-We apply a functional-discrete method with convergence rate not worse than that of a geometric series for an eigenvalue transmission problem with periodic boundary conditions. It is shown that the convergence rate increases with an increase in the ordinal number of trial eigenvalue. Based on asymptotic behavior of eigenvalues of the basic problem and the functional-discrete method, qualitative results concerning the arrangement of the eigenvalues of the original eigenvalue transmission problem with periodic boundary conditions are proved. A number of numerical examples are given to support the theory.