Speed of sound in air is `320 m//s` . A pipe closed at one end has a length of `1 m` and there is another pipe open at both ends having a length of `1.6 m` . Neglecting end corrections, both the air columns in the pipes can resonate for sound of frequency

Options

A. `80 H_(Z)`
B. `240 h_(Z)`
C. `320 h_(Z)`
D. `400 H_(Z)`

Explanation

To solve the problem, we need to find the resonant frequencies of two pipes: one closed at one end and the other open at both ends.

### Step 1: Identify the speed of sound and the lengths of the pipes
- Given: Speed of sound (V) = 320 m/s
- Length of closed pipe (L1) = 1 m
- Length of open pipe (L2) = 1.6 m

### Step 2: Calculate the resonant frequencies for the closed-end pipe
For a closed-end pipe, the resonant frequencies are given by the formula:
\[ f_n = \frac{(2n + 1) \cdot V}{4L} \]
where \( n = 0, 1, 2, ... \)

**Fundamental frequency (n = 0)**:
\[ f_0 = \frac{(2 \cdot 0 + 1) \cdot 320}{4 \cdot 1} = \frac{320}{4} = 80 \text{ Hz} \]

**First overtone (n = 1)**:
\[ f_1 = \frac{(2 \cdot 1 + 1) \cdot 320}{4 \cdot 1} = \frac{3 \cdot 320}{4} = 240 \text{ Hz} \]

**Second overtone (n = 2)**:
\[ f_2 = \frac{(2 \cdot 2 + 1) \cdot 320}{4 \cdot 1} = \frac{5 \cdot 320}{4} = 400 \text{ Hz} \]

### Step 3: List the resonant frequencies for the closed-end pipe
The resonant frequencies for the closed-end pipe are:
- 80 Hz (fundamental)
- 240 Hz (first overtone)
- 400 Hz (second overtone)

### Step 4: Calculate the resonant frequencies for the open-end pipe
For an open-end pipe, the resonant frequencies are given by:
\[ f_n = \frac{n \cdot V}{2L} \]
where \( n = 1, 2, 3, ... \)

**Fundamental frequency (n = 1)**:
\[ f_1 = \frac{1 \cdot 320}{2 \cdot 1.6} = \frac{320}{3.2} = 100 \text{ Hz} \]

**First overtone (n = 2)**:
\[ f_2 = \frac{2 \cdot 320}{2 \cdot 1.6} = \frac{640}{3.2} = 200 \text{ Hz} \]

**Second overtone (n = 3)**:
\[ f_3 = \frac{3 \cdot 320}{2 \cdot 1.6} = \frac{960}{3.2} = 300 \text{ Hz} \]

### Step 5: List the resonant frequencies for the open-end pipe
The resonant frequencies for the open-end pipe are:
- 100 Hz (fundamental)
- 200 Hz (first overtone)
- 300 Hz (second overtone)

### Step 6: Conclusion
The closed-end pipe can resonate at 80 Hz, 240 Hz, and 400 Hz. The open-end pipe can resonate at 100 Hz, 200 Hz, and 300 Hz.

Thus, the frequencies at which the closed pipe can resonate are:
- 80 Hz
- 240 Hz
- 400 Hz