  In this paper we solve the inviscid inertial wave solutions in a circular
pipe or annulus rotating constantly about its axis with moderate angular speed.
The solutions are constructed by the so-called helical wave functions. We
reveal that the mean velocity profiles must satisfy certain conditions to
accommodate the inertial waves at the bulk region away from boundary. These
conditions require the axial and azimuthal components of the mean velocity take
the shapes of the zeroth and first order Bessel functions of the first kind,
respectively. The theory is then verified by data obtained from direct
numerical simulations for both rotating pipe and circular annulus, and
excellent agreement is found between theory and numerical results. Large scale
vortex clusters are found in the bulk region where the mean velocity profiles
match the theoretical predictions. The success of the theory in rotating pipe,
circular annulus, and streamwise rotating channel suggests that such inertial
waves are quite common in wall bounded flow with background rotation.
