- Show that Equation (15.12) can be written in the form

where

Let

Demonstrate that the oblique shock solution only exists for [i.e., when Equation (15.207) possesses three real roots.] Show that the strong shock solution, , and the weak shock solution, , are given by

where

Here, if , and if . - Assuming that information propagates with respect to a two-dimensional supersonic flow pattern at the local sound speed, show that, in order for the
flow at some point
to affect the flow at some other point
, the latter point must lie between the
and
characteristics that pass through
.
- Show that for a weak oblique shock with
,

where , et cetera, and - Show that for a weak oblique shock

where . Here, and are the Mach angles upstream and downstream of the shock front, respectively. Moreover, is the wave angle, and the deflection angle. Hence, deduce that the shock front subtends the same angle, , with the Mach lines upstream and downstream of it. In other words, the shock position is the ``average'' of the Mach line positions on either side of it. Consider supersonic flow incident on a wedge of small nose angle, with an afterbody, as illustrated in Figure 15.9(a). Assume that the shock front is attached to the apex of the wedge, and that the flow downstream of the shock is supersonic. Use the result just proved to show that the shape of the shock front in the region of attenuation by expansion (i.e., in the region in which the shock front is intersected by Mach lines emanating from the shoulder) is parabolic. [Hint: Use the well-known optical result that a parabolic mirror perfectly focuses a parallel beam of light rays.] (Leipmann & Roshko 1957.) - Show that, to second order in the deflection angle,
, the relative change in pressure across a weak oblique shock is written
- An ideal gas of pressure, density, temperature, and Mach number
,
,
, and
, respectively, flows over a convex corner that turns through an
angle
, as shown in Figure 15.16. Consider a particular Mach line in the Prandtl-Mayer expansion fan that subtends an angle
with the continuation of the upstream wall, as shown in the figure. Let
,
,
,
,
,
, and
be the pressure, density, temperature, Mach number, magnitude of the deflection angle, Mach angle, and Prandtl-Mayer function, respectively, on the Mach line in question.
Furthermore, let

where is the ratio of specific heats. Show that, inside the fan,

where . Here, is defined implicitly by . Demonstrate that the fan extends over the range of angles , where

Show that , , , ,

Assuming that , show that

and

where , et cetera. Of course, the previous four relations are the same as those for a weak shock. (See Exercise iii.) Why is this not surprising? (Hint: The jump in specific entropy across a weak shock is third order in the deflection angle.) Deduce that to second order in the deflection angle, - If an oblique shock is intercepted by a wall then it is reflected, as illustrated in Figure 15.17. Calculate
, assuming
that the shocks are sufficiently weak that the approximate expressions of Section 15.4 can be used. Demonstrate that, in this
limit,
**Figure:**Merging of two oblique shocks of the same family produced by successive concave corners of deflection angles and . Here, , , et cetera, are Mach numbers. - Figure 15.18 shows a situation in which two oblique shocks of the same family [in this case, the
family], produced by successive concave corners in a wall, merge together to
form a single stronger shock [of the
family]. Assuming that the shocks are sufficiently weak that the approximate expressions of Section 15.4 can be used (which implies that
and
),
demonstrate that

and is the ratio of specific heats. Show that

Demonstrate that the strength of the merged shock is approximately the sum of the strengths of the two component shocks, and, hence, that the pressures on either side of the slipstream shown in the figure are equal (at least, to first order in and ). Finally, show that

where is specific entropy, and the specific gas constant. It is, thus, clear that the specific entropy is not quite the same on either side of the slipstream.**Figure:**Crossing of two oblique shocks of different families. Here, , , et cetera, are Mach numbers, and , , et cetera, are deflection angles. - If two shocks of opposite families intersect then they pass through one another, but are slightly bent in the process, as illustrated in Figure 15.19.
Assuming that the shocks are sufficiently weak that the approximate expressions of Section 15.4 can be used (which implies that
,
, et cetera), show that

where denotes pressure. Hence, deduce that

Show, that the respective strengths of the two shocks are unaffected by the intersection (at least, to first order in the deflection angles). - Show that Equation (14.66) can be written in the form

(Leipmann & Roshko 1957.) - Show that for a thin, symmetrical airfoil, with zero angle of attack, whose profile is a lens defined by two circular arcs, the drag coefficient is
- Prove that on a supersonic swept-back wing of infinite span the thin-airfoil pressure coefficient, (15.50), is multiplied by the
*sweepback factor*, - Consider the problem of subsonic flow past a wave-shaped wall that was discussed in Section 15.13. Show that if the
flow is bounded by a second wall that lies at
(where
) then