Problem 3.13

Given:
\( T = \) 80^oF,
\(M = \) 1.9.

To calculate: \( V=?, \mu=? \)

The temperature can be converted to standard units 'Kelvin' as,

\[ T=80^{\circ}\,\text{F}=\left(80-32\right)\times5/9+273=299.6667\,\text{K}\ . \]

The speed of sound in air, at a given temperature, can be calculated as,

\[ a=\sqrt{\gamma\,R\,T} \]

The velocity of the flowing air can be calculated using the Mach number as,

\[ V=M\,a=M\times\sqrt{\gamma\,R\,T}=1.9\times\sqrt{1.4\times287\times299.6667} \]

\[ \boxed{V=659.292\,\text{m/s}}\ . \]

The Mach angle is given by the equation,

\[ \sin\left(\mu\right)=\frac{1}{M} \]

\[ \mu=\sin^{-1}\left(\frac{1}{M}\right)=\arcsin\left(\frac{1}{1.9}\right) \]

\[ \boxed{\mu=31.757^{\circ}}\ . \]