Page:Threeshockwaves.pdf/5

Returning to the configuration of shocks depicted in Fig. 3, since $\zeta _{1,2}$ is assumed known,

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

(3)

Hence, for the region in front of the shock

$|\sigma |=\sigma _{\perp }^{(1,2)}\mathrm {cosec} \alpha .$

(4)

For the Mach shock the appropriate value of $\sigma _{\perp }^{(1,4)}$ is $|\alpha |$.

$\left.{\begin{array}{lll}\zeta _{1,4}&={\frac {1}{\gamma +1}}\left\{2\gamma \left[\sigma _{\perp }^{(1,2)}\right]^{2}\mathrm {cosec} ^{2}\alpha -(\gamma -1)\right\}\\&={\frac {1}{\gamma +1}}\left\{\left[(\gamma +1)\zeta _{1,2}+(\gamma -1)\right]\mathrm {cosec} ^{2}\alpha -(\gamma -1)\right\}.\end{array}}\right.$

(5)

But the equality of the pressures in regions $3$ and $4$ requires that

$\zeta _{1,4}=\zeta _{1,2}\zeta _{2,3}.$

(6)

The quantity $\zeta _{2,3}$ has been plotted by von Neumann as a function of the angle of incidence $\alpha$.^[1] Accordingly the intersection of the curve $\zeta _{2,3}(\alpha )$ with $\zeta _{1,4}/\zeta _{1,2}$ determines the angle $\alpha _{M}$ at which we have what we might call the stationary Mach-effect. In this manner it is found that

$\alpha _{M}=21.75^{\circ },\quad 31.25^{\circ },\quad 42.5^{\circ }\quad {\text{for}}\quad \zeta _{1,2}=\infty ,\quad 7.0225,\quad 2$

(7)

respectively.

1. We may note here that what von Neumann calls $\zeta '$ is our $\zeta _{2,3}$ while his $\zeta$ is our $1/\zeta _{1,2}$.