Behavior of waveguide modes in a neighborhood of the critical cross section

A curved waveguide of variable thickness filled with an inhomogeneous medium is considered. Shortwave asymptotic equations are obtained which describe the waveguide modes near the critical cross section. The amplitudes of waves reflected from the critical cross section and also of waves filtering through the postcritical section of the waveguide are obtained.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 42, pp. 181–188, 1974.