We consider one-dimensional longitudinal vibrations of a rigid rod with a nonuniform cross-section, fixed at its ends with lumped masses and springs. The cross-section inertia effects are taken into account on the basis of the Rayleigh theory. The equation of motion and the boundary conditions are derived from Hamilton’s variational principle. The characteristic equation is constructed and the eigenvalues for the harmonic vibrations of the rod are calculated. It is shown that the eigenvalues are bounded from above. Two types of the orthogonality of the eigenfunctions corresponding to the eigenvalues are discussed. The Green function is constructed for the problem of forced vibrations of the rod governed by a linear fourth-order partial differential equation, which involves mixed derivatives. Exact solutions of the rod vibration problems are found for rods with constant and conical cross-sections. Rigid isotropic waveguides are often used for generating, transmitting, and amplifying mechanical vibrations, for example, in acoustic transducers. Theoretical investigation of acoustic, mechanical, and electromagnetic waveguides is usually based on the analysis of second-order wave equations. This approach is justified in descriptions of the wave propagation in relatively thin and long rigid rods. As was shown by Rayleigh [1], the error due to the neglect of the transverse motion of the rod is proportional to the square of the ratio of the characteristic section radius to the length of the rod (aspect ratio). For a more accurate analysis of the longitudinal vibrations of a relatively thick and short rod, the rod deformation in the transverse direction must also be taken into account. The approach to the analysis of the vibrations of a thick and short rod used in this study is based on the theory of longitudinal vibrations of a rod, in which the effects due to the transverse motion are taken into account (the corresponding mathematical model is called the Rayleigh rod). The equation of motion and the boundary conditions for the onedimensional longitudinal vibrations of the Rayleigh rod with variable cross section and ends fixed by means of lumped masses and springs are derived from Hamilton’s variational principle. As a result, we arrive at a linear fourth-order partial differential equation with variable coefficients, which involves mixed derivatives. Previously, approximate analytical methods, such as the Galerkin method [2] and the method based on the expansion of the solution in a power series in the Poisson coefficient [3], were used for solving this equation. The frequencies of the natural vibrations of a cylindrical rod with rigidly fixed ends were determined in [4, pp. 159, 160]. In this study we use the method of the separation of variables based on the exact solutions of the equations of motion of the Rayleigh rod, which makes it possible to construct the Green function. A similar approach to an analysis of the longitudinal vibrations of stepped rigid waveguides described by second-order wave equations was applied in [5, 6].