## 1 Introduction
[17] studied this problem using a wave approach for the reflection, transmission and propagation of longitudinal waves for uniform, linear and conical change of area. [PERSON] et al. [3] developed a transfer matrix method to understand the propagation characteristics of a rod with variable cross-section. They utilized [PERSON]'s principle to derive and solve the equations of motion for a rod with an exponential and a polynomial shape and applied this analysis to higher order rod theory. [PERSON] et al. [18] derived asymptotic approximations for solutions of Bloch waves for finite and infinite length structures and applied this analysis to slender elastic structures. [PERSON] et al. [19] also investigated transfer function matrix methods for systems with structural interfaces and this method could be applied to a rod with a piece-wise constant cross-section. Asymptotic methods have been extensively applied to this problem [20] using variational and asymptotic approaches. These methods have been applied to a rod with circular [21] cross section and the resulting dispersion curve and spectrum were calculated. They were also applied to Timoshenko type rods [22] to determine the shear coefficients of circular, annular, elliptical and square rods. These variational methods have also been extended to transverse motion of isotropic centrally symmetric beams [23] and anisotropic closed cross section beams [24], however, transverse motion is beyond the scope of this paper and is not investigated herein. Finally, it is noted [25] that the theory of an area change can be extended to the theory of a change in the material properties of the medium. This paper derives a modal solution to the displacement field of a finite length rod whose area varies with respect to its length. This new analysis method is explicitly designed to calculate series solutions to this problem where no analytical solution currently exists. This allows the displacement field to be calculated for any problem where the area and derivative of the area can be represented as analytical functions. The specific problem of a linear change of area is shown in Figure 1 with the boundary condition at \(x=0\) fixed and the boundary condition at \(x=L\) free. Different combinations of these boundary conditions and different functional forms for change of area are possible and these are discussed and analyzed in detail below. The solution begins by writing the displacement field as an \(n\)-indexed series solution of an unknown indexed wave propagation coefficient multiplied by the eigenfunction of the uniform area rod. This expression is inserted into the differential equation of the non-uniform area rod that contains a restoring force term, an inertial force term and an area change force term. The resulting expression is then multiplied by a \(p\)-index expression that is the eigenfunction of the uniform area rod and integrated over the length of the rod. This results in an infinite set of linear algebraic equations that can be truncated and solved to yield the wave propagation coefficients. Once these are known, the system displacements can be determined. Five different example problems are generated and four of them are checked using other analytical or numerical methods.