$A$ 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

$A$ standing wave $y = A \sin \left( \frac{20}{3} \pi x \right) \cos (1000 \pi t)$ is maintained in a taut string,where $y$ and $x$ are expressed in meters. The distance between the successive points oscillating with the amplitude $A/2$ across a node is equal to ... $cm$.

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 $C^{\prime}$.

$A$ string of length $1\,m$ and linear mass density $0.01\,kg/m$ is stretched to a tension of $100\,N$. When both ends of the string are fixed,the three lowest frequencies for standing waves are $f_1, f_2$,and $f_3$. When only one end of the string is fixed,the three lowest frequencies for standing waves are $n_1, n_2$,and $n_3$. Then:

Energy is not carried by which of the following waves?

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