To determine the length of an open-pipe resonator at its fundamental frequency, we will use principles from the physics of sound waves. Specifically, we will use the relationship between the speed of sound, the frequency of the sound wave, and the wavelength of the wave.
Here are the steps to follow:
1. Determine the Wavelength:
The speed of sound (v) and frequency (f) are related by the equation for a wave,
[tex]\[ v = f \times \lambda \][/tex]
where [tex]\( \lambda \)[/tex] is the wavelength of the sound wave.
Given the speed of sound (v) is 331 m/s and the fundamental frequency (f) is 377 Hz, we can rearrange the formula to solve for [tex]\( \lambda \)[/tex],
[tex]\[ \lambda = \frac{v}{f} \][/tex]
2. Calculate Wavelength:
Plugging the given values into this formula, we have
[tex]\[ \lambda = \frac{331 \, \text{m/s}}{377 \, \text{Hz}} \][/tex]
3. Carry out the Division:
[tex]\[ \lambda \approx \frac{331}{377} \, \text{m} \][/tex]
[tex]\[ \lambda \approx 0.878 \, \text{m} \][/tex]
4. Find the Length of the Resonator:
An open-pipe resonator has a length that is one-half the wavelength ([tex]\(\lambda/2\)[/tex]) for the fundamental frequency. Therefore,
[tex]\[ L = \frac{\lambda}{2} \][/tex]
[tex]\[ L = \frac{0.878 \, \text{m}}{2} \][/tex]
[tex]\[ L \approx 0.439 \, \text{m} \][/tex]
5. Report the Answer to 3 Significant Figures:
Rounding [tex]\( L \)[/tex] to at least three significant figures, we get:
[tex]\[ L \approx 0.439 \, \text{m} \][/tex]
Therefore, the length of the open-pipe resonator, rounded to three significant figures, is approximately 0.439 meters.
