On the Properties of Acoustic Waves in a Compressible Ideal Stratified Fluid

The problem of small motions of a compressible ideal stratified fluid was previously studied by the author. This problem was reduced to the Cauchy problem for a differential-operator equation of the second order in the orthogonal sum of some Hilbert spaces. An equation with a closed operator was associated with the resulting equation. The application of the operator block matrix method, as well as the theory of abstract differential equations enables us to find sufficient conditions for the existence of a solution to the corresponding problem. In the present work, the corresponding problem of normal oscillations of this hydraulic system is investigated. It is assumed that the square of the oscillation frequency of this hydraulic system exceeds the square of the Weisala–Brent frequency. This case in the classification from the monograph [2] is called the “case of acoustic waves”. The problem is investigated on the base of an approach associated with the application of the so-called spectral theory of operator bundles (operator functions). Using the factorization of the operator bundle (to which the original problem is reduced) with respect to a circle of some radius, using M. V. Keldysh’s theorem, a statement is obtained about the completeness and minimality of the system of root elements. It is further proved that the corresponding system forms the so-called Riesz basis in the Hilbert space. Also, the nature of the asymptotic behavior of the branches of the eigenvalues of the operator pencil under study is clarified.