$A$ transverse wave is represented by $y = 2 \sin(\omega t - kx) \ cm$. The value of wavelength (in $cm$) for which the wave velocity becomes equal to the maximum particle velocity is:

$A$ traveling wave is described by the equation $y(x,t) = [0.05 \sin(8x - 4t)] \text{ m}$. The velocity of the wave is (All the quantities are in $SI$ units). (in $\text{ m/s}$)

Given below are some functions of $x$ and $t$ to represent the displacement of an elastic wave.
$(i) \, y = 5 \cos (4x) \sin (20t)$
$(ii) \, y = 4 \sin (5x - t/2) + 3 \cos (5x - t/2)$
$(iii) \, y = 10 \cos (252\pi t) \cos (250\pi t)$
$(iv) \, y = 100 \cos (100\pi t + 0.5x)$
State which of these represent:
$(a)$ a travelling wave along $-x$ direction
$(b)$ a stationary wave
$(c)$ beats
$(d)$ a travelling wave along $+x$ direction.
Give reasons for your answers.

If $\vec{u}$ is the instantaneous velocity of a particle and $\vec{v}$ is the velocity of the wave,then:

$A$ plane progressive wave is represented by the equation $y = 0.1 \sin \left( 200\pi t - \frac{20\pi x}{17} \right)$,where $y$ is displacement in $m$,$t$ is in seconds,and $x$ is the distance from a fixed origin in meters. The frequency,wavelength,and speed of the wave respectively are:

The equation of the wave is $Y = 10 \sin \left(\frac{2 \pi t}{30} + \alpha\right)$. If the displacement is $5 \text{ cm}$ at $t = 0$,then the total phase at $t = 7.5 \text{ s}$ will be (Given: $\sin 30^{\circ} = 0.5$)