Let us investigate shocks in MHD fluids. Since information in such fluids is
carried via three different waves--namely, *fast*
or compressional-Alfvén waves, *intermediate* or shear-Alfvén waves, and *slow* or magnetosonic waves (see Sect. 5.4)--we might expect MHD fluids to support
three different types of shock, corresponding to disturbances traveling
faster than each of the aforementioned waves. This is indeed the case.

In general, a shock propagating through an MHD fluid produces a significant difference in plasma properties on either side of the shock front.
The thickness of the front is determined by a balance between convective
and dissipative effects. However, dissipative effects in high temperature
plasmas are only comparable to convective effects when the spatial gradients
in plasma variables become extremely large. Hence, MHD shocks in such plasmas tend to be *extremely narrow*, and are well-approximated
as *discontinuous* changes in plasma parameters. The MHD equations,
and Maxwell's equations, can be integrated across a shock
to give a set of *jump conditions* which relate plasma properties on
each side of the shock front. If the shock is sufficiently narrow then these relations become *independent* of its detailed structure. Let us derive the
jump conditions for a narrow, planar, steady-state, MHD shock.

Maxwell's equations, and the MHD equations, (681)-(684),
can be written in the following convenient form:

where

(924) |

(925) |

(926) |

Let us move into the *rest frame* of the shock. Suppose that the
shock front coincides with the - plane.
Furthermore, let the regions
of the plasma upstream and downstream of the shock, which are termed
regions 1 and 2, respectively, be *spatially uniform* and *non-time-varying*. It follows
that
. Moreover,
, except in the immediate vicinity of the shock.
Finally, let the velocity
and magnetic fields upstream and downstream of the shock
all lie in the - plane. The situation under discussion is illustrated in Fig. 28. Here, , , , and are
the downstream mass density, pressure, velocity, and magnetic field,
respectively, whereas , , , and
are the corresponding upstream quantities.

In the immediate vicinity of the shock, Eqs. (919)-(923) reduce to

(927) | |||

(928) | |||

(929) | |||

(930) | |||

(931) | |||

(932) |

Integration across the shock yields the desired jump conditions:

where . These relations are often called the