A tube is open only at one end. A certain harmonic produced by the tube has a frequency of 445.0 Hz. The next higher harmonic has a frequency of 525.9 Hz. The speed of sound in air is 343 m/s.
(a) What is the integer n that describes the harmonic whose frequency is 445.0 Hz?
(b) What is the length of the tube?
m
(a) The ratio of frequencies is the ratio of harmonic dumbers. In your case it is 525.9/445.0 = 1.1818, which is very nearly 13/11. Therefore 13 and 11 are the two "n" numbers involved. Tubes open at one end only have only odd numbered hamonics. The number "n" corresponds to the number of quarter-waves in the pipe.
(b) If a note corresponds to a n'th harmonic, the length of the pipe is n quarter waves. Get the wavelength from the sound speed and frequency.
wavelength = (sound speed)/frequency
(a) To find the integer n that describes the harmonic with a frequency of 445.0 Hz, we can use the equation:
frequency = (speed of sound)/(2L) * n
where L is the length of the tube and n is the harmonic number.
Let's rearrange the equation to solve for n:
n = 2L * (frequency)/(speed of sound)
For the harmonic with a frequency of 445.0 Hz:
n = 2L * (445.0 Hz)/(343 m/s)
(b) To find the length of the tube, we can use the equation:
wavelength = (speed of sound)/frequency
Let's rearrange the equation to solve for the length of the tube:
L = (speed of sound) * (1/frequency)
For the frequency of 445.0 Hz:
L = 343 m/s * (1/445.0 Hz)
To find the integer "n" that describes the harmonic with a frequency of 445.0 Hz, we need to use the ratio of frequencies. The ratio of the next higher harmonic (525.9 Hz) to the given frequency (445.0 Hz) is 525.9/445.0 = 1.1818.
Since tubes open at one end only have odd-numbered harmonics, we know that the harmonic "n" involved is the odd number closest to 1.1818. In this case, the closest odd numbers are 11 and 13.
However, the given frequency is closer to 11, so we can conclude that the integer "n" describing the harmonic with a frequency of 445.0 Hz is 11.
To calculate the length of the tube, we need to determine the wavelength. The wavelength can be found using the formula:
wavelength = (speed of sound) / (frequency)
Given that the speed of sound in air is 343 m/s and the frequency of the harmonic is 445.0 Hz, we can substitute these values into the formula:
wavelength = 343 m/s / 445.0 Hz
This will give us the wavelength in meters.
To find the length of the tube, we need to convert the wavelength to the length of the pipe. Since the length of the pipe is equal to a quarter-wavelength (n quarter waves), we can use the formula:
length = wavelength * n
Substituting the values, we have:
length = (343 m/s / 445.0 Hz) * 11
Calculating this expression will give us the length of the tube in meters.