Two open pipes `A` and `B` are sounded together such that beats are heard between the first overtone of `A` and second overtone of `B`. If the fundamental frequency of `A` and `B` is `256 Hz` and `170 Hz` respectively , then the beat frequency heard is

Options

A. `4 Hz`
B. `3 Hz`
C. `2 Hz`
D. `1 Hz`

Explanation

To solve the problem, we need to determine the beat frequency between the first overtone of pipe A and the second overtone of pipe B. Here are the steps to find the solution:

### Step 1: Understand the definitions
- The **fundamental frequency** (first harmonic) of a pipe is denoted as \( F_0 \).
- The **first overtone** (second harmonic) of a pipe is \( 2F_0 \).
- The **second overtone** (third harmonic) of a pipe is \( 3F_0 \).

### Step 2: Identify the frequencies
For pipe A:
- Fundamental frequency \( F_0(A) = 256 \, \text{Hz} \)
- First overtone \( F_1(A) = 2F_0(A) = 2 \times 256 = 512 \, \text{Hz} \)

For pipe B:
- Fundamental frequency \( F_0(B) = 170 \, \text{Hz} \)
- Second overtone \( F_2(B) = 3F_0(B) = 3 \times 170 = 510 \, \text{Hz} \)

### Step 3: Calculate the beat frequency
The beat frequency is calculated as the absolute difference between the two frequencies:
\[ \text{Beat frequency} = |F_1(A) - F_2(B)| = |512 \, \text{Hz} - 510 \, \text{Hz}| \]
\[ \text{Beat frequency} = |2 \, \text{Hz}| = 2 \, \text{Hz} \]

### Step 4: Conclusion
The beat frequency heard is \( 2 \, \text{Hz} \).

### Final Answer:
The beat frequency is \( 2 \, \text{Hz} \).

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