The seventh harmonic of a one end closed pipe organ has a frequency of 732 Hz, on a day when the temperature is 10 degrees Celsius. How long is the pipe? How long would it be for 732 to be the first harmonic?
speed of sound in air at atmosphere
v = 331 m/s + (0.6 m/s/C)•T
v=wavelength * Frequency
=> wavelength= v/Frequency = 0.46 m if 732 were first harmonic
=> Length being 7th harmonic
= 7 * 0.46 m = 3.22 m
To find the length of the pipe, we can use the formula for the harmonic series of a closed pipe organ:
fn = (2n - 1) * v / (4L)
Where:
- fn is the frequency of the nth harmonic,
- n is the harmonic number,
- v is the speed of sound, which is approximately 343 meters per second at room temperature,
- L is the length of the pipe.
Let's start by finding the length of the pipe for the seventh harmonic, given that the frequency is 732 Hz.
7th harmonic frequency formula:
732 = (2 * 7 - 1) * v / (4L)
To find L, we can rearrange the equation as follows:
L = (2 * 7 - 1) * v / (4 * 732)
Now, let's calculate the length of the pipe at 10 degrees Celsius using the given speed of sound:
v = v₀ * sqrt(T₀ / T)
Where:
- v₀ is the speed of sound at a reference temperature,
- T₀ is the reference temperature, which is typically 20 degrees Celsius,
- T is the current temperature.
v₀ can be calculated as follows:
v₀ = 343 m/s
Now, let's calculate the speed of sound at 10 degrees Celsius:
v = 343 * sqrt(293 / (273 + 10)) = 343 * sqrt(293 / 283) ≈ 349.14 m/s
Now, we can substitute the values into the length equation to find the length of the pipe for the seventh harmonic:
L = (2 * 7 - 1) * 349.14 / (4 * 732)
L ≈ 3.0066 meters
So, the length of the pipe for the seventh harmonic is approximately 3.0066 meters.
To find the length of the pipe for 732 Hz as the first harmonic, we can rearrange the frequency formula:
732 = (2 * 1 - 1) * v / (4L)
Simplifying the equation:
732 = v / (4L)
Now, we can solve for L:
L = v / (4 * 732)
Substituting the value of v:
L = 349.14 / (4 * 732)
L ≈ 0.1197 meters
So, the length of the pipe for 732 Hz to be the first harmonic is approximately 0.1197 meters.