A one dimensional model describing the longitudinal vibrations of a linearly elastic rod composed of multiple sections made of different materials is derived. The proposed theory can be applied to many technical problems, including analysis of low frequency underwater transducers and mechanical wave guides. Cylindrical, conical and exponential sections are considered. The hypothesis that lateral displacements and axial shear stresses in the bar are zero is rejected. The lateral displacements are assumed to be proportional to the axial strain and the Poisson ratio, resulting in the so-called Rayleigh-Bishop equation as opposed to the classical one dimensional wave equation. The Rayleigh-Bishop equation is a fourth order partial differential equation, not resolved with respect to the highest order time derivative. A system of partial differential equations, as well as the corresponding boundary conditions and continuity conditions at the junctions between the sections, are derived using Hamilton's variational principle. An analytical solution is obtained in terms of Green's functions.
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