Far Eastern Mathematical Journal

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Longitudinal finite-amplitude wave in nonlinear homogeneous elastic medium. The equations of Landau-Murnaghan


Guzev M.A., Molotkov I.V.

2016, issue 2, Ñ. 160-168


Abstract
High-frequency asymptotic solution of the equations of motion for waves in nonlinear and homogeneous elastic mediumis is obtained, with predominantly longitudinal polarization. The main part of the solution is known from the consideration of the linear problem. The general solution except the main part contains two completely new part describing the excitation of the transverse wave and wave with the double frequency. These effects result in distortion of wave fronts, as well as to the weak attenuation of the primary longitudinal wave along the way. The inclusion of these nonlinear effects are important in the analysis of seismic waves.

Keywords:
longitudinal wave, high frequency asymptotics, transverse wave, a wave with the double frequency

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References

[1] J.W. Strutt (Rayleigh), “Aerial plane waves of finite amplitude”, Proc. Roy. Soc., A-84 (1910), 247–284.
[2] R.D. Fay, “Plane sound waves of finite amplitude”, J. Acoust. Soc. Amer, 3:2 part 1 (1996), 222–241.
[3] L. Landau, G. Rumer, “Ueberschall absorption in festern koerpern”, Zs. Sov. Phys., 3 (1937), 18–27.
[4]. E.D. Murnaghan, Finite deformation of an elastic solid, John Wiley, NY, 1951.
[5] L.D. Landau, E.M. Lifshits, Teoriia uprugosti, Nauka, M., 1987.
[6] P. Debye, Vortraege ueber die kinetische theorie der materie und der electrizitaet, Teubner, Berlin, 1914.
[7] Dzh. Uizem, Lineinye i nelineinye volny., Nauka, M., 1977.
[8] I.A. Molotkov, Analiticheskie metody v teorii nelineinykh voln, Fizmatlit, M., 2003.
[9] I.Ia. Pomeranchuk, “O teploprovodnosti dielektrikov pri temperaturakh bol'she de-baevskoi”, ZhETF, 11 (1941), 246.
[10] S.K. Godunov, I.M. Peshkov, “Simmetricheskie giperbolicheskie uravneniia nelineinoi teorii uprugosti”, Zh. vychisl. matem. i matem. fiziki, 48:6 (2008), 1034–1055.
[11] A.A. Sheina, A.I. Aleksandrovich, “Reshenie prostranstvennykh zadach nelineinoi teorii uprugosti metodami mnogomernogo kompleksnogo analiza”, Vestnik Nizhegor. univ., 4 (2011).
[12] I.G. Kuleev, I.I. Kuleev, “Relaksatsiia vazipoperechnykh fononov v mekhanizme Kherringa i pogloshchenie ul'trazvuka v kubicheskikh kristallakh s polozhitel'noi i otritsatel'noi anizotropiei uprugikh modulei vtorogo poriadka”, Fizika tverdogo tela, 51:11 (2009), 2211–2223.
[13] C. Payan, V. Garnier, J. Mogsan, P.A. Johnson, “Determination on third order elastic constants in a complex solid applying coda wave interferometry”, Appl. Phys. Lett., 98 (2009), 011904.
[14] V.M. Babich, A.S. Alekseev, “Luchevoi metod rascheta intensivnosti volnovykh frontov”, Izv. AN SSSR, ser. geof., 1 (1958), 17–31.
[15] V. Cerveny, I. A. Molotkov, I. Psencik, Ray method in seismology, Prague, 1977.

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