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Equations of motion for harmonic waves in multilayered cylindrical structures are given in a compact form. This compact form helps us to efficiently form a boundary condition matrix of any N-layer structure based on material parameters for all cylindrical layers. Fluid-fluid, fluid-solid, and solid-fluid boundaries are supported. The search algorithm automatically searches for zeros of a boundary-condition determinant that yields the borehole velocity dispersion for a given frequency band. We describe how to obtain radial displacements and stress amplitudes for a chosen frequency that can be useful in estimating frequency dependent radial depth of investigation for the borehole Stoneley, flexural, and quadrupole modes in the presence of a casing and tool effects on sonic data. An efficient formulation for calculating borehole dispersions plays an important role in the analysis and interpretation of measured borehole dispersions in the presence of a sonic tool structure and radially heterogeneous formation. Applications of this formulation in the analysis of field data will be presented.