Acoustic radiation from stiffened double concentric large cylindrical shells with periodic cavities is analytically examined in the medium and high frequency range using the Sommerfeld–Watson transform. Creeping wave acoustic model of the stiffened double cylindrical shells in the shadow, penumbra, and illuminated regions is established by the residue theorem. An asymptotic expression of far-field acoustic pressure is derived in the geometrical acoustic zone according to the stationary phase method. Sound field of the bare or stiffened double cylindrical shells with annular fluid is determined by the creeping wave poles in the broad circumferential region. Mechanisms of creeping wave propagation through the stiffened double cylindrical shells are shown. There are a great many elastic and acoustic waves that cannot transmit through the annular fluid, annular bulkheads, and periodic cavities with a moderate gap between the double circular cylindrical shells. Acoustic boundary layer theories are proposed to describe sound propagation through the annular fluid and bulkheads for the creeping waves.