Conjugate flows and smooth bores in a weakly stratified fluid

The problem of steady-state flows in a layer of a continuously stratified fluid is considered. The sufficient condition of existence of families of shear flows that are consistent with the meaning of the laws of conservation of mass, momentum, and energy with a uniform flow is given. Approximate solutions of the smooth-bore type, which describe the wave transitions for pairs of conjugate flows of the first spectral mode, are obtained.