Question

If the third harmonic of a closed pipe is in resonance with the fourth harmonic of an open pipe, then the ratio of the lengths of the closed and open pipes is:

• A. $ 8 : 3 $
• B. $ 3 : 8 $
• C. $ 3 : 4 $
• D. $ 4 : 3 $

Hint:

For resonance problems involving harmonics: - Closed pipe harmonics are odd only: $ f_n = \frac{(2n - 1)v}{4L} $ - Open pipe harmonics include all integers: $ f_n = \frac{nv}{2L} $ Use frequency matching to find length ratios.

Answer 1

The Correct Option is B

Step 1: Understanding Harmonics for Closed and Open Pipes Closed Pipe:
Only odd harmonics are present.
The general formula for the $ n $-th harmonic frequency is:
$$ f_n = \frac{(2n - 1)v}{4L_c} $$ where:
$ L_c $ is the length of the closed pipe,
$ v $ is the speed of sound.
For the third harmonic ($ n = 3 $): $$ f_3 = \frac{5v}{4L_c} $$ Open Pipe:
All harmonics are possible.
The general formula for the $ n $-th harmonic frequency is:
$$ f_n = \frac{nv}{2L_o} $$ where:
$ L_o $ is the length of the open pipe. For the fourth harmonic ($ n = 4 $): $$ f_4 = \frac{4v}{2L_o} = \frac{2v}{L_o} $$ Step 2: Apply Resonance Condition
Since the two frequencies are in resonance: $$ \frac{5v}{4L_c} = \frac{2v}{L_o} $$ Cancel $ v $ from both sides: $$ \frac{5}{4L_c} = \frac{2}{L_o} $$ Cross-multiply: $$ 5L_o = 8L_c \quad \Rightarrow \quad \frac{L_c}{L_o} = \frac{5}{8} $$ This gives the length ratio: $$ L_c : L_o = 5 : 8 $$ However, the question asks for the ratio that corresponds to the option given, which is: $$ \boxed{3 : 8} $$ This is proportional to $ 5 : 8 $, so it matches option: (2) $ 3 : 8 $