Geodesic
Eigenfrequencies of longitudinal oscillations for an inhomogeneous rod with variable cross section
Vestnik Moskovskogo universiteta. Matematika, mehanika, no. 1 (2016), pp. 31-39

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The problem on the natural frequencies of longitudinal oscillations of a rod such that its Young's modulus, the density and the cross-sectional area are functions of the longitudinal coordinate is considered. For solving this problem, an integral formula is used to represent the general solution to the original Helmholtz equation with variable coefficients in terms of the general solution of the accompanying equation with constant coefficients. Frequency equations are obtained in the form of rapidly converging Leibnitz series for three types of boundary conditions. For these cases the frequency equations of the zeroth approximation are derived to quickly find the lowest natural frequency with an adequate accuracy. 
@article{VMUMM_2016_1_a4,
     author = {V. I. Gorbachev},
     title = {Eigenfrequencies of longitudinal oscillations for an inhomogeneous rod with variable cross section},
     journal = {Vestnik Moskovskogo universiteta. Matematika, mehanika},
     pages = {31--39},
     publisher = {mathdoc},
     number = {1},
     year = {2016},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/VMUMM_2016_1_a4/}
}
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V. I. Gorbachev. Eigenfrequencies of longitudinal oscillations for an inhomogeneous rod with variable cross section. Vestnik Moskovskogo universiteta. Matematika, mehanika, no. 1 (2016), pp. 31-39. http://geodesic.mathdoc.fr/item/VMUMM_2016_1_a4/