In a large room,a person receives direct sound waves from a source $120 \ m$ away from him. He also receives waves from the same source which reach him after being reflected from the $25 \ m$ high ceiling at a point halfway between them. The two waves interfere constructively for wavelengths of

$A$ 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:

$A$ vibrating string of length $\ell$ under a tension $T$ resonates with a mode corresponding to the first overtone (third harmonic) of an air column of length $75 \,cm$ inside a tube closed at one end. The string also generates $4$ beats per second when excited along with a tuning fork of frequency $n$. Now, when the tension of the string is slightly increased, the number of beats reduces to $2$ per second. Assuming the velocity of sound in air to be $340 \,m/s$, the frequency $n$ of the tuning fork in $Hz$ is:

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 $; mm$)

$A$ wire of linear mass density $\mu = 9.8 \times 10^{-3} \, kg \, m^{-1}$ passes over a frictionless light pulley fixed on the top of a frictionless inclined plane which makes an angle of $30^{\circ}$ with the horizontal. Masses $m$ and $M$ are tied at the two ends of the wire such that $m$ rests on the plane and $M$ hangs freely vertically downwards. The entire system is in equilibrium and a transverse wave propagates along the wire with a velocity of $100 \, m \, s^{-1}$. Choose the correct option for $m$ in $kg$.

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
$(r)$ $\lambda_{f} = L$
$(s)$ $\lambda_{f} = 2L$
$(t)$ $\lambda_{f} = 4L$