$A$ wave $y = a \cos (kx + \omega t)$ superimposes on another wave to form a stationary wave having a node at $x = 0$. What is the equation of the other wave?

For the stationary wave $y = 4 \sin \left(\frac{\pi x}{15}\right) \cos (96 \pi t)$,the distance between a node and the next antinode is

Two waves are approaching each other with a velocity of $16 \ m/s$ and frequency $n$. The distance between two consecutive nodes is

Two waves of same frequency $(n)$ approaching each other with same velocity of $18\text{ m/s}$ interfere. The distance between two consecutive antinodes is

$(i)$ For the wave $y(x, t) = 0.06 \sin \left(\frac{2 \pi}{3} x\right) \cos (120 \pi t)$,where $x$ and $y$ are in $m$ and $t$ is in $s$. The length of the string is $1.5 \, m$ and its mass is $3.0 \times 10^{-2} \, kg$. Do all the points on the string oscillate with the same $(a)$ frequency,$(b)$ phase,$(c)$ amplitude? Explain your answers.
$(ii)$ What is the amplitude of a point $0.375 \, m$ away from one end?

Two travelling waves of equal amplitudes and equal frequencies move in opposite directions along a string. They interfere to produce a stationary wave whose equation is given by $y = (10 \cos \pi x \sin \frac{2 \pi t}{T}) \, cm$. The amplitude of the particle at $x = \frac{4}{3} \, cm$ will be ........ $cm$.