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

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Home Editorial board Publication ethics Peer review guidelines Latest issue All issues Rules for authors Online submission system's guidelines Submit manuscript Contacts Address: Vatutina st. 53, Vladikavkaz, 362025, RNO-A, Russia Phone: (8672)23-00-54 E-mail: rio@smath.ru	DOI: 10.46698/i0037-3667-8880-p On the Properties of Acoustic Waves in a Compressible Ideal Stratified Fluid Tsvetkov, D. О. Vladikavkaz Mathematical Journal 2025. Vol. 27. Issue 2. Abstract: 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. Keywords: stratification effects in ideal fluids, compressible fluid, spectral problem Language: Russian Download the full text Download JATS XML For citation: Tsvetkov, D. О. On the Properties of Acoustic Waves in a Compressible Ideal Stratified Fluid, Vladikavkaz Math. J., 2025, vol. 27, no. 2, pp. 136-147 (in Russian). DOI 10.46698/i0037-3667-8880-p + References 1. Kopachevsky, N. D. and Krein, S. G. Operator Approach to Linear Problems of Hydrodynamics. Vol. 1: Self-Adjoint Problems for an Ideal Fluid, Basel, Boston, Berlin, Birkhauser Verlag, 2001, 384 p. DOI:10.1007/978-3-0348-8342-9. 2. Gabov, S. A. and Sveshnikov, A. G. Zadachi dinamiki stratifitsirovannykh zhidkostey [Problems in the dynamics of stratified fluids], Moscow, 1986, 288 p. (in Russian). 3. Brekhovskikh, L. М. and Goncharov, V. V. Vvedenie v mekhaniku sploshnykh sred [Introduction to the mechanics of continuous media], Moscow, 1982, 325 p. (in Russian). 4. Shatov, A. K. Types of Kelvin Waves in a Compressible Stratified Fluid, USSR Computational Mathematics and Mathematical Physics, 1987, vol. 27, no. 1, pp. 201-204. DOI: 10.1016/0041-5553(87)90144-3. 5. Kholodova, S. E. Wave Motions in a Compressible Stratified Rotating Fluid, Computational Mathematics and Mathematical Physics, 2007, vol. 47, no. 12, pp. 2014-2022. DOI: 10.1134/S0965542507120111. 6. Zakora, D. A. Model of the Maxwell Compressible Fluid, Journal of Mathematical Sciences, 2020, vol. 250, pp. 593-610. DOI: 10.1007/s10958-020-05030-6. 7. Kholodova, S. E. Matematicheskoe modelerivanie i analiz techeniy i voln vo vrashchayushcheysya i elektroprovodimykh zhidkikh sredakh [Mathematical modeling and analysis of flows and waves in rotating and electrically conducting liquid media], Doctor's Thesis, Saint Petersburg, 2018, 451 p. (in Russian). 8. Tsvetkov, D. O. On an Initial-Boundary Value Problem Which Arises in the Dynamics of a Compressible Ideal Stratified Fluid, The Bulletin of Irkutsk State University. Series Mathematics, 2024, vol. 48, pp. 49-63. DOI: 10.26516/1997-7670.2024.48.49. 9. Gohberg, I. C. and Krein, M. G. Introduction to the Theory of Linear Nonselfadjoint Operators in Hilbert Space, Translations of Mathematical Monographs, 1969, vol. 18, 448 p. 10. Markus, A. S. Introduction to the Spectral Theory of Polynomial Operator Pencils, Translations of Mathematical Monographs, 1988, vol. 71, 260 p. 11. Kopachevsky, N. D., Azizov, T. Ya., Zakora, D. A. and Tsvetkov, D. O. Operatornye metody v prikladnoy matematike. T. 2. Osnovnye kursy [Operator methods in applied mathematics. Vol. 2. Basic courses], Simferopol, Arial, 2022, 536 p. (in Russian). The sole copyright holder of the published work is the Founder of the Vladikavkaz Mathematical Journal. The terms of use of this work are governed by an open license (Creative Commons Attribution-NonCommercial 4.0 International). The use of metadata of the scientific article, including the title, abstract, author information, references, identifiers, and other bibliographic description elements for subsequent unrestricted use, is carried out under the terms of the CC BY or CC0 open licenses. ← Contents of issue


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