A piccolo and a flute can be approximated as cylindrical tubes with both ends open. The lowest fundamental frequency produced by one kind of piccolo is 508.7 Hz, and that produced by one kind of flute is 220.3 Hz. What is the ratio of the piccolo's length to the flute's length?
To find the ratio of the piccolo's length to the flute's length, we need to use the equation for the fundamental frequency of a cylindrical pipe:
f = (nv) / (2L)
where:
- f is the fundamental frequency,
- n is the harmonic number (usually 1 for the fundamental frequency),
- v is the speed of sound (we can assume it to be constant), and
- L is the length of the cylindrical pipe.
We can rewrite this equation to solve for the length:
L = (nv) / (2f)
Let's find the lengths of the piccolo and the flute:
For the piccolo:
- f = 508.7 Hz
- n = 1 (fundamental frequency)
- v = the speed of sound (constant)
For the flute:
- f = 220.3 Hz
- n = 1 (fundamental frequency)
- v = the speed of sound (constant)
Since the speed of sound is constant, we can ignore it for finding the ratio of the lengths. Therefore, we only need to find the ratio of the fundamental frequencies:
Ratio = (Length of Piccolo) / (Length of Flute)
= (n * f piccolo) / (n * f flute)
= f piccolo / f flute
Now let's calculate the ratio:
Ratio = (508.7 Hz) / (220.3 Hz)
Ratio ≈ 2.31
Therefore, the ratio of the piccolo's length to the flute's length is approximately 2.31.