To seek measures to weaken the collar wave signals so that the formation arrivals are observable, it is important to make theoretical analysis of separate collar wave and formation arrivals in the acoustic logging-while-drilling environment. However, until now, the collar wave signal and the formation P- and S-arrivals have never been separately calculated. We have obtained individual component waves using the residue theorem and the branch-cut integral method, including residues at leaky poles. The waveform summed up from all individual waves is shown to agree well with the full waveform calculated by real-axis integration. In particular, the formation P-wave is obtained by summing the formation leaky mode and the compressional branch-cut integral for slow formations. The collar wave is found to propagate in the borehole and the formation as well as in the collar. Although the traveling speed of the collar wave is almost irrelevant to the formation, the attenuation and excitation spectrum of the collar wave are significantly affected by the formation, which reveals that an effective collar wave weakening design should be based on a model with the formation being taken into consideration.

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