A string of length L secured at both ends vibrates in its...

For a string of length $L$ secured at both ends, the fundamental mode of vibration consists of a single loop.

The distance between two consecutive nodes is $\dfrac{\lambda}{2}$.

Since the string is fixed at both ends, the length of the string is equal to half the wavelength.

$L = \dfrac{\lambda}{2}$

$\lambda = 2L$

The wave number $k$ is defined as:

$k = \dfrac{2\pi}{\lambda}$

Substituting $\lambda = 2L$:

$k = \dfrac{2\pi}{2L} = \dfrac{\pi}{L}$

Thus, the wavelength is $2L$ and the wave number is $\dfrac{\pi}{L}$.