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

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

$A$ sinusoidal progressive wave is generated in a string. Its equation is given by $y = (2 \text{ mm}) \sin(2\pi x - 100\pi t + \pi/3)$. The time when the particle at $x = 4 \text{ m}$ first passes through the mean position will be:

The displacement of a plane progressive wave in a medium, travelling towards the positive $x$-axis with a velocity of $4 \ m/s$, at $t=0$ is given by $y=3 \sin 2 \pi (-x/3)$. The expression for the displacement at a later time $t=4 \ s$ will be:

The equation of a progressive wave is given by $x = 1.2 \sin(314t + 12.56y)$. What are the wavelength and the direction of the wave?

$A$ 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?