# 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 ($\rho_O$), we know that oxygen is 16 times denser than hydrogen: $$ \rho_O = 16 \rho_H $$ Applying this to the speed of sound formulas: - **For hydrogen:** $$ V_H = \sqrt{\frac{\gamma P}{\rho_H}} $$ - **For oxygen:** $$ V_O = \sqrt{\frac{\gamma P}{\rho_O}} = \sqrt{\frac{\gamma P}{16 \rho_H}} $$ From equations (1) and (2), we find: $$ V_O = \frac{1}{4} V_H $$ Hence, the speed of sound in oxygen is four times slower than in hydrogen. ## 2. Pressure At a **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, this equation can be rewritten in terms of temperature ($T$) and molar mass ($M$): $$ V = \sqrt{\frac{\gamma R T}{M}} $$ Since temperature ($T$) is held constant, a change in pressure does not affect the speed of sound. ## 3. Moisture The speed of sound is inversely proportional to the square root of the air's density ($\rho$). When moisture (water vapor) is added to air, its density **decreases**. 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 is blowing **with** the sound, the effective speed increases. - If the wind is blowing **against** the sound, the effective speed decreases. ## 5. Temperature The speed of sound is directly proportional to the **square root of the absolute temperature** ($T$) in Kelvin. $$ V = \sqrt{\frac{\gamma R T}{M}} $$ This simplifies to: $$ V \propto \sqrt{T} $$ 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 $T_1 = 0^\circ C = 273 \, \text{K}$, and $V_2$ be the speed at a new temperature $T_{new}$ (in Celsius), where $T_2 = (T_{new} + 273) \, \text{K}$. 2. The ratio of the speeds is: $$ \frac{V_2}{V_1} = \sqrt{\frac{T_2}{T_1}} = \sqrt{\frac{T_{new} + 273}{273}} = \left(1 + \frac{T_{new}}{273}\right)^{1/2} $$ 3. Using the binomial approximation for small $T_{new}$: $$ \left(1 + \frac{T_{new}}{273}\right)^{1/2} \approx 1 + \frac{1}{2} \left(\frac{T_{new}}{273}\right) = 1 + \frac{T_{new}}{546} $$ 4. Therefore: $$ V_2 \approx V_1 \left(1 + \frac{T_{new}}{546}\right) = V_1 + \frac{V_1 T_{new}}{546} $$ 5. Knowing that the speed of sound in air at $0^\circ C$ ($V_1$) is approximately $331.3 \, \text{m/s}$: $$ V_2 \approx 331.3 + \frac{331.3 \times T_{new}}{546} \approx 331.3 + 0.61 T_{new} $$ Hence, for every $1^\circ C$ increase in temperature, the speed of sound increases by approximately **0.61 m/s**. ## Summary of Effects on Speed of Sound | Factor | Effect on Speed of Sound | Reason | | :--- | :--- | :--- | | **Density** | Speed decreases as density increases. | $V \propto 1/\sqrt{\rho}$ | | **Pressure** | No effect at constant temperature. | The ratio $P/\rho$ remains constant. | | **Moisture** | Speed increases as moisture increases. | Moist air is less dense than dry air. | | **Wind** | Speed increases with the wind, decreases against it. | The medium itself is moving. | | **Temperature**| Speed increases as temperature increases. | $V \propto \sqrt{T}$ | ## Quick Revision Table | **Factor** | **Formula/Relation** | **Effect on Speed of Sound** | | :--- | :--- | :--- | | **Density** | $V \propto 1/\sqrt{\rho}$ | Speed decreases as density increases | | **Pressure** | $V = \sqrt{\gamma P/\rho}$ | No effect at constant temperature | | **Moisture** | $V \propto 1/\sqrt{\rho}$ | Moisture decreases density, so speed increases | | **Wind** | Varies with direction | Faster with wind, slower against wind | | **Temperature**| $V \propto \sqrt{T}$ | Speed increases by 0.61 m/s per 1°C rise |