A uniform rope of length $L$ and mass $m_1$ hangs vertically from a rigid support. A block of mass $m_2$ is attached to the free end of the rope. A transverse pulse of wavelength $\lambda _1$, is produced at the lower end of the rope. The wave length of the pulse when it reaches the top of the rope is $\lambda _2$. The ratio $\lambda _2\,/\,\lambda _1$ is 

A sound is produced by plucking a string in a musical instrument, then

A rope of length $L$ and uniform linear density is hanging from the ceiling. A transverse wave pulse, generated close to the free end of the rope, travels upwards through the rope. Select the correct option.

Spacing between two successive nodes in a standing wave on a string is $x$ . If frequency of the standing wave is kept unchanged but tension in the string is doubled, then new spacing between successive nodes will  become

If the initial tension on a stretched string is doubled, then the ratio of the initial and final speeds of a transverse wave along the string is :

A horizontal stretched string, fixed at two ends, is vibrating in its fifth harmonic according to the equation, $y(x$, $t )=(0.01 \ m ) \sin \left[\left(62.8 \ m ^{-1}\right) x \right] \cos \left[\left(628 s ^{-1}\right) t \right]$. Assuming $\pi=3.14$, the correct statement$(s)$ is (are) :

$(A)$ The number of nodes is $5$ .

$(B)$ The length of the string is $0.25 \ m$.

$(C)$ The maximum displacement of the midpoint of the string its equilibrium position is $0.01 \ m$.

$(D)$ The fundamental frequency is $100 \ Hz$.