Figure 11.5 shows four cases for the flow over the same airfoil wherein $\boldsymbol{M}_{\infty}$ is progressively increased from 0.3 to $M_{\mathrm{cr}}=0.61$. Have you wondered where the numbers on Fig. 11.5 came from? Here is your chance to find out. Point $\boldsymbol{A}$ on the airfoil is the point of minimum pressure (hence maximum $M$ ) on the airfoil. Assume that the minimum pressure (maximum Mach number) continues to occur at this same point as $M_{\infty}$ is increased. In part (a) of Fig. 11.5, for $M_{\infty}=0.3$, the local Mach number at point $A$ was arbitrarily chosen as $M_A=0.435$, this arbitrariness is legitimate because we have not specified the airfoil shape, but rather are stating that, whatever the shape is, a maximum Mach number of 0.435 occurs at point $A$ on the airfoil surface. However, once the numbers are given for part $(a)$, then the numbers for parts (b), (c), and (d) are not arbitrary. Rather, $M_A$ is a unique function of $M_{\infty}$ for the remaining pictures. With all this as background information, starting with the data shown in Fig. $11.5(a)$, calculate $M_A$ when $M_{\infty}=0.61$. Obviously, from Fig. $11.5(d)$, your result should turn out to be $M_A=1.0$ because $M_{\infty}=0.61$ is said to be the critical Mach number. Said in another way, you are being asked to prove that the critical Mach number for this airfoil is 0.61 .