Geodesic
Two types of waves in a two-layer stratified fluid
Vestnik KRAUNC. Fiziko-matematičeskie nauki, Tome 43 (2023) no. 2, pp. 111-125
Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

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 $\frac{\Delta\rho}\rho$, i.e. it is very small.
Keywords: two-layer liquid, potential oscillations.
Mots-clés : Boussinesq approximation, dispersion relation 
@article{VKAM_2023_43_2_a7,
     author = {A. I. Rudenko},
     title = {Two types of waves in a two-layer stratified fluid},
     journal = {Vestnik KRAUNC. Fiziko-matemati\v{c}eskie nauki},
     pages = {111--125},
     year = {2023},
     volume = {43},
     number = {2},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/VKAM_2023_43_2_a7/}
}
TY  - JOUR
AU  - A. I. Rudenko
TI  - Two types of waves in a two-layer stratified fluid
JO  - Vestnik KRAUNC. Fiziko-matematičeskie nauki
PY  - 2023
SP  - 111
EP  - 125
VL  - 43
IS  - 2
UR  - http://geodesic.mathdoc.fr/item/VKAM_2023_43_2_a7/
LA  - en
ID  - VKAM_2023_43_2_a7
ER  - 
%0 Journal Article
%A A. I. Rudenko
%T Two types of waves in a two-layer stratified fluid
%J Vestnik KRAUNC. Fiziko-matematičeskie nauki
%D 2023
%P 111-125
%V 43
%N 2
%U http://geodesic.mathdoc.fr/item/VKAM_2023_43_2_a7/
%G en
%F VKAM_2023_43_2_a7
A. I. Rudenko. Two types of waves in a two-layer stratified fluid. Vestnik KRAUNC. Fiziko-matematičeskie nauki, Tome 43 (2023) no. 2, pp. 111-125. http://geodesic.mathdoc.fr/item/VKAM_2023_43_2_a7/

[1] Ovsyannikov L. V., et. al., Nelineynye problemy teorii poverkhnostnykh i vnutrennikh voln [Nonlinear problems of the theory of surface and internal waves]., Nauka, Novosibirsk, 1985 | MR

[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., 3 (2021), 95-106

[3] Dias F., Il’ichev A., “Interfacial waves with free-surface boundary conditions: an approach via model equation”, Physica D, 23 (2001), 278-300 | DOI | MR

[4] Sretenskiy L. N., Teoriya volnovykh dvizheniy zhidkosti [Theory of wave motions in a fluid]., Nauka, Moscow, 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, 9 (2002.), 221-235 | DOI

[6] Hashizume Y., “Interaction between Short Surface Waves and Long Internal waves”, Journal of the Physical Society of Japan, 48:2 (2006), 631-638 | DOI

[7] Matsuno Y. A., “Unified theory of nonlinear wave propagation in two-layer fluid system”, J. Phys. Soc. Japan, 62:6 (1993), 1902-1916 | DOI | MR

[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 | MR

[10] Landau L.D., Lifshitz E.M., Teoreticheskaya fizika. Gidrodinamika [Theoretical physics. Hydrodynamics]., v. 6, Moscow. Nauka, 1986

[11] Whitham G., Lineyniye I nelineynye volny [Linear and nonlinear waves]., Moscow: Mir, 1977 | MR

[12] Lyapidevskiy V.Yu., Teshukov V.M., Matematicheskie modeli rasprostraneniya dlinnykh voln v neodnorodnoy zhidkosti [Mathematical models of propagation of long waves in a non-homogeneous 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, 56 (1977), 241-266 | DOI | MR | Zbl

[14] Jamali M., Seymour B., Lawrence G. A., “Asymptotic analysis of a surface-interfacial wave interaction”, Phys. Fluids, 15:1 (2003), 47-55 | DOI | MR

[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., 12 (1978), 157-165