Abstract
This chapter is devoted to analysis of nonlinear elastic plane transverse harmonic waves. It starts with basic nonlinear wave equations for these waves, and subsequently, the statement is divided into two parts. In the first part the quadratically nonlinear waves are discussed. The second part includes an analysis of the following nonlinear approach—simultaneous allowance for quadratic and cubic nonlinearities. Part 1 is devoted to solving the second and third standard problems for which the method of successive approximations is used. The first two approximations are considered, and the corresponding wave effects are described and discussed. In Part 2, cubically nonlinear waves are considered. Here, the method of slowly varying amplitudes as applied to the study of the plane elastic harmonic transverse waves is applied, and the problem of self-switching of two transverse elastic harmonic plane waves is presented in sequence.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Cattani, C., Rushchitsky, J.J.: Plane waves in cubically nonlinear elastic media. Int. Appl. Mech. 38(11), 1361–1365 (2002)
Cattani, C., Rushchitsky, J.J.: Generation of the third harmonic by the plane waves in materials with Murnaghan potential. Int. Appl. Mech. 38(12), 1482–1487 (2002)
Cattani, C., Rushchitsky, J.J.: Cubically nonlinear elastic waves versus quadratically ones. Int. Appl. Mech. 39(12), 1361–1399 (2003)
Cattani, C., Rushchitsky, J.J.: Wavelet and Wave Analysis as Applied to Materials with Micro or Nanostructure. World Scientific, Singapore (2007)
Cattani, C., Rushchitsky, J.J., Sinchilo, S.V.: Cubic nonlinearity in elastic materials: theoretical prediction and computer modelling of new wave effects. Math. Comput. Model. Dyn. Syst. 10(3–4), 331–352 (2004)
Rushchitsky, J.J.: Self-switching of waves in materials. Int. Appl. Mech. 37(11), 1492–1498 (2001)
Rushchitsky, J.J.:Self-switching of displacement waves in elastic nonlinearly deformed materials. Comptes Rendus de l'Academie des Sciences, Serie IIb Mecanique, Tome 330(2), 175–180 (2002)
Rushchitsky, J.J.: Fragments of the theory of transistors: switching the plane transverse hypersound wave in nonlinearly elastic nanocomposite materials. J. Math. Sci. (Matematychni metody ta fizyko-mekhanichni polia—Mathematical Methods and Physical-Mechanical Fields) 51(3), 186–192 (2008)
Rushchitsky, J.J.: On phenomenom of self-switching of hypersound waves in quadratically nonlinear elastic nanocomposite materials. Int. Appl. Mech. 45(1), 73–93 (2009)
Rushchitsky, J.J.: Certain class of nonlinear hyperelastic waves: classical and novel models wave equations, wave effects. Int. J. Appl. Math. Mech. 8(6), 400–443 (2012)
Rushchitsky, J.J., Cattani, C., Sinchilo, S.V.: Comparative analysis of evolution of the elastic harmonic wave profile due to generation of the second and third harmonics. Int. Appl. Mech. 40(2), 183–189 (2004)
Rushchitsky, J.J., Savelieva, E.V.: On the interaction of transverse cubically nonlinear waves in elastic materials. Int. Appl. Mech. 42(6), 661–668 (2006)
Rushchitsky, J.J., Savelieva, E.V.: Self-switching of transverse plane wave when is being passed in the elastic composite material. Int. Appl. Mech. 43(7), 763–781 (2007)
Rushchitsky, J.J., Tsurpal, S.I.: Khvyli v materialakh z mikrostrukturoiu (Waves in Materials with the Microstructure). SP Timoshenko Institute of Mechanics, Kiev (1998)
Author information
Authors and Affiliations
Exercises
Exercises
-
1.
See in textbooks the definition of modulated wave. In which sense is wave (7.13) the modulated wave?
-
2.
Sketch separately the plots of the three longitudinal waves from (7.19): first harmonic, second harmonic, and composite wave. Collect together all three waves in one plot and comment on the evolution of wave (7.19).
-
3.
Sketch separately the plots of the two transverse waves from (7.20): first harmonic and composite wave. Collect together the two waves in one plot and comment on the evolution of wave (7.20).
-
4.
Compare the shorten equation (7.27) for the cubically nonlinear approach and the similar equation (5.65) for the quadratically nonlinear approach. How are they different and similar?
-
5.
Compare the evolution equation (7.30) for the cubically nonlinear approach and the similar equation (5.66) for the quadratically nonlinear approach. How are they different and similar?
-
6.
Compare (7.31) (cubically nonlinear equations) and (5.70) (quadratically nonlinear equations) Analyze the procedure for obtaining from these different equations identical Manley–Rowe relations (but with a different S ….)
Rights and permissions
Copyright information
© 2014 Springer International Publishing Switzerland
About this chapter
Cite this chapter
Rushchitsky, J.J. (2014). Nonlinear Plane Transverse Waves in Elastic Materials (Murnaghan Model, Five-Constant Model). In: Nonlinear Elastic Waves in Materials. Foundations of Engineering Mechanics. Springer, Cham. https://doi.org/10.1007/978-3-319-00464-8_7
Download citation
DOI: https://doi.org/10.1007/978-3-319-00464-8_7
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-00463-1
Online ISBN: 978-3-319-00464-8
eBook Packages: EngineeringEngineering (R0)