Suppose that the range of output frequencies is from 47.4 Hz to 14.8 kHz for a pipe organ. Take 343 m/s for the speed of sound. (a) What is the length (in units of m) of the longest pipe open at both ends and producing sound at its fundamental frequency? (b) What is the length (in units of m) of the shortest pipe open at both ends and producing sound at its fundamental frequency?
(a) Ah, the longest pipe! Well, to find its length, we first need to find the wavelength of the fundamental frequency. The formula for the wavelength of a sound wave is given by λ = v/f, where v is the speed of sound (343 m/s) and f is the frequency (the fundamental frequency in this case). So, let's plug in the values:
λ = (343 m/s)/(47.4 Hz)
But we need the length of the pipe, right? Well, lucky for us, the fundamental frequency corresponds to a quarter of a wavelength for a pipe open at both ends. So, we'll divide the wavelength by 4:
Length = λ/4 = [(343 m/s)/(47.4 Hz)]/4
And if we simplify:
Length = 1.816 m
So, the length of the longest pipe open at both ends and producing sound at its fundamental frequency is approximately 1.816 m.
(b) Now, for the shortest pipe! The process is similar. We'll use the same formula:
λ = v/f = (343 m/s)/(14.8 kHz)
Again, we divide the wavelength by 4 to get the length of the pipe:
Length = λ/4 = [(343 m/s)/(14.8 kHz)]/4
Simplifying further:
Length ≈ 0.573 m
So, the length of the shortest pipe open at both ends and producing sound at its fundamental frequency is approximately 0.573 m.
To find the length of the longest and shortest pipe open at both ends producing sound at their fundamental frequencies, we can use the formula for the speed of sound in a pipe:
v = f * λ
where:
v is the speed of sound (343 m/s),
f is the fundamental frequency of the pipe, and
λ is the wavelength of the sound.
For a pipe open at both ends, the fundamental frequency is given by:
f = v / (2L)
where L is the length of the pipe.
(a) To find the length of the longest pipe:
Given:
Minimum frequency (f_min) = 47.4 Hz,
Speed of sound (v) = 343 m/s.
Using the formula for the fundamental frequency:
f_min = v / (2L)
Rearranging the formula to solve for L:
L = v / (2 * f_min)
Substituting the given values:
L = 343 m/s / (2 * 47.4 Hz)
L ≈ 1.44 m
Hence, the length of the longest pipe open at both ends and producing sound at its fundamental frequency is approximately 1.44 meters.
(b) To find the length of the shortest pipe:
Given:
Maximum frequency (f_max) = 14.8 kHz = 14,800 Hz,
Speed of sound (v) = 343 m/s.
Using the formula for the fundamental frequency:
f_max = v / (2L)
Rearranging the formula to solve for L:
L = v / (2 * f_max)
Substituting the given values:
L = 343 m/s / (2 * 14,800 Hz)
L ≈ 0.0116 m
Hence, the length of the shortest pipe open at both ends and producing sound at its fundamental frequency is approximately 0.0116 meters.
To find the length of the longest and shortest pipes producing sound at their fundamental frequencies, we can use the formula for the fundamental frequency of an open pipe:
f = v / (2L)
where:
f is the frequency,
v is the speed of sound, and
L is the length of the pipe.
(a) The longest pipe will produce the lowest frequency, which is equal to 47.4 Hz.
To find the length (L) of this pipe, rearrange the formula:
L = v / (2f)
Substitute the given values:
v = 343 m/s
f = 47.4 Hz
L = 343 m/s / (2 * 47.4 Hz)
Convert Hz to s⁻¹:
L = 343 m/s / (2 * 47.4 s⁻¹)
L ≈ 3.62 m
So, the length of the longest pipe open at both ends and producing sound at its fundamental frequency is approximately 3.62 meters.
(b) The shortest pipe will produce the highest frequency, which is equal to 14.8 kHz.
Using the same formula as above, but substituting the new frequency value:
v = 343 m/s
f = 14.8 kHz = 14,800 Hz
L = 343 m/s / (2 * 14,800 Hz)
Convert Hz to s⁻¹:
L = 343 m/s / (2 * 14,800 s⁻¹)
L ≈ 0.0116 m
So, the length of the shortest pipe open at both ends and producing sound at its fundamental frequency is approximately 0.0116 meters (or 11.6 mm).