$A$ pipe closed at one end has a length of $0.8 \ m$. At its open end,a $0.5 \ m$ long uniform string is vibrating in its second harmonic and it resonates with the fundamental frequency of the pipe. If the tension in the wire is $50 \ N$ and the speed of sound is $320 \ m/s$,the mass of the string is: (in $g$)

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:
$(a)$ $y = 2 \cos(3x) \sin(10t)$
$(b)$ $y = 2 \sqrt{x - vt}$
$(c)$ $y = 3 \sin(5x - 0.5t) + 4 \cos(5x - 0.5t)$
$(d)$ $y = \cos x \sin t + \cos 2x \sin 2t$

$A$ wave pulse on a string has the dimensions shown in the figure. The wave speed is $v = 1 \, cm/s$. If point $O$ is a free end,what is the shape of the wave at time $t = 3 \, s$?

$A$ 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 right-angled 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 point $D$ where a detector is placed. If a maxima is formed at the detector,then the possible values for the wavelength $\lambda$ of the wave produced are given by:

Fill in the blanks:
$(i)$ For humans,the minimum wavelength of a sound wave in the audible range is nearly ...... .
$(ii)$ Only ...... waves can be propagated by a fluid medium.
$(iii)$ The distance travelled by a travelling wave having wavelength $\lambda$ and periodic time $T$,in one second is ...... .

The frequency of a stretched uniform wire under tension is in resonance with the fundamental frequency of a closed tube. If the tension in the wire is increased by $8 \ N$,it is in resonance with the first overtone of the closed tube. The initial tension in the wire is .... $N$.