Longitudinal vibrations of elastic rods of variable cross-section (concentrators)

We consider longitudinal vibrations of elastic rods of variable cross-section (concentrators) of conical, exponential, and catenoidal types and obtain analytic expressions for the gain factors of the concentrators in the cases of boundary conditions of the first and second kinds. We numerically study various profiles and compare the gain factors depending on the type of boundary conditions and the eigenvalue number. We note that as the eigenvalue number increases, the curves for the gain factors of both the first and the second kind tend to the limit curves.