The knowledge of acoustic wave propagation in a borehole embedded in gas hydrate-bearing sediments is of great significance for the exploitation of gas hydrate. A gas hydrate-bearing sediment is a typical three-phase porous medium containing two solids and one fluid. However, until now, the borehole acoustic wavefield and its component waves within such a porous medium have never been calculated. In this work, a real-axis integration method is proposed to calculate the borehole acoustic field embedded in a three-phase porous medium based on the Biot-type three-phase theory. Meanwhile, a component wave approach, combined with the branch-cut integral method and the residue theorem, including residues at leaky poles is proposed to study the borehole wave propagation of a three-phase porous medium. The branch points and poles of the potential acoustic wave function are obtained, which correspond to the normal and leaky modes on various Riemann sheets. On this basis, the excitation intensity and waveforms of each component are obtained. The result shows that the waveform summed up from all individual waves agrees well with the full waveform calculated by real-axis integration, which provides a theoretical basis for the subsequent inversion of reservoir parameters by using the information of various mode waves.