$A$ wave travelling in the $-z$ direction having displacement along the $x$ direction as $1 \ m$,wavelength $\pi \ m$,and frequency $\frac{1}{\pi} \ Hz$ is represented by:

The equation of a wave is given by $y = 10 \sin \left( \frac{2 \pi}{45} t + \alpha \right)$. If the displacement is $5 \text{ cm}$ at $t = 0$,then the total phase at $t = 7.5 \text{ s}$ is

$A$ transverse wave is given by $y = A \sin 2\pi \left( \frac{t}{T} - \frac{x}{\lambda} \right)$. The maximum particle velocity is equal to $4$ times the wave velocity when:

The amplitude of a wave disturbance propagating in the positive $x$-direction is given by $y = \frac{1}{1+x^2}$ at time $t=0$ and $y = \frac{1}{1+(x-2)^2}$ at $t=1 \text{ s}$,where $x$ and $y$ are in meters. The shape of the wave does not change during the propagation. The velocity of the wave will be $... \text{ m/s}$.

The displacement $y$ of a wave travelling in the $x$-direction is given by $y = 10^{-4} \sin(600t - 2x + \frac{\pi}{3})$ metres,where $x$ is expressed in metres and $t$ in seconds. The speed of the wave motion,in $m \ s^{-1}$,is

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.