Waves in heavy stratified gas: splitting into acoustic and gravity waves subproblems

Two-dimensional linearized hydrodynamic equations describing the wave propagation in a stratified heavy gas are considered. The hydrodynamic equation system is reformulated as a single Schr?odinger type operator equation. The waves with $\beta \frac{l_z}{l_x}\ll 1$ are considered, where $l_z$ and $l_x$ are the characteristic vertical and horizontal scales, respectively, and the asymptotic behavior of solutions as $\beta \to 0$. It is shown that the set of solutions depending on β form two disjoint classes. For solutions from each of the selected classes, its own, asymptotic as $\beta \to 0$ , approximate equation system is proposed. The selected classes of solutions are acoustic and internal gravity waves. It is shown that the hydrodynamic variables of acoustic and gravity waves are related by certain stationary relationships, different for each class. This allows to pose the problem of separating the contributions of acoustic and gravity waves in the initial condition. The existence of a solution to this wave separation problem is shown. Examples of solving the problem of dividing the general problem into subproblems on the propagation of acoustic and gravity waves are given. Estimates for the division of the energy of the initial disturbance by wave type are obtained.