$A$ progressive wave is given by $Y = 12 \sin (5t - 4x)$. On this wave,how far away are the two points having a phase difference of $90^{\circ}$?

The figure given below shows four progressive waves $A, B, C$ and $D$ with their phases expressed with respect to the wave $A$. It can be calculated from the figure that

Which of the following is not true for the progressive wave $y = 4\sin 2\pi \left( \frac{t}{0.02} - \frac{x}{100} \right)$,where $y$ and $x$ are in $cm$ and $t$ is in $sec$?

$A$ 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?

$A$ wave is represented by $x = 0.4 \cos \left(8t - \frac{y}{2}\right)$ where $x$ and $y$ are in meters and $t$ is in seconds. The speed of the wave is

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