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

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 $C^{\prime}$.

$A$ standing wave pattern of amplitude $A$ in a string of length $L$ shows $2$ nodes (plus those at two ends). If one end of the string corresponds to the origin and $v$ is the speed of the progressive wave,the disturbance in the string could be represented (with appropriate phase) as:

$A$ stretched string is vibrating in its $5^{th}$ harmonic as shown. Consider a particle $1$ (figure). At an instant,this particle is at its mean position and is moving towards its negative extreme. Which of the following sets of particles are in the same phase as particle $1$?

At nodes in stationary waves,

The equation of a stationary wave along a stretched string is given by $y = 5 \sin \left( \frac{\pi x}{3} \right) \cos (40 \pi t)$. Here $x$ and $y$ are in $cm$ and $t$ is in seconds. The separation between two adjacent nodes is: (in $cm$)