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Consider the situation illustrated in Figure 14.2 in which, at
, a tightfitting piston is suddenly pushed into a stationary gas, contained in a uniform tube, at the
steady speed
, generating a shock front that propagates away from the piston, and into the gas, at the speed
. Suppose that the
piston is located at
at
, and moves in the
direction.
The
gas can be divided into two regions. In region
, which lies to the right of the shock wave (i.e.,
), the gas remains
undisturbed. Hence, its velocity is
. Let its pressure, density, and temperature be
,
, and
, respectively.
In region
, which lies between the piston and the shock wave (i.e.,
), the gas is comoving with the piston, so its velocity is
.
Let its pressure, density, and temperature, be
,
, and
, respectively. Thus, the system is formally
the same as that discussed in the final paragraph of the previous section, provided that we make the identifications
and
.
Figure 14.2:
A pistongenerated shock wave.

Let
be our conventional measure of shock strength. It follows from Equations (14.111) and (14.112)
that
where
is the gas's ratio of specific heats.
We can define the Mach number of the shock wave as
, where
is the sound speed in the unperturbed gas. According to
Equation (14.110),

(14.129) 
Finally, we can define the Mach number of the piston as
. Thus,

(14.130) 
where use has been made of Equation (14.111). Equations (14.127) and (14.129) can be combined with the previous equation to give

(14.131) 
Thus, Equations (14.127)(14.129), and (14.131), fully determine the parameters of the shocked gas, as well as the
speed of the shock front, in terms of the parameters of the unperturbed gas and the piston speed. In particular,
for a weak shock (i.e.,
),
whereas for a strong shock (i.e.,
),
Thus, a weak shock is associated with a subsonic piston (i.e.,
), whereas a strong shock is associated with a
supersonic piston (i.e.,
).
Next: PistonGenerated Expansion Wave
Up: OneDimensional Compressible Inviscid Flow
Previous: Normal Shocks
Richard Fitzpatrick
20160331