Аннотация:A thin two-layered waveguide is considered. The governing equations for this waveguide is amatrix Klein–Gordon equation of dimension 2. A formal solution of this system in the form ofa double integral can be obtained by using Fourier transformation. Then, the double integralcan be reduced to a single integral with the help of residue integration with respect to the timefrequency. However, such an integral can be dicult to estimate since it involves branchingand oscillating functions. This integral is studied asymptotically. A zone diagram technique isproposed to represent the set of possible asymptotic formulae. The zone diagram generalizesthe concept of far-field and near-field zones.