Question

A standing wave is produced on a string clamped at one end and free at the other. The length of the string
(a) must be an integral multiple of $\lambda /4$
(b) must be an integral multiple of $\lambda /2$
(c) must be an integral multiple of $\lambda$
(d) must be an integral multiple of $\lambda /2$.

Answer 1

(a) must be an integral multiple of $\lambda /4$

A standing wave is produced on a string clamped at one end and free at the other. Its fundamental frequency is given by
$$\begin{gathered} \nu =\left(n+\frac{1}{2}\right)\frac{v}{2L} \\ \Rightarrow v=\nu \lambda \\ \Rightarrow \nu =\left(n+\frac{1}{2}\right)\frac{\nu \lambda }{2L} \\ \Rightarrow L=\left(\frac{2n+1}{4}\right)\lambda \\ \Rightarrow L=\frac{\lambda }{4},\frac{3\lambda }{4},... \end{gathered}$$