Plane elastic wave interaction. Considering of quadratically and cubically nonlinearity

Authors

  • K. V. Savelieva S. P. Timoshenko Institute of Mechanics NAS of Ukraine, 03057, Kyiv, P. Nesterov str., 3
  • O. G. Dashko S. P. Timoshenko Institute of Mechanics NAS of Ukraine, 03057, Kyiv, P. Nesterov str., 3 https://orcid.org/0000-0002-4775-418X

DOI:

https://doi.org/10.17721/1812-5409.2022/1.6

Keywords:

harmonic wave, cubic nonlinearity, elastic potential, plane wave, quadruplet, four-wave interaction

Abstract

The interaction of elastic plane harmonic waves in the material, the nonlinear properties of which are described by the elastic potential of Murnaghan, is investigated theoretically. The displacement vector is depended of only one spatial variable and time, a record of the complete system of equations for plane waves moves along the abscissa axis is recorded and used. The interaction of longitudinal waves with a separate considering cubic nonlinearity is investigated. On the basis of the cubic equation of motion, the interaction of four harmonic waves is studied. The method of slowly variable amplitudes is used. Firstly the two-wave interaction is investigated, then the interaction of four waves is described. Shorten and evolutionary equations are obtained, the first integrals of these equations and the record of the law of conservation for a set of four interacting waves are obtained. An analogy is made between the triplets studied when taking into account the interaction of three waves and the triplets investigated in the case under consideration, taking into account the four-wave interaction, quadruplets.

Pages of the article in the issue: 50 - 53

Language of the article: Ukrainian

References

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Published

2022-04-26

How to Cite

Savelieva, K. V., & Dashko, O. G. (2022). Plane elastic wave interaction. Considering of quadratically and cubically nonlinearity. Bulletin of Taras Shevchenko National University of Kyiv. Physical and Mathematical Sciences, (1), 50–53. https://doi.org/10.17721/1812-5409.2022/1.6

Issue

Section

Differential equations, mathematical physics and mechanics