$A$ clamped string is oscillating in the $n^{th}$ harmonic,then

An object of density $2000 \ kg \ m^{-3}$ is hung from a thin light wire. The fundamental frequency of the transverse waves in the wire is $200 \ Hz$. If the object is immersed in water such that half of its volume is submerged,then the fundamental frequency of the transverse waves in the wire is (in $Hz$)

Given below are some functions of $x$ and $t$ to represent the displacement (transverse or longitudinal) of an elastic wave. State which of these represent $(i)$ a travelling wave,$(ii)$ a stationary wave or $(iii)$ none at all:
$(a)$ $y = 2 \cos(3x) \sin(10t)$
$(b)$ $y = 2 \sqrt{x - vt}$
$(c)$ $y = 3 \sin(5x - 0.5t) + 4 \cos(5x - 0.5t)$
$(d)$ $y = \cos x \sin t + \cos 2x \sin 2t$

$A$ 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:

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 ...... .

An air column in a tube $32 \,cm$ long, closed at one end, is in resonance with a tuning fork. The air column in another tube, open at both ends, of length $66 \,cm$ is in resonance with another tuning fork. When these two tuning forks are sounded together, they produce $8$ beats per second. Then the frequencies of the two tuning forks are (Consider fundamental frequencies only):