9.6 Factors Affecting the Speed of Sound

The speed of sound ($V$) is influenced by factors such as density, pressure, moisture, wind, and temperature.

1. Density

The speed of sound is inversely proportional to the square root of the medium's density ($\rho$): $V=\sqrt{\frac{\gamma P}{\rho }}$

This means as density ($\rho$) increases, the speed of sound decreases: $V\propto \frac{1}{\sqrt{\rho }}$

For example, comparing hydrogen (${\rho }_H$) and oxygen (${\rho }_O$), oxygen is 16 times denser than hydrogen: ${\rho }_O=16{\rho }_H$

• For hydrogen: $V_H=\sqrt{\frac{\gamma P}{{\rho }_H}}$
• For oxygen: $V_O=\sqrt{\frac{\gamma P}{16{\rho }_H}}=\frac{1}{4}{V}_H$

Hence, the speed of sound in oxygen is four times slower than in hydrogen.

2. Pressure

At constant temperature, the ratio $\frac{P}{\rho }$ remains constant. Therefore, the speed of sound does not depend on pressure under these conditions.

Starting with the general equation: $V=\sqrt{\frac{\gamma P}{\rho }}$

Using the ideal gas law ($pV=nRT$), this can be rewritten in terms of temperature ($T$) and molar mass ($M$): $V=\sqrt{\frac{\gamma RT}{M}}$

Since temperature ($T$) is held constant, a change in pressure does not affect the speed of sound. This follows from Boyle's Law: increasing pressure at constant temperature increases density proportionally, keeping $P/\rho$ constant.

3. Moisture

The speed of sound is inversely proportional to the square root of the air's density ($\rho$). When moisture (water vapour) is added to air, its density decreases because water vapour ($M=18\text{ g/mol}$) has a lower molar mass than dry air ($M\approx 29\text{ g/mol}$). This results in an increase in the speed of sound: $V\propto \frac{1}{\sqrt{\rho }}$

Since moist air is less dense than dry air, sound travels faster in moist conditions.

4. Wind

Wind affects the speed of sound relative to a stationary observer:

• If the wind blows with the sound, the effective speed increases.
• If the wind blows against the sound, the effective speed decreases.

The medium itself is moving, so the observed speed of sound = $V_{\text{sound}}\pm V_{\text{wind}}$.

5. Temperature

The speed of sound is directly proportional to the square root of the absolute temperature ($T$ in Kelvin): $V=\sqrt{\frac{\gamma RT}{M}}⟹V\propto \sqrt{T}$

Thus, an increase in temperature results in an increase in the speed of sound.

Proof: Speed of Sound Increases by 0.61 m/s per 1°C Increase

1. Let $V_1$ be the speed of sound at $T_1=0^{\circ }\text{C}=273\text{ K}$, and $V_2$ be the speed at temperature $T_{\text{new}}$ (in °C), where $T_2=(T_{\text{new}}+273)\text{ K}$.
2. The ratio of speeds: $\frac{V_2}{V_1}=\sqrt{\frac{T_2}{T_1}}=\sqrt{\frac{T_{\text{new}}+273}{273}}={\left(1+\frac{T_{\text{new}}}{273}\right)}^{1/2}$
3. Using the binomial approximation for small $T_{\text{new}}$: ${\left(1+\frac{T_{\text{new}}}{273}\right)}^{1/2}\approx 1+\frac{T_{\text{new}}}{546}$
4. Therefore: $V_2\approx V_1+\frac{V_1{T}_{\text{new}}}{546}$
5. With $V_1=331.3\text{ m/s}$ at $0^{\circ }\text{C}$: $V_2\approx 331.3+\frac{331.3\times T_{\text{new}}}{546}\approx 331.3+0.61T_{\text{new}}$

Hence, for every $1^{\circ }\text{C}$ increase in temperature, the speed of sound increases by approximately 0.61 m/s.

Summary of Effects on Speed of Sound