Page:Threeshockwaves.pdf/8

Let the shock OB be of a given intensity $\zeta _{1,2}$. Then according to the Rankine-Hugoniot equations in the forms (1) and (2) the velocity of the gas in region 1 normal to the shock front OB in units of the velocity of sound in this region is given by equation (3) and accordingly

$\sigma _{\parallel }^{(1,2)}=\sigma _{\perp }^{(1,2)}\cot \alpha .$

(8)

The velocity of the gas in the region 2 both in magnitude and in direction can therefore be readily determined through the equations (1) and (2). For crossing the shock OC inclined at an angle $\beta$ to the direction of OB we can again use the Rankine-Hugoniot equations with

$\left.{\begin{array}{lll}\sigma _{\perp }^{(2,3)}&={\frac {1}{c_{2}}}\left(\tau _{\perp }^{(1,2)}\cos \beta +\sigma _{\parallel }^{(1,2)}\sin \beta \right)\\\sigma _{\parallel }^{(2,3)}&={\frac {1}{c_{2}}}\left(\tau _{\perp }^{(1,2)}\sin \beta -\sigma _{\parallel }^{(1,2)}\cos \beta \right).\end{array}}\right.$

(9)

Thus with the foregoing values for $\sigma _{\perp }$ and $\sigma _{\parallel }$ we can deduce the conditions behind OC. In particular we shall arrive in region 3 with a definite direction for the motion of the gas in this region. Moreover

$\zeta _{2,3}={\frac {P_{3}}{P_{2}}}={\frac {1}{\gamma +1}}\left\{2\gamma \left[\sigma _{\perp }^{(2,3)}\right]^{2}-(\gamma -1)\right\}.$

(10)

Now the shock OE must be so inclined to the direction of motion OA in region 1 that the pressure behind the shock OE is the same as in region 3. In other words the equation determining the angle