$A$ string is vibrating in its fifth overtone between two rigid supports $2.4 \ m$ apart. The distance between successive node and antinode is (in $m$)

Two progressive waves $Y_1 = \sin 2 \pi \left( \frac{t}{0.4} - \frac{x}{4} \right)$ and $Y_2 = \sin 2 \pi \left( \frac{t}{0.4} + \frac{x}{4} \right)$ superpose to form a standing wave. $x$ and $y$ are in $SI$ units. The amplitude of the particle at $x = 0.5 \ m$ is $\left[ \sin 45^{\circ} = \cos 45^{\circ} = \frac{1}{\sqrt{2}} \right]$.

Explain the reflection of a wave at a rigid support.

$Assertion :$ For the formation of stationary waves,the medium must be bounded having definite boundaries.
$Reason :$ In the stationary wave,some particles of the medium remain permanently at rest.

The pattern of standing waves formed on a stretched string at two instants of time is shown in the figure. The velocity of the two waves superimposing to form stationary waves is $360 \ m/s$ and their frequencies are $256 \ Hz$.
$(a)$ Calculate the time at which the second curve is plotted.
$(b)$ Mark nodes and antinodes on the curve.
$(c)$ Calculate the distance between $A^{\prime}$ and $C^{\prime}$.

The wave pattern on a stretched string is shown in the figure. Interpret what kind of wave this is and find its wavelength.