Question

For an ideal gas obtain an expression for the ratio of the speed of sound where \(\mathrm{Ma}=1\) to the speed of sound based on the stagnation temperature, \(c^{*} / c_{0}\).

Short answer

Answer: The expression for the ratio of the speed of sound for a Mach number of 1 to the speed of sound based on the stagnation temperature is \(\frac{c^*}{c_0} = \frac{\sqrt{T_0}}{\sqrt{T_0 - \frac{c^2}{2C_p}}}\).

Step-by-step solution

Write down the equations for the speed of sound, Mach number, and stagnation temperature

For an ideal gas, the speed of sound \(c\) is given by: \(c = \sqrt{\gamma RT}\) where \(\gamma\) is the specific heat ratio, \(R\) is the specific gas constant, and \(T\) is the temperature. The Mach number \(\mathrm{Ma}\) is defined as: \(\mathrm{Ma} = \frac{v}{c}\) where \(v\) is the velocity of the gas. For \(\mathrm{Ma} = 1\), we have: \(v = c\) The stagnation temperature \(T_0\) is defined as the temperature of the gas when it is brought to rest isentropically: \(T_0 = T + \frac{v^2}{2C_p}\) where \(C_p\) is the specific heat at constant pressure. Now, let's find the expression for the speed of sound based on the stagnation temperature, denoted as \(c^*\).

Express \(c^*\) in terms of \(T_0\)

To find the expression for \(c^*\), substitute the value of \(T_0\) back into the speed of sound equation: \(c^* = \sqrt{\gamma R T_0}\)

Express \(c\) in terms of the stagnation temperature \(T_0\)

From the relationship for the stagnation temperature, we can obtain the expression for the actual temperature \(T\): \(T = T_0 - \frac{v^2}{2C_p}\) Now we substitute this expression for \(T\) into the speed of sound formula: \(c = \sqrt{\gamma R \left( T_0 - \frac{v^2}{2C_p} \right)}\)

Use the relationship between \(c\) and \(v\) for a Mach number of 1

Since we are considering the case where \(\mathrm{Ma} = 1\), we know that \(v = c\). We can use this relationship to eliminate \(v\) from the equation above: \(c = \sqrt{\gamma R \left( T_0 - \frac{c^2}{2C_p} \right)}\)

Find the expression for the ratio \(c^*/c_0\)

We are now ready to find the expression for the ratio of the speed of sound for a Mach number of 1 to the speed of sound based on the stagnation temperature. Divide the expression for \(c^*\) by the expression for \(c\) in terms of \(T_0\): \(\frac{c^*}{c_0} = \frac{\sqrt{\gamma R T_0}}{\sqrt{\gamma R \left( T_0 - \frac{c^2}{2C_p} \right)}}\) Upon careful observation, we see that \(\gamma R\) is common in the numerator and the denominator, therefore it can be canceled out: \(\frac{c^*}{c_0} = \frac{\sqrt{T_0}}{\sqrt{T_0 - \frac{c^2}{2C_p}}}\) Thus, we have derived an expression for the ratio of the speed of sound for a Mach number of 1 to the speed of sound based on the stagnation temperature: \(\boxed{\frac{c^*}{c_0} = \frac{\sqrt{T_0}}{\sqrt{T_0 - \frac{c^2}{2C_p}}}}\)