$A$ narrow tube is bent in the form of a circle of radius $R,$ as shown in the figure. Two small holes $S$ and $D$ are made in the tube at positions at a right angle to each other. $A$ source placed at $S$ generates a wave of intensity $I_0$ which is equally divided into two parts: one part travels along the longer path,while the other travels along the shorter path. Both the waves meet at the point $D$ where a detector is placed. If a minimum is formed at the detector,then the magnitude of the wavelength $\lambda$ of the wave produced is given by:

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

$A$ transverse sinusoidal wave of amplitude $a,$ wavelength $\lambda,$ and frequency $n$ is travelling on a stretched string. The maximum speed of any point on the string is $v/10,$ where $v$ is the speed of propagation of the wave. If $a = 10^{-3} \ m$ and $v = 10 \ m/s,$ then $\lambda$ and $n$ are given by:

Column $I$ shows four systems,each of the same length $L$,for producing standing waves. The lowest possible natural frequency of a system is called its fundamental frequency,whose wavelength is denoted as $\lambda_{f}$. Match each system with statements given in Column $II$ describing the nature and wavelength of the standing waves.
Column $I$:
$(A)$ Pipe closed at one end
$(B)$ Pipe open at both ends
$(C)$ Stretched wire clamped at both ends
$(D)$ Stretched wire clamped at both ends and at mid-point
Column $II$:
$(p)$ Longitudinal waves
$(q)$ Transverse waves
$(r)$ $\lambda_{f} = L$
$(s)$ $\lambda_{f} = 2L$
$(t)$ $\lambda_{f} = 4L$

Fill in the blanks:
$(i)$ If the fundamental frequency of a given closed pipe is $50 \ Hz$,then the frequency for the second overtone is ...... .
$(ii)$ The speed of sound in air at $STP$ is ...... .
$(iii)$ In the case of sound waves,in order to experience beats quite clearly,the value of the beat frequency $|f_1 - f_2|$ should not be greater than ...... .

Speed of a transverse wave on a straight wire (mass $6.0\; g$,length $60\; cm$,and area of cross-section $1.0\; mm^{2}$) is $90\; ms^{-1}$. If the Young's modulus of the wire is $16 \times 10^{11}\; Nm^{-2}$,the extension of the wire over its natural length is: (in $; mm$)