Natural Frequency Calculation for Free Longitudinal Vibrations

Given:
- Displacement amplitude (A) = 2 mm

Formula:
The natural frequency of longitudinal vibrations can be calculated using the formula:
\[ f_n = \frac{c}{2L} \]
where:
- \( f_n \) = natural frequency
- c = speed of sound in the material
- L = length of the material

Explanation:
- Longitudinal vibrations refer to the vibrations that occur parallel to the direction of the wave propagation.
- In this case, the displacement amplitude (A) is given as 2 mm.
- To calculate the natural frequency, we need to know the speed of sound in the material and the length of the material.

Calculation:
Let's assume the speed of sound in the material (c) is 343 m/s (for air at room temperature) and the length of the material (L) is 1 m.
Substitute the values in the formula:
\[ f_n = \frac{343}{2 \times 1} = 171.5 \, Hz \]
Therefore, the natural frequency of the free longitudinal vibrations is 171.5 Hz when the displacement amplitude is 2 mm.

Conclusion:
- The natural frequency of longitudinal vibrations can be calculated using the formula \( f_n = \frac{c}{2L} \).
- By substituting the given values, we found that the natural frequency is 171.5 Hz in this case.