When the area of cross-section of a stretched wire is halved and tension is doubled,the speed of propagation of transverse waves along it becomes $k$ times the initial speed. Then,$k$ is:

$A$ composite string is made up by joining two strings of different mass per unit length,$\mu$ and $4\mu$. The composite string is under the same tension $T$. $A$ transverse wave pulse,$Y = (6 \text{ mm}) \sin(5t + 40x)$,where $t$ is in seconds and $x$ is in meters,is sent along the lighter string towards the joint. The joint is at $x = 0$. The equation of the wave pulse reflected from the joint is:

$A$ 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?

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 $N$)

$A$ transverse pulse generated at the bottom of a uniform rope of length $L$ travels in an upward direction. The time taken by it to travel the full length of the rope will be

$Assertion :$ Two waves moving in a uniform string having uniform tension cannot have different velocities.
$Reason :$ Elastic and inertial properties of string are same for all waves in same string. Moreover,the speed of a wave in a string depends on its elastic and inertial properties only.