Question

The fundamental frequency of sound produced in an open pipe of length \(L_1\) is same as the frequency of the 3rd harmonic of the sound produced in the closed pipe of length \(L_2\). Then the ratio of \(\frac{L_1}{L_2}\) is:

• A. \(\frac{L_1}{L_2} = \frac{3}{1}\)
• B. \(\frac{L_1}{L_2} = \frac{1}{3}\)
• C. \(\frac{L_1}{L_2} = \frac{2}{3}\)
• D. \(\frac{L_1}{L_2} = \frac{3}{2}\)

Hint:

Open pipe fundamental: \(f = \frac{v}{2L}\); Closed pipe odd harmonics: \(f = \frac{nv}{4L}\) where \(n=1,3,5,\dots\).

Answer 1

The Correct Option is C

Step 1: Write fundamental frequency of open pipe.
For an open pipe, fundamental frequency is:
\[ f_{\text{open}} = \frac{v}{2L_1} \]

Step 2: Write frequency of 3rd harmonic in closed pipe.
For a closed pipe, only odd harmonics are present:
\[ f_n = \frac{nv}{4L_2}, \quad n = 1,3,5,\dots \]
So, for 3rd harmonic:
\[ f_{\text{closed}} = \frac{3v}{4L_2} \]

Step 3: Equate the two frequencies.
\[ \frac{v}{2L_1} = \frac{3v}{4L_2} \]

Step 4: Cancel common terms.
\[ \frac{1}{2L_1} = \frac{3}{4L_2} \]

Step 5: Cross multiply.
\[ 4L_2 = 6L_1 \]

Step 6: Find ratio.
\[ \frac{L_1}{L_2} = \frac{4}{6} \]
\[ \frac{L_1}{L_2} = \frac{2}{3} \]

Step 7: Final answer.
\[ \boxed{\frac{2}{3}} \]