Step 1: Understanding the Concept:
When a string is fixed at both ends, the ends are "nodes" because they cannot move.
The different stable patterns of vibration are called "harmonics."
The fundamental mode (1st harmonic) has the string vibrating as a single loop with nodes at the ends.
The second harmonic has the string vibrating in two loops, with an additional node in the middle.
One full "wavelength" (\(\lambda\)) consists of two such loops (one crest and one trough).
Key Formula or Approach:
For a string of length \(L\) vibrating in the \(n\)-th harmonic:
\[ L = n \left( \frac{\lambda}{2} \right) \]
Rearranging for wavelength:
\[ \lambda = \frac{2L}{n} \]
Step 2: Detailed Explanation:
Given:
Length of string, \(L = 1\) m.
Harmonic number, \(n = 2\) (second harmonic).
Substituting these into our wavelength formula:
\[ \lambda = \frac{2 \times 1}{2} \]
\[ \lambda = \frac{2}{2} = 1 \text{ m} \]
Visualization: In the 2nd harmonic, the string is divided into two equal loops. Each loop is \(L/2 = 0.5\) m. Since a wavelength is the distance of two loops, \(\lambda = 0.5 + 0.5 = 1.0\) m.
This means exactly one full cycle of the wave fits within the physical length of the string.
Step 3: Final Answer:
The wavelength of the vibration is 1 m, which matches option (A).