Dissociating isothermal shock ($2{\chi }^2-({\mathcal{M}}_1^2+1)\chi +{\mathcal{M}}_1^2=0$)

Suppose molecular hydrogen is fully dissociated into atoms across a steady planar shock, with the same far-upstream and far-downstream temperature and negligible binding-energy mass correction. The upstream isothermal sound speed obeys $a_1^2=k_{B}T/(2m_H)$, while $a_2^2=2a_1^2$. Conservation of mass gives $\chi ={\rho }_2/{\rho }_1=u_1/u_2$. Momentum conservation then gives $u_1^2+a_1^2=u_1^2/\chi +2a_1^2\chi$, producing the displayed quadratic with ${\mathcal{M}}_1=u_1/a_1$.

The high root approaches ${\mathcal{M}}_1^2/2$ at large upstream Mach number. Its real compressive domain is ${\mathcal{M}}_1\ge 1+\sqrt{2}$. At the endpoint the downstream flow is sonic; above it the high root is subsonic downstream. Completion of dissociation and maintenance of equal temperatures additionally require a consistent chemical and radiative energy budget. The two flux equations alone do not establish that physical realizability.