Speed of Sound — Revision Notes

⚡ 30-Second Revision

General Formula: — $v = \sqrt{\text{Elasticity}/\text{Density}}$
Gases (Laplace's): — $v = \sqrt{\gamma P/\rho} = \sqrt{\gamma RT/M}$
Temperature Effect (Gases): — $v \propto \sqrt{T}$ (absolute temp); $v_T \approx v_0 + 0.61T$
Pressure Effect (Gases, constant T): — Independent of pressure ($P/\rho$ is constant)
Density Effect (Gases, same T, P): — $v \propto 1/\sqrt{M}$ or $v \propto 1/\sqrt{\rho}$
Humidity Effect: — Moist air is less dense, so $v_{\text{moist}} > v_{\text{dry}}$
Medium Comparison: — $v_{\text{solids}} > v_{\text{liquids}} > v_{\text{gases}}$
Independence: — Speed is independent of frequency and amplitude.

2-Minute Revision

The speed of sound is how fast sound waves travel through a medium. It's a mechanical wave, so it needs a medium (solid, liquid, or gas) and cannot travel in a vacuum. The fundamental factors determining its speed are the medium's elasticity (stiffness) and density (inertia).

Generally, sound is fastest in solids, then liquids, and slowest in gases, because solids are much stiffer, allowing quicker energy transfer despite higher density. For gases, the speed is given by Laplace's formula, $v = \sqrt{\gamma P/\rho}$, where $\gamma$ is the adiabatic index.

Temperature significantly affects speed in gases: $v \propto \sqrt{T}$ (absolute temperature), meaning speed increases with temperature. For small temperature changes, $v_T \approx v_0 + 0.61T$. Crucially, the speed of sound is independent of pressure (at constant temperature), frequency, and amplitude of the sound wave itself.

Humidity slightly increases the speed of sound in air because moist air is less dense than dry air.

5-Minute Revision

Let's quickly review the core concepts of the speed of sound. Sound is a mechanical wave, meaning it requires a material medium to propagate. Its speed, $v$, is fundamentally determined by the medium's elastic properties (how resistant it is to deformation) and its inertial properties (its density). The general formula is $v = \sqrt{\text{Elasticity}/\text{Density}}$.

For gases, the process of sound propagation is adiabatic (no heat exchange), as corrected by Laplace. The formula becomes $v = \sqrt{\gamma P/\rho}$, where $\gamma$ is the adiabatic index ($C_p/C_v$), $P$ is pressure, and $\rho$ is density. Using the ideal gas law, this can also be written as $v = \sqrt{\gamma RT/M}$, where $R$ is the gas constant, $T$ is absolute temperature, and $M$ is molar mass.

Key factors influencing speed in gases:

1
Temperature (T): — $v \propto \sqrt{T}$. An increase in temperature increases molecular kinetic energy, leading to faster sound. For small changes, $v_T \approx v_0 + 0.61T$.
2
Pressure (P): — At constant temperature, speed is independent of pressure because $P/\rho$ remains constant.
3
Density ($\rho$) / Molar Mass (M): — For different gases at the same T and P, $v \propto 1/\sqrt{\rho}$ or $v \propto 1/\sqrt{M}$. Lighter gases transmit sound faster.
4
Humidity: — Moist air is less dense than dry air (water vapor is lighter than average air molecules), so sound travels slightly faster in humid air.

Comparison across media: $v_{\text{solids}} > v_{\text{liquids}} > v_{\text{gases}}$. Solids have the highest elasticity, which outweighs their higher density, leading to the fastest speeds.

What DOES NOT affect speed: Frequency, amplitude, and wavelength of the sound wave itself. The speed is a property of the medium, not the wave.

Example: If sound travels at $331,\text{m/s}$ at $0^circ\text{C}$, what is its speed at $20^circ\text{C}$? Using $v_T \approx v_0 + 0.61T$: $v_{20} \approx 331 + 0.61 \times 20 = 331 + 12.2 = 343.2,\text{m/s}$. Using $v_T = v_0 \sqrt{(273+T)/273}$: $v_{20} = 331 \sqrt{(273+20)/273} = 331 \sqrt{293/273} \approx 331 \times 1.035 \approx 342.5,\text{m/s}$. Both methods give similar results, with the approximation being quicker for NEET.

Prelims Revision Notes

Speed of Sound: NEET Quick Recall

1
Nature of Sound: — Mechanical wave, requires a medium. Cannot travel in vacuum.
2
General Formula: — $v = \sqrt{\text{Elasticity}/\text{Density}}$.

* Elasticity (e.g., Bulk Modulus $B$, Young's Modulus $Y$) represents stiffness. * Density ($\rho$) represents inertia.

1
Speed in Different Media:

* $v_{\text{solids}} > v_{\text{liquids}} > v_{\text{gases}}$. * Example: Steel ($~5100,\text{m/s}$) > Water ($~1480,\text{m/s}$) > Air ($~331,\text{m/s}$ at $0^circ\text{C}$). * Reason: High elasticity in solids/liquids dominates over their higher density.

1
Speed in Gases (Laplace's Formula):

* Sound propagation in gases is an adiabatic process (no heat exchange). * $v = \sqrt{\gamma P/\rho}$, where $\gamma = C_p/C_v$ (adiabatic index). * Using Ideal Gas Law ($P/\rho = RT/M$): $v = \sqrt{\gamma RT/M}$.

1
Factors Affecting Speed in Gases:

* Temperature (T): $v \propto \sqrt{T}$ (absolute temperature in Kelvin). * Approximation: $v_T \approx v_0 + 0.61T$ (for $T$ in Celsius, $v_0$ at $0^circ\text{C}$). For every $1^circ\text{C}$ rise, speed increases by $0.

61, ext{m/s}$. * **Pressure (P):** Independent of pressure if temperature is constant. (Because$P/\rho$remains constant). * **Density ($\rho$) / Molar Mass (M):** For different gases at same T and P,$v \propto 1/\sqrt{\rho}$or$v \propto 1/\sqrt{M}$.

Lighter gases (smaller M) have higher speeds. * Humidity: Moist air is less dense than dry air (water vapor $M=18$, dry air $M\approx 29$). Thus, $v_{\text{moist air}} > v_{\text{dry air}}$.

1
Factors NOT Affecting Speed:

* Frequency of the sound wave. * Amplitude (loudness) of the sound wave. * Wavelength of the sound wave. * These are wave properties, not medium properties. Speed is intrinsic to the medium.

Vyyuha Quick Recall

To remember factors affecting speed of sound in air: Thirsty People Hate Dry Air.

Temperature: Increases speed ($v \propto \sqrt{T}$)
Pressure: No effect (at constant T)
Humidity: Increases speed (moist air is less dense)
Density (of gas): Decreases speed ($v \propto 1/\sqrt{\rho}$)
Amplitude/Frequency: No effect