At a shock front propagating in a magnetized fluid at an angle 0 with respect to the magnetic induction B, the jump conditions are 13,14
(4) B,| = %
Here U and V are components of the fluid velocity normal and tangential to the front in the shock frame; p = 1/v is the mass density; p is the pressure; Bl = B sin By = B cos 9; p is the magnetic permeability (p = 4tt in cgs units); and the specific enthalpy is w = e + pv, where the specific internal energy e satisfies de = Tds — pdv in terms of the temperature T and the specific entropy s. Quantities in the region behind (downstream from) the front are distinguished by a bar. If B = 0, then15
(9) w — w = ^(p — p)(v + v);
In what follows we assume that the fluid is a perfect gas with adiabatic index 7 = 1 + 2/n, where n is the number of degrees of freedom. Then p = pRT/m, where R is the universal gas constant and m is the molar weight; the sound speed is given by Cs2 = (dp/dp)s = 7pv\ and w — ^e — 'ypv/(7 — 1). For a general oblique shock in a perfect gas the quantity X = r-1 (U/Va)2 satisfies14
(11) (X-(3/a)(X-cos2 6)2 = X sin2 0 j[l + (r - l)/2a] X - cos2 6>|, where r = p/p, a = \ [7 + 1 - (7 - l)r], and (3 = CS2/VA2 = 4tv^P/B2.
The density ratio is bounded by
(12) 1 < r < (7 + l)/(7- I)-
If the shock is normal to B (i.e., if 0 = tt/2), then
(13) U2 = (r/a) [Cs2 + VA2 [1 + (1 - 7/2)(r - 1)]};
If 6 = 0, there are two possibilities: switch-on shocks, which require (3 < 1 and for which
(19) B2 = 2B2(r - l)(a - /3);
and acoustic (hydrodynamic) shocks, for which
(24) V = B± = 0;
(25) p = p + pU2(l - r-1).
For acoustic shocks the specific volume and pressure are related by
(26) v/v = [(7 + 1)P + (7 - l)p] / [(7 - 1)P + (7 + l)p]- In terms of the upstream Mach number M = U/Cs,
(28) p/p = (27M2 -7+ l)/(7 + 1);
(29) T/T = [(7 - 1)M2 + 2](27M2 -7 + l)/(7 + 1)2M2;
(30) M2 = [(7 - 1)M2 + 2]/[27M2 -7 + I]. The entropy change across the shock is
(31) As = s - s = cv ln[(p/p)(p/p)7],
where cv = — l)m is the specific heat at constant volume; here R is the
gas constant. In the weak-shock limit (M —► 1),
zQOx A 27(7-1) 2 3 167^ 3
(32) As ^ c, (M -1) ^ (M - 1) .
3(7 + 1) 3(7 + 1 )m
The radius at time t of a strong spherical blast wave resulting from the explosive release of energy E in a medium with uniform density p is
where Co is a constant depending on 7. For 7 = 7/5, Co = 1.033.