What are the first, second, and third frequencies audible from a 20 cm long organ pipe when (A) only one end is open and when (B) both ends are open. The speed of sound through the air inside the organ pipe is 343m/s. (Hint: if a harmonic does not exist, it would not be heard.)
To find the frequencies audible from the organ pipe, we need to consider the different harmonics that can be produced. In the case of an open-ended organ pipe, the harmonics are given by the equation:
f = (2n - 1) v/4L
Where:
- f is the frequency of the harmonic
- n is the harmonic number (1, 2, 3, ...)
- v is the speed of sound in air
- L is the length of the organ pipe
Now let's consider the two cases:
(A) When only one end is open:
In this case, the pipe acts as a closed cylinder, and only odd-numbered harmonics are produced. So, we will calculate the frequencies for n = 1, 2, and 3.
For n = 1:
f1 = (2*1 - 1) * 343/(4 * 0.2)
f1 = 342.75 Hz
For n = 2:
f2 = (2*2 - 1) * 343/(4 * 0.2)
f2 = 685.5 Hz
For n = 3:
f3 = (2*3 - 1) * 343/(4 * 0.2)
f3 = 1028.25 Hz
So, the first, second, and third frequencies audible from the 20 cm long organ pipe with one end open are approximately 342.75 Hz, 685.5 Hz, and 1028.25 Hz respectively.
(B) When both ends are open:
In this case, the pipe acts as an open cylinder, and all the harmonics are produced. So, we will calculate the frequencies for n = 1, 2, and 3.
For n = 1:
f1 = (2*1 - 1) * 343/(4 * 0.2)
f1 = 342.75 Hz
For n = 2:
f2 = (2*2 - 1) * 343/(4 * 0.2)
f2 = 685.5 Hz
For n = 3:
f3 = (2*3 - 1) * 343/(4 * 0.2)
f3 = 1028.25 Hz
So, the first, second, and third frequencies audible from the 20 cm long organ pipe with both ends open are also approximately 342.75 Hz, 685.5 Hz, and 1028.25 Hz respectively.
Therefore, in both cases, the frequencies audible are the same because the length of the organ pipe does not change.