$A$ string of mass $M$ is under a tension $T$. The length of the string is $L$. $A$ transverse wave starts at one end of the string. The time required for the disturbance to reach the other end is

$A$ rope of length $L$ and mass $M$ hangs freely from the ceiling. If the time taken by a transverse wave to travel from the bottom to the top of the rope is $T$,then the time taken to cover the first half length is:

$A$ 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 wave of wavelength $\lambda_1$ is produced at the lower end of the rope. The wavelength of the wave when it reaches the top of the rope is $\lambda_2$. The ratio $\frac{\lambda_1}{\lambda_2}$ is

$A$ wire is under tension of $2 \text{ kg wt}$ and a wave is travelling through it with some speed. Tension in the wire is so increased that the wave travels through it with thrice the original speed. The increase in tension is (in kg wt)

Consider a system of three connected strings,$S_1, S_2$ and $S_3$ with uniform linear mass densities $\mu \text{ kg/m}$,$4\mu \text{ kg/m}$ and $16\mu \text{ kg/m}$,respectively,as shown in the figure. $S_1$ and $S_2$ are connected at the point $P$,whereas $S_2$ and $S_3$ are connected at the point $Q$,and the other end of $S_3$ is connected to a wall. $A$ wave generator $O$ is connected to the free end of $S_1$. The wave from the generator is represented by $y = y_0 \cos(\omega t - kx) \text{ cm}$,where $y_0, \omega$ and $k$ are constants of appropriate dimensions. Which of the following statements is/are correct:
$(A)$ When the wave reflects from $P$ for the first time,the reflected wave is represented by $y = \alpha_1 y_0 \cos(\omega t + kx + \pi) \text{ cm}$,where $\alpha_1$ is a positive constant.
$(B)$ When the wave transmits through $P$ for the first time,the transmitted wave is represented by $y = \alpha_2 y_0 \cos(\omega t - kx) \text{ cm}$,where $\alpha_2$ is a positive constant.
$(C)$ When the wave reflects from $Q$ for the first time,the reflected wave is represented by $y = \alpha_3 y_0 \cos(\omega t - kx + \pi) \text{ cm}$,where $\alpha_3$ is a positive constant.
$(D)$ When the wave transmits through $Q$ for the first time,the transmitted wave is represented by $y = \alpha_4 y_0 \cos(\omega t - 4kx) \text{ cm}$,where $\alpha_4$ is a positive constant.

$A$ uniform thin rope of length $12 \, m$ and mass $6 \, kg$ hangs vertically from a rigid support and a block of mass $2 \, kg$ is attached to its free end. $A$ transverse short wavetrain of wavelength $6 \, cm$ is produced at the lower end of the rope. What is the wavelength of the wavetrain (in $cm$) when it reaches the top of the rope?