How to define Mach Subsonic by the Mach Supersonic? I read the book Mechanic of fluids shames and I find this relationship: $$\frac{1+kM_1^2}{1+kM_2^2} =\frac{M_1}{M_2} \left ( \frac{1+\dfrac{(k-1)}{2}M_1^2}{1+\dfrac{(k-1)}{2}M_2^2} \right )^{0.5}$$ where $M_1$ is the Mach number of supersonic flow and $M_2$ is the Mach number for subsonic flow. How can I find $M_2$ as a function of $M_1$, say $M_2 = f(M_1)$? Sorry for my English.

A small amount of work will show that squaring the expression above gives:

$$(m_1-m_2)(m_1+m_2)(2km_1^2 m_2^2-k (m_2^2+m_1^2)+m_1^2+m_2^2-2) = 0.$$ Two solutions are obvious (and presumably uninteresting): $m_2 = \pm m_1$. The other two are also straightforward (assuming I haven't made a mistake): $$m_2 = \pm \sqrt{\frac{1+\frac{k-1}{2} m_1^2}{k m_1^2+\frac{1-k}{2}}}.$$ The original equation rules out the negative solutions.