### Understanding Harmonics
**Harmonics** are specific frequencies at which a system can naturally oscillate. For a vibrating air column, harmonics refer to the different frequencies that the air column can support. These frequencies are integral multiples of the fundamental frequency.
### Harmonics in a Pipe Closed at One End
A pipe closed at one end supports only odd harmonics. This is because the closed end of the pipe must be a node (point of no displacement), while the open end must be an antinode (point of maximum displacement). The allowed wavelengths for the harmonics must fit this boundary condition.
**Fundamental Frequency (1st Harmonic):**
- For a pipe closed at one end, the fundamental frequency has a wavelength that is four times the length of the pipe: \( \lambda_1 = 4L \).
**Second Harmonic:**
- The second harmonic (which is the third harmonic in terms of frequency) has a wavelength that is four-thirds the length of the pipe: \( \lambda_3 = \frac{4L}{3} \).
**Third Harmonic:**
- The third harmonic (which is the fifth harmonic in terms of frequency) has a wavelength that is four-fifths the length of the pipe: \( \lambda_5 = \frac{4L}{5} \).
**General Rule:**
- For a pipe closed at one end, the wavelengths of the harmonics are given by: \( \lambda_n = \frac{4L}{n} \), where \( n \) is an odd integer (1, 3, 5, …).
Thus, only odd harmonics are present.
### Calculating the Velocity of Sound in Air
We need to find the velocity of sound given the beat frequency and the wavelengths of two sound waves.
**Given:**
- Wavelengths of the two sound waves: \( \lambda_1 = \frac{81}{173} \text{ m} \) and \( \lambda_2 = \frac{81}{170} \text{ m} \)
- Beat frequency: \( 10 \text{ beats/second} \)
**Step-by-Step Solution:**
1. **Calculate the Frequencies:**
Using the formula \( v = f \lambda \), where \( v \) is the speed of sound and \( f \) is the frequency.
Let \( v \) be the velocity of sound in air.
For the first wavelength:
\[
f_1 = \frac{v}{\lambda_1} = \frac{v}{\frac{81}{173}} = \frac{173v}{81}
\]
For the second wavelength:
\[
f_2 = \frac{v}{\lambda_2} = \frac{v}{\frac{81}{170}} = \frac{170v}{81}
\]
2. **Calculate the Difference in Frequencies:**
The beat frequency is the difference between the two frequencies:
\[
|f_1 - f_2| = \left| \frac{173v}{81} - \frac{170v}{81} \right| = \frac{3v}{81} = \frac{v}{27}
\]
Given the beat frequency is \( 10 \text{ beats/second} \):
\[
\frac{v}{27} = 10
\]
3. **Solve for the Velocity:**
\[
v = 10 \times 27 = 270 \text{ m/s}
\]
**Answer:**
The velocity of sound in air is \( 270 \text{ m/s} \).
