Speed of Sound Formula

The speed of sound (\(v\)) in an ideal gas depends on its properties. A common formula used is:

\( v = \sqrt{\frac{\gamma RT}{M}} \)

Where:

• \(\gamma\) represents the adiabatic index (the ratio of specific heats, \(C_p/C_v\)).
• \(R\) is the universal ideal gas constant.
• \(T\) is the absolute temperature of the gas.
• \(M\) is the molar mass of the gas.

Temperature Constant Impact

The problem specifies that the temperature (\(T\)) of the ideal gas is held constant.

Based on the formula \(v = \sqrt{\frac{\gamma RT}{M}}\), the speed of sound (\(v\)) is directly dependent on the square root of the absolute temperature (\(T\)). Since \(\gamma\), \(R\), and \(M\) are constants for a specific gas, and \(T\) is kept constant, the speed of sound does not change.

The change in pressure, even doubling it, does not affect the speed of sound if the temperature remains constant. This is because, according to the ideal gas law (\(PV=nRT\)), if pressure (\(P\)) doubles and temperature (\(T\)) is constant, the density (\(\rho\)) must also double to maintain the relationship \(\frac{P}{\rho} = \frac{RT}{M}\). Substituting this into \(v = \sqrt{\frac{\gamma P}{\rho}}\), we get \(v = \sqrt{\frac{\gamma (2P_1)}{(2\rho_1)}} = \sqrt{\frac{\gamma P_1}{\rho_1}}\), which is the original speed.

Ratio Calculation

Let the initial speed of sound be \(x\) and the final speed of sound be \(y\).

Given that the temperature (\(T\)) is constant:

• Initial speed: \(x\)
• Final speed: \(y\)

Since the speed of sound depends only on temperature (when \(\gamma\), \(R\), \(M\) are constant), and the temperature did not change:

\( x = y \)

Therefore, the ratio of \(x\) to \(y\) is:

\( \frac{x}{y} = \frac{x}{x} = 1 \)