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 of the first, second, and third harmonics in an organ pipe, we use the formula:
f = (n * v) / (2L),
where f is the frequency, n is the harmonic number, v is the speed of sound, and L is the length of the pipe.
(A) When only one end of the organ pipe is open:
1. The first harmonic (n = 1) is the fundamental frequency:
f1 = (1 * v) / (2L)
2. The second harmonic (n = 2):
f2 = (2 * v) / (2L)
3. The third harmonic (n = 3):
f3 = (3 * v) / (2L)
(B) When both ends of the organ pipe are open:
1. The first harmonic (n = 1) is the fundamental frequency:
f1' = (1 * v) / (2L)
2. The second harmonic (n = 2):
f2' = (2 * v) / (2L)
3. The third harmonic (n = 3):
f3' = (3 * v) / (2L)
Now, we can substitute the given values into the formulas to find the frequencies.
Given:
L = 20 cm = 0.20 m
v = 343 m/s
(A) When only one end is open:
f1 = (1 * 343) / (2 * 0.20)
f2 = (2 * 343) / (2 * 0.20)
f3 = (3 * 343) / (2 * 0.20)
Simplifying:
f1 = 1715 Hz
f2 = 3430 Hz
f3 = 5145 Hz
(B) When both ends are open:
f1' = (1 * 343) / (2 * 0.20)
f2' = (2 * 343) / (2 * 0.20)
f3' = (3 * 343) / (2 * 0.20)
Simplifying:
f1' = 1715 Hz
f2' = 3430 Hz
f3' = 5145 Hz
So, the first, second, and third frequencies audible from a 20 cm long organ pipe are the same for both cases: 1715 Hz, 3430 Hz, and 5145 Hz.
To determine the frequencies audible from a 20 cm long organ pipe, we need to consider the different harmonics that can be produced. Let's look at each case separately:
A) When only one end of the pipe is open:
In this case, the organ pipe acts as a closed-open tube. The first harmonic is the fundamental frequency, and it is given by the formula:
f₁ = (v / 2L)
where f₁ is the frequency, v is the speed of sound (343 m/s), and L is the length of the pipe (20 cm = 0.2 m).
Substituting the values into the formula, we get:
f₁ = (343 m/s) / (2 * 0.2 m) = 855 Hz
So, the first frequency audible from the 20 cm long organ pipe is 855 Hz.
The second harmonic (first overtone) is not possible in a closed-open tube, so we move on to the third harmonic.
For a closed-open tube, the third harmonic is given by the formula:
f₃ = (3v / 4L)
Substituting the values into the formula, we get:
f₃ = (3 * 343 m/s) / (4 * 0.2 m) = 1287.75 Hz
Since we are dealing with sound frequencies, we round it to the nearest whole number:
f₃ ≈ 1288 Hz
So, the third frequency audible from the 20 cm long organ pipe when only one end is open is approximately 1288 Hz.
B) When both ends of the pipe are open:
In this case, the organ pipe acts as an open-open tube. The harmonics follow a different pattern compared to the closed-open tube.
The formula for the frequencies of an open-open tube is:
fₙ = (nv / 2L)
where n is the harmonic number.
For the open-open tube, the first harmonic (fundamental frequency) is given by:
f₁ = (v / 2L)
Substituting the values into the formula, we get:
f₁ = (343 m/s) / (2 * 0.2 m) = 855 Hz
So, the first frequency audible from the 20 cm long organ pipe when both ends are open is 855 Hz, which is the same as in the closed-open case.
The second harmonic for an open-open tube is:
f₂ = (2v / 2L) = (v / L)
Substituting the values into the formula, we get:
f₂ = (343 m/s) / (0.2 m) = 1715 Hz
So, the second frequency audible from the 20 cm long organ pipe when both ends are open is 1715 Hz.
The third harmonic for an open-open tube is:
f₃ = (3v / 2L)
Substituting the values into the formula, we get:
f₃ = (3 * 343 m/s) / (2 * 0.2 m) = 2572.5 Hz
Since we are dealing with sound frequencies, we round it to the nearest whole number:
f₃ ≈ 2573 Hz
So, the third frequency audible from the 20 cm long organ pipe when both ends are open is approximately 2573 Hz.
In summary:
When only one end of the pipe is open:
- First frequency: 855 Hz
- Third frequency: 1288 Hz
When both ends of the pipe are open:
- First frequency: 855 Hz
- Second frequency: 1715 Hz
- Third frequency: 2573 Hz