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.

(N/A) The wave equation $y = 100 \cos (100\pi t + 0.5x)$ represents a wave propagating along the $-x$ direction because the coefficients of $t$ and $x$ have the same sign.
$(b)$ The stationary wave is $y = 5 \cos (4x) \sin (20t)$ because it is in the form $y = A \cos (kx) \sin (\omega t)$,which represents a standing wave.
$(c)$ The equation $y = 10 \cos (252\pi t) \cos (250\pi t)$ represents beats,as it is the product of two cosine functions with slightly different frequencies,which is the characteristic form of the beat phenomenon.
$(d)$ The equation $y = 4 \sin (5x - t/2) + 3 \cos (5x - t/2)$ represents a travelling wave along the $+x$ direction because the coefficients of $t$ and $x$ have opposite signs.