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 ([tex]\(\lambda/2\)[/tex]).
- 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 ([tex]\(6 \times \lambda/2 = 3 \lambda\)[/tex]).
- Therefore, the wavelength ([tex]\(\lambda\)[/tex]) can be derived from the length of the pipe:
[tex]\[
\lambda = \frac{2 \times (\text{Length of pipe})}{(\text{Number of half-wavelengths})}
\][/tex]
Substituting the given values:
[tex]\[
\lambda = \frac{2 \times 767 \text{ cm}}{6}
= 255.67 \text{ cm}
\][/tex]
4. Convert the Wavelength to Meters:
- Given that [tex]\(1 \text{ m} = 100 \text{ cm}\)[/tex]:
[tex]\[
\lambda = \frac{255.67 \text{ cm}}{100}
= 2.5567 \text{ m}
\][/tex]
5. Calculate the Frequency:
- The frequency ([tex]\(f\)[/tex]) of the sound wave can be derived using the relationship between speed, wavelength, and frequency:
[tex]\[
f = \frac{v}{\lambda}
\][/tex]
where [tex]\(v\)[/tex] is the speed of sound and [tex]\(\lambda\)[/tex] is the wavelength.
- Substituting the values:
[tex]\[
f = \frac{343 \text{ m/s}}{2.5567 \text{ m}}
\approx 134.16 \text{ Hz}
\][/tex]
So, the frequency of the sound wave in the pipe is approximately 134.16 Hz.
