Investigation of the direct and inverse solutions for torsional waves propagating in a cylinder

We investigate the direct and inverse problem associated with the torsional waves propagating in a cylinder. We analyse the usual wave equation as well as the damped wave equation and consider the problem of recovering the initial profile from the observations of the final profile. This inverse problem arises when experimental measurements are taken at any given time, and it is desired to calculate the initial profile. An integral representation for the problem is found, from which a formula for initial disturbance is derived using Picard's criterion and the singular system of the associated operators.