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APPENDIX I | 42 
4. SHOCK FRONTS 
IV. SHOCK FRONTS 
Let P be a plane dividing the medium into two parts, and let the total pres- 
sure in the medium be p, on one side of this plane, and pez on the other side. Let the 
corresponding densities of the medium be »p, and pg. Let the medium on the first side 
be moving with velocity u,, and that on the second side with velocity we, the motion 
being perpendicular to the plane and the positive direction for u being taken from 
medium 2 toward medium 1 (Figure 17). Thus at Pa discontinuity may exist not only in 
the pressure and the density, but also in 
the particle velocity. It was shown by Rie- 
mann that the laws of the conservation of 
2 | matter and of momentum could be satisfied by 
assuming that the discontinuity at P propa- 
gates itself from medium 2 into medium 1 at 
Poy Up ae, ue at a velocity U given by Equation [11], pro- 
<3 vided uw, and wz have values such that Equa- 
Po Pp tion [12] is satisfied. Such a self-propaga- 
ting discontinuity is called a shock front. 
As the shock front advances, suc- 
cessive portions of the medium undergo a 
P discontinuous change from density p, and 
pressure p, to p2 and pe, at the same time 
being accelerated from velocity u, towo. 
Figure 17 It was pointed out by Hugoniot that a certain 
change in the energy of the medium would also 
be required by the law of the conservation of energy. He showed that if E,, E2 denote 
the internal energy per unit mass of the medium in the two regions, then the difference, 
E2— E,, must have the value given by Equation [13]. In ordinary sound waves FE varies 
with p according to the law that holds for adiabatic changes of density, the change of 
E representing the work done by the pressure in compressing or rarefying the medium. 
To satisfy Equation [13], E must vary with p more rapidly than according to the edia- 
batic law. 
Now in the phenomena of viscosity and of the conduction of heat we are fa- 
miliar with irreversible processes by which the internal energy of a medium can be in- 
creased, with an accompanying increase in its entropy. No process can be imagined by 
which the energy might be decreased; probably such a process would violate the second 
law of thermodynamics. Hence it is assumed that a continuous irreversible conversion 
of mechanical energy into heat occurs in the shock front, of sufficient magnitude to 
make Equation [13] hold. The energy thus converted is brought up to the shock frent 
as it progresses by the ordinary processes of mechanical transmission of energy through 
the mediun. 
It can be shown that positive amounts of energy will be delivered to the shock 
front only if p2>p,, and hence p2>p,. Thus only shock fronts of compression can occur. 
