An organ pipe is open at both ends. It is producing sound at its third harmonic, the frequency of which is 406 Hz. The speed of sound is 343 m/s. What is the length of the pipe?
Read http://www.studyphysics.ca/newnotes/20/unit03_mechanicalwaves/chp141516_waves/lesson51.htm
to convince yourswelf that the third harmonic wavelength is in this case (3/2)L, where L is the pipe length.
In this case, the wavelength is
lambda = V/f = 343/406 = 0.8448 m
The tube length must be 2/3 of that which would be 0.563 m
Well, to find the length of the pipe, we'll have to pipe up with a formula. The formula for the frequency of a closed-open organ pipe is:
f = (2n - 1)(v / 4L)
Where f is the frequency, n is the harmonic number (in this case 3), v is the speed of sound (343 m/s), and L is the length of the pipe.
Now, let's solve the equation for L, shall we?
L = (2n - 1)(v / 4f)
Plugging in the values:
L = (2 * 3 - 1)(343 / (4 * 406))
Calculating the length...
L = 5 * (343 / 1624) meters
And the answer is...
L = *drumroll* ... approximately 1.055 meters! Voila!
To find the length of the organ pipe, we can use the formula for the wavelength of a sound wave in a pipe.
The length of the organ pipe is given by the formula:
L = (2n - 1) * (v / 2f)
Where:
L is the length of the pipe
n is the harmonic number
v is the speed of sound
f is the frequency of the sound wave
In this case, the harmonic number is 3, the frequency is 406 Hz, and the speed of sound is 343 m/s.
Substituting the given values into the formula:
L = (2 * 3 - 1) * (343 / (2 * 406))
L = (6 - 1) * (343 / 812)
L = 5 * 0.422
L = 2.11 meters
Therefore, the length of the organ pipe is 2.11 meters.
To find the length of the pipe, we can use the formula:
L = (m * λ) / 4
where:
L is the length of the pipe,
m is the harmonic number, and
λ is the wavelength of the sound wave.
First, we need to find the wavelength of the sound wave. Since the pipe is open at both ends, the open pipe has antinodes at both ends. For the third harmonic, there are two antinodes, so we can represent the wavelength as:
λ = 2L / (m)
Substituting the given values:
frequency (f) = 406 Hz and speed of sound (v) = 343 m/s
We can use the formula:
v = f * λ
Rearranging it to solve for λ:
λ = v / f
Substituting the values:
λ = 343 m/s / 406 Hz
Calculating the wavelength:
λ = 0.846 m
Now, we can substitute this value of λ into the formula for the length of the pipe:
L = (m * λ) / 4
= (3 * 0.846 m) / 4
= 0.635 m
Therefore, the length of the pipe is approximately 0.635 meters.