ИСТИНА

Transient processes in relatively simple waveguides still constitute a challenging problem. Even a 2-layer 2D scalar waveguide is an interesting subject of a research. Beside other tasks, we are particularly interested in an asymptotic description of forerunners and of rays in layered and stratified waveguides. We develop a consistent approach based on the classical paper by Randles and Miklowitz (1971). The essence of the approach is that the dispersion diagram (understood as a function k(ω) linking the longitudinal wavenumber with the temporal circular frequency) is considered as a multivalued complex analytical function of a complex variable, i.e., an analytical continuation of a usual “real” dispersion diagram. The integral representation of a transient field is considered as a sum of contour integrals, and the integration path is deformed. As the result, one can get rid of exponentially small waves and derive simple expressions for the forerunners (the waves traveling faster than any modal group velocity). To follow this scheme in all details, one should develop a version of the saddle point method that works for a big (or infinite) set of integration contours drawn on a complicated Riemann surface. For this, we represent the non-stationary wave field as a double Fourier integral in the (ω, k) plane. The dispersion diagram forms a polar set in the (ω, k) complex space. This polar set is, generally, an analytic manifold of real dimension 2 in the space with real dimension 4. After taking the residual integral, the field becomes reduced to a set of contour integrals over the Leray residual form on the dispersion diagram. Then, the saddle point method can be applied to functions defined on a dispersion diagram. A procedure of rebuilding of integration contours, which can be performed only in the multi-contour version of the saddle point method, is described. As the result of applying this method, the (x, t) plane of the longitudinal coordinate and time becomes separated into zones of different wave structure. These zones correspond to presence of different wave fronts. Asymptotic formulae for some wave components are obtained.