Page:Threeshockwaves.pdf/4

$\left.{\begin{array}{lll}\sigma _{\perp }^{(1,2)}={\frac {u^{(1)}-D}{c_{1}}}\\\zeta _{1,2}={\frac {P_{2}}{P_{1}}}={\frac {1}{\gamma +1}}\left\{2\gamma \left[\sigma _{\perp }^{(1,2)}\right]^{2}-(\gamma -1)\right\}\\\eta _{1,2}={\frac {\rho _{1}}{\rho _{2}}}={\frac {(\gamma -1)\left[\sigma _{\perp }^{(1,2)}\right]^{2}+2}{(\gamma +1)\left[\sigma _{\perp }^{(1,2)}\right]^{2}}}\\\tau _{\perp }={\frac {u_{\perp }^{(2)}-D}{c_{1}}}={\frac {1}{\gamma +1}}\left\{(\gamma -1)\sigma _{\perp }^{(1,2)}+2\left[\sigma _{\perp }^{(1,2)}\right]^{-1}\right\}\\c_{2}=c_{1}{\sqrt {\zeta _{1,2}\eta _{1,2}}}\end{array}}\right.$

(1)

where $c_{1}$ and $c_{2}$ denote the velocities of sound in the two regions respectively and the rest of the symbols have their usual meanings. Finally

$u_{\parallel }^{(1)}=u_{\parallel }^{(2)}$

(2)

.

In other words, if we measure the velocities in units of the velocity of sound in region $1$ then all the relevant physical quantities (including the velocity of sound in region $2$) can be expressed parametrically in terms of $\sigma _{\perp }^{(1,2)}$. As we have already stated, we shall measure the intensity of a shock wave by the ratio of the pressures on either side of the shock front, i.e., $\zeta _{1,2}$. We may note that if in a fixed frame of reference the shock is moving in the direction $2\rightarrow 1$, $\zeta _{1,2}>1$ and $\sigma _{\perp }^{(1,2)}>1$; conversely the direction of motion of a shock front can be inferred from whether $\sigma _{\perp }^{(1,2)}$ is greater than or less than unity.