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:
\( \frac{x}{y} = \frac{x}{x} = 1 \)
Similar Questions
Which one among the following is true for the speed of sound in a given medium?
Consider the following statements in respect of sound waves:
I. The speed of sound waves decreases with an increase in temperature if the atmospheric pressure remains the same.
II. The speed of sound waves increases with increasing humidity if the atmospheric pressure remains the same.
III. The speed of sound waves is not affected by a change in atmospheric pressure, provided the temperature is kept constant.
Which of the statements given above are correct?
Important Questions from The Speed of Sound
An object is 10.64 km below the sea level. A research team sends down a sonar signal to confirm this depth. After how long can it expect to get the echo? Take speed of sound in sea water = 1520 m/s.
A boy stands 83 m in front of a high wall and then blows a whistle, calculate the time interval when he hears an echo. Speed of sound is 332 m/s.
A man claps his hands in front of a wall and he hears an echo after 1.6s. He walks. 33m towards wall and hears the echo 1.4 s after clapping. Then velocity of sound in air is:
For hearing a distinct sound, the time interval between the original sound and the reflected one must be atleast ______ seconds.
The speed of a longitudinal wave in a solid bar is given by v = √(X/ρ), where 'ρ' is density of the medium. What is the unknown term 'X'?
