Velocity of secnod in air is 320 m/s. Neglecting end correctionss, the air column in the pipe can resonate for sound of frequency

Options

A. 80 Hz
B. 240 Hz
C. 320 Hz
D. 400 Hz

Answer

To solve the problem of finding the resonant frequencies of an air column in a closed pipe, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Problem**: We know the speed of sound in air (V) is 320 m/s. We need to determine the resonant frequencies of the air column in a closed organ pipe.

2. **Identify the Fundamental Frequency**: For a closed organ pipe, the fundamental frequency (first harmonic) occurs when there is one node at the closed end and one antinode at the open end. The length of the pipe (L) is equal to one quarter of the wavelength (λ) of the sound wave:
\[ L = \frac{\lambda}{4} \]

3. **Express Wavelength in Terms of Length**: Rearranging the equation gives:
\[ \lambda = 4L \]

4. **Use the Frequency-Wavelength Relationship**: The frequency (f) of a wave is related to its speed and wavelength by the equation:
\[ f = \frac{V}{\lambda} \]
Substituting the expression for λ from the previous step:
\[ f = \frac{V}{4L} \]

5. **Calculate the Fundamental Frequency**: If we assume the length of the pipe (L) is 1 meter (as an example), we can calculate the fundamental frequency:
\[ f_1 = \frac{320 \, \text{m/s}}{4 \times 1 \, \text{m}} = \frac{320}{4} = 80 \, \text{Hz} \]

6. **Identify Higher Harmonics**: The frequencies of the higher harmonics (odd harmonics) can be expressed as:
\[ f_n = \frac{(2n + 1)V}{4L} \]
where \( n = 0, 1, 2, \ldots \)

7. **Calculate the First Few Frequencies**:
- For \( n = 0 \):
\[ f_1 = 80 \, \text{Hz} \]
- For \( n = 1 \):
\[ f_3 = \frac{3 \times 320}{4 \times 1} = \frac{960}{4} = 240 \, \text{Hz} \]
- For \( n = 2 \):
\[ f_5 = \frac{5 \times 320}{4 \times 1} = \frac{1600}{4} = 400 \, \text{Hz} \]

8. **Conclusion**: The resonant frequencies of the air column in the pipe are 80 Hz, 240 Hz, and 400 Hz, corresponding to the first, third, and fifth harmonics, respectively.