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.