ABSTRACT
In the previous chapters on hydrodynamics and magnetohydrodynamics (MHD), we covered wave propagation in the presence of tangential discontinuities in the magnetic fields, presence of flows, etc. In the studies mentioned above, the corresponding dispersion relation was derived with the assumption that the total pressure and the vertical component of the velocity field are continuous across the tangential discontinuity. In the case of the fluid or plasma being compressible, the velocity of sound plays an important role. There are situations in several branches of physics, such as MHD, space plasmas, and gas dynamics in which the pressure develops a discontinuity, and in the absence of dissipative effects such as viscosity and heat conductivity, propagation and convection of compressional disturbances, with speeds increasing with compression, lead to a continual steepening of waveforms that eventually can no longer be expressed by single-valued functions of position. In such cases, we introduce a discontinuity in the flow to get over this difficulty. A shock wave
may be defined as a surface in a flow field across which the flow variables change discontinuously. It is important that shock waves exist in order that one may admit certain types of boundary conditions that could not be satisfied in a continuous flow pattern. In practical situations, a real fluid cannot sustain an actual discontinuity so the latter is only an idealization of the sharp gradients in the flow variables that occur in reality in a shock wave. A typical situation that one encounters in everyday life is the sonic boom that is heard at the wake of a supersonic aircraft with a Mach number larger than unity. The Mach number is a nondimensional number which relates to the ratio of the flow speed to the sound speed. For M > 1, the flow is supersonic, However, for M < 1 and M ≈ 1, the flows are termed as subsonic and transonic, respectively. In this chapter, we shall study different types of shocks, such as normal and oblique shocks. We shall briefly discuss weak shocks, waves in a polytropic gas and relaxing medium. We will also briefly mention the Sedov blast wave solution. The Burger’s equation and the KdV equation in shocks will also be dealt with, to include nonlinear effects. Application of shocks in MHD and space plasmas will also be included.
