Reconstructing the cross–sectional area of an axially vibrating non–uniform rod from one of its mode shapes

Abstract

It is shown that the cross–sectional area of a non–uniform axially vibrating rod can be reconstructed from one of its mode shapes, and that the solution, up to a scale factor, is unique. The conditions, which allow construction of a physical rod with positive smooth cross–sectional area, are given. The problem is addressed for the discrete and the continuous models of the rod. Examples demonstrate the various results.