In the experiment for the determination of the speed of sound in air using the resonance column meth

Q: Why does the displacement antinode form outside the open end of the tube?

At the open end, the air molecules are not abruptly released into free space; they experience a transition zone where they can vibrate slightly beyond the physical boundary of the tube. This inertia of the moving air column extending outside the tube creates the end correction.

Q: Why is the first overtone of a closed pipe the third harmonic instead of the second?

A pipe closed at one end must have a node at the closed end and an antinode at the open end. This boundary condition only allows odd-integer multiples of a quarter-wavelength ($1, 3, 5, \dots$). Therefore, the second harmonic cannot exist, and the first overtone is the third harmonic ($3\lambda/4$).

Q: Does the end correction depend on the frequency of the tuning fork?

No, the end correction is primarily a geometric property of the tube. For a cylindrical tube, it depends on the radius ($e \approx 0.6r$) and is independent of the frequency or wavelength of the sound wave.

Q: What would happen if we neglected the end correction in calculating the speed of sound?

If we neglected $e$, our calculated wavelength would be smaller than the actual wavelength, leading to an underestimation of the speed of sound ($v = f\lambda$).

Q: Can we find the radius of the resonance tube from this data?

Yes! Since the end correction is $e \approx 0.6r$, we can estimate the radius of the tube as $r = \frac{e}{0.6} = \frac{0.025}{0.6} \approx 0.0417\text{ m}$ or $4.17\text{ cm}$.

Question

In the experiment for the determination of the speed of sound in air using the resonance column method, the length of the air column that resonates in the fundamental mode, with a tuning fork is $0.1	ext{ m}$. When this length is changed to $0.35	ext{ m}$, the same tuning fork resonates with the first overtone. Calculate the end correction.

Options:

(A) $0.012	ext{ m}$

(B) $0.025	ext{ m}$

(C) $0.05	ext{ m}$

(D) $0.024	ext{ m}$

Sigma Percentile · Accepted Answer

### Introduction to the Resonance Column Experiment

The resonance column experiment is a classic physics laboratory setup used to determine the **speed of sound** in air.

It consists of a vertical tube partially filled with water, where the water level can be adjusted to change the length of the air column above it.

When a vibrating tuning fork is held near the open top end of the tube, it sends longitudinal sound waves down the air column.

These waves reflect off the water surface (which acts as a rigid boundary, creating a displacement node) and travel back up the tube.

At specific lengths of the air column, the incident and reflected waves interfere constructively to form **standing waves**, resulting in a dramatic increase in the loudness of the sound—a phenomenon known as **resonance**.

### The Physics of End Correction

In ideal theoretical calculations, we assume that the open end of the tube behaves as a perfect displacement antinode.

However, in reality, the air molecules at the open boundary of the tube are not completely free on one side and constrained on the other; they can spill out slightly into the surrounding atmosphere.

Consequently, the boundary condition for the displacement antinode is met slightly outside the physical open end of the tube.

This extra distance is called the **end correction**, denoted by $e$.

For a cylindrical tube of radius $r$, Lord Rayleigh experimentally and theoretically determined that the end correction is approximately:

$$e \approx 0.6r$$

Therefore, the effective length of the vibrating air column is always greater than its physical length.

For a tube of physical length $l$, the effective length $L$ is given by:

$$L = l + e$$

### Analyzing the Resonance Modes

Let us analyze the two resonance states described in the problem.

#### 1. The Fundamental Mode

In the fundamental mode (the lowest frequency resonance), the standing wave pattern has a single node at the water surface and a single antinode at the effective open end.

This corresponds to a quarter of a wavelength:

$$L_1 = l_1 + e = \frac{\lambda}{4}$$

Given that the physical length of the air column for the fundamental mode is $l_1 = 0.1	ext{ m}$, we write:

$$0.1 + e = \frac{\lambda}{4} \quad 	ext{--- (Equation 1)}$$

#### 2. The First Overtone

When the water level is lowered, the next resonance occurs at the first overtone.

For a tube closed at one end, only odd harmonics can exist.

Thus, the first overtone corresponds to the **third harmonic**, where the effective length of the air column is three-quarters of a wavelength:

$$L_2 = l_2 + e = \frac{3\lambda}{4}$$

Given that the physical length of the air column for this mode is $l_2 = 0.35	ext{ m}$, we write:

$$0.35 + e = \frac{3\lambda}{4} \quad 	ext{--- (Equation 2)}$$

### Solving for the End Correction

Since the same tuning fork is used for both measurements, the frequency $f$ and the speed of sound $v$ remain constant.

Consequently, the wavelength $\lambda$ is identical in both equations.

To eliminate the unknown wavelength $\lambda$, we divide Equation 2 by Equation 1:

$$\frac{0.35 + e}{0.1 + e} = \frac{\frac{3\lambda}{4}}{\frac{\lambda}{4}}$$

$$\frac{0.35 + e}{0.1 + e} = 3$$

Now, we solve this simple linear equation for $e$:

$$0.35 + e = 3(0.1 + e)$$

$$0.35 + e = 0.3 + 3e$$

Subtracting $e$ from both sides:

$$0.35 = 0.3 + 2e$$

Subtracting $0.3$ from both sides:

$$2e = 0.05$$

$$e = 0.025	ext{ m}$$

Thus, the end correction of the resonance tube is **$0.025	ext{ m}$** (or $2.5	ext{ cm}$).

This matches **Option (b)** perfectly.

Select Answer:

Visualized Solution

Understanding the Resonance Column Setup

Analyzing the Fundamental Resonance

Analyzing the First Overtone Resonance

Relating the Two Modes

Substituting the Given Values

Solving the Linear Equation

Isolating the End Correction $e$

Calculating the Final Value of $e$

Exploring Further Applications

The Sigma Insight: Standing Waves in Strings and Organ Pipes

Introduction to the Resonance Column Experiment

The Physics of End Correction

Analyzing the Resonance Modes

# 1

# 2

Solving for the End Correction

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