In other words, it is convenient to transform to a frame which moves at the local velocity of the plasma. It immediately follows from the jump condition (934) that

or . Thus, in the de Hoffmann-Teller frame, the upstream plasma flow is

Equations (965) and (966) can be combined with the
general jump conditions (933)-(938)
to give

where is the component of the upstream velocity normal to the shock front, and is the angle subtended between the upstream plasma flow and the shock front normal. Finally, given the compression ratio, , the square of the normal upstream velocity, , is a real root of a cubic equation known as the

(973) | |||

As before, the second law of thermodynamics mandates that .

Let us first consider the weak shock limit
. In this case, it is easily seen that the three roots of the
shock adiabatic reduce to

(974) | |||

(975) | |||

(976) |

However, from Sect. 5.4, we recognize these velocities as belonging to slow, intermediate (or Shear-Alfvén), and fast waves, respectively, propagating in the normal direction to the shock front. We conclude that slow, intermediate, and fast MHD shocks degenerate into the associated MHD waves in the limit of small shock amplitude. Conversely, we can think of the various MHD shocks as

(977) |

When is slightly larger than unity it is easily demonstrated that the conditions for the existence of a slow, intermediate, and fast shock are , , and , respectively.

Let us now consider the strong shock limit,
. In this case, the shock
adiabatic yields
, and

(978) |

Consider the special case in which both the plasma flow and the
magnetic field are normal to the shock front. In this case, the three roots of the shock adiabatic are

(979) | |||

(980) | |||

(981) |

We recognize the first of these roots as the hydrodynamic shock discussed in Sect. 5.19--

Let us, finally, consider the special case . As is easily demonstrated, the three roots of the
shock adiabatic are

(982) | |||

(983) | |||

(984) |

The first of these roots is clearly a fast shock, and is identical to the perpendicular shock discussed in Sect. 5.20, except that there is no plasma flow across the shock front in this case. The fact that the two other roots are zero indicates that, like the corresponding MHD waves, slow and intermediate MHD shocks do not propagate perpendicular to the magnetic field.

MHD shocks have been observed in a large variety of situations. For instance, shocks are
known to be formed by supernova explosions, by strong stellar winds, by solar flares, and
by the solar wind upstream of planetary magnetospheres.^{}