In stationary waves,the distance between a node and its nearest antinode is $20 \ cm$. The phase difference between two particles having a separation of $60 \ cm$ will be

$(i)$ 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 ........ $cm$.

The given diagram shows a detector placed between a transmitter of sound waves and a metal plate. At three adjacent points,$R, S$ and $T$,the meter shows zero intensity. Which of the following is the frequency of the emitted wave in $Hz$? (Take the velocity of sound as $v = 300 \, m/s$)

In stationary waves,

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