Vestnik КRAUNC. Fiz.-Mat. nauki. 2023. vol. 43. no. 2. P. 111-125. ISSN 2079-6641

PHYSICS
https://doi.org/10.26117/2079-6641-2023-43-2-111-125
Research Article
Full text in English
MSC 35Q35

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Two Types of Waves in a Two-Layer Stratified Fluid

A. I. Rudenko^\ast

Department of applied mathematics and information technology, Kaliningrad State Technical University, 236022, Kaliningrad, Sovetsky prospect 1, Russia

Abstract. Within the framework of the linear theory of small potential oscillations the article studies the structure and characteristics of internal and surface waves in a two-layer stably stratified fluid. Dispersion relations have been obtained and analyzed. The Boussinesq approximation has been studied. We have determined the existence of two types of waves: a fast wave and a slow wave. The fast wave differs little from a surface wave in a homogeneous fluid. The main results of studying the case of n = 1 are as follows: obtaining expressions for fluid particles velocity components and hydrodynamic pressure in the upper layer of the considered two-layer fluid, as well as obtaining relations, which connect the oscillation amplitudes of the free surface and the interface of the considered two-layer fluid. The main results of studying the case of n = 2 are as follows: obtaining expressions for fluid particles velocity components and hydrodynamic pressure in the lower layer of the considered two-layer fluid, as well as obtaining dispersion relations in the considered two-layer fluid. In the case of implementing the Boussinesq approximation, the fast wave differs little from a surface wave in a homogeneous fluid, andthe velocity of slow waves is proportional to the square root of the squared ratio \dfrac{\Delta\rho}{\rho}, i.e. it is very small.

Key words: two-layer liquid, Boussinesq approximation, dispersion relation, potential oscillations

Received: 11.04.2023; Revised: 08.06.2023; Accepted: 10.06.2023; First online: 30.06.2023

For citation. Rudenko A. I. Two types of waves in a two-layer stratified fluid. Vestnik KRAUNC. Fiz.-mat. nauki.
2023, 43: 2, 87-110. EDN: YGKXHT. https://doi.org/10.26117/2079-6641-2023-43-2-87-110.

Funding. The work was carried out without the support of funds.

Competing interests. There are no conflicts of interest regarding authorship and publication.

Contribution and Responsibility. The author participated in the writing of the article and is fully responsible for
submitting the final version of the article to the press.

^\astCorrespondence: E-mail: alex-rudenko@bk.ru

The content is published under the terms of the Creative Commons Attribution 4.0 International License

© Rudenko A. I., 2023

© Institute of Cosmophysical Research and Radio Wave Propagation, 2023 (original layout, design, compilation)

References

1. Ovsyannikov L. V., et. al. Nelineynye problemy teorii poverkhnostnykh i vnutrennikh voln [Nonlinear problems of the theory of surface and internal waves].. Novosibirsk: Nauka, 1985.
2. Zaytsev A. A., Rudenko A. I., Alekseeva S. M. Rasprostranenie sinusoidal’nykh voln v dvusloynoy stratifitsirovannoy po plotnosti zhidkosti [Propagation of sine waves in a two-layer density-stratified fluid]., IKBFU’s Vestnik., 2021. vol. 3, pp. 95-106.
3. Dias F., Il’ichev A. Interfacial waves with free-surface boundary conditions: an approach via model equation, Physica D, 2001. vol. 23, pp. 278-300.
4. Sretenskiy L. N. Teoriya volnovykh dvizheniy zhidkosti [Theory of wave motions in a fluid].. Moscow: Nauka, 1977.
5. Grimshaw R., Pelinovsky E., Poloukhina O. Highen-order Korteweg – de Vries models for internal solitary wave in a stratified she ar flow with a free surface, Nonlinear Processes Geophysical, 2002. vol. 9, pp. 221-235.
6. Hashizume Y. Interaction between Short Surface Waves and Long Internal waves, Journal of the Physical Society of Japan, 2006. vol. 48, no. 2, pp. 631-638.
7. Matsuno Y. A. Unified theory of nonlinear wave propagation in two-layer fluid system, J. Phys. Soc. Japan, 1993. vol. 62, no. 6, pp. 1902-1916.
8. Lamb G. Gidrodinamika [Hydrodynamics]. / Moscow: Gostekhizdat, 1947.
9. Kochin N.E., Kibel’ N.A., Roze N.A. Teoreticheskaya gidromekhanika [Theoretical hydromechanics]. / Moscow: Fizmatgiz, 1963.
10. Landau L.D., Lifshitz E.M. Teoreticheskaya fizika. Gidrodinamika [Theoretical physics. Hydrodynamics]., vol. 6 / Moscow. Nauka, 1986.
11. Whitham G. Lineyniye I nelineynye volny [Linear and nonlinear waves]. / Moscow: Mir, 1977.
12. Lyapidevskiy V.Yu., Teshukov V.M. Matematicheskie modeli rasprostraneniya dlinnykh voln v neodnorodnoy zhidkosti [Mathematical models of propagation of long waves in a nonhomogeneous fluid]. / Siberian Branch of Russian Academy of Sciences. Novosibirsk, 2000.
13. Grimshaw R. H. J. The modulation of an internal gravity-wave packet, and the resonance with the mean motion, Studies in Applied Mathematics, 1977. vol. 56, pp. 241-266.
14. Jamali M., Seymour B., Lawrence G. A. Asymptotic analysis of a surface-interfacial wave interaction, Phys. Fluids, 2003. vol. 15, no. 1, pp. 47-55.
15. Kubota T., Ko D. R. S., Dobbs L. D. Propagation of weakly nonlinear internal waves in a stratified fluid of finite depth, AIAA. J. Hydrodyn., 1978. vol. 12, pp. 157-165.

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