u201d Geophysics,\r\nvol. 52, no. 1, pp. 60\u201371, 1987.\r\n[8] B. Chen, D.-G. Fang, and B.-H. Zhou, \u201cModified Berenger PML\r\nabsorbing boundary condition for FD-TD meshes,\u201d IEEE Microwave\r\nand Guided Wave Letters, vol. 5, no. 11, pp. 399\u2013401, 1995.\r\n[9] D. Komatitsch and J. Tromp, \u201cA perfectly matched layer absorbing\r\nboundary condition for the second-order seismic wave equation,\u201d\r\nGeophysical Journal International, vol. 154, no. 1, pp. 146\u2013153, 2003.\r\n[10] J.-P. Berenger, \u201cA perfectly matched layer for the absorption of\r\nelectromagnetic waves,\u201d Journal of computational physics, vol. 114,\r\nno. 2, pp. 185\u2013200, 1994.\r\n[11] W. A. Strauss, Partial differential equations - An introduction, second\r\nedition ed.","publisher":"World Academy of Science, Engineering and Technology","index":"Open Science Index 128, 2017"}