A tuning fork of frequency 480 Hz is used in an experiment for measuring speed of sound (v) in air b

Q: Why is the water surface considered a node?

Because the water is a much denser medium than air, it restricts the longitudinal displacement of air molecules, forcing the displacement amplitude to be strictly zero at the boundary.

Q: Why do we subtract $l_1$ from $l_2$ instead of just using $l_1$?

Subtracting the lengths elegantly eliminates the end correction $e$, which is a small extra length outside the tube where the antinode actually forms. Mathematically, $(l_2 + e) - (l_1 + e) = l_2 - l_1 = \lambda/2$.

Q: What exactly is an end correction?

The antinode at the open end doesn't form exactly at the physical rim of the tube but slightly above it. This extra distance is called the end correction, and it is approximately $0.6$ times the radius of the tube.

Q: Why is the second resonance at $3\lambda/4$ and not $2\lambda/4$?

A tube closed at one end only supports odd harmonics. The next possible standing wave pattern must have a node at the water and an antinode at the open end, which corresponds to $3\lambda/4$.

Q: Can we find the speed of sound using just the first resonance?

Yes, but only if we know the exact end correction. Without it, using just $v = f(4l_1)$ would give an inaccurate, slightly lower value for the speed of sound.

Question

A tuning fork of frequency $480	ext{ Hz}$ is used in an experiment for measuring speed of sound ($v$) in air by resonance tube method. Resonance is observed to occur at two successive lengths of the air column $l_1 = 30	ext{ cm}$ and $l_2 = 70	ext{ cm}$. Then, $v$ is equal to

Options:

(A) $332	ext{ ms}^{-1}$

(B) $384	ext{ ms}^{-1}$

(C) $338	ext{ ms}^{-1}$

(D) $379	ext{ ms}^{-1}$

Sigma Percentile · Accepted Answer

### The Resonance Tube Experiment

Imagine you are in a physics lab, standing in front of a tall, partially water-filled glass tube. You strike a tuning fork and hold it over the open end. As you slowly lower the water level, the sound suddenly amplifies, booming through the room. You have just found a **resonance**!

In this setup, the water surface acts as a rigid boundary, creating a **displacement node** for the sound waves. The open end of the tube, however, allows the air molecules to oscillate freely, forming an **antinode**.

### The First and Second Resonances

For the first resonance, the length of the air column ($l_1$) corresponds to the fundamental mode. In this mode, exactly one-fourth of a wavelength fits inside the tube:

$$l_1 = \frac{\lambda}{4}$$

As we continue to lower the water level, the sound fades and then booms again. This second resonance occurs at the first overtone. Now, the air column ($l_2$) accommodates three-fourths of a wavelength:

$$l_2 = \frac{3\lambda}{4}$$

### The Genius of Subtraction

You might wonder, why do we need two resonances? Why not just use the first one? The open end of the tube has a slight imperfection called the **end correction** ($e$). The antinode actually forms slightly above the tube's rim.

By subtracting the first resonant length from the second, we brilliantly eliminate this hidden error!

$$l_2 - l_1 = \frac{3\lambda}{4} - \frac{\lambda}{4} = \frac{\lambda}{2}$$

This tells us that the distance between two successive resonances is exactly half a wavelength.

### Calculating the Wavelength

Let's plug in the measurements from our experiment. We observed the first resonance at $30 	ext{ cm}$ and the second at $70 	ext{ cm}$.

$$\lambda = 2(l_2 - l_1)$$

$$\lambda = 2(70 - 30) 	ext{ cm}$$

$$\lambda = 2(40) 	ext{ cm} = 80 	ext{ cm}$$

Converting this to standard SI units, our wavelength $\lambda$ is **$0.8 	ext{ m}$**.

### The Speed of Sound

Finally, we can determine the speed of sound ($v$) using the fundamental wave equation. The speed of a wave is simply the product of its frequency ($f$) and its wavelength ($\lambda$):

$$v = f\lambda$$

We know the tuning fork vibrates at a frequency of $480 	ext{ Hz}$. Let's substitute our values:

$$v = 480 	imes 0.8$$

$$v = 384 	ext{ m/s}$$

And there we have it! The speed of sound in the air during this experiment is **$384 	ext{ m/s}$**. This elegant method not only gives us a precise value but also demonstrates the beautiful geometry of standing waves.

Select Answer:

Visualized Solution

Visual Anchor

First Resonance

Second Resonance

Difference in Lengths

Substituting Values

Calculating Wavelength

Speed of Sound Formula

Final Calculation

The Way Forward

The Sigma Insight: Standing Waves in Strings and Organ Pipes

The Resonance Tube Experiment

The First and Second Resonances

The Genius of Subtraction

Calculating the Wavelength

The Speed of Sound

Similar Questions

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