In the first overtone, an organ pipe that is open at one end and closed at the other end produces the note concert A, of 440 Hz. (a) find the frequency and wavelength of the fundamental mode. (b) What will the fundamental frequency be if (i) the open end is now closed; (ii) both ends are now open?
To solve this problem, we need to understand the principles of the overtone series and the equation that relates frequency, wavelength, and the speed of sound in a medium.
(a) Finding the frequency and wavelength of the fundamental mode:
The fundamental mode is the first harmonic or the first overtone. In this case, the note produced by the organ pipe is concert A, with a frequency of 440 Hz.
We know that the fundamental frequency (f1) is the first harmonic, which is given by:
f1 = (v / λ1)
Here, v is the speed of sound in air, and λ1 is the wavelength of the first mode.
The speed of sound in air is approximately 343 meters per second.
To find the wavelength (λ1), we can use the formula:
λ1 = 2L
For an organ pipe that is open at one end and closed at the other end, the length of the pipe (L) is equal to a quarter of the wavelength of the note produced by this pipe.
Let's substitute the given values into the formulas:
f1 = (343 m/s / λ1)
440 Hz = (343 m/s / λ1)
Now, we can isolate λ1:
λ1 = 343 m/s / 440 Hz
λ1 ≈ 0.7795 m
So, the wavelength of the fundamental mode is approximately 0.7795 meters.
(b) Determining the fundamental frequency in different scenarios:
(i) If the open end is now closed:
When the open end of the pipe is closed and the other end remains closed, the pipe's length becomes half the wavelength (L = λ/2). In this case, we need to find the new fundamental frequency (f1').
Using the formula for frequency, we have:
f1' = (v / λ')
f1' = (343 m/s / λ/2)
f1' = 2 * (343 m/s / λ)
Since λ is the wavelength of the fundamental mode in scenario (a) and L = λ/2:
f1' = 2 * (343 m/s / λ1)
f1' = 2 * (343 m/s / 0.7795 m)
f1' ≈ 881.60 Hz
So, if the open end is closed, the new fundamental frequency will be approximately 881.60 Hz.
(ii) If both ends are now open:
When both ends of the pipe are open, the length of the pipe becomes equal to half the wavelength (L = λ/2), just like in scenario (i). Therefore, we can use the same formula as in scenario (i) to find the new fundamental frequency (f1'').
Using the formula for frequency, we have:
f1'' = (v / λ'')
f1'' = (343 m/s / λ/2)
f1'' = 2 * (343 m/s / λ)
Since λ is the wavelength of the fundamental mode in scenario (a) and L = λ/2:
f1'' = 2 * (343 m/s / λ1)
f1'' = 2 * (343 m/s / 0.7795 m)
f1'' ≈ 881.60 Hz
So, if both ends are open, the new fundamental frequency will also be approximately 881.60 Hz.