Question

Two pipes of lengths $\text{L}_1$ and $\text{L}_2$, open at both ends are joined in series. If ' $f_1$ ' and ' $f_2$ ' are the fundamental frequencies of two pipes, then the fundamental frequency of series combination will be (neglect end correction)}

• A. $\frac{f_1 f_2}{f_1 - f_2}$
• B. $\text{f}_1 + \text{f}_2$
• C. $\frac{f_1 f_2}{f_1+f_2}$
• D. $\sqrt{\text{f}_1^2 + \text{f}_2^2}$

Hint:

When you add lengths, the frequencies combine like resistors in parallel.

Answer 1

The Correct Option is C

Step 1: Concept
For an open pipe, the fundamental frequency is $f = \frac{v}{2L}$.

Step 2: Meaning
Total length $L = L_1 + L_2$. We know $L_1 = \frac{v}{2f_1}$ and $L_2 = \frac{v}{2f_2}$.

Step 3: Analysis
New frequency $F = \frac{v}{2(L_1 + L_2)}$.
$1/F = \frac{2(L_1 + L_2)}{v} = \frac{2L_1}{v} + \frac{2L_2}{v} = 1/f_1 + 1/f_2$.
$1/F = \frac{f_1 + f_2}{f_1 f_2} \implies F = \frac{f_1 f_2}{f_1 + f_2}$.

Step 4: Conclusion
The combined frequency is $\frac{f_1 f_2}{f_1+f_2}$. Final Answer: (C)