The overall length of a piccolo is 32.0 cm. The resonating air column vibrates as in a pipe that is open at both ends. (a) Find the frequency of the lowest note a piccolo can play. (b) Opening holes in the side effectively shortens the length of the resonant column. If the highest note a piccolo can sound is 4,000 Hz, find the distance between adjacent antinodes for this mode of vibration.
The Correct Answer and Explanation is :
To analyze the piccolo’s resonance and sound properties, we can use the following concepts from wave physics related to open pipes.
(a) Frequency of the Lowest Note
For a pipe open at both ends, the fundamental frequency (first harmonic) can be calculated using the formula:
[
f = \frac{v}{\lambda}
]
Where:
- ( f ) = frequency of the sound
- ( v ) = speed of sound in air (approximately 343 m/s at room temperature)
- ( \lambda ) = wavelength
In a pipe that is open at both ends, the wavelength of the fundamental frequency is twice the length of the pipe. Thus:
[
\lambda = 2L
]
For a piccolo with a length ( L = 32.0 \, \text{cm} = 0.32 \, \text{m} ):
[
\lambda = 2 \times 0.32 \, \text{m} = 0.64 \, \text{m}
]
Now, substituting ( \lambda ) back into the frequency formula:
[
f = \frac{343 \, \text{m/s}}{0.64 \, \text{m}} \approx 535.16 \, \text{Hz}
]
Therefore, the frequency of the lowest note a piccolo can play is approximately 535 Hz.
(b) Distance Between Adjacent Antinodes for 4,000 Hz
For the highest note (4,000 Hz), we again use the wave relationship. The wavelength for this frequency can be calculated as follows:
[
\lambda = \frac{v}{f} = \frac{343 \, \text{m/s}}{4000 \, \text{Hz}} \approx 0.08575 \, \text{m}
]
In a standing wave, there are specific points called nodes (points of no displacement) and antinodes (points of maximum displacement). The distance between adjacent antinodes is half of the wavelength:
[
\text{Distance between adjacent antinodes} = \frac{\lambda}{2} = \frac{0.08575 \, \text{m}}{2} \approx 0.042875 \, \text{m} \approx 4.29 \, \text{cm}
]
Conclusion
In summary, the lowest frequency of a piccolo is approximately 535 Hz, and the distance between adjacent antinodes at the highest note of 4,000 Hz is approximately 4.29 cm. These calculations illustrate the relationship between pipe length, frequency, and wavelength, which are essential for understanding how musical instruments produce sound.