An organ pipe is open at both ends. It is producing sound at its sixth harmonic, the frequency of which is 257 Hz. The speed of sound is 343 m/s. What is the length of the pipe
REASONING The frequency of a pipe open at both ends is given by Equation 17.4 as , where n is an integer specifying the harmonic number, v is the speed of sound, and L is the length of the pipe. This relation can be used to find L, since all the other variables are known.
SOLUTION Solving the equation above for L, and recognizing that n = 6 for the 6th harmonic, we have
L=n((v)/(2(fn)))
L=6((343m/s)/(2(257Hz)))
L=4.0038911m
To find the length of the pipe, we can use the formula:
\(L = \frac{v}{2f}\),
where L is the length of the pipe, v is the speed of sound, and f is the frequency.
Let's plug in the given values:
\(v = 343 \, \text{m/s}\),
\(f = 257 \, \text{Hz}\).
\(L = \frac{343}{2 \times 257}\).
Calculating the expression, we find:
\(L = \frac{343}{514}\).
Finally, simplifying the expression, we get:
\(L = 0.668 \, \text{m}\).
Therefore, the length of the pipe is approximately 0.668 meters.
To find the length of the organ pipe, we can use the formula:
Length = (n * λ) / 2
Where:
- Length is the length of the pipe
- n is the harmonic number
- λ (lambda) is the wavelength of the sound wave
In this case, the organ pipe is producing sound at the sixth harmonic, so n = 6. We know the frequency of the sound wave is 257 Hz.
To find the wavelength, we can use the formula:
wavelength (λ) = speed of sound / frequency
Given that the speed of sound is 343 m/s, we can substitute these values into the formula:
wavelength (λ) = 343 m/s / 257 Hz
We can now calculate the wavelength:
wavelength (λ) ≈ 1.334 m
Now, we can substitute the values of n and λ into the first formula to find the length of the pipe:
Length = (6 * 1.334 m) / 2
Length ≈ 4.002 m
Therefore, the length of the organ pipe is approximately 4.002 meters.