Abstract <p>The aim of this work is to study the modal structure of solutions describing the generation of internal gravity waves (IGWs) in stratified media with model distributions of the buoyancy frequency and background shear currents, allowing us to determine the main qualitative characteristics of the behavior of dispersion relations at small wave numbers depending on the modal number. The problem of constructing solutions describing the generation of linear IGWs in a layer of a stratified medium of finite depth with model distributions of the buoyancy frequency and background shear current is considered. Under the assumption of the Miles–Howard stability for the Richardson number, the corresponding dispersion dependences are studied. It is shown that, depending on the parameters of the linear shear current, the dispersion curves of the wave modes can have qualitatively different asymptotic representations at small wave numbers. The dispersion curves of a finite number of modes describing waves with a limited length, at small values of the wave number, admit expansions in even powers of a small parameter. The dispersion curves of the remaining modes, corresponding to waves with an arbitrarily large length, are expanded in a series in odd powers of small wave numbers. The phase structure of the wave fields is studied depending on the mode number and the main characteristics of the shear currents. Estimates have been obtained analytically that allow one, depending on the parameters of the model flow, to find the wave mode number that divides the entire existing set of wave modes into limited and longwave ones.</p>