Question:
The lengths of the two pipes open at both ends are L and \((L+L_1)\). If they are sounded together, the beat frequency will be
(\(v\) = velocity of sound in air)
The lengths of the two pipes open at both ends are L and \((L+L_1)\). If they are sounded together, the beat frequency will be
(\(v\) = velocity of sound in air)
(\(v\) = velocity of sound in air)
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Fundamental frequency of an open pipe is v/2L; subtract the two frequencies.
Updated On: Oct 1, 2026
- \(\frac{vL_1}{L(L+L_1)}\)
- \(\frac{2vL_1}{L(L+L_1)}\)
- \(\frac{2L(L+L_1)}{vL_1}\)
- \(\frac{vL_1}{2L(L+L_1)}\)
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The Correct Option is D
Solution and Explanation
Step 1: Understanding the Concept:
The fundamental frequency of an open pipe of length \(L\) is \(f = \frac{v}{2L}\). Beat frequency is the difference between the two frequencies.
Step 2: Key Formula or Approach:
\(f_1 = \frac{v}{2L}\) and \(f_2 = \frac{v}{2(L + L_1)}\). The shorter pipe has the higher frequency.
Step 3: Detailed Explanation:
\[ f_b = f_1 - f_2 = \frac v2\left[\frac1L - \frac{1}{L + L_1}\right] = \frac v2 \cdot\frac{L_1}{L(L + L_1)} = \frac{vL_1}{2L(L + L_1)} \]
Option A is missing the factor of \(2\) that comes from the open pipe formula. Option B has an extra factor of \(2\) in the numerator, which does not come out of the algebra.
Final Answer:
The beat frequency is \(\frac{vL_1}{2L(L + L_1)}\), option (D). \[ \boxed{\frac{vL_1}{2L(L+L_1)}} \]
The fundamental frequency of an open pipe of length \(L\) is \(f = \frac{v}{2L}\). Beat frequency is the difference between the two frequencies.
Step 2: Key Formula or Approach:
\(f_1 = \frac{v}{2L}\) and \(f_2 = \frac{v}{2(L + L_1)}\). The shorter pipe has the higher frequency.
Step 3: Detailed Explanation:
\[ f_b = f_1 - f_2 = \frac v2\left[\frac1L - \frac{1}{L + L_1}\right] = \frac v2 \cdot\frac{L_1}{L(L + L_1)} = \frac{vL_1}{2L(L + L_1)} \]
Option A is missing the factor of \(2\) that comes from the open pipe formula. Option B has an extra factor of \(2\) in the numerator, which does not come out of the algebra.
Final Answer:
The beat frequency is \(\frac{vL_1}{2L(L + L_1)}\), option (D). \[ \boxed{\frac{vL_1}{2L(L+L_1)}} \]
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