An organ pipe P1 closed at one end vibrating in its first harmonic and another pipe P2 open at both

Q: Why does the closed pipe only have odd harmonics, and does that affect this problem?

A closed pipe indeed only supports odd harmonics ($1, 3, 5, \dots$) because of its boundary conditions (node at one end, antinode at the other). In this problem, we are specifically told the closed pipe is in its first harmonic (fundamental), which is supported. The open pipe supports all integer harmonics, including the third harmonic. Thus, the physics is perfectly consistent.

Q: Does the speed of sound change between the closed and open pipe?

No, the speed of sound $v$ depends only on the properties of the medium (like temperature and molecular mass of air). Since both pipes are in the same room under the same conditions, $v$ is identical and cancels out of the equation.

Q: What is the difference between a harmonic and an overtone?

A harmonic is an integer multiple of the fundamental frequency. An overtone is any resonant frequency above the fundamental. For an open pipe, the first overtone is the second harmonic ($2f_0$). For a closed pipe, the first overtone is the third harmonic ($3f_0$) because the second harmonic does not exist.

Q: Why did we not include end corrections in this calculation?

In standard JEE problems, unless the diameter of the pipe is specified or end correction is explicitly mentioned, we assume the pipes are thin enough that end corrections are negligible ($e \approx 0$).

Q: How would the ratio change if we included end corrections?

If end corrections were included, the effective length of the closed pipe would be $l_1 + 0.6r$ and the open pipe would be $l_2 + 1.2r$ (where $r$ is the radius). The frequency equation would become $\frac{v}{4(l_1 + 0.6r)} = \frac{3v}{2(l_2 + 1.2r)}$, making the ratio of physical lengths dependent on the radius of the pipes.

Question

An organ pipe $P_1$ closed at one end vibrating in its first harmonic and another pipe $P_2$ open at both ends vibrating in its third harmonic are in resonance with a given tuning fork. The ratio of the length of $P_1$ and $P_2$ is

Options:

(A) $\frac{8}{3}$

(B) $\frac{3}{8}$

(C) $\frac{1}{6}$

(D) $\frac{1}{3}$

Sigma Percentile · Accepted Answer

### Introduction to Resonance in Air Columns

Imagine standing in a room with a tuning fork vibrating gently.

If you hold it over an empty tube, suddenly, at a very specific length, the sound swells into a rich, booming hum.

This is the magic of **resonance**—the physical phenomenon where an external driving force matches the natural frequency of a system, causing it to vibrate with maximum amplitude.

In this problem, we explore a classic JEE scenario: two different organ pipes, one closed at one end ($P_1$) and one open at both ends ($P_2$), vibrating in harmony with the exact same tuning fork.

Let's dive deep into the physics of standing waves to find the precise geometric relationship between their lengths.

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### Analyzing the Closed Pipe ($P_1$)

Let's first look at the closed organ pipe, $P_1$.

Because one end is closed, the air molecules at the bottom are restricted and cannot move; this forms a **displacement node**.

At the open top, however, the air is free to rush in and out, creating a **displacement antinode**.

For the fundamental mode (the first harmonic), this node-to-antinode transition spans the entire length of the pipe $l_1$.

Since the distance from a node to the nearest antinode is exactly a quarter of a wavelength ($\frac{\lambda_1}{4}$), we can write:

$$l_1 = \frac{\lambda_1}{4} \implies \lambda_1 = 4l_1$$

Using the wave speed equation $v = f \lambda$, the fundamental frequency $f_1$ of this closed pipe is:

$$f_1 = \frac{v}{4l_1}$$

where $v$ is the speed of sound in air.

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### Analyzing the Open Pipe ($P_2$)

Now, let's turn our attention to the open organ pipe, $P_2$.

Since both ends are open to the atmosphere, the air molecules at both boundaries are free to vibrate maximally, forming **displacement antinodes** at both ends.

For an open pipe of length $l_2$, the fundamental frequency is given by:

$$f_0 = \frac{v}{2l_2}$$

The harmonics of an open pipe are simple integer multiples of this fundamental frequency:

$$f_n = n \left(\frac{v}{2l_2}ight)$$

We are told that $P_2$ is vibrating in its **third harmonic** ($n = 3$). Substituting this value gives:

$$f_3 = \frac{3v}{2l_2}$$

This mode contains three displacement nodes and four displacement antinodes inside the pipe.

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### The Resonance Condition and Final Calculation

Since both pipes are in resonance with the same tuning fork, their frequencies of vibration must be identical to the tuning fork's frequency $f$:

$$f_1 = f_3 = f$$

Now, we substitute our expressions for $f_1$ and $f_3$ into this resonance condition:

$$\frac{v}{4l_1} = \frac{3v}{2l_2}$$

Since both pipes are filled with air at the same temperature, the speed of sound $v$ is identical in both cases and cancels out beautifully:

$$\frac{1}{4l_1} = \frac{3}{2l_2}$$

To find the ratio of the length of $P_1$ to $P_2$, we rearrange the terms:

$$\frac{l_1}{l_2} = \frac{2}{4 	imes 3} = \frac{2}{12} = \frac{1}{6}$$

Thus, the ratio of the lengths of the two pipes is **$1:6$**, which corresponds to **Option (c)**.

This elegant result shows that an open pipe must be six times longer than a closed pipe to resonate at its third harmonic with the same frequency!

Animated Solution for Physics - Waves: An organ pipe $P_{1}$ closed at one end vibrating in its first harmonic and another pipe $P_{2}$ open at both ends vibrating in its third harmonic are in resonance with a given tuning fork. The ratio of the length of $P_{1}$ and $P_{2}$ is

Select Answer:

Visualized Solution

Visualizing the Resonance Setup

Fundamental Frequency of Closed Pipe $P_{1}$

Third Harmonic of Open Pipe $P_{2}$

The Resonance Condition

Equating the Frequencies

Cancelling Common Terms

Solving for the Ratio $\frac{l _{1}}{l _{2}}$

Conclusion

Exploring Variations

The Sigma Insight: Standing Waves in Strings and Organ Pipes

Introduction to Resonance in Air Columns

Analyzing the Closed Pipe ( $P_{1}$ )

Analyzing the Open Pipe ( $P_{2}$ )

The Resonance Condition and Final Calculation

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