# “Fundamental of Aerodynamics” 5th editon, example about oblique shock. Is the author's solution wrong?

I am reading "Fundamentals of Aerodynamics" 5th edition, J.D.Anderson. If you have the book, go to chapter 9: Oblique shock and expansion waves. Please look at example 9.2, page 618:

Consider a supersonic flow with M = 2, p = 1 atm, and T = 288 K. This flow is deflected at a compression corner through 20◦. Calculate M, p, T, $p_0$, and $T_0$ behind the resulting oblique shock wave.(I only have problem with $p_0$).

Before see the solution, try to solve this example by yourself, you might need to see the appendices, here it is: https://drive.google.com/file/d/1MX6MASeP8pU-u_4MDjInfKIzPJfUfMaB/view?usp=sharing

And this is the solution from author:

The interesting thing here is: I was reading the note carefully, I still see something wrong here: And this is my solution for $p_0,2$:

$p_2$ = 2.82 atm

From $M_2$ = 1.21 $\Rightarrow$ $p_0,2$/$p_2$ = 2.489 (isentropic flow, appendix A)

So $p_{0,2}$ = 2.82 x 2.489 = 7.019 atm

You may think the result is the same as the author's solution, but this is just coincident, look at example 9.5, page 620:

Consider a Mach 3 flow. It is desired to slow this flow to a subsonic speed. Consider two separate ways of achieving this: (1) the Mach 3 flow is slowed by passing directly through a normal shock wave; (2) the Mach 3 flow first passes through an oblique shock with a 40◦ wave angle, and then subsequently through a normal shock. These two cases are sketched in Figure 9.14. Calculate the ratio of the final total pressure values for the two cases, that is, the total pressure behind the normal shock for case 2 divided by the total pressure behind the normal shock for case 1. Comment on the significance of the result. Again, try to solve this example by your own. Author's solution: My solution:

Case 1: same as author's solution

Case 2:

$M_{n,1}$ = 1.93 $\Rightarrow$ $p_2$/$p_1$ = 4.224 (1)

$M_1 = 3$ $\Rightarrow$ $p_{0,1}$/$p_1$ = 36.73 (2)

(1) and (2) $\Rightarrow$ $p_{0,1}$/$p_2$ = 36.73/4.224 (3)

$M_2 = 1.9$ $\Rightarrow$ $p_{0,2}$/$p_2$ = 6.701 (4)

(3) and (4) $\Rightarrow$ $p_{0,2}$/$p_{0,1}$ = 6.701x4.224/36.73 = 0.771 (different from author's solution $p_{0,2}$/$p_{0,1}$ = 0.7535

$p_{0,3}$/$p_{0,2}$ is same as author's solution

In both examples, you can see what is going on. Am I wrong or Anderson wrong ?? You might think: "Oh, Anderson said that changes across an oblique shock wave are governed only by the component of velocity normal to the wave." But this holds only for p, T, $\rho$, $M_{n,1}$, $M_{n,2}$. For $p_{0,1}$ and $p_{0,2}$, we must consider the tangential components of velocities.

• Great question, we definitely have a few guys around here who can help out! – Lnafziger Jun 22 '18 at 19:56
• I’d suggest contacting Dr Anderson directly with that question as he could probably address it best. – Carlo Felicione Jun 24 '18 at 11:30