The length of a pipe open at both ends is 40 cm. A 1.275 kHz source will resonantly excite which harmonic mode of pipe ? If we close one end of the pipe, will resonance again occur with same source ? Speed of sound in air is 340 m/s.

Answer

To solve the problem, we will follow these steps:

### Step 1: Understand the Harmonics in an Open Pipe
For a pipe open at both ends, the harmonic frequencies are given by:
\[ f_n = \frac{n \cdot v}{2L} \]
where:
- \( f_n \) is the frequency of the nth harmonic,
- \( n \) is the harmonic number (1, 2, 3, ...),
- \( v \) is the speed of sound in air (340 m/s),
- \( L \) is the length of the pipe (in meters).

### Step 2: Convert Length of the Pipe to Meters
The length of the pipe is given as 40 cm. We need to convert this to meters:
\[ L = 40 \, \text{cm} = 0.40 \, \text{m} \]

### Step 3: Calculate the Harmonic Number for the Given Frequency
The frequency of the source is given as 1.275 kHz, which we convert to Hz:
\[ f = 1.275 \, \text{kHz} = 1275 \, \text{Hz} \]

Now, we can rearrange the harmonic frequency formula to find \( n \):
\[ n = \frac{2Lf}{v} \]
Substituting the values:
\[ n = \frac{2 \cdot 0.40 \cdot 1275}{340} \]

### Step 4: Calculate the Value of \( n \)
Calculating the above expression:
\[ n = \frac{2 \cdot 0.40 \cdot 1275}{340} = \frac{1020}{340} \approx 3 \]
Thus, the harmonic mode of the pipe is the **3rd harmonic**.

### Step 5: Analyze the Case When One End of the Pipe is Closed
When one end of the pipe is closed, the harmonics are given by:
\[ f_n = \frac{(2n-1) \cdot v}{4L} \]
where \( n \) is a natural number (1, 2, 3, ...).

### Step 6: Check for Resonance with the Same Frequency
We rearrange the formula to find \( n \):
\[ 2n - 1 = \frac{4Lf}{v} \]
Substituting the known values:
\[ 2n - 1 = \frac{4 \cdot 0.40 \cdot 1275}{340} \]
Calculating:
\[ 2n - 1 = \frac{2040}{340} \approx 6 \]
Thus:
\[ 2n = 7 \implies n = 3.5 \]
Since \( n \) must be a natural number, **resonance will not occur** when one end of the pipe is closed.

### Final Answers:
1. The harmonic mode of the pipe when open at both ends is the **3rd harmonic**.
2. When one end of the pipe is closed, resonance will **not occur** with the same source.

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