In an experiment to measure the speed of sound by a resonating air column, a tuning fork of frequenc

Q: Why does the difference between successive resonance lengths equal half a wavelength?

In a closed organ pipe, resonances occur at odd multiples of quarter-wavelengths: $\frac{\lambda}{4}, \frac{3\lambda}{4}, \frac{5\lambda}{4}, \dots$. The difference between any two consecutive odd multiples is always $\frac{2\lambda}{4} = \frac{\lambda}{2}$.

Q: Does the end correction affect the calculation of the wavelength?

No! Because the end correction $e$ is present at the open end for both resonance lengths, it cancels out when we subtract the two equations: $(l_2 + e) - (l_1 + e) = l_2 - l_1 = \frac{\lambda}{2}$.

Q: Can an end correction be negative?

Physically, for an ideal open end, the end correction is positive ($e \approx 0.6r$). A negative end correction in calculations usually indicates experimental error or specific acoustic impedance variations at the boundary.

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

Harmonics are integer multiples of the fundamental frequency. Overtones are the resonant frequencies above the fundamental. For a closed pipe, the 1st overtone is the 3rd harmonic.

Q: Why is the water surface considered a node?

Water is much denser than air, so air molecules cannot vibrate freely at the water surface. This boundary restricts displacement, creating a displacement node.

Question

In an experiment to measure the speed of sound by a resonating air column, a tuning fork of frequency $500	ext{ Hz}$ is used. The length of the air column is varied by changing the level of water in the resonance tube. Two successive resonances are heard at air columns of length $50.7	ext{ cm}$ and $83.9	ext{ cm}$. Which of the following statements is (are) true?

Options:

(A) The speed of sound determined from this experiment is $332	ext{ m s}^{-1}$.

(B) The end correction in this experiment is $0.9	ext{ cm}$.

(C) The wavelength of the sound wave is $66.4	ext{ cm}$.

(D) The resonance at $50.7	ext{ cm}$ corresponds to the fundamental harmonic.

Sigma Percentile · Accepted Answer

### Analyzing the Setup

Imagine standing in a physics lab, holding a vibrating tuning fork over a tall cylindrical tube filled with water. As you slowly lower the water level, the air column inside the tube lengthens. Suddenly, at a very specific height, the sound amplifies into a rich, resonant roar. This is the magic of **standing waves**.

In this problem, we are given a tuning fork vibrating at a frequency of $f = 500	ext{ Hz}$. The water surface acts as a rigid boundary, creating a displacement node at the bottom of the air column, while the open top of the tube behaves as a displacement antinode. This system is a classic example of an organ pipe closed at one end.

We are told that two successive resonances are heard when the air column lengths are:

$$l_1 = 50.7	ext{ cm}$$

$$l_2 = 83.9	ext{ cm}$$

Our goal is to analyze these resonances to determine the wavelength of the sound wave, the speed of sound, the harmonic order, and the end correction of the tube.

---

### The Master Equation: Successive Resonances

For any closed organ pipe, the resonance lengths correspond to odd multiples of quarter-wavelengths:

$$L = (2n-1)\frac{\lambda}{4}$$

Because successive resonances occur at consecutive odd integers (e.g., $1, 3, 5, \dots$), the difference in length between any two successive resonances is always exactly **half a wavelength**:

$$\Delta l = l_2 - l_1 = \frac{\lambda}{2}$$

This is an incredibly elegant shortcut because it completely bypasses the end correction at the open end! Let's substitute our given values into this relation:

$$\frac{\lambda}{2} = 83.9	ext{ cm} - 50.7	ext{ cm} = 33.2	ext{ cm}$$

Solving for the wavelength $\lambda$:

$$\lambda = 2 	imes 33.2	ext{ cm} = 66.4	ext{ cm}$$

This immediately confirms that **Option (c)** is correct!

---

### Calculating the Speed of Sound

Now that we have the wavelength, we can easily find the speed of sound using the fundamental wave equation:

$$v = f \lambda$$

First, let's convert the wavelength into standard SI units (meters):

$$\lambda = 0.664	ext{ m}$$

Now, substitute the frequency $f = 500	ext{ Hz}$:

$$v = 500	ext{ Hz} 	imes 0.664	ext{ m} = 332	ext{ m s}^{-1}$$

This matches **Option (a)** perfectly, making it correct as well!

---

### Identifying the Harmonic Order

To check if the resonance at $50.7	ext{ cm}$ corresponds to the fundamental harmonic, let's calculate the theoretical fundamental length (one-quarter of a wavelength):

$$\frac{\lambda}{4} = \frac{66.4	ext{ cm}}{4} = 16.6	ext{ cm}$$

If $l_1$ were the fundamental resonance, its length would be close to $16.6	ext{ cm}$. However, our first resonance occurs at $50.7	ext{ cm}$. Let's check the next harmonic (the 3rd harmonic or 1st overtone):

$$3 	imes \frac{\lambda}{4} = 3 	imes 16.6	ext{ cm} = 49.8	ext{ cm}$$

Since $50.7	ext{ cm}$ is very close to $49.8	ext{ cm}$, the resonance at $50.7	ext{ cm}$ corresponds to the **3rd harmonic** (first overtone), not the fundamental. Thus, **Option (d)** is false.

---

### Finding the End Correction

Finally, let's calculate the end correction $e$. The acoustic length of the air column for the 3rd harmonic is:

$$l_1 + e = \frac{3\lambda}{4}$$

Substituting our values:

$$50.7	ext{ cm} + e = 49.8	ext{ cm}$$

$$e = 49.8 - 50.7 = -0.9	ext{ cm}$$

Since the end correction is mathematically negative (which can happen due to experimental variations or specific tube geometries, though physically it is typically positive), **Option (b)** is false because it states the end correction is $+0.9	ext{ cm}$.

### Summary of Correct Options

* **Option (a)** is **True** ($v = 332	ext{ m s}^{-1}$).
* **Option (c)** is **True** ($\lambda = 66.4	ext{ cm}$).

Select Answer:

Visualized Solution

Visualizing the Resonance Column

The Principle of Successive Resonances

Calculating the Wavelength

Verifying Wavelength (Option C)

Calculating the Speed of Sound

Verifying Speed of Sound (Option A)

Identifying the Harmonics

Verifying the 3rd Harmonic (Option D)

Calculating the End Correction

Verifying End Correction (Option B)

Final Conclusion

The Sigma Insight: Standing Waves in Strings and Organ Pipes

Analyzing the Setup

The Master Equation

Calculating the Speed of Sound

Identifying the Harmonic Order

Finding the End Correction

Summary of Correct Options

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