318 
6 
continued beyond some finite ft = U, , and the singularity at this t. 
ox Ohe 
sets in by ; x becoming infinite. At the next instant y 
Oat FalGenes 
can only be kept one-valued by permitting first-order discontinuities in 
exe Ox 
Da ee 
It has been established experimentally that this strange behavior 
of the solutions corresponds to a certain extent to the facts: In typical 
situations like those referred to above, real substances do indeed develop 
"discontinuities",- i.e. rapid changes of the specific volume v = Ox and 
Da. 
of the velocity V = 2x , Which are discontinuities in the same approxi- 
mation in which viscosity and heat conduction can be neglected, and are 
called shocks. These shocks appear even if the state of the substance at 
the initial instant t = os was perfectly continuous, they develop as far 
as can be observed at the times t+ = te at which the solution of (7), (8) 
becomes discontinuous, and as far as the motion stays continuous (7), (8) 
seem to be satisfied. 
§4. After shocks have formed, the motion is still governed by 
the same conservation principles .(of momentum and energy) on which (7), (8) 
are based. Hence (7), (8) still hold in the regions of space which contain 
no shocks, but beyond this it is necessary to apply those same conservation 
principles to the shocks themselves. 
Let a shock at a = &(t) be formed by the states 22 SoS 
a. 
ax = WW) = V : 2 Sy iC at @ = G— Oand ox = Wie V2 
ul 
é 
+ 
e) 
ali at The sition of 
zy Se heSe, U a a 
2 
ee 
the shock is x = x(2(+), t) jits velocity ip) AX, Ae ital 
aeon tes 
the flow of mass across the shock. Then the conservation theorems of mass 
