Analytical properties of the spectral problem for the internal gravity waves equation with shear flows under critical wave generation modes

We consider issues related to the formulation of problems of describing the dynamics of linear internal gravity waves in stratified media with horizontal shear flows under critical wave generation modes. In a plane setting, we discuss new model physical formulations of the problems where critical modes may occur. For arbitrary distributions of the buoyancy frequency and shear flows satisfying the Miles–Howard conditions and natural regularity conditions, we study analytical properties of solutions of the main spectral problem of the internal gravity waves equation with shear flows under critical wave generation modes for the cases of simlpe and multiple eigenvalues.