15.13 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) $\mathrm{y}=2\mathrm{cos}\left(3\mathrm{x}\right)\mathrm{sin}\left(10\mathrm{t}\right)$

(b) $\mathrm{y}=2\sqrt{\mathrm{x}-\mathrm{vt}}$

(c) $\mathrm{y}=3\mathrm{sin}\left(5\mathrm{x}–0.5\mathrm{t}\right)+4\mathrm{cos}\left(5\mathrm{x}–0.5\mathrm{t}\right)$

(d) $\mathrm{y}=\mathrm{cosxsint}+\mathrm{cos}2\mathrm{xsin}2\mathrm{t}$

(a) The given equation represents a stationary wave because this equation is similar to $\mathrm{y}=\mathrm{asin}\left(\mathrm{kx}\right)\mathrm{cos}\left(\mathrm{\omega t}\right).$

(b) The given equation is not a periodic function. 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 'kx' and 'ωt' are in the combination of (kx–ωt).

(d) The given equation represents a stationary wave because this equation actually represents the superposition of two stationary waves similar to $\mathrm{y}=\mathrm{asin}\left(\mathrm{kx}\right)\mathrm{cos}\left(\mathrm{\omega t}\right).$