Pochhammer–Chree equation solver for dispersion correction of elastic waves in a (split) Hopkinson bar

Abstract

A robust algorithm for solving the Bancroft version of the Pochhammer–Chree (PC) equation is developed based on the iterative root-finding process. The formulated solver not only obtains the conventional n-series solutions but also derives a new series of solutions, named m-series solutions. The n-series solutions are located on the PC function surface that relatively gradually varies in the vicinity of the roots, whereas the m-series solutions are located between two PC function surfaces with (nearly) positive and negative infinity values. The proposed solver obtains a series of sound speeds at exactly the frequencies necessary for dispersion correction, and the derived solutions are accurate to the ninth decimal place. The solver is capable of solving the PC equation up to n = 20 and m = 20 in the ranges of Poisson’s ratio ( ν) of 0.02 [Formula: see text]  ν [Formula: see text] 0.48, normalised frequency ( F) of F [Formula: see text] 30, and normalised sound speed ( C) of C [Formula: see text] 300. The developed algorithm was implemented in MATLAB®, which is available in the Supplemental Material (accessible online).