## 3 Results
The previously derived exact analytical solution is given by  
\[\hat{U}(x)=C_{1}J_{0}\bigg{[}\kappa\bigg{(}\frac{b}{m}+x\bigg{)}\bigg{]}+C_{2} Y_{0}\bigg{[}\kappa\bigg{(}\frac{b}{m}+x\bigg{)}\bigg{]}, \tag{22}\]  
where \(J_{0}\) denotes a zero order Bessel function of the first kind, \(Y_{0}\) denotes a zero order Bessel function of the second kind and  
\[\kappa=\frac{\omega}{\sqrt{E/\rho}}. \tag{23}\]  
The constants in Equation (22) are  
\[C_{1}=\frac{-Y_{0}\Big{[}\kappa\bigg{(}\frac{b}{m}\bigg{)}\Big{]}F_{0}}{2 wE \kappa(mL+b)\Big{\{}Y_{0}\Big{[}\kappa\bigg{(}\frac{b}{m}\bigg{)}\Big{]}J_{1 }\Big{[}\kappa\Big{(}\frac{b}{m}+L\Big{)}\Big{]}-J_{0}\Big{[}\kappa\Big{(} \frac{b}{m}\Big{)}\Big{]}Y_{1}\Big{[}\kappa\Big{(}\frac{b}{m}+L\Big{)}\Big{]} \Big{\}}} \tag{24}\]  
and  
\[C_{2}=\frac{J_{0}\Big{[}\kappa\bigg{(}\frac{b}{m}\bigg{)}\Big{]}F_{0}}{2 wE \kappa(mL+b)\Big{\{}Y_{0}\Big{[}\kappa\Big{(}\frac{b}{m}\Big{)}\Big{]}J_{1 }\Big{[}\kappa\Big{(}\frac{b}{m}+L\Big{)}\Big{]}-J_{0}\Big{[}\kappa\Big{(} \frac{b}{m}\Big{)}\Big{]}Y_{1}\Big{[}\kappa\Big{(}\frac{b}{m}+L\Big{)}\Big{]} \Big{\}}}. \tag{25}\]  
The baseline model was calculated with a standard finite sine transform where the height of the rod was a constant 0.018 m which produced a rod volume of 0.00324 m\({}^{3}\). Convergence of the series in Equation (6) is an open issue as the rate of convergence is dependent on the model parameters, shape and change of shape of the rod. However, for the modeled systems presented in this paper, generalized statements can be deduced based on comparison of various  
Figure 2: Displacement versus frequency for a rod with a linear variation of cross-sectional area. The solid line is the series solution, the circular markers are the analytical solution and the dashed line is the baseline model with uniform cross-sectional area. solutions. Figure 3 is a plot of the convergence of the system versus number of summation terms and frequency. This convergence metric was calculated using the equation  
\[C(N,f)=20\log_{10}\left[\frac{1}{J}\sum_{j=1}^{J}\left|\frac{\hat{\Omega}(x_{j})- U(x_{j})}{\hat{\Omega}(x_{j})}\right|\right], \tag{26}\]  
where \(\hat{\Omega}(x_{j})\) is the analytical solution, \(U(x_{j})\) is the series solution calculated using \(N\) terms, \(x_{j}\) is the location of the \(j\)th calculation, \(J\) was equal to \(15\) and the spatial locations were equally spaced across the length of the rod. The region in white is \(-40\) dB (or lower) and this corresponds to a one percent (or less) normalized difference between the two solutions. For this problem, one percent difference occurred when the normalized higher order wave propagation coefficients were \(-15.3\) dB down on average across all frequency bins for this specific shape. In general, as frequency increases, more terms are needed for the series solution to converge. The second problem analyzed is a rod that has a triangular variation in cross-sectional area, as shown in Figure 4. For this problem, the area is approximated with a Fourier series and is written as  
\[H_{p,n}=2 wE\bigg{[}\frac{(2n-1)\pi}{2L}\bigg{]}\bigg{\{}\frac{mL[\cos(n\pi)\cos(p \pi)-2n\cos(n\pi)\cos(p\pi)+2p-1]}{2\pi(-n^{2}+n+p^{2}-p)}\bigg{\}}\qquad p\
eq n \tag{20}\]0  
where  
\[c_{q}=\left\{ \begin{array}{ll}\frac{1}{2}&q=0\\ \frac{-[\cos(\pi q)-1]^{2}}{2\pi^{2}q^{2}}&q\
eq 0,\end{array} \right. \tag{28}\]  
and the derivative of the area with respect to the longitudinal variable is  
\[\frac{\mathrm{d}A(x)}{\mathrm{d}x}=2 wm\sum_{q=-Q}^{q=+Q}c_{q}\left(\frac{ \mathrm{i}2\pi q}{L}\right)\exp\left(\frac{\mathrm{i}2\pi qx}{L}\right), \tag{29}\]  
where \(2Q+1\) is the total number of terms in the Fourier series and was set equal to \(25\) for this analysis. It is noted that \(m\) is not the slope as defined in the first example, rather it is the additional height of the rod corresponding to the triangular region.