Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X



Dissipation of energy for magnetoelastic waves in a conductive medium

Authors: Elias Andreou and George Dassios
Journal: Quart. Appl. Math. 55 (1997), 23-39
MSC: Primary 73R05; Secondary 73D15
DOI: https://doi.org/10.1090/qam/1433749
MathSciNet review: MR1433749
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Abstract: We consider the propagation of magnetoelastic waves within a homogeneous and isotropic elastic medium exhibiting finite electric conductivity. An appropriate physical analysis leads to a decoupling of the governing system of equations which in turn effects an irreducible factorization of the ninth-degree characteristic polynomial into a product of first, third, and fifth-degree polynomials. Regular and singular perturbation methods are then used to deduce asymptotic expansions of the characteristic roots which reflect the low and the high frequency dependence of the frequency on the wave number. Dyadic analysis of the spacial spectral equations brings the general solution into its canonical dyadic form. Extensive asymptotic analysis of the quadratic forms that define the kinetic, the strain, the magnetic and the dissipation energy provides the rate of dissipation of these energies as the time variable approaches infinity. The rate of dissipation obtained coincides with the corresponding rate for thermoelastic waves. Therefore, a similarity between the dissipative effects of thermal coupling and that of finite conductivity upon the propagation of elastic waves is established.

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  • [1] A. Banos, Jr., Normal modes characterizing magnetoelastic plane waves, Phys. Rev. 104, 300-305 (1956) MR 0086592
  • [2] J. Bazer, Geometrical magnetoelasticity, Geophys. J. R. Astr. Soc. 25, 207-237 (1971)
  • [3] C. M. Bender and S. A. Orszag, Advanced Mathematical Methods for Scientists and Engineers, McGraw-Hill, Singapore, 1984 MR 538168
  • [4] P. Boulanger, Inhomogeneous magnetoelastic plane waves, Elastic Wave Propagation, Proc. Second I.U.T.A.M.-I.U.P.A.P. Symp. (M. F. McCarthy and M. A. Hayes, eds.), Galway, 1989, pp. 601-606
  • [5] L. Brand, Vector and Tensor Analysis, John Wiley, New York, 1947 MR 0021449
  • [6] P. Chadwick, Elastic Wave Propagation in a Magnetic Field, 9th Internat. Congr. Appl. Mech., Brussels, Proc. III, 1957, pp. 143-153
  • [7] S. Chander, Phase velocity and energy loss in magneto-thermo-elastic plane waves, Internat. J. Engrg. Sci. 6, 409-424 (1968)
  • [8] G. Dassios, Equipartition of energy in elastic wave propagation, Mech. Res. Comm. 6, 45-50 (1979) MR 524233
  • [9] G. Dassios, Energy theorems for magnetoelastic waves in a perfectly conducting medium, Quart. Appl. Math. 39, 479-490 (1982) MR 644102
  • [10] G. Dassios and M. Grillakis, Dissipation rates and partition of energy in thermoelasticity, Arch. Rational Mech. Anal. 87, 49-91 (1984) MR 760319
  • [11] J. W. Dunkin and A. C. Eringen, On the propagation of waves in an electromagnetic elastic solid, Internat. J. Engrg. Sci. 1, 461-495 (1963) MR 0160379
  • [12] A. C. Eringen and G. A. Maugin, Electrodynamics of Continua I, II, Springer-Verlag, New York, 1989 MR 1031714
  • [13] M. Hermite, Sur la Résolution de l'Équation du Cinquieme Degré, Comptes Rendus 46, 508-515 (1858)
  • [14] A. B. Jerrard, On Certain Transformations Connected with the Finite Solution of Equations of the Fifth Degree, Philos. Mag. 7, 202-203 (1835)
  • [15] F. John, Partial Differential Equations, Springer-Verlag, New York, 1982
  • [16] S. Kaliski, Vibrations and Waves. Part A: Vibrations. Part B: Waves, Polish Scientific Publishers, Warsaw, 1992 MR 1185326
  • [17] V. I. Keilis-Borok and A. S. Munin, Magnetoelastic Waves and the Boundary of the Earth's Core, Izv. Geophys. Ser., 1959, pp. 1529-1541; English translation, 1960, pp. 1089-1095
  • [18] L. Knopoff, The interaction between elastic wave motions and a magnetic field in electrical conductors, J. Geophys. Res. 60, 441-456 (1955)
  • [19] F. E. M. Lilley and D. E. Smylie, Elastic wave motion and a nonuniform magnetic field in electrical conductors, J. Geophys. Res. 73, 6527-6533 (1968)
  • [20] F. C. Moon, Magneto-Solid Mechanics, John Wiley, New York, 1984
  • [21] W. Nowacki, Dynamic Problems in Thermoelasticity, Noordhoff, Warsaw, 1975
  • [22] G. Paria, Magneto-elasticity and magneto-thermo-elasticity, Adv. Appl. Mech. 10, 78-112 (1967)
  • [23] A. J. Willson, The propagation of magneto-thermo-elastic plane waves, Proc. Cambridge Philos. Soc. 59, 483-488 (1963) MR 0148294

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DOI: https://doi.org/10.1090/qam/1433749
Article copyright: © Copyright 1997 American Mathematical Society

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