Interaction of longitudinal nonlinear elastic waves
DOI:
https://doi.org/10.17721/1812-5409.2024/1.15Keywords:
harmonic wave, cubic nonlinearity, planar wave, quadruplet, four-wave interactionAbstract
The theoretical investigation of the interaction of elastic planar harmonic waves in a material whose nonlinear properties are described by the Murnaghan elastic potential is presented. A review of the methods for analytical study of the wave process is provided. The methodology for studying transverse and longitudinal waves is described. Using the perturbation method for transverse waves, results are presented for the simultaneous propagation of two types of waves: vertically and horizontally polarized. The corresponding equations are written, and the distortion of the respective wave profiles is analyzed. It has been established that because of the nonlinear wave interaction, the transverse waves gradually transform into their third harmonics. With different initial intensities of waves of different polarization, energy is transferred from the powerful wave to the weaker wave. Numerical studies were conducted using values of effective constants for a range of nanocomposite materials. For longitudinal waves, the simultaneous propagation of waves with separate consideration of quadratic and cubic nonlinearity was investigated. Various cases of harmonic wave interaction were studied based on cubic equations of motion. The method of slowly varying amplitudes was sequentially used. The obtained equations, the first integrals of these equations, and the conservation law for four interacting waves are analyzed. Truncated and full evolutionary equations were obtained, and the Manley-Rowe relations were recorded. This research method assumes weak variability of the amplitudes and phases of waves over one period of the oscillatory process. The field of application of such wave research includes several problems in nonlinear optics and plasma physics. Considering cubic nonlinearity is also necessary for the study of internal and surface waves in a fluid. Similarly to how wave triplets can form in quadratically nonlinear media through three-wave interaction, four-wave interaction occurs in cubically nonlinear media, with the formation of wave quadruplets under certain conditions.
Pages of the article in the issue: 78 - 81
Language of the article: Ukrainian
References
Rushchitsky, J.J. & Tsurpal, S.I. (1998) Waves in Materials with Microstructure. Kyiv, S.P. Timoshenko Institute of Mechanics.
Achenbach, J.D. (1973) Wave Propagation in Elastic Solids. Amsterdam: North Holland Publishing Company.
Cattani, C. & Rushchitsky, J.J. (2007) Wavelet and Wave Analysis as Applied to Materials with Micro or Nanostructure. Singapore-London: World Scientific.
Rushchitsky, J.J. (2009) On the Self–Switching Hypersonic Waves in Cubic Nonlinear Hyperelastic Nanocomposites Int. Appl. Mech. 45 (1). p. 73–93.
Guz, A.N. & Rushchitsky, J.J. & Guz, I.A. (2010) Introduction to mechanics of nanocomposites. Kiev: Akademperiodika.
Shen Y.R. (1984) The principles of nonlinear optics. New York: John Wiley.
Rushchitsky, J.J. (2014) Nonlinear Elastic Waves in Materials. Heidelberg: Springer.
Rushchitsky, J.J. (2005) Quadratically nonlinear cylindrical hyperelastic waves – derivation of wave equations. Plane strain state. Int. Appl. Mech. 41 (5). p. 701–712.
Rushchitsky, J.J. (2005) Quadratically nonlinear cylindrical hyperelastic waves – derivation of wave equations. Axisymmetric and other states. Int. Appl. Mech. 41 (6). p. 831–840.
Rushchitsky, J.J. (2005) Quadratically nonlinear cylindrical hyperelastic waves – primary analysis of evolution Int. Appl. Mech. 41 (7). p. 825–833.
Rushchitsky, J.J. (2009) On the Self–Switching Hypersonic Waves in Cubic Nonlinear Hyperelastic Nanocomposites Int. Appl. Mech. 45 (1). p. 73–93.
Downloads
Published
How to Cite
Issue
Section
License
Copyright (c) 2024 Katerina Savelieva, Olha Dashko
This work is licensed under a Creative Commons Attribution 4.0 International License.
Authors who publish with this journal agree to the following terms:
- Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a Creative Commons Attribution License that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.
- Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in this journal.
- Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work (See The Effect of Open Access).
