The stationary wave $y = 2a \sin kx \cos \omega t$ in a stretched string is the result of the superposition of $y_1 = a \sin(kx - \omega t)$ and

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 $\left[ \sin 45^{\circ} = \cos 45^{\circ} = \frac{1}{\sqrt{2}} \right]$.

$A$ wave represented by the equation $y_1 = a \cos(kx - \omega t)$ is superimposed with another wave to form a stationary wave such that the point $x = 0$ is a node. The equation for the other wave is

The distance between the successive node and anti-node is

$Assertion :$ 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.

$A$ standing wave having $3$ nodes and $2$ antinodes is formed between two atoms having a distance of $1.21 \; \mathring{A}$ between them. The wavelength of the standing wave is .... $\mathring{A}$