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.