Isentropic Plane Waves in Magnetohydrodynamics

Citation

Lynn, Yen-Mow (1961) Isentropic Plane Waves in Magnetohydrodynamics. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/K49C-2N76. https://resolver.caltech.edu/CaltechETD:etd-12082005-134709

Abstract

NOTE: Text or symbols not renderable in plain ASCII are indicated by [...]. Abstract is included in .pdf document. Plane waves propagating in a perfectly electrically conducting polytropic gas of otherwise uniform state in the presence of an arbitrarily oriented uniform magnetic field are studied; they correspond to plane simple waves in magnetohydrodynamics. Riemann invariants across finite amplitude waves in ordinary gasdynamics are generalized herein to take into account all possible magnetohydrodynamics effects. There exist totally seven types of waves, namely, contact surfaces, forward and backward facing transverse simple waves and forward and backward facing coupled (fast and slow) simple waves. But of these only coupled waves are genuinely nonlinear and receive most of our attention. The mathematical theory of simple waves is discussed first to give a general picture of the underlying structure of solutions. Contact surfaces and transverse simple wave solutions are given next with particular emphasis on the case of the contact surface adjacent to a vacuum, region. An exact analytical solution of coupled waves for gases of arbitrary value of [...] is obtained in terms of generalized Riemann invariants; some of these invariants are expressed in terms of definite integrals of a parameter [...]. The invariant relations among several physical quantities are thus expressed in a parametric form. An alternative method of solving coupled waves by graphical means is proposed and some detailed calculations are presented. General properties of physical variables across coupled waves are mentioned. For the special case of gas in a purely transverse magnetic field, a scheme of solving arbitrary flow problems is discussed briefly. The corresponding case of coupled wave solutions is given in terms of a hypergeometric function. Finally, some examples are shown to illustrate the application of the solutions to actual physical problems.

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