Higher-order rod approximations for the propagation of longitudinal stress waves in elastic bars

The elementary one-dimensional wave theory is often used to describe the propagation of longitudinal stress waves along a slender bar. However, for relatively large diameter bars and high-frequency waves, geometrical wave dispersion due to lateral inertia occurs, rendering this single-mode theory inaccurate. In this paper, an approximate four-mode rod equation for a circular bar that takes wave dispersion into account is presented. Little difference is observed between the dispersion curves of the four-mode equation and the Pochhammer–Chree equation that gives the exact propagation coefficient for an infinite circular bar. The advantage of the former is that its solution can be computed directly whereas the latter requires an iterative procedure. Also presented are different orders of rod approximation equations derived from the Pochhammer equation, each of which can also be solved directly. Applications of the more accurate rod equations include correcting for wave dispersion in a split Hopkinson pressure bar (SHPB) test, and more accurately determining the frequency-dependent elastic modulus of a viscoelastic bar from an experimentally measured propagation coefficient.