The range of human hearing is roughly from twenty hertz to twenty kilohertz. Based on these limits and a value of 343 m/s for the speed of sound, what are the lengths of the longest and shortest pipes (open at both ends and producing sound at their fundamental frequencies) that you expect to find in a pipe organ?
open at both ends? then the pipe is 1/2 wavelength long.
wavelength= 343/f
compute wavelength, then pipe lengths.
To find the lengths of the longest and shortest pipes in a pipe organ, we can use the formula for the fundamental frequency of a pipe that is open at both ends:
f = (2n - 1) * v / (4L)
Where:
f is the fundamental frequency
n is the harmonic number (usually set to 1 for the fundamental frequency)
v is the speed of sound
L is the length of the pipe
We are given that the range of human hearing is from 20 Hz to 20 kHz, which corresponds to a range of frequencies. We can use this information to find the corresponding range of pipe lengths.
For the shortest pipe:
Using the lower limit of human hearing (20 Hz):
20 = (2 * 1 - 1) * 343 / (4L)
Simplifying this equation and solving for L, we get:
L = (2 * 1 - 1) * 343 / (4 * 20)
L = 8.575 meters
For the longest pipe:
Using the upper limit of human hearing (20 kHz):
20,000 = (2 * 1 - 1) * 343 / (4L)
Simplifying this equation and solving for L, we get:
L = (2 * 1 - 1) * 343 / (4 * 20000)
L = 0.042875 meters (or 42.875 mm)
Therefore, the lengths of the longest and shortest pipes (open at both ends and producing sound at their fundamental frequencies) that you would expect to find in a pipe organ are approximately 8.575 meters and 42.875 millimeters, respectively.