What happens to the speed of sound in air when the pressure increases at a constant temperature

Analysis

The question asks how the speed of sound in air is affected by a change in pressure, provided that the temperature remains constant.

Concept Explanation

To understand this, we look at the formula for the speed of sound ($v$) in a gas, which is derived from the Laplace correction to Newton's formula:

Where:

• $\gamma$ = adiabatic index (ratio of specific heats)
• $P$ = pressure of the gas
• $\rho$ = density of the gas

According to Boyle’s Law, for a fixed mass of gas at a constant temperature, pressure ($P$) is directly proportional to density ($\rho$) ($P \propto \rho$). This means that if the pressure of the gas increases, the density increases by the exact same factor.

When you substitute this relationship into the formula, the increase in pressure is perfectly offset by the increase in density. Mathematically, the ratio $\frac{P}{\rho}$ remains constant as long as the temperature does not change. Therefore, the speed of sound is independent of pressure.

Step-by-Step Solution

1. Identify the variables: The speed of sound depends on the medium's elasticity and density.
2. Apply the Gas Law: Since temperature is constant, the relationship $P = \rho RT/M$ holds. Substituting this into the speed formula shows that $v = \sqrt{\frac{\gamma RT}{M}}$.
3. Observe the result: In the final expression (v = \sqrt{\frac{\gamma RT}{M}), the pressure ($P$) term has been cancelled out.
4. Conclusion: Because the pressure and density change proportionally, the speed of sound remains unaffected by changes in pressure at a constant temperature.

Exam Tip

Always remember: Temperature is the only factor that significantly changes the speed of sound in a gas. Pressure and density changes cancel each other out, so they do not affect the speed. If you see a question about pressure, remember it is a "distractor"—the answer is almost always that it remains unchanged.