Question 250: Waves

Question

An open organ pipe is closed such that the third overtone of the closed pipe is found to be higher in frequency by 200 Hz than the second overtone of the original pipe. The fundamental frequency of the open pipe is (Neglect end correction)

Accepted Answer

For an open organ pipe, the frequency of the nth overtone is given by the formula:
$ f_n = (n + 1) 	imes f_1 $
where $ f_1 $ is the fundamental frequency. For an open pipe, $ f_1 = \frac{v}{2L} $, where $ v $ is the speed of sound and $ L $ is the length of the pipe.
Let's denote the fundamental frequency of the open pipe as $ f_o $.
For a closed organ pipe, the frequency of the nth overtone is given by the formula:
$ f_n = \left(2n + 1ight) 	imes f_c $
where $ f_c $ is the fundamental frequency for the closed pipe, which is half of that for the open pipe $ f_c = \frac{f_o}{2} $.
Now, according to the problem:
The third overtone of the closed pipe: this is actually the 4th harmonic, since harmonics for a closed pipe go as 1st, 3rd, 5th, etc. So the frequency is:
$ f_3^{	ext{closed}} = (2 	imes 3 + 1) 	imes f_c = 7 	imes \frac{f_o}{2} = \frac{7f_o}{2} $
The second overtone of the original open pipe: this is the 3rd harmonic of the open pipe:
$ f_2^{	ext{open}} = (2 + 1) 	imes f_o = 3f_o $
According to the problem, the third overtone of the closed pipe is 200 Hz higher than the second overtone of the open pipe:
$ \frac{7f_o}{2} = 3f_o + 200 $
Let's solve this equation:
$ \frac{7f_o}{2} = 3f_o + 200 $
Multiplying everything by 2 to get rid of the fraction:
$ 7f_o = 6f_o + 400 $
Subtract $ 6f_o $ from both sides:
$ f_o = 400 $
The fundamental frequency of the open pipe is 400 Hz.
Thus, the correct answer is:
Option C: 400 Hz

Answer

150 Hz

Answer

200 Hz

Answer

400 Hz

Answer

500 Hz

Waves

An open organ pipe is closed such that the third overtone of the closed pipe is found to be higher in frequency by 200 Hz than the second overtone of the original pipe. The fundamental frequency of the open pipe is (Neglect end correction)

Choose the correct answer: