In this work, the Bishop and Love models for longitudinal vibrations are
adopted to study the dynamics of isotropic rods with conical and exponential crosssections.
Exact solutions of both models are derived, using appropriate transformations.
The analytical solutions of these two models are obtained in terms of generalised
hypergeometric functions and Legendre spherical functions respectively. The
exact solution of Love model for a rod with exponential cross-section is expressed as a
sum of Gauss hypergeometric functions. The models are solved numerically by using
the method of lines to reduce the original PDE to a system of ODEs. The accuracy of
the numerical approximations is studied in the case of special solutions.