  A thin two-layered waveguide is considered. The governing equations for this
waveguide is a matrix Klein--Gordon equation of dimension~2. A formal solution
of this system in the form of a double integral can be obtained by using
Fourier transformation. Then, the double integral can be reduced to a single
integral with the help of residue integration with respect to the time
frequency. However, such an integral can be difficult to estimate since it
involves branching and oscillating functions. This integral is studied
asymptotically. A zone diagram technique is proposed to represent the set of
possible asymptotic formulae. The zone diagram generalizes the concept of
far-field and near-field zones.
