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The speed of sound prior to completion of the shock is $M_1 * a$, where a is the speed of sound at that point.
The solution is then that the speed of the air after the shock has propagated is $(M_1 * a) - M_2 * a * \sqrt{T_2/T_1} $. I am not sure where this relationship comes from.
Where does this equation in the solution come from? And, what is the physical interpretation of it?
So I am going to make some assumptions since the parameters were not defined with regard to the shock wave.
I assume $M_{j}$ is the Mach number in the jth region (i.e., 1 for upstream [pre-shock] and 2 for downstream [shocked gas]). I also assume that $a$ is not just the speed of sound at a specific location related to $M_{j}$, but rather the asymptotic upstream sound speed. So it really should be labeled with a subscript 1. Finally, I assume that $T_{j}$ is the thermodynamic temperature in the jth region.
The Mach number is defined as the bulk flow speed, $U_{n,j}$, along the outward unit normal vector, $\hat{\mathbf{n}}$, of the local shock surface divided by the relevant speed of communication. Here that would be the sound speed, $a_{j}$. Note that the Mach number only really makes sense, physically, if we discuss things in the shock rest frame. That is because $U_{n,j}$ is relative to the shock itself, not an arbitrary speed in an arbitrary frame of reference.
The second relationship, about which you are asking, provides clues as to why I made the above assumptions. That is, the term $M_{2} \ a \ \tfrac{ T_{2} }{ T_{1} }$ would correspond to $U_{n,2}$ if $a$ is indeed $a_{1}$ and $T_{j}$ is the region averaged temperature. That is, $a_{1} \ \tfrac{ T_{2} }{ T_{1} }$ = $a_{2}$.
So in the shock rest frame, the bulk flow speed of the air along the shock normal in the downstream region is $U_{n,2}$ = $M_{2} \ a_{1} \ \tfrac{ T_{2} }{ T_{1} }$. The bulk flow speed of the air along the shock normal in the upstream region is $U_{n,2}$ = $a_{1} \ M_{1}$.
The relationship stating the speed of the air is the difference of these two quantities implies the answer is with respect to the downstream rest frame, which is not entirely clear to me.
In general, one should define the Rankine–Hugoniot relations with respect to some lab frame or stationary, external frame (LAB). Then the bulk flow velocity of the air in the shock rest frame (SHF) is defined as: $$ \mathbf{V}_{j}^{SHF} = \mathbf{V}_{j}^{LAB} - (\mathbf{V}_{sh}^{LAB} \cdot \hat{\mathbf{n}}) \ \hat{\mathbf{n}} \tag{0} $$ where $\mathbf{V}_{sh}^{LAB}$ is the 3-vector shock velocity in the lab frame and $\mathbf{V}_{j}^{LAB}$ is the bulk flow 3-vector medium velocity in the lab frame in the jth region (i.e., the shock has its own speed and the medium flows at $\mathbf{V}_{j}^{LAB}$ in the jth region). Then the shock normal speed in the shock rest frame is given by: $$ U_{n,j} = \mathbf{V}_{j}^{SHF} \cdot \hat{\mathbf{n}} \tag{1} $$
In short, without a proper reference frame, the original question is impossible to answer without a lot of assumptions. Presumably, the question is asking the reader to find $\mathbf{V}_{2}^{LAB}$ but that is not given by $U_{n,1} - U_{n,2}$, as suggested, unless $\mathbf{V}_{1}^{LAB}$ = 0.
