DOI: https://doi.org/10.15587/1729-4061.2018.128641

### Analysis of energy of internal waves in a three­layer semi­infinite hydrodynamic system

Olga Avramenko, Maria Lunyova

#### Abstract

Energy characteristics of waves propagation along the contact surfaces in a hydrodynamics system "liquid half-space – layer –layer with a rigid lid" are explored. Based on the solutions of first approximation to a weakly non-linear model, the integral relations for wave motion energy in each layer and for the total energy of the system were obtained. An analysis of energy of wave processes revealed that an increase in wave number causes a decrease in energy of wave motion of the upper layers, and energy of wave motion of the lower half-space at some values of wave number reaches extreme values. In this case, total energy of the system is descending in nature and rather quickly approaches its limit value.

The numerical values of energy for three different cases of propagation of progressive waves were obtained: only along the upper contact surface, only along the lower contact surface, along both surfaces simultaneously. Comparison of the obtained values of energy revealed that in the case of waves propagation along both contact surfaces simultaneously, the total energy of the system is close to the sum of energies of the system at waves propagation along one of the surfaces

#### Keywords

weakly non-linear model; three-layer hydrodynamic system; internal waves; wave motion energy

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#### References

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Hsu, H.-C., Tsai, C.-C. (2016). Lagrangian approach to interfacial water waves with free surface. Applied Ocean Research, 59, 616–637. doi: 10.1016/j.apor.2016.08.001

Jo, T.-C., Choi, Y.-K. (2014). Dynamics of strongly nonlinear internal long waves in a three-layer fluid system. Ocean Science Journal, 49 (4), 357–366. doi: 10.1007/s12601-014-0033-6

Zhu, H., Wang, L., Avital, E. J., Tang, H., Williams, J. J. R. (2016). Numerical simulation of interaction between internal solitary waves and submerged ridges. Applied Ocean Research, 58, 118–134. doi: 10.1016/j.apor.2016.03.017

Smith, S., Crockett, J. (2014). Experiments on nonlinear harmonic wave generation from colliding internal wave beams. Experimental Thermal and Fluid Science, 54, 93–101. doi: 10.1016/j.expthermflusci.2014.01.012

Massel, S. R. (2016). On the nonlinear internal waves propagating in an inhomogeneous shallow sea. Oceanologia, 58 (2), 59–70. doi: 10.1016/j.oceano.2016.01.005

Avramenko, O. V., Naradovyi, V. V., Selezov, I. T. (2018). Energy of Motion of Internal and Surface Waves in a Two-Layer Hydrodynamic System. Journal of Mathematical Sciences, 229 (3), 241–252. doi: 10.1007/s10958-018-3674-7

Avramenko, O., Lunyova, M., Naradovyi, V. (2017). Wave propagation in a three-layer semi-infinite hydrodynamic system with a rigid lid. Eastern-European Journal of Enterprise Technologies, 5 (5 (89)), 58–66. doi: 10.15587/1729-4061.2017.111941

Nayfeh, A. H. (1976). Nonlinear Propagation of Wave-Packets on Fluid Interfaces. Journal of Applied Mechanics, 43 (4), 584. doi: 10.1115/1.3423936

Tarapov, I. E. (2005). Continuum Mechanics. Vol. 3. Mechanics of Inviscid Liquid. Kharkiv: Zolotye Stranitsy.

Avramenko, O. V., Hurtovyi, Yu. V., Naradovyi, V. V. (2014). Analiz enerhiyi khvylovoho rukhu v dvosharovykh hidrodynamichnykh systemakh. Naukovi zapysky. Seriya: Matematychni nauky, 73, 3–8.

#### GOST Style Citations

Bona J. L., Lannes D., Saut J.-C. Asymptotic models for internal waves // Journal de Mathématiques Pures et Appliquées. 2008. Vol. 89, Issue 6. P. 538–566. doi: 10.1016/j.matpur.2008.02.003

Wang Y., Tice I., Kim C. The Viscous Surface-Internal Wave Problem: Global Well-Posedness and Decay // Archive for Rational Mechanics and Analysis. 2013. Vol. 212, Issue 1. P. 1–92. doi: 10.1007/s00205-013-0700-2

Hsu H.-C., Tsai C.-C. Lagrangian approach to interfacial water waves with free surface // Applied Ocean Research. 2016. Vol. 59. P. 616–637. doi: 10.1016/j.apor.2016.08.001

Jo T.-C., Choi Y.-K. Dynamics of strongly nonlinear internal long waves in a three-layer fluid system // Ocean Science Journal. 2014. Vol. 49, Issue 4. P. 357–366. doi: 10.1007/s12601-014-0033-6

Numerical simulation of interaction between internal solitary waves and submerged ridges / Zhu H., Wang L., Avital E. J., Tang H., Williams J. J. R. // Applied Ocean Research. 2016. Vol. 58. P. 118–134. doi: 10.1016/j.apor.2016.03.017

Smith S., Crockett J. Experiments on nonlinear harmonic wave generation from colliding internal wave beams // Experimental Thermal and Fluid Science. 2014. Vol. 54. P. 93–101. doi: 10.1016/j.expthermflusci.2014.01.012

Massel S. R. On the nonlinear internal waves propagating in an inhomogeneous shallow sea // Oceanologia. 2016. Vol. 58, Issue 2. P. 59–70. doi: 10.1016/j.oceano.2016.01.005

Avramenko O. V., Naradovyi V. V., Selezov I. T. Energy of Motion of Internal and Surface Waves in a Two-Layer Hydrodynamic System // Journal of Mathematical Sciences. 2018. Vol. 229, Issue 3. P. 241–252. doi: 10.1007/s10958-018-3674-7

Avramenko O., Lunyova M., Naradovyi V. Wave propagation in a three-layer semi-infinite hydrodynamic system with a rigid lid // Eastern-European Journal of Enterprise Technologies. 2017. Vol. 5, Issue 5 (89). P. 58–66. doi: 10.15587/1729-4061.2017.111941

Nayfeh A. H. Nonlinear Propagation of Wave-Packets on Fluid Interfaces // Journal of Applied Mechanics. 1976. Vol. 43, Issue 4. P. 584. doi: 10.1115/1.3423936

Tarapov I. E. Continuum Mechanics. Vol. 3. Mechanics of Inviscid Liquid. Kharkiv: Zolotye Stranitsy, 2005.

Avramenko O. V., Hurtovyi Yu. V., Naradovyi V. V. Analiz enerhiyi khvylovoho rukhu v dvosharovykh hidrodynamichnykh systemakh // Naukovi zapysky. Seriya: Matematychni nauky. 2014. Issue 73. P. 3–8.

Copyright (c) 2018 Olga Avramenko, Maria Lunyova