This paper examines the axisymmetric type vortex breakdown in a swirling pipe flow, in order to clarify the growing process of a bubble. The unsteady flow field is investigated based on the inviscid linear theory of the internal wave (inertia wave). An expression of the time-dependent flow field with the internal wave propagation, which satisfies the condition of arbitrary velocity change at the upstream boundary of the circular pipe, is derived. Using this expression, the transition of the flow field, the streamlines and the streaklines calculated by computer are examined. The unsteady flow field, as the result of a sudden azimuthal velocity change at the upstream boundary, clearly shows the birth of a bubble and its growing process into a stationary state.