Let's consider the problem step by step.
1. Understand the Physical Situation:
- The pipe has a length of 767 cm.
- The air column in the pipe forms seven nodes when vibrating.
- The speed of sound in air is 343 m/s.
2. Determine the Number of Half-Wavelengths:
- Each node represents a point where the sound wave has zero amplitude.
- The distance between two consecutive nodes is half of the wavelength (\(\(\lambda/2\)\)).
- With seven nodes, there are six segments between consecutive nodes, which equals six half-wavelengths.
3. Calculate the Wavelength:
- The total length of the pipe is equivalent to six half-wavelengths (\(\(6 \times \lambda/2 = 3 \lambda\)\)).
- Therefore, the wavelength (\(\(\lambda\)\)) can be derived from the length of the pipe:
\(\[
\lambda = \frac{2 \times (\text{Length of pipe})}{(\text{Number of half-wavelengths})}
\]\)
Substituting the given values:
\(\[
\lambda = \frac{2 \times 767 \text{ cm}}{6}
= 255.67 \text{ cm}
\]\)
4. Convert the Wavelength to Meters:
- Given that \(\(1 \text{ m} = 100 \text{ cm}\)\):
\(\[
\lambda = \frac{255.67 \text{ cm}}{100}
= 2.5567 \text{ m}
\]\)
5. Calculate the Frequency:
- The frequency (\(\(f\)\)) of the sound wave can be derived using the relationship between speed, wavelength, and frequency:
\(\[
f = \frac{v}{\lambda}
\]\)
where \(\(v\)\) is the speed of sound and \(\(\lambda\)\) is the wavelength.
- Substituting the values:
\(\[
f = \frac{343 \text{ m/s}}{2.5567 \text{ m}}
\approx 134.16 \text{ Hz}
\]\)
So, the frequency of the sound wave in the pipe is approximately 134.16 Hz.
