x
0
4 ¼ x 4 þ c 0 þ
γ þ 1
2
u 4
h
i
t,
so that the new separation between x 3 and x 4 is
x
0
4 À x
0
3 ¼ x 4 À x 3
ð
Þþ
γ þ 1
2
u 4 À u 3
ð
Þ t:
However, in this case u 4 0
4 À x
0
3
À
Á
<(x 4 À x 3 ) and the negative sloping
edge becomes steeper: Fig. 2.2(b) summarises the change to the shape of the profile
after a time t.
Eventually, the regions of compression will start to overtake the regions of
rarefaction so that the wave profile tends to the form shown in Fig. 2.2(c). This
would imply that the velocity, density or pressure would have three values at some
point x which is physically impossible; realistically, a discontinuity is formed and the
wave profile takes the form of the broken line as shown in Fig. 2.2(c). Prior to the
onset of the discontinuity, however, the velocity, pressure and density gradients
become so large that viscosity and heat conduction (which have been neglected in
our previous equations) come into play to counteract the gradients. A balance is
achieved between the competing effects of viscosity and heat conduction, on the one
hand, and the tendency of the compressive parts to overtake the expansive parts, on
the other hand. When this balance is achieved, the compressive portion of the wave
profile propagates without further distortion resulting in the formation of a shock
front or shock wave. In reality, the shock front is not a mathematical discontinuity
but a very thin transition region of the order of a few molecular mean-free paths
[10, 11] and across which there is an almost discontinuous jump in the mechanical
and thermodynamic variables.
2.4 Formation of a Normal Shock Wave
Let us now investigate a little further how the disturbance speed depends on
amplitude by considering the propagation of a series of disturbances in air due to
the motion of a piston in a tube. Suppose the piston undergoes a uniform acceleration
Fig. 2.2 Change in the
shape of the wave profile
(see text)
2.4 Formation of a Normal Shock Wave
51
0
4 ¼ x 4 þ c 0 þ
γ þ 1
2
u 4
h
i
t,
so that the new separation between x 3 and x 4 is
x
0
4 À x
0
3 ¼ x 4 À x 3
ð
Þþ
γ þ 1
2
u 4 À u 3
ð
Þ t:
However, in this case u 4 0
4 À x
0
3
À
Á
<(x 4 À x 3 ) and the negative sloping
edge becomes steeper: Fig. 2.2(b) summarises the change to the shape of the profile
after a time t.
Eventually, the regions of compression will start to overtake the regions of
rarefaction so that the wave profile tends to the form shown in Fig. 2.2(c). This
would imply that the velocity, density or pressure would have three values at some
point x which is physically impossible; realistically, a discontinuity is formed and the
wave profile takes the form of the broken line as shown in Fig. 2.2(c). Prior to the
onset of the discontinuity, however, the velocity, pressure and density gradients
become so large that viscosity and heat conduction (which have been neglected in
our previous equations) come into play to counteract the gradients. A balance is
achieved between the competing effects of viscosity and heat conduction, on the one
hand, and the tendency of the compressive parts to overtake the expansive parts, on
the other hand. When this balance is achieved, the compressive portion of the wave
profile propagates without further distortion resulting in the formation of a shock
front or shock wave. In reality, the shock front is not a mathematical discontinuity
but a very thin transition region of the order of a few molecular mean-free paths
[10, 11] and across which there is an almost discontinuous jump in the mechanical
and thermodynamic variables.
2.4 Formation of a Normal Shock Wave
Let us now investigate a little further how the disturbance speed depends on
amplitude by considering the propagation of a series of disturbances in air due to
the motion of a piston in a tube. Suppose the piston undergoes a uniform acceleration
Fig. 2.2 Change in the
shape of the wave profile
(see text)
2.4 Formation of a Normal Shock Wave
51