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Analytical solutions for the vibration of beams with variable cross-sections are, in general, complex and, in many cases, impossible. On the other hand, approximate methods, such as the weighted residual, Rayleigh–Ritz and finite difference methods, also have their own shortcomings, such as a limited number of natural frequencies and low accuracy. In this paper, using the wave propagation method, the beam is partitioned into several continuous segments, each with a uniform cross-section, for which there exists an exact analytical solution. Waves entering a segment in positive and negative directions are calculated from waves that entered the initial segment. Then, by satisfying the boundary conditions, the characteristic equation is obtained and all natural frequencies are calculated. Also, using the sum of waves at each point that are moving in positive and negative directions, the mode shapes are obtained. To verify this modified method, frequencies whose mode shapes are in a polynomial cross-sectioned beam having an exact analytical solution are compared and thereby proven to be highly accurate. Therefore, this method can also be used to calculate natural frequencies and their mode shapes in beams with variable cross-sections without any analytical solution

Modified wave approach for calculation of natural frequencies and mode shapes in arbitrary non-uniform beams