$A$ string of length $L$ and mass $M$ hangs freely from a fixed point. The velocity of transverse waves along the string at a distance $x$ from the free end is

The linear mass density of a vibrating string is $1.3 \times 10^{-4} \ kg/m$. $A$ transverse wave is propagating on the string and is described by the equation $y = 0.021 \sin(x + 30t)$,where $x$ and $y$ are measured in meters and $t$ in seconds. The tension in the string is approximately: (in $N$)

$A$ uniform metal wire of density $\rho$,cross-sectional area $A$,and length $L$ is stretched with a tension $T$. The speed of a transverse wave in the wire is given by:

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.

$A$ transverse wave is travelling with velocity $V$ through a metal wire of length $L$ and density $\rho$. The tensile stress in the wire 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