$A$ wire of length $50 \ cm$ and weighing $10 \ g$ is attached to a spring at one end and to a fixed wall at the other end. The spring has a spring constant of $50 \ N/m$ and is stretched by $1 \ cm$. If a wave pulse is produced on the string near the wall, then how much time will it take to reach the spring (in $s$)?

$A$ string of mass $0.1 \, kg$ is under a tension $1.6 \, N$. The length of the string is $1 \, m$. $A$ transverse wave starts from one end of the string. The time taken by the wave to reach the other end is (in $s$.)

$A$ sitar wire is replaced by another wire of same length and material but of three times the earlier radius. If the tension in the wire remains the same,by what factor will the frequency change?

$A$ 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:

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.

The equation of a wave on a string of linear mass density $0.02 \text{ kg m}^{-1}$ is $Y = 0.01 \sin [2\pi (t/0.02 - x/0.50)] \text{ m}$. The tension in the string is (in $\text{ N}$)