Polynomial mode approximation for longitudinal wave dispersion in circular rods

The dispersive nature of the longitudinal wave propagation in rods has been studied extensively theoretically as well as experimentally by many authors (see e.g. [1] or [2]). Split Hopkinson Pressure Bars (SHPB) is employed to dynamically characterize materials in the intermediate range of strain rate and the post-processing of the SHPB tests requires the knowledge of the dispersion in the bars [3] [4] [5]. Simplified models of longitudinal wave propagation have been proposed by Love [6] and later by Mindlin et al. [7]. In his paper published in 2006, Anderson [8] developed a rod theory for the propagation of longitudinal waves in slender rods of circular cross section and it is an improvement from the previous works in [7]. In the approximation of the longitudinal and radial displacements he uses only the first two modes of the given basis for each displacement, this leads to the four-mode equation. The four-mode equation is a polynomial whose root is the wave longitudinal velocity. This equation is therefore easier to solve than the Pochhammer equation [1] because the latter is transcendental. However we noticed an error in one of the coefficients of the four-mode equation [8], and this error is not a mere typo. This error was also repeated in [9]. The aim of this paper is threefold. First we want to give the correct four-mode equation and the associated dispersion curve. Secondly, we wanted to examine the convergence of the approximation at higher orders then given by Anderson. Thirdly, this is the occasion to give dimensionless equations. The paper is organised as follows: Section 2 details how to obtain the dimensionless approximation for longitudinal waves. Section 3 recalls the Pochhammer dispersion equation for slender rods -which is later used to compute the reference dispersion curve- and details the way it is solved. Section 4 then gives the correct dispersion curve for the four-mode equation and examines the convergence of the approximate dispersion curves with an increasing number of degrees of freedom (modes). Lastly, Section 5 contains a discussion on the advantages and drawbacks of using either directly the Pochhammer equation or the Jacobi polynomial approximation, we also discuss the computation cost of both methods.