Two-scales expansions of quasinormal waves in irregular refractive wave guides. Conditions of their applicability for high frequencies

Using the method of two-scale's expansions the problem of the waveguide propagation investigated for the case on a nonuniform halfspace, the refractive index with one maximum depending slowly on the horizontal coordinate.
This method is shown to be valuable for high frequencies if $\varepsilon p\ll1$, $p$ being the frequency and $\varepsilon$ being small parameter specifeing the gradient of the refraction index in horizontal direction. In the case of the surface wave guide the method is also applicable without restrictions on the gradient of the refraction index if $m$ is small, where $m$ – is the number of zeros of the eigen function.