This paper investigates some particularities related with the influence of the magnitude of the initial twisting of the axisymmetric wave propagation in the initially twisted circular bi-material compounded cylinder. The investigation is carried out within the scope of the piecewise homogeneous body model with the use of the three-dimensional linearized theory of elastic wave propagation in an initially stressed body. The mathematical formulation of the problem is presented and the corresponding solution method is proposed and developed. The numerical results are further presented and discussed. In particular, the mechanism of the arising of the new type modes caused by the initial twisting of the circular compounded cylinders is established.

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