Question

A string of length \( L \) is fixed at both ends and vibrates in its fundamental mode. If the speed of waves on the string is \( v \), then the angular wave number of the standing wave is:

• A. \(\frac{2}{L}\)
• B. \(\frac{\pi}{L}\)
• C. \(\frac{2\pi}{L}\)
• D. \(\frac{\pi}{L}\)
• E. \(\frac{\pi}{2L}\)

Hint:

The fundamental mode of a fixed string corresponds to the simplest standing wave pattern, one-half of a wavelength along the string's length.

Answer 1

The Correct Option is D

For a string fixed at both ends, the fundamental frequency corresponds to the first harmonic, where the wavelength \( \lambda = 2L \). 
The wave number \( k \) is given by: \[ k = \frac{2\pi}{\lambda} = \frac{2\pi}{2L} = \frac{\pi}{L} \]