If the tension of sonometer’s wire increases four times then the fundamental frequency of the wire will increase by .... $ times$

The transverse displacement of a string clamped at its both ends is given by 

$y\left( {x,t} \right) = 2\,\sin \,\left( {\frac{{2\pi }}{3}x} \right)\,\cos \,\left( {100\,\pi t} \right)$ 

where $x$ and $y$ are in $cm$ and $t$ is in $s$. Which of the following statements is correct ?

A tuning fork of frequency $392 Hz,$ resonates with $50 cm $ length of a string under tension ($T$). If length of the string is decreased by $2\%$, keeping the tension constant, the number of beats heard when the string and the tuning fork made to vibrate simultaneously is

Explain the reflection of wave at free support.

Two uniform strings of mass per unit length $\mu$ and $4 \mu$, and length $L$ and $2 L$, respectively, are joined at point $O$, and tied at two fixed ends $P$ and $Q$, as shown in the figure. The strings are under a uniform tension $T$. If we define the frequency $v_0=\frac{1}{2 L} \sqrt{\frac{T}{\mu}}$, which of the following statement($s$) is(are) correct?

$(A)$ With a node at $O$, the minimum frequency of vibration of the composite string is $v_0$

$(B)$ With an antinode at $O$, the minimum frequency of vibration of the composite string is $2 v_0$

$(C)$ When the composite string vibrates at the minimum frequency with a node at $O$, it has $6$ nodes, including the end nodes

$(D)$ No vibrational mode with an antinode at $O$ is possible for the composite string

A string fixed at both the ends is vibrating in two segments. The wavelength of the corresponding wave is