Wave Motion 36 (2002) 169–184 Axially symmetric modes in finite cylinders—the wave guide solution Leif Kari ∗ MWL/Department of Vehicle Engineering, Royal Institute of Technology, 100 44 Stockholm, Sweden Received 11 December 2000; received in revised form 28 May 2001; accepted 10 December 2001 Abstract Theresonancefrequenciesandmodeshapesforaxiallysymmetricmodesinelastic,finite,solidcylindersareexaminedviaa waveguidemodel,whereinfluencesofhigherordermodes,structure-bornesounddispersion,cylinderdiameterandlengthare investigated. Based on the dispersion relation for an infinite cylinder with boundary conditions at the lateral surfaces satisfied by mode matching, results are shown to converge properly. Comparisons with alternative and approximate approaches are made. The presented model agrees fully with three-dimensional models and simplified slender rod models while diverging from the latter for increased diameter-to-length ratios. To a great extent, the increased influences of higher order modes and dispersion explain the discrepancies reported for approximate approaches. © 2002 Elsevier Science B.V. All rights reserved. 1. Introduction By the late 19th century Pochhammer [1] and Chree [2] developed the dispersion relations for an infinite, elastic cylinder. Love [3] pointed out that the real dispersion relation eigenvalues are finite in number at a given frequency, asinthecaseofimaginaryeigenvalues.Thus,thecorrespondingeigenmodescannotthemselvesformacompleteset, inwhichtoexpandanarbitrarycylinderendboundarycondition.Therefore,dispersionrelationsarebelievedtoplay a role only for infinite cylinder related solutions. However, Adem [4] shows that there are also an infinite number of complexeigenvalues,renderingacompletesetpossible.Yetittookmorethan80yearsbeforeOnoeetal.[5]resolved the complete eigenvalue spectrum, mainly by establishing cut-off frequencies, asymptotic behaviour and boundary line grids. Owing to dispersion relation complexity, lacking awareness of complex eigenvalues plus the difficulties of simultaneously satisfying boundary conditions at the lateral and radial surfaces, a number of approximate models have been developed for finite cylinders within the context of the Pochhammer–Chree approach, containing only a few modes approximating to current eigenvalue spectra covering a limited range only. Nevertheless, they have been extensively applied over the years with particular emphasis on the simplest. In particular, numerous models are developed for axisymmetric, non-torsional, waves in finite cylinders. The classical long rod [6] assumes that lateral, plane cross-sections remain plane and lateral, and that uniaxial and uniform stress exists, while radial expansions and contractions arising from axial stress are neglected. Love [7] extends the long rod by taking into account the radial displacement, thus exhibiting dispersion lacking in the long rod. Bishop [8] extends the Love ∗ Tel.: +46-8-7907974; fax: +46-8-7906122. E-mail address: leifk@fkt.kth.se (L. Kari). 0165-2125/02/$ – see front matter © 2002 Elsevier Science B.V. All rights reserved. PII:S0165-2125(02)00008-2