The vibrations of a string of length $60 \, cm$ fixed at both ends are represented by the equation $y = 2 \sin \left( \frac{4 \pi x}{15} \right) \cos (96 \pi t)$,where $x$ and $y$ are in $cm$. The maximum number of loops that can be formed in it is

The stationary wave produced on a string is represented by the equation $y = 5 \cos (\pi x / 3) \sin (40 \pi t)$. Where $x$ and $y$ are in $cm$ and $t$ is in seconds. The distance between consecutive nodes is .... $cm$.

$A$ standing wave is represented by $Y = A \sin(100t) \cos(0.01x)$,where $Y$ and $A$ are in millimetres,$t$ is in seconds,and $x$ is in metres. The velocity of the wave is ..... $m/s$.

$A$ stationary wave is represented by $y = 12 \cos \left(\frac{\pi}{6} x\right) \sin (8 \pi t)$,where $x$ and $y$ are in $cm$ and $t$ is in seconds. The distance between two successive antinodes is (in $cm$)

$(i)$ 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?

$A$ standing wave with a number of loops is produced on a string fixed at both ends. Which one of the following statements is correct?