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(A). What is the fourth harmonic frequency (in Hz) of an open organ pipe with a length of 78.6 cm?

(B). What is the first overtone (or the n = 3 harmonic) frequency (in Hz) of an organ pipe, closed at one end, with a length of 68.5 cm?
(C). If you have the second-shortest organ pipe length (open at both ends) that will resonate at 539 Hz, its fundamental frequency must be how many Hertz?

1 Answer

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Answer: 0.318 Hz.

Explanation: (A). In an open organ pipe, the frequencies of the harmonics are given by the formula:

f = (nv) / (2L)

Where:

f is the frequency of the harmonic,

n is the harmonic number,

v is the speed of sound in air, which is approximately 343 m/s at room temperature,

and L is the length of the pipe in meters.

Converting the length from centimeters to meters, we have:

L = 78.6 cm = 78.6 / 100 = 0.786 m

To find the fourth harmonic frequency (n = 4) of the open organ pipe, we plug in the values into the formula:

f = (4 * 343) / (2 * 0.786)

Calculating the above expression, we find:

f ≈ 869.18 Hz

Therefore, the fourth harmonic frequency of the open organ pipe is approximately 869.18 Hz.

(B). For a closed organ pipe, the frequencies of the harmonics are given by the formula:

f = (nv) / (4L)

Using the given values:

L = 68.5 cm = 68.5 / 100 = 0.685 m

n = 3 (the first overtone or second harmonic)

Substituting the values into the formula, we have:

f = (3 * 343) / (4 * 0.685)

Computing the above expression, we find:

f ≈ 500.00 Hz

Therefore, the first overtone (or n = 3 harmonic) frequency of the closed organ pipe is approximately 500.00 Hz.

(C). For an open organ pipe resonating at a specific frequency, the fundamental frequency is given by the formula:

f = (v) / (2L)

Using the given resonant frequency of 539 Hz, we can solve for the fundamental frequency by rearranging the formula as follows:

2L = v / f

L = (v / f) / 2

Substituting the known values:

v = 343 m/s

f = 539 Hz

Calculating the above expression, we find:

L = (343 / 539) / 2

L ≈ 0.318 m

Therefore, the fundamental frequency of the organ pipe with the second-shortest length (open at both ends) is approximately 0.318 Hz.

User Raj Jagani
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