Acoustic wave propagation in axisymmetric periodic waveguides concerns many different application fields: acoustic waves in ducts, acousto‐mechanic waves in bore hole, and noise propagation in towed arrays. A periodic guide with an intricate elementary cell and coupling with an external medium (possibility of leaky waves) makes the theoretical analysis of such waveguides more difficult. The present work is based on the combination of three analysis tools: finite element analysis for the elementary cell dynamics, Bloch‐Floquet wave decomposition for the periodic structure processing, and the perturbation method applied to harmonic space in order to take into account the external medium interactions. Dynamic matrices partitioning and the introduction of input‐output propagation conditions lead, in the case of zero coupling (with the external medium), to the typical secular equation: [P + ΓQ + Γ2R]x = 0, where Γ is the unknown propagator and x is the associated eigenvector; P, Q, and R are matrices deduced from the initial dynamic matrices. The external medium boundary condition introduces an additional nonpolynomial term and leads to a new nonlinear secular equation that is difficult to solve. The perturbation method allows, on the one hand, this difficulty to be overcome and, on the other hand, leads to a more physical approach, particularly in the method state where coupling leads to a Bloch‐Floquet harmonic wave degeneracy overtaking. Preliminary numerical tests on well‐known waveguides are presented. The results agree very well with analytical solutions. [Work supported by GERDSM.]Acoustic wave propagation in axisymmetric periodic waveguides concerns many different application fields: acoustic waves in ducts, acousto‐mechanic waves in bore hole, and noise propagation in towed arrays. A periodic guide with an intricate elementary cell and coupling with an external medium (possibility of leaky waves) makes the theoretical analysis of such waveguides more difficult. The present work is based on the combination of three analysis tools: finite element analysis for the elementary cell dynamics, Bloch‐Floquet wave decomposition for the periodic structure processing, and the perturbation method applied to harmonic space in order to take into account the external medium interactions. Dynamic matrices partitioning and the introduction of input‐output propagation conditions lead, in the case of zero coupling (with the external medium), to the typical secular equation: [P + ΓQ + Γ2R]x = 0, where Γ is the unknown propagator and x is the associated eigenvector; P, Q, and R are matrices deduced f...