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Wave motion in a conducting fluid with a layer adjacent to the boundary, II. Eigenfunction expansions

Published online by Cambridge University Press:  17 February 2009

William V. Smith
William V. Smith
Affiliation:
Mathematics Department, Brigham Young University, Provo, Utah 84602, USA.
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Abstract

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The wave motion of magnetohydrodynamic (MHD) systems can be quite complicated. In order to study the motion of waves in a perfectly conducting fluid under the influence of an external magnetic field in a stratified medium, we make the simplifying assumption that the pressure is constant (to first order). This is the simplest form of the equations with variable coefficients and is not strongly propagative. Alfven waves are still present. The system is further simplified by assuming that the external field is parallel to the boundary. The Green's function for the operator is constructed and then the spectral family is constructed in terms of generalized eigenfunctions, giving four families of propagating waves, including waves “trapped” in the boundary layer. These trapped waves are interesting, since they are not the relics of surface waves, which do not exist in this context when the boundary layer shrinks to zero thickness no matter what (maximal energy preserving) boundary condition is chosen. We conjecture a similar structure for the full MHD problem.

Type
Research Article
Information
The ANZIAM Journal , Volume 43 , Issue 2 , October 2001 , pp. 195 - 236
Copyright
Copyright © Australian Mathematical Society 2001

References

[1]Dunford, N. and Schwartz, J. T., Linear Operators. Part II (Interscience, New York, 1963).Google Scholar
[2]Gilliam, D. S. and Schulenberger, J. R., “Electromagnetic waves in a three-dimensional half-space with a dissipative boundary”, J. Math. Anal. Appl. 89 (1982) 129–185.CrossRefGoogle Scholar
[3]Ikebe, T., “Eigenfunction expansions associated with the Schrödinger operator and their application to scattering theory”, Arch. Rational Mech. Anal. 5 (1960) 1–34.CrossRefGoogle Scholar
[4]Lax, P. and Phillips, R. S., “Local boundary conditions for dissipative symmetric linear differential operators”, Comm. Pure Appl. Math. 13 (1960) 427–455.CrossRefGoogle Scholar
[5]Lifschitz, A. E., Magnetohydrodynamics and Spectral Theory (Kluwer, Dordrecht, 1989).CrossRefGoogle Scholar
[6]Smith, W. V., “Waves in a perfectly conducting fluid filling a half-space”, IMA J. Appl. Math. 43 (1989) 47–69.CrossRefGoogle Scholar
[7]Smith, W. V., “The analysis of a model for wave motion in a liquid semiconductor: boundary interaction and variable conductivity”, SIAM J. Math. Anal. 22 (1991) 352–378.Google Scholar
[8]Smith, W. V., “Wave motion in a conducting fluid with a boundary layer, I. Hilbert space formulation”, J. Math. Anal. Appl. 188 (1994) 680–699.CrossRefGoogle Scholar
[9]Wilcox, C. H., “Transient wave propagation in homogeneous anisotropic media”, Arch. Rational Mech. Anal. 37 (1970) 323–343.CrossRefGoogle Scholar
[10]Wilcox, C. H., “Asymptotic wave functions and energy distributions in strongly propagative anisotropic media”, J. Math. Pures et Appl. 57 (1978) 275–321.Google Scholar