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) $y = 2 c o s (3 x) sin (10 t)$
(b) $y = 2 \sqrt{x - v t}$
(c) $y = 3 sin (5 x - 0.5 t) + 4 cos (5 x - 0.5 t)$
(d) $y = c o s x s i n t + c o s 2 x s i n 2 t$

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Solution

(a) The given equation represents a stationary wave because the harmonic terms $k x$ and $ω t$ appear separately in the equation.
(b) The given equation does not contain any harmonic term. Therefore, it does not represent either a travelling wave or a stationary wave.
(c) The given equation represents a travelling wave as the harmonic terms $k x$ and $ω t$ are in the combination of $k x - ω t$ .
(d) The given equation represents a stationary wave because the harmonic terms $k x$ and $ω t$ appear separately in the equation. This equation actually represents the superposition of two stationary waves.

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