Calculating dispersion relations for waveguide immersed in perfect fluid
Conventional ﬁnite element method (FEM) is capable of obtaining wave solutions, but large discretized structures at high frequency require high computational resources, the computational domain can be reduced by combining FEM with analytical assumption for guided wave. Semi Analytical Finite Element (SAFE) formulation for immersed waveguide in perfect ﬂuid is used for acquiring propagating wave modes as dynamic equilibrium states. Modes are solutions to eigenvalue problem and provide with important characteristic features of the guided waves – phase velocity, attenuation, wave structure, etc. The eﬀect of surrounding leaky medium is modeled via traction boundary condition, which is based on assumption of the continuity of stresses at solid-ﬂuid interface. The boundary condition causes wave attenuation due to energy leakage into outer medium. The derivation of the eigen-problem takes into account complex wavenumbers of leaky wave in ﬂuid and guided wave in a three-dimensional waveguide. Linearization procedure for solving nonlinear eigenvalue problem is used. Dispersion relations for immersed waveguide with Rayleigh damping are obtained. The limits of applications of Rayleigh damping and convergence analysis of immersed waveguide model are discussed.
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 I. Bartoli, A. Marzani, F. Lanza di S. and E. Viola. Modeling wave propagation in damped waveguides of arbitrary cross-section. Journal of Sound and Vibration, 295(3-5):685–707, 2006. ISSN 0022-460X. https://doi.org/10.1016/j.jsv.2006.01.021
 M. Bilodeau, N. Quaegebeur and P. Masson. Design of a guided wave absorber for Structural Health Monitoring system development. NDT & E International, 88:33–41, 2017. ISSN 0963–8695. https://doi.org/10.1016/j.ndteint.2017.03.003
 M. Castaings and M. Lowe. Finite element model for waves guided along solid systems of arbitrary section coupled to inﬁnite solid media. The Journal of the Acoustical Society of America, 123(2):696–708, February 2008. ISSN 0001–4966. https://doi.org/10.1121/1.2821973 . Available from Internet: 10.1121/1.2821973
 K.F. Graﬀ. Wave motion in elastic solids. Courier Corporation, 2012.
 H. Gravenkamp, C. Birk and C. Song. Computation of dispersion curves for embedded waveguides using a dashpot boundary condition. The Journal of the Acoustical Society of America, 135(3):1127–1138, 2014. https://doi.org/10.1121/1.4864303
 H. Gravenkamp, C. Birk and C. Song. Numerical modeling of elastic waveguides coupled to inﬁnite ﬂuid media using exact boundary conditions. Comput. Struct., 141:36–45, August 2014. ISSN 0045–7949. https://doi.org/10.1016/j.compstruc.2014.05.010. Available from Internet: 10.1016/j.compstruc.2014.05.010
 H. Gravenkamp, C. Birk and J. Van. Modeling ultrasonic waves in elastic waveguides of arbitrary cross-section embedded in inﬁnite solid medium. Comput. Struct., 149:61–71, March 2015. ISSN 0045–7949. https://doi.org/10.1016/j.compstruc.2014.11.007
 H. Gravenkamp, H. Man, C. Song and J. Prager. The computation of dispersion relations for three-dimensional elastic waveguides using the Scaled Boundary Finite Element Method. Journal of Sound and Vibration, 332(15):3756–3771, 2013. ISSN 0022-460X. https://doi.org/10.1016/j.jsv.2013.02.007
 H. Gravenkamp, C. Song and J. Prager. A numerical approach for the computation of dispersion relations for plate structures using the Scaled Boundary Finite Element Method. Journal of Sound and Vibration, 331(11):2543–2557, 2012. ISSN 0022-460X. https://doi.org/10.1016/j.jsv.2012.01.029
 T. Hayashi and D. Inoue. Calculation of leaky Lamb waves with a semi-analytical ﬁnite element method. Ultrasonics, 54(6):1460–1469, aug 2014. https://doi.org/10.1016/j.ultras.2014.04.021
 T. Hayashi, W.-J. Song and J.L. Rose. Guided wave dispersion curves for a bar with an arbitrary cross-section, a rod and rail example. Ultrasonics, 41(3):175–183, 2003. ISSN 0041-624X. https://doi.org/10.1016/S0041-624X(03)00097-0. Available from Internet: http://www.sciencedirect.com/science/article/pii/S0041624X03000970
 T. Hayashi, W.J. Song and J.L. Rose. Guided wave dispersion curves for a bar with an arbitrary cross-section, a rod and rail example. Ultrasonics, 41(3):175–183, May 2003. ISSN 0041-624X. https://doi.org/10.1016/s0041-624x(03)00097-0
 M. Mazzotti, I. Bartoli, A. Marzani and E. Viola. A coupled SAFE-2.5D BEM approach for the dispersion analysis of damped leaky guided waves in embedded waveguides of arbitrary cross-section. Ultrasonics, 53(7):1227–1241, 2013. ISSN 0041-624X. https://doi.org/10.1016/j.ultras.2013.03.003
 A.H. Nayfeh and P.B. Nagy. Excess attenuation of leaky Lamb waves due to viscous ﬂuid loading. The Journal of the Acoustical Society of America, 101(5):2649–2658, 1997. https://doi.org/10.1121/1.418506
 F.H. Quintanilla, Z. Fan, M.J.S. Lowe and R.V. Craster. Guided waves’ dispersion curves in anisotropic viscoelastic single-and multi-layered media. In Proc. R. Soc. A, volume 471, p. 20150268. The Royal Society, 2015.
 J.M. Renno and B.R. Mace. On the forced response of waveguides using the wave and ﬁnite element method. Journal of Sound and Vibration, 329(26):5474–5488, 2010. ISSN 0022-460X. https://doi.org/10.1016/j.jsv.2010.07.009
 J.-F. Semblat. Rheological interpretation of Rayleigh damping. arXiv preprint arXiv:0901.3717, 2009.
 D. O. Thompson and D. E. Chimenti. Preface: Review of progress in quantitative nondestructive evaluation. AIP Conference Proceedings, 1430(1):3–5, 2012. https://doi.org/10.1063/1.4716208. Available from Internet: http://aip.scitation.org/doi/abs/10.1063/1.4716208
 F. Treyssede. Spectral element computation of high-frequency leaky modes in three-dimensional solid waveguides. Journal of Computational Physics, 314:341–354, 2016. https://doi.org/10.1016/j.jcp.2016.03.029
 F. Treyssede and L. Laguerre. Numerical and analytical calculation of modal excitability for elastic wave generation in lossy waveguides. Journal of the Acoustical Society of America, 113:827–3837, 2014.
 I.A. Viktorov. Rayleigh and Lamb waves: physical theory and applications. Plenum press, 1967.
 E. Viola and A. Marzani. Mechanical Vibration: Where do we Stand?, volume 488 of International Centre for Mechanical Sciences, chapter Exact Analysis of Wave Motions in Rods and Hollow Cylinders, pp. 83–104. Springer Vienna, 2007. ISBN 978-3-211-68586-0.
 O.C. Zienkiewicz, R.L. Taylor, O.C. Zienkiewicz and R.L. Taylor. The ﬁnite element method, volume 3. McGraw-hill London, 1977.