15.5 You have learnt that a travelling wave in one dimension is represented by a function y = f(x, t) where x and t must appear in the combination $\left(\mathrm{x}–\mathrm{vt}\right)$ or $\left(\mathrm{x}+\mathrm{vt}\right)$, i.e. $\mathrm{y}=\mathrm{f}\left(\mathrm{x}±\mathrm{vt}\right)$. Is the converse true? Examine if the following functions for y can possibly represent a travelling wave :

The converse of the given statement is not true. The essential requirement for a function to represent a travelling wave is that it should remain finite for all values of x and t.
Explanation:

(a) For x = 0 and t = 0, the function becomes 0.

Hence, for x = 0 and t = 0, the function represents a point and not a wave.
(b) For x = 0 and t = 0,

Since the function does not converge to a finite value for x = 0 and t = 0, it does not represent a travelling wave.

(c) For x = 0 and t = 0,

$\frac{1}{\mathrm{x}+\mathrm{vt}}=\frac{1}{0}=\infty$

Since the function does not converge to a finite value for x = 0 and t = 0, it does not represent a travelling wave.