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43 | APPENDIX I 
4. SHOCK FRONTS 
In such a shock front the medium undergoes a sudden compression, and its temperature 
rises by an amount greater than the rise of temperature due to an adiabatic compres- 
sion of the same magnitude. 
A further condition for the existence of a shock front may be derived from 
Equation [12], in which the positive square root is meant and hence it is necessary 
that ua. 
Thus we have for the velocity U with which the shock front travels in the 
direction toward medium 1, the change in internal energy of the medium produced by its 
passage, and the necessary conditions for its existence: 
= V2 eels ee Py) Ps = Py {11] 
De ab pippeteie “2 Vibe Pe Pi 
Ug — = Vals (Pp — Py) (P2 — Pi) [12] 
at Ae aera 
E, — BE, = > (P, + Pr) F ra) {13] 
U2 >U1, P2> Pir P2 > Pi [14] 
Equation [13] is known as the Hugoniot relation. 
From Equation [11] it can be shown that the shock front advances through 
medium 1 faster than does an ordinary sound wave in that medium, whereas its speed 
relative to medium 2 is less than the speed of sound in that medium, i.e., if ci, ca 
are the respective speeds of waves of small amplitude in the two mediums, 
It follows that no effects from the shock front can be propagated into medium 1, and 
the values of p1yfi, U1, Will therefore be determined by conditions elsewhere in that 
medium. Effects of conditions elsewhere in medium 2, on the other hand, propagated 
with the speed of sound, can overtake the shock front. We can regard these effects as. 
furnishing one condition for fixing the values of pz and wz Just behind the front. 
Equations [12] and [13] furnish two other conditions for the determination of the 
four quantities p,, uj, p>. and E,; and a fourth relation is furnished by the function- 
al relation between E, p and p that is characteristic of the medium. 
Since the existence of a shock front involves a continual dissipation of 
energy, it may be expected that shock fronts will usually weaken as they advance and 
ultimately disappear. As the ratio p2/p, or pa/p, approaches unity, a shock front 
approximates to an ordinary sound wave, and its velocity of propagation U reduces to 
the speed of sound. 
In a physical medium, actual discontinuities are doubtless impossible. If 
the theory is amplified so as to allow for the influence of viscosity and of heat con- 
duction, which are ignored in the ordinary theory of compressive waves, it is found 
