Q.A standing wave is produced on a string fixed at one end and free at other. The length of string must be an _______. (A) odd integral multiple of $\lambda/4$
(B) integral multiple of $\lambda/2$
(C) integral multiple of $\lambda$
(D) integral multiple of $\lambda/4$

A stationary (or standing) wave is the special resultant produced when TWO waves of EQUAL amplitude and EQUAL frequency travel through the SAME medium along the SAME path but in EXACTLY OPPOSITE directions -- the case arising, for instance, when a wave reflects off a fixed boundary and interferes with its own oncoming, still-arriving self. Adding $y_1=a\sin\{2\pi(nt-x/\lambda)\}$ and $y_2=a\sin\{2\pi(nt+x/\lambda)\}$ via the sum-to-product identity gives $y=\left(2a\cos\dfrac{2\pi x}{\lambda}\right)\sin(2\pi nt)$ -- since position x and time t appear SEPARATELY here (x only inside the amplitude factor, never combined with t into one travelling argument), this is NOT a progressive wave: every particle oscillates with the same frequency n, but with a fixed AMPLITUDE that varies periodically with its own position, $A(x)=2a\cos(2\pi x/\lambda)$. …