The equation of a progressive wave is given by $y = a \sin \pi [\frac{t}{2} - \frac{x}{4}]$,where $t$ is in seconds and $x$ is in meters. The distance through which the wave moves in $8 \ s$ is .... $(m)$

Which of the following statements is incorrect during the propagation of a plane progressive mechanical wave?

Write the definition of the speed of a progressive wave and derive $v = \frac{\omega}{k}$ and $v = \frac{\lambda}{T}$.

$A$ transverse harmonic wave on a string is described by $y(x, t) = 3.0 \sin (36t + 0.018x + \pi/4)$,where $x$ and $y$ are in $cm$ and $t$ is in $s$. The positive direction of $x$ is from left to right.
$(a)$ Is this a travelling wave or a stationary wave? If it is travelling,what are the speed and direction of its propagation?
$(b)$ What are its amplitude and frequency?
$(c)$ What is the initial phase at the origin?
$(d)$ What is the least distance between two successive crests in the wave?

The equation of a progressive wave is given by $y = 0.5 \sin(10t + x) \text{ m}$. What is the wave velocity in $\text{m/s}$?

For a progressive wave,$y = 5 \sin (0.01x - 2t)$ (where $x$ and $y$ are in $cm$ and $t$ is in $s$). What will be its speed of propagation (in $cm/s$)?