$A$ standing wave exists in a string of length $150 \ cm$,which is fixed at both ends with rigid supports. The displacement amplitude of a point at a distance of $10 \ cm$ from one of the ends is $5\sqrt{3} \ mm$. The nearest distance between two points,within the same loop and having a displacement amplitude equal to $5\sqrt{3} \ mm$,is $10 \ cm$. Find the maximum displacement amplitude of the particles in the string (in $mm$).

In case of a stationary wave pattern,which of the following statements is $CORRECT$?

The displacement of a string is given by,
$y(x, t) = 10 \sin \left(\frac{2 \pi}{3} x\right) \cos (120 \pi t)$
where $x$ and $y$ are in $m$ and $t$ is in $sec$. The length of the string is $1.5 \ m$ and its mass is $3 \times 10^{-2} \ kg$.
Select the correct statement$(s)$ below:
$(A)$ It represents a progressive wave of frequency $60 \ Hz$.
$(B)$ It represents a standing wave of frequency $60 \ Hz$.
$(C)$ It is the result of two waves of wavelength $3 \ m$,frequency $60 \ Hz$ each travelling with a speed of $180 \ m/s$ in opposite directions.
$(D)$ Reflection occurs from a free end.

One end of a taut string of length $3 \ m$ along the $x$-axis is fixed at $x=0$. The speed of the waves in the string is $100 \ m/s$. The other end of the string is vibrating in the $y$-direction so that stationary waves are set up in the string. The possible waveform$(s)$ of these stationary waves is (are):
$(A)$ $y(x,t) = A \sin \frac{\pi x}{6} \cos \frac{50 \pi t}{3}$
$(B)$ $y(x,t) = A \sin \frac{\pi x}{3} \cos \frac{100 \pi t}{3}$
$(C)$ $y(x,t) = A \sin \frac{5 \pi x}{6} \cos \frac{250 \pi t}{3}$
$(D)$ $y(x,t) = A \sin \frac{5 \pi x}{2} \cos 250 \pi t$

Explain the reflection of a wave at a free support.

The standing wave in a medium is expressed as $y = 0.2 \sin(0.8 x) \cos(3000 t) \, m$. The distance between any two consecutive points of minimum or maximum displacement is