$A$ 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 $kg$.

Explain why (or how):
$(a)$ in a sound wave,a displacement node is a pressure antinode and vice versa,
$(b)$ bats can ascertain distances,directions,nature,and sizes of the obstacles without any eyes,
$(c)$ a violin note and sitar note may have the same frequency,yet we can distinguish between the two notes,
$(d)$ solids can support both longitudinal and transverse waves,but only longitudinal waves can propagate in gases,and
$(e)$ the shape of a pulse gets distorted during propagation in a dispersive medium.

Four different independent waves are represented by $(i)$ $y_1 = a_1 \sin \omega t$,$(ii)$ $y_2 = a_2 \sin 2\omega t$,$(iii)$ $y_3 = a_3 \cos \omega t$,and $(iv)$ $y_4 = a_4 \sin (\omega t + \frac{\pi}{3})$. With which two waves is interference possible?

$A$ 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:

If a microwave and an ultrasonic sound wave have the same wavelength, the ratio of their frequencies is approximately:

When the tense wire of a sitar is pulled slightly from the middle and then released,which type of waves are produced?