The stress relief associated with the drilling of a borehole may induce a mechanically damaged zone with radial transverse isotropy (RTI), where the properties in the radial direction differ from those in the axial and tangential directions. The effect of such a zone on multipole acoustic full-waveform logging is investigated using a numerical model based on the frequency-axial-wavenumber method. Calculations of the spectral behavior show that the fundamental mode associated with the multipole source behaves the same way in an RTI zone as it does in a damaged zone with isotropic properties. In a slow virgin formation, calculations of full waveforms show that the presence of a damaged zone with RTI is more difficult to detect than a damaged zone with isotropic elasticity because the refracted P-wave encounters an isotropic zone but not an RTI zone. The results indicate that a damaged zone with RTI, which is a precursor to destructive events such as borehole instability and sand production, can be detected only by analyzing the spectral behavior of the fundamental mode.