Mechanics of Fluids

velocity into the undisturbed fluid. A second tiny wave follows immediately,
but the fluid into which this second wave moves has already been traversed
by the first wave. That fluid is therefore at a slightly higher pressure, density
and temperature than formerly. If for simplicity a perfect gas is assumed
so that a = � (γ RT) (although the conclusion may be shown to apply to
any fluid) it is clear that the second wave travels slightly faster than the
first. Similarly the third wave is propagated with a slightly higher velocity
than the second and so on. The pressure distribution at a certain time might
be as shown in Fig. 11.5a, and a little while later would become like that
shown at b. In practice, separate small waves would not be distinguishable
but, instead, a gradual rise of pressure which becomes steeper as it advances.
Before long the wave front becomes infinitely steep and a sharp discontinuity
of pressure results. This is known as a shock wave, and it will be studied in
the next section.
A similar argument applied to a rarefaction wave shows that this becomes
less steep as shown in Figs. 11.5c and d. The foremost edge of the wave
proceeds into the undisturbed fluid with the sonic velocity in that fluid, but
all other parts of the wave have a lower velocity. Consequently no effect
of a pressure decrease, no matter how large or how sudden it is initially, is
propagated at more than the sonic velocity in the undisturbed fluid. A pressure
increase, on the other hand, may be propagated at a velocity greater
than that of sound.
11.5 SHOCK WAVES
Whereas in Section 11.4 we considered a pressure change of infinitesimal
size, we now turn attention to an abrupt finite pressure change known as a
shock. The possibility of such an abrupt change in a compressible fluid was
envisaged by the German mathematician G. F. Bernhard Riemann (1826–66)
Fig. 11.5
Shock waves 499