Speed of Sound in Different Media

1. The Fundamental Principles: Stiffness vs. Inertia

Before we jump into the formulas, let's build a strong conceptual foundation. The speed of any mechanical wave, including sound, is determined by a tug-of-war between two properties of the medium:

1. Elasticity (or Stiffness): This is a measure of how strongly the particles of the medium are connected and how quickly they "snap back" to their original position after being disturbed. A stiffer medium transmits disturbances faster.
2. Inertia: This is a measure of the medium's resistance to motion, which is related to its density. A denser, more "sluggish" medium transmits disturbances more slowly.

This relationship is captured in a general formula:
$v=\sqrt{\frac{\text{Elastic Property}}{\text{Inertial Property}}}$

To get a better feel for this interplay, the following short video provides an excellent explanation.

As the video explained, the much higher stiffness of solids and liquids more than compensates for their greater density, leading to a clear trend:

Speed of Sound: Solids > Liquids > Gases

This bar chart provides a great visual confirmation of this rule across various materials.

2. The Formulas for Calculating Sound Speed

Now, let's equip ourselves with the specific formulas needed for problem-solving. We will replace the general "Elastic Property" and "Inertial Property" terms with their precise physical quantities for each state of matter.

Speed of Sound in Solids and Liquids

For solids and liquids, the formulas are quite direct.

• In Solids (longitudinal waves in a thin rod): The elastic property is the Young's Modulus ($Y$), and the inertial property is the density ($\rho$).
  $v_{\text{solid}}=\sqrt{\frac{Y}{\rho }}$

• In Liquids: The elastic property is the Bulk Modulus ($B$), which measures resistance to compression. The inertial property is again the density ($\rho$).
  $v_{\text{liquid}}=\sqrt{\frac{B}{\rho }}$

The following video demonstrates how to apply these formulas with straightforward numerical examples.

Speed of Sound in Gases

Calculating the speed of sound in gases is more nuanced and a very important topic for NEET.

Newton's Formula and Laplace's Correction

Initially, Newton derived a formula assuming that the compressions and rarefactions of a sound wave were slow enough for the process to be isothermal (constant temperature). This gave $v=\sqrt{P/\rho }$, which was off by about 16%.

Laplace later corrected this by pointing out that the process is actually adiabatic (no heat exchange) because the oscillations are too rapid. This introduced the adiabatic constant, $\gamma$ ($\gamma =C_p/C_v$), into the formula, giving the correct expression:

$v_{\text{gas}}=\sqrt{\frac{\gamma P}{\rho }}$

For air, which is mostly diatomic, $\gamma \approx 1.4$. This correction yields a value that matches experimental results.

To understand this crucial correction in more detail, you can review the following article.

A More Practical Formula for Gases

While $v=\sqrt{\gamma P/\rho }$ is correct, it's often more useful to express the speed in terms of temperature and molar mass. By using the ideal gas equation ($PV=nRT$) and the relationship $\rho =m/V$, we can transform the formula into:

$v_{\text{gas}}=\sqrt{\frac{\gamma RT}{M}}$

where:

• $\gamma$ is the adiabatic constant ($C_p/C_v$)
• $R$ is the universal gas constant ($\approx 8.314\text{J/(mol⋅K)}$)
• $T$ is the absolute temperature in Kelvin
• $M$ is the molar mass of the gas in kilograms per mole (a common source of error is using g/mol!)

This form of the equation reveals the key dependencies for gases:

• Effect of Temperature: $v\propto \sqrt{T}$. Sound travels faster at higher temperatures.
• Effect of Molar Mass: $v\propto 1/\sqrt{M}$. Sound travels faster in lighter gases (e.g., hydrogen vs. oxygen).
• Effect of Pressure: At a constant temperature, pressure has no effect on the speed of sound in an ideal gas. This is because as you increase pressure, density increases proportionally, keeping the ratio $P/\rho$ constant. This is a classic conceptual trap in exams.
• Effect of Humidity: Humid air is less dense than dry air because water molecules (M $\approx$ 18 g/mol) are lighter than the average nitrogen/oxygen molecules they displace (M $\approx$ 29 g/mol). Since $v\propto 1/\sqrt{\rho }$, sound travels faster in humid air.

3. NEET-Style Problem Solving

Now, let's put these formulas into practice. This is the most crucial part of your preparation. Try to solve each problem on your own before looking at the solution.

For further practice, I highly recommend working through the "Practice problems" in the Aakash article (LINK, section 5) and the multiple-choice questions in the exercises of the Disha Publication notes (LINK, sections 3 & 4).

Conclusion

Today we have uncovered the physics behind the speed of sound. By understanding the balance between a medium's elastic and inertial properties, we can now calculate this speed in solids, liquids, and, most importantly for NEET, in gases under various conditions.

Key Takeaways:

• The general formula for wave speed is $v=\sqrt{\text{Elasticity}/\text{Inertia}}$.
• Sound travels fastest in solids, then liquids, then gases.
• For Solids: $v=\sqrt{Y/\rho }$
• For Liquids: $v=\sqrt{B/\rho }$
• For Gases: The most useful formula is $v=\sqrt{\frac{\gamma RT}{M}}$. From this, remember:
  • Speed is independent of pressure (at constant T).
  • $v\propto \sqrt{T}$ (T must be in Kelvin).
  • $v\propto 1/\sqrt{M}$ (faster in lighter gases).

Preview of the Next Lesson:
We now have a complete picture of a wave's properties: its equation, its speed, and how it behaves in bounded media. The final piece of this module is to consider what happens when the source of the sound or the listener is moving. This apparent change in frequency is known as the Doppler Effect, and it will be the focus of our next lesson.