A closed organ pipe (closed at one end) has a length of 8.67 x 10^-1 m.
(a) What is the fundamental frequency of this pipe?
(b) What is the frequency of the first overtone of this pipe?
(c) What is the frequency of the second overtone of this pipe?
closed: lambda/2=8.67e-1
then f=speed sound/lambad
the fundamental wavelength λ1 = 2L . The first overtone adds one node as here, where the vibrating length is one wavelength. Thus, the wavelength here is: λ2 = L . The second overtone (third harmonic) has one further node: Thus, λ3 = 2/3 L.
To find the fundamental frequency of a closed organ pipe, we can use the formula:
f1 = v / (4L)
Where f1 is the fundamental frequency, v is the speed of sound, and L is the length of the pipe.
(a) To find the fundamental frequency:
Step 1: Determine the speed of sound.
The speed of sound depends on the medium through which it is traveling, such as air or water. For example, in air at room temperature, the speed of sound is approximately 343 m/s.
Step 2: Substitute the values into the formula.
f1 = v / (4L)
f1 = 343 m/s / (4 * 8.67 x 10^-1 m)
f1 ≈ 343 m/s / 3.47 m
f1 ≈ 98.85 Hz
Therefore, the fundamental frequency of the closed organ pipe is approximately 98.85 Hz.
(b) The first overtone is the second harmonic frequency. To find the frequency of the first overtone:
Step 1: Calculate the frequency of the fundamental tone (f1).
Using the formula from part (a), we found that f1 ≈ 98.85 Hz.
Step 2: Multiply the frequency of the fundamental tone by 2.
f2 = 2 * f1
f2 ≈ 2 * 98.85 Hz
f2 ≈ 197.7 Hz
Therefore, the frequency of the first overtone of the closed organ pipe is approximately 197.7 Hz.
(c) The second overtone is the third harmonic frequency. To find the frequency of the second overtone:
Step 1: Calculate the frequency of the fundamental tone (f1).
Using the formula from part (a), we found that f1 ≈ 98.85 Hz.
Step 2: Multiply the frequency of the fundamental tone by 3.
f3 = 3 * f1
f3 ≈ 3 * 98.85 Hz
f3 ≈ 296.55 Hz
Therefore, the frequency of the second overtone of the closed organ pipe is approximately 296.55 Hz.