The equation of wave prodeced in a string with both the end fixed is y = cos⁡ 2πt sin ⁡2πx. Then what is the minimum length of the string?

Least length of a vibrating clamped string will be obtained when one loop is formed in the string. In that case, we have:

$\frac{\lambda }{2}=l\dots \left(\mathrm{i}\right)$

Also,

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

( k is the wave number, it is the coefficient of x in a wave equation)

$\begin{array}{l}\lambda =\frac{2\pi }{2\pi }=1\mathrm{m}(\because k=2\pi )\\ \therefore l=\frac{1}{2}=\text{ Minimum length (from (i))}\end{array}$