A tube that is open on both ends is 4 meters long. Assuming that the speed of sound is 340m/s, find the following:
a) What is the wavelength of the fundamental standing wave (the first harmonic)
b) What is the frequency of the fundamental wave?
c) What is the frequency of the second harmonic?
a. Wavelength = 2 * 4m = 8 Meters.
b. Wavelength = V/F1 = 8m.
340/F = 8, F1 = ?.
F1 = Frequency of 1st harmonic.
c. F2 = 2 * F1.
To find the answers to these questions, we need to use the formula that relates the speed of sound, the wavelength, and the frequency of a wave. The formula is:
v = λ * f
Where:
v is the speed of sound,
λ is the wavelength, and
f is the frequency.
Let's go step by step:
a) To find the wavelength of the fundamental standing wave (the first harmonic), we can use the formula above. The tube is open on both ends, so the fundamental standing wave corresponds to half a wavelength. Hence, we need to find the half-wavelength.
The length of the tube is 4 meters, so the whole wavelength (λ) will be twice that, or 8 meters. Since we need the half-wavelength, we divide the whole wavelength by 2.
Therefore, the half-wavelength (λ/2) is 8 meters / 2 = 4 meters.
b) To find the frequency of the fundamental wave, we rearrange the formula:
f = v / λ
Using the given speed of sound, 340 m/s, and the half-wavelength we found above, 4 meters, we can substitute these values into the formula:
f = 340 m/s / 4 meters
Hence, the frequency of the fundamental wave (first harmonic) is f = 85 Hz.
c) The second harmonic corresponds to the first overtone, which is a wave with twice the frequency of the fundamental wave. So, to find the frequency of the second harmonic, we multiply the frequency of the fundamental wave by 2.
Therefore, the frequency of the second harmonic is 85 Hz * 2 = 170 Hz.