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from Part I - 1-D MHD in Ten Weeks
Published online by Cambridge University Press: 05 June 2025
By linearising the equations of HD developed in Chapter 1, the wave equation for the propagation of sound is derived. This is examined from two approaches: direct solution of the wave equation and examining normal modes to convert the problem to one of linear algebra. This introduces the very important concepts of eigenvalues (characteristic speeds) and eigenkets (right eigenvectors) along with the role they play in examining fluid dynamics in terms of waves. From the 1-D, non-linearised, steady-state equations, the Rankine–Hugoniot jump conditions are derived from which the conditions for tangential/contact discontinuities and shocks are developed. An optional section considers the phenomenon of bores and hydraulic jumps, while the last section introduces concepts such as streamlines and stream tubes culminating with Bernoulli’s theorem applied to an incompressible fluid, a subsonic compressible fluid, and a supersonic compressible fluid.
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