Given below are some functions of $x$ and $t$ to represent the displacement (transverse or longitudinal) of an elastic wave. State which of these represent $(i)$ a travelling wave,$(ii)$ a stationary wave or $(iii)$ none at all:
$(a)$ $y = 2 \cos(3x) \sin(10t)$
$(b)$ $y = 2 \sqrt{x - vt}$
$(c)$ $y = 3 \sin(5x - 0.5t) + 4 \cos(5x - 0.5t)$
$(d)$ $y = \cos x \sin t + \cos 2x \sin 2t$

(A) $y = 2 \cos(3x) \sin(10t)$ represents a stationary wave because the spatial part $k x$ and temporal part $\omega t$ appear as separate factors.
$(b)$ $y = 2 \sqrt{x - vt}$ does not represent a wave because it does not satisfy the general wave equation $\frac{\partial^2 y}{\partial x^2} = \frac{1}{v^2} \frac{\partial^2 y}{\partial t^2}$ and is not a periodic function.
$(c)$ $y = 3 \sin(5x - 0.5t) + 4 \cos(5x - 0.5t)$ represents a travelling wave because it is a linear combination of functions of the form $f(kx - \omega t)$.
$(d)$ $y = \cos x \sin t + \cos 2x \sin 2t$ represents a stationary wave because it is a superposition of two stationary waves,where each term has separate spatial and temporal components.